GROUP REPRESENTATIONS AND
THE QUANTUM STATISTICS OF
SPINS
Jonathan Michael Harrison
Scho ol of Mathematics
Septemb er
A dissertation submitted to the University of Bristol
in accordance with the requirements of the degree
of Doctor of Philosophy in the Faculty of Science
Abstract
An intuitive explanation of the connection b etween the quantum statistics of par
ticles and their spins has b een sought since the early pro ofs of the spin statistics
theorem were published in the s Recently Berry and Robbins have sug
gested a new approach They construct a p osition dep endent spin basis in which
exchanging the p ositions of identical particles automatically exchanges their spins
In this basis the spin statistics connection can b e derived from the singlevaluedness
of the wavefunction The p osition dep endent basis for n particles is constructed
using the Schwinger representation of spin which can b e regarded as a choice of
representation of the group SU n
In this thesis we generalise the construction to include all representations of
SU n For n vectors that can b e used to construct the p osition dep endent
basis are assembled directly using Young tableau and the sign of these vectors under
the exchange of the two particles determined We nd that for a typical represen
tations of SU there are several subspaces of vectors with di erent spins that can
b e used to construct the p osition dep endent basis The sign of vectors in these sub
spaces under the exchange of the particles is determined not only by the spin but
also by the symmetry conditions recorded in the Young tableau which lab els the
representation of SU
For n particles the decomp osition into subspaces that can b e used in the con
struction is achieved using the characters of the relevant groups We see that typi
cal representations admit parastatistics The numb er of subspaces of spin s which
transform according to a given irreducible representation of S is written in terms
n
of Littlewo o d Richardson and Clebsch Gordan co e cients iii
Acknowledgements
There are many p eople to whom I owe a great deal of thanks for their supp ort
and encouragement over the past four years Firstly and most imp ortantly to my
sup ervisor Jonathan Robbins I have learnt much from him and his ideas interest
and go o d humour have b een an inspiration Those sentiments also apply to the
memb ers of the quantum chaos group which it has b een a pleasure to b e part of
I have enjoyed talking to p eople ab out this work and in particular Michael Berry s
and Aidan Scho eld s comments were much appreciated Sp ecial thanks should also
go to Gregory Berkolaiko for his friendship and many useful conversations some
even ab out maths I would also like to thank Suhasini Scott and all my friends
amongst the departments graduate students who were always the b est company
These acknowledgements would not b e complete without mentioning the mem
b ers of the Bristol University physics department who expanded my horizons whilst
I was an undergraduate for which I am truly grateful
Finally I would like to say a big thank you to my family without whom as the
saying go es none of this would have b een p ossible When my parents showed me
maths puzzles as a child I think they little exp ected how far it could go
This research was made p ossible through the nancial supp ort of the EPSRC v
Author s Declaration
I declare that the work in this thesis was carried out in ac
cordance with the Regulations of the University of Bristol
The work is original except where indicated by sp ecial ref
erence in the text and no part of the dissertation has b een
submitted for any other degree Any views expressed in the
dissertation are those of the author and do not necessarily
represent those of the University of Bristol The thesis has
not b een presented to any other university for examination
either in the United Kingdom or overseas
Jonathan Michael Harrison
Date th Septemb er vii
Contents
Abstract iii
Acknowledgements v
Author s Declaration vii
Spin Statistics and al l that
The discovery of spin statistics
Pauli s exclusion principle
Spin and statistics
The symmetrisation p ostulate
Relativistic quantum eld theory
Pauli s pro ofs
Axiomatic pro ofs
Criticisms of the eld theory pro ofs
Feynman and the elementary pro ofs
Geometric rotation
Feynman s Dirac lecture
Recent candidates for an elementary pro of
Parastatistics
Parab osons and Parafermions
Quons
Exp erimental tests
Conclusions ix
Contents
Preliminaries
Groups and representations
The symmetry group of an equilateral triangle
Cosets and the quotient group
Classes and characters
The symmetric group
The irreducible representations of S
The group SU n
SU
Ro ots and weights of Lie algebras
The highest weight classi cation of irreducible representations
Representations of the symmetric group and Young tableau
Partitions graphs and tableau
Characteristic units of the group algebra
Characteristic units of S
n
Constructing representations of SU n with characteristic units
Characters of U n and SU n
Characters of S
n
The Littlewo o d Richardson theorem
Multiplying Young tableau
Applying the Littlewo o d Richardson theorem to representations
Decomp osing representations of SU m n into representa
tions of SU m SU n
Quantum indistinguishability
Intro duction
The p osition dep endent spin basis
Quantum mechanics in the p osition dep endent spin basis
The Schwinger representation of spin
U R for two particles
The exchange sign is indep endent of U r x
Contents
Constructing U r
Smo othness of jM r i
Parallel transp ort of jM r i
The Schwinger representation of SU n
U R for n particles
Maps from con guration space to SU n
Smo othness and parallel transp ort for n particles
Alternative constructions
Parastatistics
The comp onents of the BR construction
Summary
Calculation of the exchange sign for the irreducible representations
of SU
The group SU
The spin subspace
Preparing the subspace W
Basis vectors of the tensor pro duct representation
Basis vectors of irreducible representations of SU
Basis vectors of a representation of SU SU
s s multiplets in representations of SU
Spin vectors with zero weight with resp ect to the exchange
algebra
s s multiplets with E eigenvalue zero
z
Results for general multiplets
Example The s s multiplets of with e
z
The exchange sign
De ning a xed exchange rotation
Selecting a vector jM i
Constructing the highest weight state of a multiplet using
Young tableau
The e ect of the symmetry conditions on the exchange sign xi
Contents
Evaluating the e ect of the symmetry conditions for a general
highest weight state
Numerical calculation of the exchange sign
Character decomp osition of SU n
Character decomp ositions
The semidirect pro duct
Classes of G n H
Irreducible representations of G n H
The Weyl group
De nition of the Weyl group
The Weyl group for SU n
The group
n
Classes of
n
Representations of
n
The action of the Weyl group on spin vectors jMi
The physical subgroup H
The n spin subgroup
De nition of H
Classes of H
The volume element of H
Representations of H
Physical representations of H
s
Characters of B H
Representations of SU n
Characters of SU n
Restricting a representation of SU n to H
The decomp osition of SU
The character of SU for elements of H
Evaluation of A
I
Evaluation of A
The numb er of physical representations of H in SU xii
Contents
The decomp osition of SU
Evaluation of A
I
Evaluation of A
Evaluation of A
Physical representations of H in SU
Example The representation of SU contains a spin
multiplet which exhibits parastatistics
The decomp osition of SU n
Evaluation of A
Results for the decomp osition of SU n
Conclusions
Bibliography xiii
List of Figures
The First Belt Trick
The Second Belt Trick
The de nition of t for jr j jr j
ij j i
multiplets of
Numerical results for representations of SU lab elled by Young
tableau with four b oxes
Numerical results for representations of SU lab elled by Young
tableau with six b oxes
Irreducible characters of S
xv
Chapter
Spin Statistics and al l that
The spin statistics theorem is a ma jor milestone in the development of quantum
mechanics Its discovery provided an explanation of the Pauli Exclusion Principle
which had successfully predicted b oth the structure of the p erio dic table and the
sp ectra of atoms As the Exclusion Principle profoundly changed our view of the
world it can b e di cult to appreciate how radical the ideas of Pauli and Dirac were
in their day In fact the interest in and controversy around the spin statistics theo
rem has never really disapp eared since its inception In this chapter we will follow
the development of our current understanding of spin statistics and the recent at
tempts to nd examples of the violation of the spin statistics theorem All this is
the physical background to the constructions of non relativistic spin statistics that
will b e made in the subsequent chapters
The discovery of spin statistics
The most complete history of the spin statistics theorem is the b o ok of Duck and
Sudarshan Their treatment repro duces the signi cant pap ers along with com
ments and explanations and is also an enjoyable read If I can recommend my version
it is only that I will b e briefer but anyone with an interest in the spin statistics the
orem will b ene t from reading their account
Chapter Spin Statistics and all that
Pauli s exclusion principle
It is the Pauli exclusion principle a consequence as we will see of the spin statistics
theorem that was rst used to explain the structure of the p erio dic table and pro
vided a stepping stone in the discovery of the spin of the electron Stoner realised
that by adding an extra spin quantum numb er to lab el electron states he could
explain the family structure of the p erio dic table This inspired Pauli to
pro duce his statement of the Exclusion Principle
There can never b e two or more equivalent electrons in an atom These
are de ned to b e electrons for which the value of all quantum numb ers
is the same If in the atom one electron o ccurs which has quantum
numb ers with these sp eci c values then the state is o ccupied
Pauli anticipated a deep er foundation for his exclusion principle which would re
quire a b etter understanding of quantum theory He continued to pursue these ideas
during his research leading eventually to the canonical relativistic argument for the
spin statistics theorem that we have to day
As a consequence of the spin statistics theorem it is still Pauli s exclusion prin
ciple that has the most signi cant and far reaching implications Quite literally
without this seemingly arbitrary rule for electrons the world as we know it would
not exist The imp ortance of the exclusion principle may explain the calibre of
physicists from Heisenb erg Dirac and Fermi to Feynman that contributed to the
development of spin statistics It certainly explains why Pauli was later awarded
the Nob el Prize for his insight
Spin and statistics
During the discovery of the exclusion principle the additional quantum numb er at
tributed to the electron had no physical interpretation Goudsmit and Uhlenb eck
rst prop osed that this quantum numb er b e assigned to an eigen rotation of
the electron They realised that a spherical rotating hollow sphere of charge would
have the required gyro magnetic ratio for spin The di culty with this view that
The discovery of spin statistics
the surface velo city would b e greater than the sp eed of light was solved later when
Dirac intro duced the p oint electron realising the spin as a purely intrinsic prop erty
of elementary particles The electron spin was immediately applied to explain the
level splitting in atomic sp ectra
With the discovery of spin we have one half of spin statistics The statistics
we refer to is the statistical mechanics of a gas of identical elementary particles
Initially it was Bose who derived the probability distribution of a photon gas
by dividing phase space into cells of volume h where h is Planck s constant Any
numb er of quanta are assigned to the states of the gas After corresp onding with
Bose Einstein was inspired to extend the ideas to an ideal gas of identical molecules
The Bose Einstein probability distribution they derive is
N
r
E
r
e
where
k T
and is the chemical p otential of the gas N is the average o ccupation numb er
r
of the state r which has energy E From this probability distribution b oth the
r
energy sp ectrum and thermo dynamic prop erties of the gas are then calculated In
the series of pap ers the phenomenon of Bose Einstein condensation
was also discovered
Indep endently b oth Fermi and Dirac solved the statistical mechanics of
an ideal gas of identical particles which ob ey the Pauli exclusion principle Each
state is now either o ccupied by a single particle or is uno ccupied The Fermi Dirac
probability distribution is
N
r
E
r
e
Again the probability distribution allows the energy sp ectrum and thermo dynamic
prop erties of the gas to b e calculated
Chapter Spin Statistics and all that
The prop erties of these two ideal gases are clearly very di erent At low temp er
atures a Fermi Dirac gas lls up all the lowest energy states while a Bose Einstein
gas condenses as the ma jority of the particles enter the ground state with zero en
ergy and zero momentum With an understanding of particle spin and statistics we
are in a p osition to state the spin statistics theorem
Bosons particles with Bose Einstein statistics all have integer spin while
all fermions have half integer spin
At the time this was a remarkable exp erimental fact There are two classes of parti
cles those that ob ey or do not ob ey the Pauli exclusion principle These two classes
p ossess very di erent prop erties However the memb ership of the classes is decided
by a quantum numb er spin which seems totally unrelated
The symmetrisation p ostulate
The symmetrisation p ostulate was discovered indep endently by b oth Dirac and
Heisenb erg in
States containing several identical elementary particles are either sym
metric or antisymmetric under p ermutations of the particles according
to the particle sp ecies Bosons are symmetric and fermions are antisym
metric States which cannot b e represented by wave functions with the
required symmetry are forbidden
We will summarise Dirac s argument which app ears in the same pap er in which he
derives the prop erties of an ideal Fermi Dirac gas His paradigm is a system of two
electrons orbiting an atom mn denotes the state in which one electron is in the
orbit lab elled by m and the other in the orbit n He then asks the question are
mn and nm two di erent states
His argument pro ceeds as follows The states mn and nm are physically
indistinguishable If b oth states corresp ond to separate rows or columns in the
matrices which op erate on the system then the amplitude for the two transitions
The discovery of spin statistics
mn m n and nm n m can b e individually calculated They would cor
resp ond to two di erent matrix elements However as the transitions are physically
indistinguishable only the combined intensity should b e able to b e determined by
exp eriment So if the theory is to enable only observable quantities to b e calculated
mn and nm must count as a single state we will see later that this may not
necessarily b e the case
Taking the states mn and nm to b e physically indistinguishable has conse
quences Only op erators that are symmetric in the p ositions and momenta of the
two electrons can b e represented by a matrices as for an op erator A
A x x mn A x x nm A x x mn
However it is p ossible to represent the physical prop erties of the system using ma
trices which dep end symmetrically on the electrons co ordinates
Turning to the eigenfunctions for the two electron system and neglecting the
interaction b etween the electrons the eigenfunction for the state mn can b e con
structed from a pro duct of single electron eigenfunctions There is
m n
however a second eigenfunction which also corresp onds to the same
m n
state and two indep endent eigenfunctions would give rise to two rows and columns
in the matrices What is required is a set of eigenfunctions of the form
mn
a b
mn mn m n mn m n
where the co e cients a and b are constants The set should contain only one
mn mn
corresp onding to the states mn and nm so applying a p ermutation of the
mn
electrons to the eigenfunction
c
mn mn
where c is a phase factor This set of eigenfunctions must b e su cient to obtain a
matrix representation of any op erator symmetric in the co ordinates so that
X
A A
mn
m n mn m n
m n
Chapter Spin Statistics and all that
is also in the given set of eigenfunctions where
m n
Dirac nds that there are only two solutions either a b or a b
mn mn mn mn
Either set of eigenfunctions gives a complete solution of the problem and this choice
cannot b e determined from the quantum theory The result extends to any numb er
of electrons the sets of eigenfunctions are then either symmetric or antisymmetric
under p ermutations of the electrons As an antisymmetric eigenfunction vanishes for
two electrons in the same orbit there can b e no more than a single electron in each
orbit and we see that the symmetrisation p ostulate predicts the exclusion principle
Dirac s argument insists that all states of identical particles b e either symmetric
or antisymmetric Later we will b e considering parastatistics in which the states of
identical particles are allowed not to b e entirely symmetric or antisymmetric As
these violate Dirac s result so we should b e clear ab out the essential requirements
of his argument The rst condition was that those states corresp onding to p er
mutations of the electrons should b e physically indistinguishable This is a strong
condition on the states but on its own it is insu cient to deduce the result The
argument also requires the assumption that indistinguishable states are represented
by a single vector up to a phase factor This was intro duced in order to require
there to b e a single eigenfunction corresp onding to b oth states
mn
Messiah and Greenb erg considered the situation where only the indistin
guishability condition applies to states Firstly they nd that all physical op erators
A on the states must b e p ermutation invariant
A
As the Hamiltonian op erator is an observable it also commutes with p ermutations
y
Consequently evolving A for a time t we obtain U t AU t which is also p ermu
tation invariant if A is They then show that for any vectors in a subspace which
transforms according to an irreducible representation of the p ermutation group the
y
exp ectation value of U t AU t is indep endent of the particular choice of vector
This is interesting as for more than two particles there are irreducible representa
Relativistic quantum eld theory
tions of the p ermutation group with a dimension of two or more In this subspace
it is then p ossible to cho ose two di erent vectors to represent a state The choice of
vector can t b e determined by physical observations
In the language of Dirac if we considered a three electron state of an atom l mn
then a wavefunction can b e written as a linear combination of all the p ermutations
of the function
m n
l
X
c
m n
l
l mn
If the constants c are chosen such that for a p ermutation
T
l mn
l mn
transforms according to where T is an irreducible representation of S then
l mn
the irreducible representation T This is a generalisation of equation In that
equation we assumed there could only b e a single vector to represent the electron
so that T is the two dimensional irreducible state If we have selected constants c
representation of S then the exp ectation values of an observable for all in the
l mn
two dimensional subspace will b e equal The indistinguishability of the electrons
forces us to cho ose vectors which b elong to irreducible representations of the p er
mutation group but there could still b e more than one vector for a given l mn in
this subspace
Relativistic quantum eld theory
Quantum eld theory was conceived by Dirac He worked from the canonical
commutation relations to de ne particle creation and annihilation op erators These
op erators add or remove quanta from states which can b e multiply o ccupied and
so the theory is therefore a eld theory of b osons In order to de ne a eld theory
for fermions Jordan and Wigner replaced the commutation relations for the
creation and annihilation op erators with anticommutation relations Using these
op erators they de ne antisymmetric states and wavefunctions which ob ey the Pauli
Chapter Spin Statistics and all that
exclusion principle So in eld theory the spin statistics relation b ecomes a connec
tion b etween the choice of commutators or anticommutators for the creation and
annihilation op erators of the eld and the spin of the particles represented by the
eld
Pauli s pro ofs
While the original eld theories were non relativistic by quantising the Klein
Gordon equation Pauli and Weisskopf pro duced the canonical relativistic quan
tum eld theory Pauli then used this relativistic quantum eld theory to attempt
his rst pro of of the spin statistics theorem The idea was to show that the relation
b etween spin and statistics is a necessary consequence of the p ostulates of relativistic
quantum eld theory In this resp ect all the eld theory pro ofs are alike although
they di er dep ending on the precise axioms of the eld theory used
Pauli s rst pro of was not conclusive He himself questioned the validity of sev
eral of the op erations he used and it was some years b efore he reached what is now
regarded as his ortho dox pro of of the spin statistics relation While Pauli b egan his
work on the spin statistics relation Iwanenko and So colow quantised the Dirac
equation using anticommuting creation and annihilation op erators They concluded
that attempting to apply Bose statistics to the Dirac equation inevitably pro duces
problems and so Bose statistics are most natural for the scalar relativistic equation
while Fermi statistics are natural for Dirac s relativistic equation
Before discussing Pauli s second pro of we should note some of the signi cant
ideas contributed by less recognised authors Fierz intro duced the notion of
representing elementary particles with irreducible relativistic spinors which Pauli
would later generalise Belinfante s unique approach was to require invariance
under the charge conjugation transformation This was not only novel but interest
ingly the argument is now used in reverse the pro of of the spin statistics theorem
b eing the foundation for the pro of of the PCT theorem DeWet also pro duced
a pro of based on canonical eld theory in which he was the rst to identify one of
Relativistic quantum eld theory
the crucial assumptions on which many pro ofs from relativistic eld theory dep end
y
x y for x y spacelike
where is a eld op erator It would b e fteen years b efore this was proved
That Pauli s name is so strongly linked to the spin statistics theorem is due to his
pro of based on a classi cation of the spinor representations of the prop er
Lorentz group Although the pro of relies on eld theory we can see the origin of the
spin statistics connection in the representations of the spinors The prop er Lorentz
group is the continuous group of linear transformations which leaves invariant the
scalar pro duct
X
k
x x x x x x
k
k
Pauli uses the spinor representations of the Lorentz group A basic spinor is de ned
from a four vector v by the relation
U v
are the Pauli matrices with de ned to b e the identity matrix The group
multiplication law is then the normal matrix multiplication for the spinor matrices
The inverse relation to return a four vector from a spinor is given by
v U
Spinor indices are raised and lowered with the alternating tensor with
etc A general spinor U can b e characterised by two angular
momentum quantum numb ers j k where there are j upp er dotted indices and
k lower indices
Pauli divides the representations into classes dep ending on their prop erties when
the representation is restricted to the subgroup of space rotations If we take a rep
resentation U j k where j k is half integral then applying a space rotation by
vectors in the representation space undergo a sign change In a representation where
Chapter Spin Statistics and all that
j k is integral vectors in the representation space don t change sign under a ro
tation Pauli refers to these representations as single or double valued The spinor
representations U which transform as double valued representations of the Lorentz
group corresp ond to particles with half integral spin while the single valued repre
sentations corresp ond to those with integral spin
The single valued representations are further classi ed into representations where
j and k are b oth integral U and those where b oth j and k are half integral U
The direct pro duct of two representations decomp oses into a sum of irreducible
representations in a particular class
U U U U U U
U U U U U
For the double valued representations U refers to representations where j is in
tegral and k half integral and U to j half integral k integral The multiplication
table for these representations is
U U U U U U
U U U U U U
Pauli considers b oth commutation and anticommutation relations for the four
classes of spinor elds In either case he p ostulates the brackets of the eld op erators
can b e expressed in terms of an invariant D function and the derivatives of that
function
U x U x D x x
where on the bracket refers to anticommutation or commutation relations resp ec
U is the complex conjugate of U The D function is given by tively and
Z
d p sin x
ip x
D x e
x is the normal three vector p osition and similarly p is the momentum The D
function is uniquely determined by the conditions
m D D x D j x
x
0
Relativistic quantum eld theory
Using the rules for multiplying the spinor representations one can nd conditions on
the brackets For half integral spins with either commutation or anticommutation
relations
U U U U U
To construct a spinor in U from D and its derivatives only the o dd derivatives of
D can app ear For integral spins the brackets must have the opp osite form
U U U U U
These brackets corresp ond to only even derivatives of D Pauli then symmetrises
these relations for p ermutations of the p ositions x x
U x U x U x X U x
X is therefore even under x x He shows that the o dd derivatives of the D
function are even under a p ermutation of the p ositions x x but o dd under
x x Consequently for integral spins the X vanishes under symmetrisation
This excludes the p ossibility of anticommutation relations for integral spin particles
b ecause at x x the expression for X would b e p ositive and could therefore only
vanish for elds which are identically zero
For half integral spins the derivatives of the D function are even under the p er
mutation x x To rule out commutation relations Pauli uses an argument due to
Fierz that the energy of the system is only p ositive when anticommutation relations
are chosen In this way he establishes the connection b etween spin and particle
statistics without it b eing necessary to x the particular spin of the particles in
question There are still problems with the pro of as with all pro ofs of the time
the manipulations of the eld are not shown to b e valid and interactions are not
included However it was a great advance and also includes a discussion of symmetry
conditions which anticipated the PCT theorem but which I have omitted from this
outline
The asp ect of Pauli s pro of of most interest in this thesis is his use of double
valued representations When we discuss the construction of Berry and Robbins in
Chapter Spin Statistics and all that
Chapter we will also encounter double valued representations where under restric
tion to a subgroup of p ermutations vectors of half integral spin change sign The
group used is however not the Lorentz group but the group of sp ecial unitary matri
ces The analogy is very weak but at least in a historical context it is an interesting
connection b etween the two approaches
Axiomatic pro ofs
While Pauli is widely credited with the derivation of the spin statistics theorem the
most complete pro ofs are due to Luders Zumino and Burgoyne The most
memorable reference for a pro of of the spin statistics theorem is however the b o ok of
Streater and Wightman Their pro of also follows the axiomatic approach which
Luders Zumino and Burgoyne intro duced
Before these axiomatic pro ofs were develop ed there were two other contributions
to the history of the spin statistics theorem that should b e noted In Feynman
published a pap er purp orting to show that only the observed spin statistics con
nection was compatible with calculations of the vacuum survival probability made
using the Feynman rules The approach was very novel but an analysis by Pauli
showed the theory required an inde nite metric on the Hilb ert space while a basic
p ostulate of the eld theory is that the metric is p ositive de nite Undeterred this
served as a basis for Feynman s ideas for an elementary pro of Schwinger pub
lished a pro of based on the requirement that relativistic quantum eld theory b e
time reversal invariant In the later pro of of Luders and Zumino and also in Streater
and Wightman s b o ok this is reversed so that b oth the spin statistics theorem and
PCT theorem rest on the basic axioms of relativistic eld theory
In their b o ok Streater and Wightman follow the pro of of Burgoyne The work of
Luders and Zumino applies only to spin and although it has the advantage of
clearly separating the PCT and spin statistics theorems Both Luders and Zumino
and Streater and Wightman make use of the Hall Wightman theorem As this
theorem is often used in pro ofs of the spin statistics connection using relativistic
Relativistic quantum eld theory
quantum eld theory I will state it here
Theorem Hall Wightman The vacuum expectation value of the product
of two elds
h j x x j i F x x
is analytic in x x and continuable to al l separations
In Streater and Wightman the discussion of such results extends over most of a
chapter including it here is only designed to give a avour of the problems to b e
tackled in order to pro duce rigorous pro ofs
As well as having a full understanding of the allowed manipulations of eld op er
ators the other essential building blo ck of these pro ofs are the axioms of relativistic
quantum eld theory The mathematical discussion of Streater and Wightman is
to o involved for an intro duction so instead I will refer to the original p ostulates of
Burgoyne
The eld theory must b e relativistically invariant
The theory contains no negative energy states This is equivalent to requiring
the vacuum state to b e the lowest energy state
The metric in Hilb ert space is p ositive de nite
Distinct elds either commute or anticommute for space like separations
He then shows that for any eld with these prop erties the wrong connection b e
tween spin and statistics implies that the eld vanishes For readers with some
knowledge of eld theory the pro of is structured as follows
From eld op erators x which transform according to an irreducible repre
sentation of the homogeneous Lorentz group indexed by we de ne temp ered eld
op erators
Z
f d xf x x
Chapter Spin Statistics and all that
where the f x are a set of test functions F and G are de ned to b e the vacuum
exp ectation values
F h j x x j i
x x j i G h j
with the relative p osition x x F and G can b e extended to functions of
j
a complex four vector z i which are analytic for z z z in the complex
j
plane cut along the p ositive real axis Burgoyne shows that for space like separations
s
G G
where s is the particle spin The wrong sign commutation relations for the eld
op erators at space like separations are
s
h j x x x x j i
We have used the fourth axiom of the eld theory that the elds commute or anti
commute for space like separations Equation implies that
s
F G
Using we see that
F G
By analyticity F G vanishes everywhere in the cut plane To evaluate the
limit the temp ered elds are used From equation
f g g j i h j f
As the Hilb ert space metric is p ositive de nite we conclude that
j g j ij j f j ij
Consequently the temp ered eld op erators f and g are identically zero for all
test functions f and g The eld is therefore zero and the wrong commutation
Feynman and the elementary pro ofs
relations are untenable
The canonical pro of of the spin statistics theorem given by Streater and Wight
man is a combination of the pro of of Burgoyne with the spinor pro of of Pauli the
spinors b eing used to prove the equivalent of statement It has the advantage
over the Pauli pro of of b eing rigorous with theorems for all the necessary manipula
tions of the eld op erators also proved It is probably for this reason that amongst
such a wide range of approaches it has achieved the status of the de nitive pro of
Criticisms of the eld theory pro ofs
In the outlines of the pro ofs of the spin statistics theorem from relativistic quantum
eld theory I hop e I have b een fair b oth to their achievements and to the degree of
the formalism that they necessarily intro duce The axiomatic approach of Streater
and Wightman can app ear to reduce the spin statistics relation to a mathemati
cal problem involving the existence of an analytic continuation of certain temp ered
distributions in convex cones of four dimensional space time If there is a physical
result obscured by this analysis the b est candidate is the connection b etween spin
and the double or single valued representations of the prop er Lorentz group This
is not however used in all the pro ofs so it is hard to see it as a physical basis for
the spin statistics connection If from these relativistic arguments we were to try
and explain the spin statistics theorem what we can say is that another connection
b etween spin and statistics is incompatible with the axioms of relativistic quantum
eld theory and we must b e satis ed with that
Feynman and the elementary pro ofs
In Feynman s lectures on physics he discusses the spin statistics connection and
asks the following question
Why is it that particles with half integral spin are Fermi particles
whereas particles with integral spin are Bose particles We ap ologise
Chapter Spin Statistics and all that
for the fact that we cannot give you an elementary explanation An ex
planation has b een worked out by Pauli from complicated arguments of
quantum eld theory and relativity He has shown that the two must
necessarily go together but we have not b een able to nd a way of re
pro ducing his arguments on an elementary level It app ears to b e one of
the few places in physics where there is a rule which can b e stated very
simply but for which no one has found a simple and easy explanation
This probably means that we do not have a complete understanding of
the fundamental principle involved
This question has inspired several attempts to formulate such an elementary argu
ment and it is in this spirit that Michael Berry and Jonathan Robbins prop osed
their non relativistic construction of the spin statistics connection with which the
rest of this thesis is concerned Here we will review other approaches that have b een
taken to the question The main critiques of these prop osals have b een provided by
Hilb orn and Sudarshan and Duck
Geometric rotation
Probably the most well known of the elementary schemes are the geometric ar
guments of Bacry and Broyles The simple situation describ ed by Bacry is
su cient to demonstrate the essential idea Unfortunately b oth arguments contain
the same critical aw
Bacry considers the two electron system where one electron is lo cated at co ordi
nates x y z a with spin and the second electron at a with
spin The single particle wavefunctions are
x a y z
A A
A B
x a y z
and the two electron wavefunction is written
AB A B A B
Feynman and the elementary pro ofs
We have not as yet chosen whether the wavefunction for the system will b e symmetric
or antisymmetric Exchanging the two particles
AB AB
A rotation by ab out the y axis leaves the two electron state unchanged z y π
x
If we act on the single electron wavefunctions with the rotation ab out the y axis
i J
y
R e
y
R R
y A B y B A
So acting on the two electron wavefunction
R
y AB AB
Bacry then makes an error which invalidates the pro of the pro of of Broyles also
fails for a similar reason As the two electron state is invariant under the rotation
R he requires the same of the wavefunction
y
R
y AB AB
If this were the case it would require the wavefunction to b e antisymmetric which is
the spin statistics relation
Unfortunately the invariance of the state under a discrete symmetry transfor
mation do es not rule out the p ossibility of a sign change in the wavefunction An
example of the phenomena is a rotation of a spin state of an electron The
state is invariant but the wavefunction changes sign As the sign of un
A B
der the rotation R cannot b e determined the argument fails to provide a spin
y
statistics connection
Chapter Spin Statistics and all that
Feynman s Dirac lecture
In Feynman despite his long illness gave the Dirac memorial lecture in which
he sketched an elementary argument for the spin statistics connection He was
inspired by the unexp ected b ehaviour of tethered classical ob jects under rotations
The lecture demonstration was rotating a full wine glass through with out spilling
the water The same idea is contained in a trick with a b elt which is easier to de
scrib e If a b elt is xed at one end while the other end is connected to an ob ject
rotating the ob ject by intro duces a twist to the b elt which cannot b e undone by
translating the ob ject However the twists in the b elt that result from rotating the
ob ject by can b e removed by translating the ob ject whilst keeping its orientation
xed Figure shows a schematic of the b elt trick though it is b est veri ed p erson
ally In the gure a straight b elt is rotated through then Finally keeping the
orientation of the black cub e xed whilst translating along the path shown removes
the twists in the b elt
Figure The First Belt Trick
This classical paradox suggests that a rotation by do es not always return an
ob ject to its original state and therefore that the change in sign of the wavefunction
of spin particles is not unreasonable In their discussion of these ideas Sudar
Feynman and the elementary pro ofs
shan and Duck warn the unwary not to b e deceived by such party tricks In their
words they are pure old fashioned snake oil p eddling However they concede that
in Feynman s hands they were mesmerising
With this mo del Feynman also included two others in which the exchange of
identical particles repro duces the spin statistics connection One mo del uses a pair
of comp osite particles each consisting of a spin zero electric charge e and a spin
zero magnetic monop ole of charge g However as an explanation of spin statistics
requiring elementary particles to have the additional unphysical prop erty of sourcing
a magnetic eld is not comp elling
The third mo del makes use of a second b elt trick see gure In the trick
when two particles connected by a ribb on are exchanged the ribb on acquires a twist
To complete the exchange one of the particles must b e rotated by The argu
ment is used to demonstrate that the op eration of exchange entails a hidden rotation
by Feynman uses this rotation to determine the e ect of exchange on particle
states from which he acquires the spin statistics relation The reasonable ob jection
raised by Duck and Sudarshan is that we are required to p ostulate that elementary
particles are connected by ribb ons a prop erty which is needed for no other purp ose
While the classical argument suggests a spin statistics like result a derivation of the
spin statistics theorem should b e based on more natural physical prop erties
In his lecture Feynman also returned to his original eld theory argument He
used the time reversal prop erty of the Dirac spinor to show that Fermi Dirac statis
tics are necessary if the S matrix is to b e unitary This use of the PCT theorem
to prove the spin statistics theorem reverses the status of the theorems and requires
that the PCT theorem b e proved without the use of a spin statistics relation In
their analysis of Feynman s argument Sudarshan and Duck conclude that the in
ternal consistency of the Feynman rules for p erturbation theory is not a su cient
foundation for the spin statistics theorem
Feynman s comments on the second b elt trick were inspired by a rigorous ge
Chapter Spin Statistics and all that
1 2 2 1 2 1
Figure The Second Belt Trick
ometric argument of Finkelstein and Rubinstein Finkelstein and Rubinstein
treat nonlinear eld theories in which there exist mo des of the eld called kinks
or solitons which can not b e deformed into each other and which p osses a con
served integer particle numb er They de ne an exchange op eration in which two
soliton antisoliton pairs are created the solitons exchanged and the new pairs anni
hilated The eigenvalues of the exchange op erator are corresp onding to even or
o dd statistics This exchange op eration is shown to b e homotopic to a rotation
of the eld which implies solitons with half integer spin have o dd statistics Those
soliton sp ecies which can undergo o dd exchanges also admit even exchanges To x
the statistics and so the spin of a given typ e of soliton they note that all solitons
of the same typ e must pro duce the same sign under exchange and the sign can not
change over time Consequently given a universe of solitons measuring the exchange
sign for a pair xes the exchange prop erty of all solitons of that typ e for all time
Parastatistics which will b e discussed later is excluded in this mo del
Recent candidates for an elementary pro of
Balacharandran et al see also for a brief account have suggested a develop
ment of the argument of Finkelstein and Rubinstein applied to p oint particles Their
pro of avoids reference to eld theory or relativity However the argument makes use
Feynman and the elementary pro ofs
of an in nite dimensional con guration space The con guration space of a single
particle is R F SO where SO is the group of rotations and F is the
set of all orthonormal frames with a xed orientation in dimensions this deter
mines the particles spin Antiparticles are describ ed by a state in R F SO
F have the opp osite orientation to the particle where the orthonormal frames in
frames For states of many identical particles the con gurations where spins and
p ositions have b een exchanged are identi ed and pairs of particles are not allowed
to o ccupy the same p osition The e ect of particle antiparticle annihilation is in
cluded by asso ciating con gurations where particles and antiparticles coincide with
the con guration space of reduced particle numb ers Assuming certain continuity
conditions on this con guration space they show that exchange for particles with
spin is homotopic to a rotation of one of the particles which is the spin statistics
connection Establishing this homotopy makes use of the creation and annihilation
of particle antiparticle pairs They suggest that the analogue of these arguments in
eld theory will involve the physics of solitons although the question is not resolved
Sudarshan and Duck in the nal chapter of their b o ok include their own
elementary argument They replace the p ostulates of relativistic quantum eld
theory with conditions on the kinematic parts of the Lagrangian for the individual
elds The Lagrangian must b e
derived from a lo cal Lorentz invariant eld theory for elds which are each a
nite dimensional representation of the Lorentz group
in the Hermitian eld basis
at most linear in the rst derivatives of the elds
at most bilinear in the elds
They then show that the wrong spin statistics connection is incompatible with
rotational invariance of the Lagrangian
This pro of also seems to fail to provide the sought after elementary physical
understanding of the spin statistics connection that Feynman requested In their
Chapter Spin Statistics and all that
discussion of their result Duck and Sudarshan say the simpli cation they intro duce
is the use of rotation invariance rather than time reversal invariance in Schwinger s
eld theory pro of However the essential structure remains the same they show
that only the observed spin statistics connection is compatible with a given set of
requirements for the eld as in the axiomatic pro ofs Although they claim the pro of
is non relativistic the requirement that the eld b e Lorentz invariant is still retained
In summary the search for an elementary pro of of the spin statistics connection
has pro duced many interesting and signi cant analogues of the spin statistics con
nection Unfortunately none are conclusive Some require additional unphysical
prop erties to b e p ostulated for elementary particles while others remain re nements
of the relativistic argument
Parastatistics
A go o d review of the theories which allow a violation of the spin statistics connec
tion is provided by Greenb erg Here we will tackle only a couple of the topics
namely Green s parastatistics which is the original example of parastatistics and
the theory of quons which has particular relevance to exp erimental attempts to lo ok
for small violations in particle statistics
Parab osons and Parafermions
Parastatistics prop osed by Green is a theoretical generalisation of Bose and
Fermi statistics Bose or Fermi statistics are de ned by the choice of commutation
or anticommutation relations for the creation and annihilation op erators of particle
states The numb er op erator for a state k can b e written
y
n a a const
k k
k
where on the bracket refers to the choice of anticommutation relations and
y
to commutation relations for the op erators a The op erator a creates a particle
k
Parastatistics
in the state k and a is the corresp onding annihilation op erator The bracket is
k
either a commutator or an anticommutator according to the typ e of statistics The
commutator of the numb er op erator and creation op erators is indep endent of the
choice of Bose or Fermi statistics
y y
n a a
k k l
l l
y
We can generalise the de nition of the numb er op erator to use the op erators a and
k
a annihilating a particle in state m and creating one in state k Then substituting
m
this numb er op erator into equation we nd Green s trilinear commutation
relations
y y
a a a a
m
l ml
k k
Selecting the commutator or anticommutator in this relation will de ne two alter
native cases of parabose or parafermi statistics As these commutation rules are
trilinear the de nition of the vacuum state
a j i
k
is insu cient to allow all states to b e calculated An additional condition on single
particle states is included to resolve this
y
a a j i p j i
k k l
l
To nd solutions of these commutation rules Green made the following ansatz
Let
p p
X X
y
y
b b a a
k
k k k
The parab ose solutions are de ned by taking the pair of op erators b b to com
k k
mute if but anticommute if The parafermi statistics are found by
swapping the use of commutation and anticommutation relations in the de nition
The ansatz provides a set of parab ose and parafermi statistics for each integer p
For parab osons p is the maximum numb er of particles which can o ccupy an antisym
metric state while for parafermions it is the maximum numb er of particles which
can o ccupy a symmetric state These parastatistics were the rst alternative to the
Chapter Spin Statistics and all that
observed statistics of Fermi Dirac or Bose Einstein to b e de ned
The irreducible representations of the symmetric group are lab elled by Young
tableau discussed in greater detail in chapter
We will see later that the tableau record symmetry conditions Single particle states
are assigned to b oxes in the tableau then a tensor pro duct of n single particle states
is symmetrised with resp ect to the states in the same row then antisymmetrised with
resp ect to those in the same column The parab ose or parafermi systems transform
according to irreducible representations of the symmetric group lab elled by tableau
with no more than p rows or columns resp ectively These corresp ond to having at
most p particles in antisymmetric or symmetric states There are other forms of
parastatistics in which all representations of the symmetric group are admissible
States with the original Fermi statistics are represented by a tableau with a single
column the state is purely antisymmetric Equivalently Boson states are repre
sented by tableau with a single row where the state is completely symmetric We
see now that the de nitions of parab osons and parafermions are particular examples
of op erators whose states transform according to more complex representations of
the symmetric group The term parastatistics is used to cover all systems in which
particle states transform according to these generalised symmetry conditions
Quons
In most exp eriments which lo ok for a violation of the spin statistics theorem the
assumption is that the exp ected statistics will b e violated by a small amount To
compare these exp eriments to a theory requires a mo del in which the statistics can
vary with a small parameter q Bounds on the size of q then give a quantitative
measure of the accuracy of the spin statistics theorem To date the b est theoretical
framework for such a violation of particle statistics is the quon
Exp erimental tests
The quon algebra is de ned by taking the convex sum of the Bose and Fermi
algebras
q q
y y
a a a a
k k k l
l l
q is in the range q The usual vacuum condition
a j i
k
is su cient to calculate matrix elements of p olynomials in the creation and anni
hilation op erators At q the statistics are b osonic while at q they are
fermionic Given the discrete representations of the symmetric group we should
b e clear ab out the sense in which q interp olates b etween these statistics Vectors
formed by p olynomials in the creation op erators acting on the vacuum are sup er
p ositions of vectors in di erent irreducible representations of the symmetric group
As we vary q the weight given to vectors in these irreducible representations varies
smo othly For example as q the weights of all representations tend to zero with
the exception of the symmetric representation leaving a completely symmetric state
The quon theory p ossesses many of the desired prop erties for a theory which
allows small violations of spin statistics The norms are p ositive cluster decom
p osition theorems and the PCT theorem hold and free elds can have relativistic
kinematics It is not however ideal as observables with space like separations don t
commute and as a consequence interacting relativistic eld theory may not b e p os
sible
Exp erimental tests
To conclude this intro duction to the spin statistics theorem we will consider the cur
rent exp erimental evidence for the symmetrisation p ostulate Gillaspy provided
a review of the literature from which most of our data will b e taken
Chapter Spin Statistics and all that
Currently there are no examples of particle b ehaviour which violate the sym
metrisation p ostulate If in fact such data were to b e seen for example an inhibited
transition in a collider exp eriment it is unlikely it could b e attributed to such a
violation The frequency of such violations would b e so low that the p ossibility of
an error in the exp eriment or detectors would prevent our gedanken investigator
publishing Instead exp eriments to test the symmetrisation p ostulate yield upp er
b ounds on the probability of nding a two particle state with unusual statistics
In terms of the quon mo del of particle statistics
q for fermions
q for b osons
The limits put on by exp eriments vary widely Gillaspy attributes this to the
trade o b etween the chance of the exp eriment observing a violation and the size of
the violation that it is capable of detecting
Before discussing particular typ es of exp eriments we should mention some of the
general problems encountered when attempting to verify such a fundamental law
of physics As elementary matter particles are fermions few exp eriments in to the
symmetrisation p ostulate are carried out on b osons There are however approxi
mate results which suggest that the scale of a violation in the two classes should
b e comparable Using the indistinguishability of identical particles it can b e shown
that it is not p ossible for a particle in an ordinary state to transfer to a state vi
olating the symmetrisation p ostulate This imp ortant sup erselection rule prevents
many symmetry violating transitions For example we might assume that if there
were a small violation of the symmetrisation p ostulate then electrons in a blo ck of
matter would slowly relax into the ground state emitting photons in the pro cess
This transfer is inhibited by the indistinguishability of electrons and so the absence
of such transitions do es not provide a test of the symmetrisation p ostulate
We can now consider some of those schemes that have b een used to provide
b ounds on
Absorbing blo cks Fresh electrons which could b e in a symmetry violating state
Exp erimental tests
are absorb ed by the blo ck They are able to bind to an excited state of the
atoms and might then decay to the ground state emitting a photon The most
precise exp eriment of this typ e was conducted by Ramb erg and Shaw yielding
.
Decaying blo cks A nuclear reaction in the blo ck ejects a fresh particle from the
blo ck Asso ciated with the ejected particle is another that could have anoma
lous symmetry and decay to the ground state The most precise data for an
exp eriment of this typ e provides a limit of .
Collisions in vacuum Individual particles are pro jected into atoms in a vacuum
The separate results can then b e analysed As there are few events the system
is less sensitive but hop efully more accurate .
Ground state accumulation In a typical exp eriment mass sp ectroscopy is used
to search for atoms with an anomalous numb er of electrons in their ground
state Whether the chemical b ehaviour of such an anomalous atom would
remain unchanged remains an op en question A current limit on violation
from this typ e of exp eriment is .
The limits from these exp eriments dep end to a great extent on the assumptions made
in their analysis Interestingly those exp eriments pro ducing the lowest b ounds are
not necessarily the most recent These exp eriments and results are for small viola
tions of the symmetrisation p ostulate of the typ e we might exp ect from the quon
theory An alternative exp erimental consideration for the symmetrisation p ostulate
are the statistics of the more exotic elementary particles For these particles with
short lifetimes it is di cult to rst create and then make measurements on two par
ticle states While for some particles like the pion the spin statistics theorem has
b een con rmed there are many elementary particles for which successful exp eriments
have yet to b e devised
Chapter Spin Statistics and all that
Conclusions
The aim of this intro duction to the spin statistics theorem has b een to summarise
the present understanding of spin statistics We have seen how the structure of the
p erio dic table and atomic sp ectra led to the discovery of Pauli s exclusion principle
and the symmetrisation p ostulate These core prop erties of nature are not a conse
quence of quantum mechanics which do es not sp ecify the symmetry prop erties of
wavefunctions The search for a theoretical explanation of these phenomena then
centred around relativistic quantum eld theory where the most complete mo dern
pro ofs show that integer spin particles with Fermi Dirac statistics or half integer
spin particles with Bose Einstein statistics are not consistent with the basic axioms
of the eld theory These pro ofs are however negative pro ofs and we have seen that
parastatistics in which wavefunctions transform according to any irreducible rep
resentation of the p ermutation group are also consistent with quantum mechanics
even though they have not b een observed by exp eriment There also exists a lo cal
algebra approach to quantum eld theory in which the algebras of b ounded op era
tors generated by observables in b ounded regions of space time are studied In this
theory the various kinds of parastatistics app ear under appropriate conditions see
Haag
A pro of of the spin statistics theorem would b e more satisfactory if it were to b e
derived from a clear physical principle which allowed a more intuitive understanding
of the theorem Unfortunately so far the suggestions for such an elementary pro of
have b een awed Current exp eriments agree with the spin statistics theorem to
high precision but despite the b est e orts of many of the century s top physicists
an understanding of the spin statistics connection has remained elusive One of the
early pro ofs of the spin statistics theorem app eared in the PhD thesis of Dewey He
b egan his physical review article on his work with the summary The problem of
the connection b etween the spin and the statistics of particles was rst tackled by
Pauli His work was not correct The rest of this thesis will b e an investigation
of an elementary non relativistic construction of spin statistics As such I hop e that
it contributes to the understanding of the spin statistics theorem The explanation
Conclusions
it provides will not b e complete although I b elieve that will still leave me in go o d
company
Chapter
Representation Theory
This chapter summarises the results in the representation theory of the symmet
ric and unitary groups that are used subsequently Anyone already familiar with
this material should skip this chapter although the nal section on the Littlewo o d
Richardson theorem is probably not widely known and will b e used extensively later
I will present the representation theory of SU n and S in parallel in order to
n
emphasise the similarities b etween the two treatments that will lead to b oth b eing
classi ed by Young tableau The examples are chosen to t with the later chapters
and so may not app ear as in a standard text
Preliminaries
An isomorphism is a map b etween two sets which preserves the structure of the
domain of the map In the sets with which we will b e concerned this structure will
in general b e a multiplication law on the set An automorphism is an isomorphism
from a set back into itself so the domain and range of the map are the same
An algebra of order n over the complex numb ers is de ned by n complex num
b ers where i j k n Elements of the algebra are sets of n complex
ij k
numb ers x x x Addition multiplication and scalar multiplication are
n
Groups and representations
de ned according to the following rules
ax by ax by ax by
n n
X X
x y x y x y
ij i j ij n i j
a and b are complex numb ers An algebra is asso ciative if x yz xy z All the
algebras we will meet are asso ciative
Groups and representations
A group G is a set of elements g closed under an asso ciative multiplication law
containing an identity element and with each element having a unique inverse So
with I the identity element g g I The order of the group G is the numb er of
elements in G and is denoted by If the multiplication law of a group is commu
G
tative as well as asso ciative the group is called Abelian
A matrix representation of the group T G is a map from G to a set of nite
dimensional complex matrices which preserves the multiplication law of the group
so that
T g T g T g g
The dimension of the representation is the dimension of the vector space acted on
by the matrices The trivial representation maps all elements of the group to unity
regarded as a one dimensional matrix
Given two representations of a group T G and S G a third representation
can b e formed by taking the direct sum of these two representations
T g
A
T g S g
S g
T and S form blo cks on the diagonal of the new representation and the rest of the
entries are zero Any representation that can b e brought into this blo ck diagonal
form by a change of basis is called a reducible representation and conversely those
Chapter Representation Theory
representations which can t b e further decomp osed into diagonal blo cks are called
the irreducible representations of the group Starting with a reducible represen
tation we can lo ok for a transformation which divides the representation into two
representations of lower dimension as in equation The same pro cess can then
b e rep eated for the constituent representations If they are reducible there exists a
transformation which brings them into blo ck diagonal form As the matrix has nite
dimension the pro cedure must end with the representation expressed as a direct sum
of a set of irreducible representations
An alternative intrinsic de nition of irreducibility can b e made by considering
the space the representation acts on We will take the representation to b e n dimen
sional Then if the exists a subspace of dimension m where m n which is invariant
under all transformations of the group the representation is reducible If there is
no invariant subspace the representation is irreducible For a deep er discussion of
reducibility see chapter
A second way to construct a new representation of a group is to use the matrix
tensor pro duct This is the direct product representation
T S g T g S g
ix j y ij xy
The matrix representation can b e constructed by taking T g and replacing the
term T g with the matrix T g S If the representations T and S are of n and
ij ij g
m dimensions resp ectively the resulting matrices are of dimension n m Clearly
the multiplication law is preserved for T S so the tensor pro duct de nes a
representation The direct pro duct representation is not in general irreducible even
when the representations used in the pro duct are irreducible
A subgroup H of G is a subset of the elements of G ob eying all the requirements
of a group So for example all subgroups contain the identity A representation of
G must also b e a representation of H as the multiplication law holds for all
g H In this way we can consider restricting a irreducible representation of G to
the elements of H The representation of H de ned in this way will in general b e
Groups and representations
reducible It is the decomp osition into irreducible representations of H of a restric
tion of a representation of G that is the central problem of this thesis
The symmetry group of an equilateral triangle
For our rst example of a group we will consider the symmetries of an equilateral triangle
1
3 2
Symmetry op erations leave the triangle unchanged but p ermute the lab els of the
vertices Re ecting in the three symmetry axes exchanges pairs of vertices The
diagram can also b e rotated by clo ckwise or anti clo ckwise which p ermutes all
three vertices Leaving the diagram unchanged is the identity op eration of this sym
metry group The group of symmetry op erations has six elements which naturally
fall into three classes the identity re ections and rotations
Cosets and the quotient group
Given a group G with a subgroup H we can de ne sets of elements of G by taking
an element g G and multiplying all the elements of H by it This set of elements
of G is called a left coset of G and is denoted g H Right cosets can also b e de ned
by multiplying by g on the right
For a nite group G divides into distinct cosets of H Take an element g not
in H none of the elements of g H are in H as otherwise g h h so g h h
and is in H contrary to the assumption All the elements of g H are also di erent
as g h g h implies h h We now have two distinct sets of elements H
H
Chapter Representation Theory
and g H in G If the group G has not b een exhausted we can continue by selecting
some element g not in H or g H In this way G breaks down into a nite numb er
of distinct cosets each of elements
H
G H g H g H
We see that the order of the subgroup must divide the order of the group
H G
An invariant subgroup of G is a group H G for which
g H g H
g H g is the set of elements g hg where h runs through the subgroup H For
an invariant subgroup H we know that g H H g from which we can de ne a
multiplication law for the cosets
aH bH a H b H a bH H ab HH ab H
where we used the fact that H is a subgroup to deduce that HH H The inverse of
a coset aH is a H so the cosets of an invariant subgroup form a group themselves
This group is called the quotient group and is denoted by G H
Classes and characters
For an element g G the class of g is the set of elements g G that can b e obtained
from g by conjugating with another element h G i e
g h g h
If an element g is conjugate to g and g is conjugate to g then g and g are also
conjugate For example if we have g h g h then
g h h g h h h h g h h
The elements of a group can b e partitioned into these disjoint conjugacy classes
The numb er of elements in each class is the order of the class The identity element
Classes and characters
forms a separate class of order one for every group It can b e shown that the numb er
of irreducible representations of a group is equal to the numb er of classes of the group
Consider the example of the symmetries of the equilateral triangle We will take
h to b e a re ection For a re ection we know that h is h If we take g to b e a
clo ckwise rotation by conjugating g by h we obtain g h g h which is an
anti clo ckwise rotation by Both rotations are in the same conjugacy class In
fact we nd that the three sets of symmetries rotations re ections and the identity
form the conjugacy classes of this group We will see later that there are also three
irreducible representations of this symmetric group
The classes of a group are imp ortant when we consider a representation of the
group For two elements in the same class
T g T h T g T h
Then taking the trace of b oth sides by summing the diagonal matrix elements we
nd that
T r T g T r T g
The trace of a representation is a function of the classes of the group and is called
the character of the representation
g T r T g
T
We have already seen that a representation can b e decomp osed in to a direct sum
of irreducible representations
T g
B C
B C
M
B C T g
B C
T g T g
j
B C
B C
A
T g
m
where T is an irreducible representation of G Taking the trace of T we see that
j
X
g g
T T
j
j
Chapter Representation Theory
Any character can b e written as a sum of the characters of the irreducible represen
tations of the group
The irreducible characters have some imp ortant prop erties which will b e used
subsequently A pro of of these relations can b e found in chapter of or any
other intro duction to group theory Firstly the irreducible characters are orthogonal
if averaged over the group
X
g g
k j k
j
G
g
j and k lab el irreducible representations of G g is the complex conjugate of the
character As the character is a function of the classes of G if we lab el the classes
of G by and take g to b e an element of we can rewrite as
X
g g
k j k
j
G
where is the order of the class If we consider the character X of a representation
L
T g T g where the T are the irreducible representations of G then from
j j
N the multiplicity of the irreducible representation T in the decomp osition
k k
of T is
X
X g g N
k k
G
g
The orthogonality of irreducible characters can b e used to decomp ose a general rep
resentation into its irreducible comp onents
The irreducible characters are also orthogonal when the characters of two classes
are averaged over the irreducible representations
X
i
i
i
X
i
i
i
If the irreducible characters are recorded in a character table the two orthogonality
relations and refer to the orthogonality of the rows and columns of the
table resp ectively
The symmetric group
The symmetric group
The set of n distinct symb ols can b e arranged in n orderings A
n
permutation acts on the symb ols by rearranging them into a new order
n
n
There is an identity p ermutation leaving the symb ols in their original p ositions and
all p ermutations have an inverse which undo es the change in order Applying two
p ermutations to the same set of symb ols is equivalent to a single p ermutation of
the symb ols so the set of p ermutations is closed under multiplication and forms
a group This group of all p ossible p ermutations of the n symb ols is called the
symmetric group S S is of order n The symmetries of the equilateral triangle
n n
corresp ond to the symmetric group S where the symb ols that are p ermuted in this
case are the vertex lab els
A cycle ij k l is a p ermutation in which the i th symb ol is moved to the j th
place the j th to the k th and so on The l th symb ol replaces the i th The order
of the cycle is the numb er of terms in the cycle Every p ermutation can b e written
as a pro duct of disjoint cycles For example the p ermutation
a b c d e f d f a c e b
is the result of applying the cycles cycles of order one are omitted by con
vention A cycle of length m can b e further factored into a pro duct of m two
cycles Consequently any p ermutation can b e written as a pro duct of these trans
p ositions A p ermutation is then referred to as even if it can b e written as the
pro duct of an even numb er of transp ositions and o dd if the numb er of transp ositions
required is o dd The pro duct of two even or two o dd p ermutations is even while
the pro duct of an even and an o dd p ermutation is o dd The sign of a p ermutation
sgn is one for o dd p ermutations and zero for even p ermutations
For the symmetric groups two p ermutations which when written as a pro duct
of disjoint cycles have the same numb er of cycles of the same order are in the same
Chapter Representation Theory
conjugacy class We can see this cycle structure of classes by considering conjugating
a cycle by as in For example if is the cycle then
The cycle is e ectively applied to a reordered set of symb ols so for the conjugate
element the length of the cycles is preserved In S the symmetry group of the
equilateral triangle the three re ections written as cycles are and
the rotations are and
The irreducible representations of S
Continuing our example we will nd the irreducible representations of S The trivial
representation where all the elements are represented by is an irreducible repre
sentation of S The alternating representation of the symmetric group is de ned
by asso ciating to the even p ermutations in this case the identity and three cycles
and to the o dd p ermutations the two cycles We can see that the alternating
representation ob eys the same multiplication law as the group a re ection followed
by a rotation is a re ection two re ections is a rotation or the identity This repre
sentation is one dimensional and so irreducible
There is also a two dimensional irreducible representation of S To nd it we use
the de ning representation of the p ermutation group The de ning representation
of a p ermutation in S is the matrix D where
n
if i j
D
ij
otherwise
The symmetric group
For S the de ning representation is
B B C C
B B C C
D I D
B B C C
A A
B C B C
B C B C
D D
B C B C
A A
B B C C
B B C C
D D
B B C C
A A
This representation is reducible multiplying a vector x x x by any of these p ermu
tation matrices will leave the vector unchanged It forms a one dimensional subspace
which is invariant under the group transformations The subspace transforms ac
cording to the trivial representation of S We can bring this representation into
blo ck diagonal form by changing the basis One p ossible choice of transformation is
B C
B C
P
B C
A
Then P DP decomp oses into the trivial representation and the irreducible two
dimensional representation T
A A
T I T
A A
T T
A A
T T
Chapter Representation Theory
We will see later that the irreducible representations of S are in one to one corre
n
sp ondence with the partitions of n and so these three representations are the only
irreducible representations of S
The group SU n
A unitary matrix is a matrix whose inverse is its Hermitian conjugate SU n is
the group of n n unitary matrices with determinant one The group SU n is a
n
connected compact Lie group acting on vectors in C Elements of the group can
b e parameterised by real numb ers where
j
X
g exp i
j j
j
The are n n hermitian traceless matrices and are called the generators of SU n
j
There are n linearly indep endent traceless hermitian matrices so the group el
ements require n co e cients to parameterise the elements
j
The generators of the group are in nitesimal group elements
exp i I i
for small These in nitesimal group generators form a vector space and so are
often easier to work with than the group elements themselves If we lo ok at the
pro duct of group elements
exp i exp i exp i exp i I
a a a
b b b
P
This pro duct is also a group element and so can b e written as exp i As
c c
we must have
i f
a c
b abc
The constants f are the structure constants of the group The structure constants
abc
determine the multiplication law of the group They ob ey the Jacobi identity
f f f f f f
bcd ade abd cde cad bde
The group SU n
The structure constants also determine a representation of the group the adjoint
representation if see chapter The set of generators with com
a ij aij
mutation relations de nes an algebra asso ciated to the group the Lie algebra su n
SU
A simple example of the structure constants for three generators is f the
abc abc
completely antisymmetric tensor The commutation relations are then the angular
momentum commutation relations The de ning representation of the Lie algebra
is the n n representation of the algebra which generates the group itself For
the angular momentum commutation relations traceless hermitian generators can
b e de ned from the Pauli matrices
i
A A A
x y z
i
These generate the group SU
Ro ots and weights of Lie algebras
A representation of the group de nes a representation of the algebra and vice versa
This corresp ondence is natural but to actually show that representations are con
nected in this way takes some work a go o d discussion is found in chapter of
We will turn now to discuss an irreducible representation T SU n asso ciated with
a representation T su n of the algebra
From the set of n generators we select a Cartan sub algebra This is a
maximal set of n commuting generators which we will lab el H The space
i
the representation T su n acts on will b e spanned by eigenvectors of the Cartan
sub algebra and the eigenvalues with resp ect to the Cartan sub algebra will b e used
to lab el these eigenvectors
H j T i j T i
i i
Chapter Representation Theory
The eigenvalues are the weights of a representation and the vector with com
i
p onents is a weight vector The order of the terms in is arbitrary the results
i
apply equally to any ordering
The adjoint representation is the representation obtained by taking the genera
tors as the basis of the space the algebra acts on A generator then acts on a basis
vector by commutation
j i j i
i j i j
The Cartan sub algebra corresp onds to a set of vectors with zero weight
H jH i
i j
Diagonalising the space acted on by the generators gives a set of states jY i where
H jY i jY i
i i
The states corresp ond to generators Y so that
H Y Y
i i
The Y are linear combinations of the generators not in the Cartan sub algebra
These weight vectors of the adjoint representation with comp onents are the
i
roots of the Lie algebra
Let us see how an op erator Y acts on a vector in a general representation
T su n
H Y j T i H Y j T i Y H j T i
i i i
Y j T i
i
We see that the vector Y j T i is lab elled by the weight vector We have
now identi ed the ro ot vectors with raising and lowering op erators for the weights
of a Lie algebra By cho osing an initial state and rep eatedly applying raising or
lowering op erators we nd that
p q
The group SU n
p and q are the numb er of times the op erator Y or Y can b e applied to b efore
reaching zero Both p and q dep end on the choice of weight and ro ot Equation
is derived in chapter of it is equivalent to the condition that angular
momentum eigenvalues b e integer or half integer
The highest weight classi cation of irreducible representa
tions
To provide a de nition of a p ositive weight we x the order of the generators in the
Cartan sub algebra This xes the order of the comp onents of the weight vector
We then de ne a weight vector to b e p ositive if its rst non zero term is p ositive
With this de nition we can order the weight vectors if The
highest weight vector of a representation is greater than other weight vectors and is
non degenerate
To provide a nontrivial example we intro duce generators of the de ning repre
sentation of SU
i
A A
S S
i i
i
I iI
A A
p p
E E
x y
I iI
I
A
p
E
z
I
are the Pauli matrices de ned in and I is the identity matrix To
i
provide all the generators of SU we must also include the commutators of these
matrices We can select the Cartan sub algebra to b e the set of diagonal matrices
E S and S in that order A weight vector then consists of the eigenvalues
z z z
Chapter Representation Theory
e s s A basis for the space the representation acts on is
z z z
B C B C B C B C
B C B C B C B C
B C B C B C B C
B C B C B C B C
x x x x
B C B C B C B C
B C B C B C B C
A A A A
As the Cartan sub algebra is diagonal these basis vectors are eigenvectors of the
sub algebra and can b e lab elled by their weights
p p
p p
Using the de nition of a p ositive weight vector
is the highest weight vector of the de ning representation of SU
By de ning which weights are p ositive we are also provided with a classi cation
of raising and lowering op erators dep ending on whether the asso ciated ro ot vector is
p ositive or negative A simple root is a p ositive ro ot which can not b e written as the
sum of two p ositive ro ots A p ositive ro ot which is not simple can b e written as the
sum of two p ositive ro ots and either these are simple or we can rep eat the pro cedure
Using this scheme we see that any p ositive ro ot can b e written as a sum of simple
ro ots with p ositive integer co e cients From condition it can b e shown that
the simple ro ots are linearly indep endent and span the space of weight vectors
chapter Therefore there are the same numb er of simple ro ots as generators in the
Cartan sub algebra Starting from the simple ro ots all other ro ots can b e found by
combining the ro ots and checking the condition
By de nition op erating on the highest weight of a representation with a raising
op erator lab elled by a p ositive ro ot must give zero As every p ositive ro ot is a sum
of simple ro ots it is su cient to consider only the simple ro ots We will lab el the
j
simple ro ots j m Substituting p for a heighest weight into we
The group SU n
have
j
j
q
j
j k
The q s are non negative integers If a highest weight is the weight of a repre
k j k
sentation where q and q for j k then is called a fundamental weight
Any highest weight can b e written as a sum of these fundamental weight vectors
X
k k
q
k
Each highest weight lab els an irreducible representation of SU n The represen
k
tations with highest weights are called the fundamental representations It can
b e shown that the irreducible representations of SU n can b e constructed from
k
tensor pro ducts of the fundamental representations where the integers q are the
multiplicity of the fundamental representations in the pro duct
The ro ots are the di erences b etween the weights and the simple ro ots are the
minimum p ositive di erences From the weights of the de ning representation of
SU the simple ro ots of the group are
p
We can see that the ro ots and raise the S and S eigenvalues by one
z z
resp ectively and so corresp ond to the op erators S and S constructed from the
is E S With the simple generators in the usual way The raising op erator Y
2
ro ots we can use to nd the fundamental weights
p
p
p
The exact form of the simple ro ots and fundamental weights dep ends on the de ni
tion of the generators and the choice of the Cartan sub algebra De ning the simple
ro ots of SU in this form will simplify the work later
Chapter Representation Theory
Representations of the symmetric group and Young
tableau
Partitions graphs and tableau
We turn now to nding the irreducible representations of the symmetric group The
construction uses tableau and it is these graphs of partitions which we will intro duce
rst Let b e a partition of an integer n into m integer parts
m
where jj n and the order of terms is immaterial If the
m
numb er of one s in is a the numb er of two s b etc then the partition can b e
a b c
written We have already seen that the class of an element of
the symmetric group is determined by the numb er of cycles of each order and so
the classes of S are lab elled by partitions of n As partitions don t dep end on the
n
order of the we can adopt the convention that Then to
i m
each partition we asso ciate a graph or tableau with b oxes in the rst row
b oxes in the second and so on the rows b eing aligned on the left So for example
the partition of is asso ciated with the graph
We will keep to the convention that a graph refers only to an empty sequence of
b oxes When the b oxes are lab elled by symb ols the gure will b e referred to as a
tableau In some cases the symb ols that distinguish the b oxes may b e suppressed
when the gure is drawn although in the text it will still b e referred to as a tableau
We will see that not only are the classes of S lab elled by graphs of n b oxes but so
n
are the irreducible representations of S
n
Characteristic units of the group algebra
The Frobenius algebra of the symmetric group is the algebra obtained by taking the
elements of the group as the basis of the algebra Then the multiplication law of the
group determines a pro duct of the basis elements If is an element of the group
j
Representations of the symmetric group and Young tableau
an element of the algebra can b e written
X
j j
j
where the are complex co e cients The group algebra has many interesting prop
j
erties see chapter We will state a few of relevance here without pro of
The group algebra is isomorphic to a direct sum of matrix sub algebras subsets
of the elements of the algebra closed under multiplication Each of these sub algebras
de nes a representation of the group as a group element can b e expressed as a sum
of elements of the sub algebra The numb er of matrix sub algebras in the group
algebra is equal to the numb er of classes of the group and the representations of S
n
de ned by the sub algebras are the irreducible representations The group algebra
is an algebra that contains all the irreducible representations of the group when
written as a sum of matrix sub algebras Each irreducible representation app ears
with multiplicity equal to the dimension of the representation
There are particular elements of the group algebra that are asso ciated with
the representations of the group These characteristic units of the algebra are the
idemp otent elements such that We can express as a sum of elements of
the irreducible matrix sub algebras
X
j
where j lab els the sub algebra As is idemp otent it can b e transformed into a
j
r l r
i i
diagonal matrix Multiplying a group element by this characteristic unit
and taking the trace we obtain a comp ound character of the group
X
X r
j j
A primitive characteristic unit is the characteristic unit asso ciated with the irre
ducible character r and r for j k From the de nition of the
j
k k
primitive characteristic units we see that any characteristic unit can b e written as a
sum of primitive characteristic units Two primitive characteristic units of the same
matrix sub algebra are transforms of each other Finally we state a lemma used
later
Chapter Representation Theory
Lemma The product of two primitive characteristic units is either zero
nilpotent or a multiple of a primitive characteristic unit
If the two primitive characteristic units are in di erent sub algebras the pro duct
is zero If they are in the same sub algebra and the pro duct is not zero then as the
primitive characteristic units b oth have rank one the pro duct must also have rank
one the rank of the matrix is the numb er of linearly indep endent rows or columns
The reduced characteristic equation of the pro duct is a quadratic with a zero ro ot
x x If the pro duct is nilp otent x otherwise it is a multiple of
a primitive characteristic unit
We have seen that the irreducible characters of the group are asso ciated with
primitive characteristic units which are linear combinations of the group elements
We will show that the characteristic units of S can b e constructed by using tableau
n
and that they are also used to construct irreducible representations of the sp ecial
unitary groups
Characteristic units of S
n
The symmetrisation op erator acting on r symb ols is the sum of the group elements
of the symmetric group that p ermute the symb ols It is an element of the group al
gebra of a symmetric group acting on the symb ols The antisymmetrisation op erator
sums the same group elements but with a minus sign attached to o dd p ermutations
We take the graph of the partition of n and assign to every b ox one of the
integers from to n this is a Young tableau intro duced by Young in The inte
gers are just one choice of a set of n symb ols and so the order that the integers are
assigned to b oxes is not signi cant
Let P b e the symmetrisation op erator for the symb ols in the i th row of the
i i
tableau Then P is de ned to b e the pro duct of the m row symmetrisation op er
ators Multiplying P by a p ermutation in P pro duces the same symmetrisation
i i
Representations of the symmetric group and Young tableau
op erator P So P is an idemp otent element of the group algebra Consequently
i i i
P is also idemp otent and therefore also a characteristic unit Similarly
m
we de ne N as the pro duct of the antisymmetrisation op erators of the symb ols in
each column N is also a characteristic unit of S is the length of the
q n i
i th column of
Neither of these characteristic units is primitive but if b oth are written as a sum
of primitive characteristic units they have only a single primitive characteristic unit
in common see chapter The primitive unit in b oth N and P is asso ciated with
the character From lemma the pro duct of two primitive characteristic
units is zero nilp otent or a multiple of a primitive characteristic unit The pro duct
contains the identity element of S It can not therefore b e N P
q m n
zero or nilp otent and so is a multiple of the primitive characteristic unit The
normalised primitive characteristic unit is
I
NP
S
n
I is the dimension of the irreducible representation We have asso ciated the
irreducible representations of S to the partitions of n via the primitive charac
n
teristic units generated by a Young tableau of shap e
As primitive characteristic units are a central feature of the later chapters it
is worth seeing an example for S An irreducible representation lab elled by the
partition is asso ciated with a tableau
1 2
3
Any other lab elling of the b oxes is equally valid We can now write down P and N
for the tableau
P I
N I
I
NP I
Chapter Representation Theory
It can b e veri ed that this is an idemp otent element of the group algebra By con
struction it is asso ciated with an irreducible representation lab elled by the partition
Constructing representations of SU n with characteristic units
We have seen that Young tableau of n b oxes lab el the irreducible representations of
S The characteristic units they generate can also b e used to construct irreducible
n
representations of the sp ecial unitary groups We will construct representations of
SU n from tensor pro ducts of the de ning representation this is the n dimensional
irreducible representation The de ning representation will have n weight vectors
n n
for its basis vectors ordered so that The di erences
b etween adjacent weights provide the simple ro ots of SU n
i i i
The generators can b e normalised so the weights all have the same length and the
angle b etween weight vectors is the same for any pair For the ro ots
i j
j i or i
i j
i j or j
This can b e veri ed for the ro ots of the SU de ned in
The fundamental weights are given by a sum of the weights of the de ning
representation
j j
X X
j k k
j k
k k
From the condition they must ob ey the relation
i j
ij
i
and as the de ning representation is a fundamental representation we can use the
results for multiplying ro ots to check
Representations of the symmetric group and Young tableau
If we consider constructing a new representation by taking the tensor pro duct of
q de ning representations it is clear that the the highest weight vector would b e the
tensor pro duct of q identical vectors all lab elled by The representation is not
irreducible but it must contain the irreducible representation with highest weight
q If the tensor pro duct is symmetrised then the tensor pro duct of q s will still
have the same highest weight vector although the dimension of the representation
has b een reduced The symmetrised tensor pro duct is an irreducible representation
of SU n as it is the representation of least dimension which has highest weight
q Symmetrising a tensor pro duct of q terms is equivalent to acting on the tensor
pro duct with the primitive characteristic unit of S lab elled by a tableau with a
q
single row
Rather than symmetrising we could consider antisymmetrising a tensor pro duct
of q de ning representations Permutations of a tensor pro duct of basis vectors of the
de ning representation are added with the vectors resulting from o dd p ermutations
acquiring a minus sign This is also the action of a primitive characteristic unit of
S lab elled by a tableau of a single column In this case it is clear that a tensor
q
pro duct of q identical basis vectors will b e zero as each o dd p ermutation will
cancel an even one To b e non zero the highest weight vector must b e a pro duct of
q
q di erent basis vectors of the de ning representation the vectors The
highest weight vector will have weight
X
i
i q
q
which is the highest weight of the fundamental representation see equation
The fundamental representations of SU n are recorded using a Young tableau
for the characteristic unit of S used to construct the highest weight of the repre
q
sentation
Chapter Representation Theory
ν1 1 ν2 2 ......
νq q
i
Rather than ll in the tableau with basis vectors just the lab els i will b e used to
denote the vectors as in the example ab ove
We have already noted that a general highest weight can b e constructed from the
fundamental weights equation We want to construct an arbitrary highest
weight vector of an irreducible representation from the de ning representation It
P
i i
should have a highest weight q This highest weight vector will b e
i
recorded in the tableau
qn-1 q2q 1
1... 1 1... 1 1... 1 1... 1 2... 2 2... 2 2... 2
. . . ......
n-1 ... n-1
There are n fundamental weights that the representation can b e constructed
i
from and each is rep eated q times in the tableau so this has the correct weight
P
n
j
q To construct the The tableau is of a partition where
k
j k
representation of SU n the primitive characteristic unit of the symmetric group
asso ciated with the tableau is constructed If there are q b oxes in the tableau this
is a characteristic unit of S The characteristic unit I N P consists of a set
q S
q
of symmetry conditions which can then b e applied to the tensor pro duct of de n
ing representations The symb ols q used to construct the characteristic unit
corresp ond to the terms in the tensor pro duct of the q basis vectors of the de ning
representation of SU n A basis vector of the irreducible representation of SU n
is constructed by assigning basis vectors of the de ning representation to b oxes of
the tableau The pro duct is symmetrised with resp ect to the terms in the same row
then antisymmetrised with resp ect to those in the same column The order of terms
in the tensor pro duct has not b een sp eci ed but neither has the lab elling of the b oxes
Characters of U n and SU n
in the tableau for the primitive characteristic unit so this ambiguity is not imp ortant
To see that has the highest weight p ossible for the given shap e of tableau
i j
we can consider replacing one of the vectors with a vector with higher weight
j
where j i As each column contains all the with j i antisymmetrising the
column gives zero no state with higher weight can b e constructed using that shap e
of tableau
This pro cedure enables us to construct irreducible representations of SU n from
any Young tableau of up to n rows Representations with all the p ossible highest
weights can b e constructed this way so we can construct all the irreducible rep
resentations of SU n If we attempted to construct an irreducible representation
of SU n using a tableau with more than n rows each column would need to b e
lled with di erent basis vectors of the de ning representation The de ning rep
resentation has only n basis vectors so this is not p ossible and no representations
can b e constructed this way A tableau with n rows can b e used to construct an
irreducible representation of SU n Each of the columns of length n will contain
all n of the basis vectors of the de ning representation These terms will b e present
in any vector of the representation and so this is the same representation as that
which is constructed using the tableau with the columns of length n removed The
irreducible representations of SU n are lab elled by and can b e constructed from
Young tableau with up to n rows Columns of n b oxes can b e added to the
tableau without changing the representation of SU n
Characters of U n and SU n
If we return to thinking of the tableau as a partition then we lab el the irreducible
representations of SU n with a string of integers f f f jf j f
n
f f is the total numb er of b oxes in the tableau f is one of the p ossible
n
partitions of jf j though not all partitions lab el irreducible representations of SU n
as some partitions could b e into more than n parts The characters of U n and
Chapter Representation Theory
SU n are functions of the classes of the groups and these functions will dep end
on the irreducible representation lab elled by f To de ne these functions we rst
parameterise the classes of the group We will deal with U n and sp ecialise the
results to SU n where necessary A full account is found in or
An element Q U n is conjugate to a diagonal element where the diagonal
terms have mo dulus one see chapter So
C B
C B
C B
C B
U QU
C B
C B
A
n
i
i
for some unitary matrix U and j j for all i Let e then the n angles
i i
parameterise the classes of U n We will use to refer to the set of diagonal
elements The class is invariant under a p ermutation of the diagonal
n
terms so is unordered
As U n is a continuous group to sum a function of the classes over the group
we asso ciate a volume element to the parameters corresp onding to the prop ortion
of the space of unitary matrices in the class This is the uniform Haar measure on
the unitary group Let
Y
i
k
i k
then the prop ortion of the group parameterised by angles lying b etween and
i
d d A class function of the unitary group is a symmetric d is
n i i
function of the angles and with this de nition of a measure on the group we can
de ne the average of a function g
Theorem The average of a class function g is
n
Z Z
g d d
n
where
Z Z
d d
n
Characters of U n and SU n
The class functions that we will average over the group are the characters of
U n and its subgroups The character is the trace of the matrix representation
T U n As it is only a function of the class of the group elements not the elements
themselves we only need evaluate it on the diagonal elements of the group fg The
subgroup of diagonal elements is Ab elian the multiplication law is commutative
T T T
n n
n n
The representation decomp oses into a sum of one dimensional representations Z
k
where jZ j as in chapter of These representations ob ey the multipli
k
cation law Solving for one dimensional functions of
k
k
1
n
Z
k
n
The characters of representations of U n are sums of monomials with integer co ef
cients
The characters are symmetric functions of the angles They can however b e
asso ciated with antisymmetric functions which app ear naturally when averaging
a pro duct of characters over the group The simplest antisymmetric functions are
alternating sums of the monomials from which all the antisymmetric functions can
b e constructed by taking linear combinations
X
sgn exp k k
n n n
k
The sum is over all p ermutations of the integers k This alternating sum can b e
written as a Vandemonde determinant
k k
k
1 1
1
n
n
k
k k
k
n n
n
n
k k
j The n rows are generated by which we will abbreviate and write as j
n
replacing k with each of the k in turn We observe that itself is one of these
i
alternating sums
n
n
j j
n n
n
Chapter Representation Theory
n
n
The abbreviation j j will always refer to the alternating sum rather
n
than some general alternating sum with co e cients n n etc This is less
elegant than Weyl s notation but will b e necessary to track the di erent
i
Integrating two monomials
Z Z
Z d d Z
n
k
k k k
k k
n
1
n
1
From this we nd the pro duct of two of the antisymmetric functions
Z Z
d d n
n
k
k k
k k
k
n
1
n
1
which also gives us the volume of the unitary group n
U n
If we take a comp ound character X of the group and consider the antisymmetric
function X it is a sum of monomials and we can order the monomials in a similar
if the rst non zero di erence k k way to that used to order weights Z Z
i
k k
i
is p ositive Ordering the terms of X in this way we take the rst term to b e
c Z As X is antisymmetric it must also contain all the other terms in c
k k
and as any p ermutation of the k s is a monomial lower that c Z we know that
k
k k k We can subtract c from X and rep eat the pro cedure
n
k
expanding X as a sum of the antisymmetric functions
X c c
k
k
X over the group Averaging X where
k
k
Z Z
X d d c c X
n
However if X is the character of an irreducible representation then the average over
the group should b e one by character orthogonality So c and the other
co e cients c etc are zero The irreducible characters of U n are of the form
k k
j j
n
n n
j j
n
where k k this is the Weyl character formula for U n The highest
n
monomial in this character is
f f
f
1 2
n
n
Characters of U n and SU n
where
f k n f k n f k f k
n n n n
and therefore f f
n
The irreducible representations of U n are lab elled by a string of decreasing
integers f f in the same way that we deduced from the highest weights that
n
the representations of SU n are labled by tableau with a string of n decreasing
integers To use the Weyl character formula to nd the irreducible characters
of SU n we restrict the characters of U n to elements of the SU n subgroup To
do this we set
n n
Irreducible representations of U n restricted to SU n are also irreducible rep
n
d d u SU n If resentations of SU n Let u U n and d Det u
n n n
T d d u is an irreducible representation of SU n then d T d d u is
an irreducible representation of U n
If we lo ok at the character formula for SU n with condition
f n f n
1 1
f n
1
n
n
f f
n n
f
n
n
n
f
n n
n
n
n
n n
f
n
and the Multiplying the rst column of the determinant in the numerator by
f
n
do esn t change the determinant Rep eating this for the other last by column by
Chapter Representation Theory
columns
f f n f f n
n n
1 1
f f n
n
1
n
n
f
n n
n
n
n
n n
The character of f f is the same as f f f f This is equiv
n n n n
alent to the statement that irreducible representations of SU n lab elled by tableau
with n rows are equivalent to the representation lab elled by the tableau with the
columns of length n removed We have not established that the character lab elled
by f f corresp onds to the same representation of SU n that is con
n
structed from the tableau f f but this can b e done see
n
Characters of S
n
There is a well known formula for the irreducible characters of the symmetric group
due to Frob enius a derivation of which is found in chapter of The character
of an element dep ends on its class which we lab el with a partition ! of n where
the class is of all elements with one cycles two cycles etc As we saw with the
characteristic units of the symmetric group the irreducible representations of S are
n
also lab elled by partitions of n The irreducible characters app ear
p
in the formula as co e cients in a p olynomial of n variables x
r
X Y Y
p p j
p
j
1 2
j
x x x x x x x
p
r s
n
r s
j n
If the left hand side is expanded once for each of the classes ! the characters of all
the irreducible representations app ear as the co e cients of certain monomials in the
expression From this formula the characters of an irreducible representation of S
n
can b e computed
The Littlewo o d Richardson theorem
The Littlewo o d Richardson theorem
The Littlewo o d Richardson theorem is a combinatorial rule for computing the co
e cients in the decomp osition of a pro duct of primitive characteristic units of the
b e a primitive characteristic unit of S acting on m symmetric group Let A
m
S
m
symb ols constructed from a tableau and A b e a primitive characteristic unit
S
n
acting on n di erent symb ols We know that S S is a subgroup of S The
m n m n
A pro duct of the two primitive characteristic units A will also b e idemp otent
S
S
m
n
and so a characteristic unit of S In general it will not corresp ond to an irre
m n
ducible representation of S but will b e a sum of primitive characteristic units of
m n
S
m n
X
A Y A A
S S
S
m
m+n
n
It is these co e cients Y that we want to compute
The Littlewo o d Richardson Theorem
To every tableau which can be constructed according to the fol lowing rule there
corresponds a primitive characteristic unit A in the decomposition of the prod
S
m+n
A uct A and this decomposition is complete
S
S
m
n
LR Take the tableau intact and add to it the symbols in the
rst row of to make a new tableau without changing the
order of the symbols After the addition no two added sym
bols may be in the same column Next add the remaining
rows of in succession according to the same rule
LR The only al lowed additions are those where each symbol in
is placed in a later row than the symbol in the same
column from the preceding row of
These rules were originally prop osed by Littlewo o d and Richardson in
however the subsequent pro ofs in and are not complete the rst correct
pro ofs app eared in the s some forty years later One version of the complete pro of
Chapter Representation Theory
is given by Macdonald in
Multiplying Young tableau
The Littlewo o d Richardson rules LR can b e more simply stated as a pro cedure for
multiplying two tableau one of shap e and the other Take the graph without
loss of generality it can b e assumed j j jj and ll the b oxes on the rst row with
a s the second row with b s etc So if we have
a1 a2
b1
Add the a s to in any way which makes a new graph with a maximum of one a in
each column In our example there are ve p ossibilities
a1 a 2a 1 a1 a2
a2
a1 a 2a 1
a2
Then rep eat the pro cedure for each of the other rows in turn with the added condition
that counting right to left and top to b ottom the numb er of a s the numb er of
b s the numb er of c s So in our example adding the single b in all p ossible
ways to get a graph we nd
a1 a 2b 1 a1a 2 a1 a 2 a1 b 1
b1 a2
b1
a1 a1 a1b 1 a1 a1 a1 b 2 a2 b1
b1 a2 a2 a2
b1
b1 b1
a1 a2 a1a 2 a1 a1b 1 a1 a1 b1 a2 a2 a2 b1 a2
b1
If we take the rst of these tableau and count the numb er of a s and b s from the
right we start with one b and no a s in the right hand column This do esn t agree
The Littlewo o d Richardson theorem
with the condition that the numb er of a s always b e greater than or equal to the
numb er of b s so we discard this tableau Checking the other tableau similarly and
counting top to b ottom as well we nish with a set of nine tableau
a1a 2 a1 a 2 a1 a1
b1 a2b 1 a2
b1 b1
a1 a1
b1 a1a 2 a1 a1
a2 a2 b1 a2b 1 a2
b1 b1
Within this set of tableau the graph is rep eated twice with di erent ar
rangements of the letters In terms of characteristic units we have two primitive
characteristic units of S which b oth corresp ond to the same irreducible repre
m n
sentation two equivalent primitive characteristic units The numb er of graphs in
so in our example the pro duct of the two tableau is the co e cient Y
Y
Applying the Littlewo o d Richardson theorem to representa
tions
As the primitive characteristic units pro ject onto irreducible representations of the
symmetric groups the Littlewo o d Richardson theorem is also a theorem ab out the
decomp osition of a representation of S into irreducible representations of the
m n
S S subgroup
m n
Consider any group G of order with a subgroup H of order An irreducible
G H
representation of G restricts to a representation of H If the irreducible characters
of G are X and the irreducible characters of H are then for h H
i j
X
X h c h
i ij j
j
c is the multiplicity of the representation j of H when the representation i of G is
ij
restricted to H
Chapter Representation Theory
The character of a group is a function of the classes of the group Let b e a
class of G of order of these elements are in H Elements in di erent classes
G H
of G must b e in di erent classes of H but a class of G can break up into several
classes of H We lab el the classes of H which are in the class of G with
r
Using the character orthogonality relation we can nd the co e cients c
ik
X
j
X c
i j
ik
k
H
H
j
The sum is over all classes of H With a formula for the co e cients c we can
ik
investigate the comp osite character of G that they de ne
X X X
j
c X X X
i i j i
ik
k
H
H
i i
j
j
X
H
j G
k
H
G
j
where we applied the orthogonality relation This can b e rearranged into a
second formula relating the characters of a group and its subgroup so we have
X
c h X h
ij j i
j
k
X X
G
H
c X
j ij i
k
H
G i
k
The multiplicity of the representation j of H when the representation i of G is re
stricted to H is the same as the multiplicity of the representation i of G when the
representation j of H is used to induce a representation of G
Returning to characteristic units of the symmetric group the Littlewo o d Richardson
theorem gives a pro cedure for calculating the co e cients of the primitive characters
of S from a primitive character of the subgroup S S These are the
m n m n
co e cients of the induced representation in so the same co e cients from the
Littlewo o d Richardson theorem also give the decomp osition of a representation of
S into irreducible representations of S S
m n m n
The Littlewo o d Richardson theorem
Decomp osing representations of SU m n into representa
tions of SU m SU n
In section we discussed how characteristic units of S are used to construct
n
irreducible representations of SU n In and the Littlewo o d Richardson the
orem for multiplying tableau is used to decomp ose a representation of SU m n
into irreducible representations of its SU m SU n subgroup We know that the
multiplicity of an irreducible representation T SU m SU n when an irreducible
j
representation R SU m n is restricted to SU m SU n is equal to the numb er
i
of representations T induced by a representation R of the subgroup
j i
A representation of SU m SU n is constructed from two characteristic units
one of S and another of S The characteristic unit of S resp ectively S act on
q p q p
the tensor pro duct of q p basis vectors of the de ning representations of SU m
resp ectively SU n The pro duct of the two primitive characteristic units is a char
acteristic unit of S and induces a representation of SU m n The Littlewo o d
q p
Richardson theorem gives the numb er of primitive characteristic units of S in the
q p
decomp osition of the pro duct of the primitive characteristic units of S and S Each
q p
primitive characteristic unit generates an irreducible representations of SU m n
The numb er of irreducible representations of SU m n induced by the represen
tation of SU m SU n is the same as the multiplicity of the representation of
SU m SU n in the decomp osition of the chosen representation of SU m n
Itzykson and Nauenb erg p oint out that this simple situation is complicated
slightly by the use of multiple graphs to lab el a single representation of SU n All
the irreducible representations of SU m SU n are lab elled by two graphs of m
and n rows resp ectively but without changing the representation we can add
columns of m b oxes to the rst graph and columns of n b oxes to the second For
example the representation corresp onds to the graphs α β ...... n . m . .
.
Chapter Representation Theory
Applying the Littlewo o d Richardson theorem we multiply the two tableau pro ducing
tableau which lab el irreducible representations of SU m n Counting the numb er
of tableau of a particular shap e gives the multiplicity of the irreducible representa
tions of SU m SU n in the decomp osition of the representation of SU n m
lab elled by a tableau of that shap e Tableau with columns of m n b oxes can b e
discarded as the results for these representations of SU m n can b e obtained by
considering the tableau with the columns of length m n removed
Let us lo ok at a simple example We will nd the multiplicity of the repre
sentation of SU SU when the representation of SU is
restricted to the SU SU subgroup The representation is lab elled by
a tableau of seven b oxes For this to b e a result of the multiplication of two tableau
the two tableau must have a total of seven b oxes b etween them To the tableau
we can add columns of three b oxes without changing the representation and
we can add columns of two b oxes to For the two tableau to consist of seven
b oxes in total we must take and We want to nd the pro duct of the
tableau
This is the tableau multiplication done in section The graph
app ears twice in the result so the representation of SU when restricted to
the SU SU subgroup contains two copies of the irreducible repre
sentation of the subgroup It is this pro cedure that will b e extended in chapter to
provide a tableau metho d achieving a similar decomp osition for a subgroup of SU
We have now reviewed the representations and characters of the symmetric and
unitary groups and discussed the relationship b etween them We have also dis
cussed results relating to the decomp osition of a representation of a group restricted
to a subgroup culminating in the Littlewo o d Richardson theorem which provides a
The Littlewo o d Richardson theorem
metho d of evaluating the multiplicity of representations b oth of subgroups of the
symmetric and the unitary groups This should provide the necessary to ols for the
subsequent investigation
Chapter
Quantum indistinguishability
Berry and Robbins BR in a series of pap ers and have prop osed an
alternative formulation of non relativistic quantum mechanics in which the spin
statistics theorem is derived from the prop erties of a p osition dep endent spin basis
of the wavefunctions In this chapter we will review this construction in order to
demonstrate the underlying group theoretical prop erties The particular use of the
Schwinger op erators in can then b e seen as a choice of a certain set of represen
tations of the groups The subsequent chapters will involve deriving the prop erties
of this construction for general representations
I will not present the entire scop e of their work here In particular the dis
cussions of the relationship b etween the construction relativity and the invariance
of the system under Lorentz or Galilean transformations has b een omitted along
with the extension to particles with additional quantum prop erties such as colour
strangeness or isospin These are discussed in
Intro duction
The construction of Berry and Robbins suggests an elementary non relativistic basis
for the spin statistics theorem As quantum mechanics alone is insu cient to derive
a spin statistics connection it is necessary to include additional physical p ostulates
To b e accepted as an explanation of spin statistics these additional requirements
The p osition dep endent spin basis
should b e more transparent than the symmetrisation p ostulate itself Berry and
Robbins suggest that the spin statistics connection could b e a consequence of two
additions to normal quantum mechanics the correct incorp oration of the indistin
guishability of identical particles so that the space of wavefunctions has built into
it the indistinguishability of states related by the exchange of particles p ositions and
spins and the singlevaluedness of wavefunctions on this space
The construction is reminiscent one of the classical b elt tricks that Feynman
found indicative of the spin statistics connection see gure In this mo del
fermions are seen as tethered ob jects where exchange intro duces a twist in the ribb on
connecting them whilst b osons corresp ond to the ordinary untethered ob jects We
will see that the construction avoids the ob jection that elementary particles have
no ribb on like top ological marker as the spin basis itself records the exchange of
fermions without a classical ribb on
The p osition dep endent spin basis
The wavefunction of a system of n particles with spin s would normally b e expanded
on a spin basis jMi where the basis vectors are lab elled by the z comp onents of the
n spins M fm m m g
n
X
j R i R jMi
M
where R fr r r g is a p oint in the con guration space of the n particles
n
n
the space R with coincident p oints removed so no two particles can o ccupy the
same p osition In this description the identical particles are still identi ed by their
lab els so for example if we exchange particles and this is a di erent p oint in the
con guration space In order to make the particles truly indistinguishable p ermuted
con gurations of particles will b e identi ed and we will insist that the wavefunction
b e singlevalued on this new con guration space So if is an element of S
n
j R i j R i
Chapter Quantum indistinguishability
The p ermuted con guration is R fr r g To exchange the particles
n
rather than just the p osition lab els we must exchange the spins of the particles
when we exchange the p ositions In BR this is done using a p osition dep endent
unitary transformation U R to generate a p osition dep endent spin basis
jM R i U R jMi
This p osition dep endent spin basis jM R i is required to have certain prop erties
The basis dep ends smo othly on R
There is a single state for all p ermutations of particles this excludes paras
tatistics
i R
j M R i e jM R i
is a p ermutation of the n particles with their spins
M fm m g
n
Spins are parallel transp orted
hM R jrM R i
M and M are arbitrary sets of spin quantum numb ers This requirement
ensures that their are no lo cal changes in phase asso ciated with travelling
around a contractible closed lo op
By combining the second and third conditions we can show that the p osition
dep endent spin basis must transform according to the trivial or alternating repre
sentation of the p ermutation group First we will take to b e an exchange of two
of the particles keeping the rest xed Applying the condition twice we nd
that
i R
jM R i j M R i e j M R i
i R R
e jM R i
The general solution of this is
R k R
The p osition dep endent spin basis
where k is an integer and
R R
Using and the result we can write
h M R jr M R i hM R jrM R i i r R hM R jM R i
hM R jrM R i i r R
MM
Applying the parallel transp ort condition we have
r R
MM
requires R to b e a constant As R To satisfy for all R M and M
changes sign under o dd p ermutations it must b e identically zero We have the result
k
j M R i jM R i
In deriving we assumed that the p ermutation was an exchange of two
particles However as all p ermutations can b e constructed from a pro duct of two
cycles this can b e rewritten for a general p ermutation
k sgn
j M R i jM R i
If k is even the states of the p osition dep endent spin basis transform according to
the trivial representation of the p ermutation group If k is o dd the states transform
according to the alternating representation
In BR provide a construction of U R for which they show that
k s
The construction for n particles dep ends on the solution of a top ological problem I
will describ e later In the two particle case the solution of this problem is simple and
the construction can b e easily explained Before lo oking at the explicit construction
of U R we will see how taking k s leads to the spin statistics connection
Chapter Quantum indistinguishability
Quantum mechanics in the p osition dep endent spin
basis
First we assert the condition that wavefunctions on the p osition dep endent basis are
singlevalued
j R i j R i
From and we know that
X
s sgn
j R i R j M R i
M
M
X
s sgn
R jM R i
M
M
So for the comp onents of the wavefunction we have
s sgn
R R
M M
This has the form of the spin statistics relation and to demonstrate that it is equiv
alent we show that these co e cients ob eys the same Schr odinger equation as the
co e cients in the usual spin basis
An op erator A in the p osition dep endent spin basis is
y
A R U R AU R
In this way the op erators in the xed and p osition dep endent spin bases ob ey the
same commutation relations If we compare the action of A in the two bases
X
y y
hM R jA R j R i R U R jM i hMjU R U R AU R
M
M
X
R jM i hMjA
M
M
hMjAj i
which is the same as the equation for the action of the op erator A in the xed spin
basis By de ning the op erators on the p osition dep endent spin basis in this way all
the usual prop erties of non relativistic quantum mechanics remain unchanged So
The Schwinger representation of spin
for a choice of U R with the required prop erties and where k s the wavefunc
tions are required to ob ey the spin statistics theorem while all the usual prop erties
of quantum mechanics are maintained
The Schwinger representation of spin
In the Schwinger representation of spin a pair of harmonic oscillators with creation
y y
and annihilation op erators a a and b b are assigned to each spinning particle
The spin op erators are de ned using the Pauli matrices
a
y y
A
S
a b
b
S is a vector of spin op erators de ned by the vector of Pauli matrices In the
Schwinger representation states are lab elled by the numb er of quanta in each har
monic oscillator From
y y
S a a b b
z
y
As a a is the numb er op erator for the a oscillator the z comp onent of spin for a
state is
m n n
a
b
where n is the numb er of quanta in oscillator a Similarly the total spin of the
a
particle is
n n s
a
b
We can think of the state as b eing represented by s quanta split b etween the
two oscillators The spin raising and lowering op erators move a quantum from one
oscillator to the other changing the z comp onent of spin by one whilst keeping the
total spin the same For example with three quanta spin a state with m
z
is describ ed schematically by the following diagram a S / S
b
Chapter Quantum indistinguishability
To construct spin states for the n spins we use n pairs of oscillators
a b a b
n n
In order for the op erator U R to exchange spins BR de ne an exchange algebra of
ij
op erators E which move quanta b etween the pairs of spin oscillators
a
i
ij
y y
A
E
a a
a
j i
a
j
Given a pair of oscillators a a this de nes a vector of three exchange op erators
i j
ij ij
E as in the case of the spin op erators A second set of op erators E is de ned
a
b
similarly then
ij
ij ij
E E E
a
b
ij
De ning the exchange algebra in this way an op erator E moves a quanta from
oscillator a to a and one from b to b We can see that including these op erations
j i j i
the total spin of the individual particles is no longer xed but the total numb er of
quanta in the whole system of oscillators is still constant Schematically a general
spin state can b e represented as a distribution of sn quanta b etween the oscillators 1 2 n a ...
b
ij
The eigenvalue of such a state with resp ect to the op erator E is e
z
ij
n n n n e
a a ij
b b
i j
i j
s s
i j
States can b e lab elled by the eigenvalues of the z comp onents of the exchange algebra
e and the z comp onents of the spins m instead of the o ccupation numb ers n n
ij j ai
bi
of the oscillators
j i je e e m m i
n n n
U R for two particles
For states j i where the spins of all the particles are equal the eigenvalues e of the
ij
exchange algebra are all zero see equation and we can return to our original
lab el of the state jMi
U R for two particles
In BR construct a unitary op erator U R for two particles using the Schwinger
representation For two particles the p osition dep endent basis is a function of the
relative p osition of the particles r r r Then exchanging the particles changes
r to r The spin basis is jM i jm m i where the state with the spins exchanged
is denoted j M i jm m i The p osition dep endent spin basis is generated by U r
so
jM r i U r jM i
From equation the exchange requirement for two particles is
k
j M r i jM r i
k
where is the exchange sign for the p osition dep endent basis generated by U r
Using the Schwinger representation of the spin basis we can write the exchanged
state j M i in terms of spin state jM i To achieve this we de ne an op erator generated
by the element E of the exchange algebra
y
Y exp i E
y
A state j i of the Schwinger representation is formed by applying creation op erators
to the ground state A state with n quanta in the oscillator a etc is written
a
1
y y y y
n n n n
a a
b b
1 2
1 2
j i a a b b j i
We can split the op erator Y into op erators Y and Y op erating on the a and b
a
b
oscillators resp ectively Y and Y commute and the op erator Y is the pro duct Y Y
a a
b b
Y exp i E
a
ay
Chapter Quantum indistinguishability
The op erator Y induces a transformation on the vector of creation op erators
a
y
y y y y
Y Y
a a a a
a
a
y y
exp i
a a
y
y y
A
a a
y y
a a
y y
Y induces a similar transformation in the op erators b and b Using these results
b
we can write the action of Y on the state j i
y y y y
n n n n n n
a a a
b b b
2 2 1
2 2 1
a Y j i a b b j i
For a state jM i where s s s we nd that
s
Y jM i j M i
We can use this exchanged state to investigate prop erties of the p osition dep endent
basis
We will assume that U r is generated by the algebra of exchange op erators so
that
U r exp ic r E
Starting from the exchange requirement we know that
k
M i U r jM i U r j
M i from equation we have Using the op erator Y to rewrite j
k s
U r Y jM i U r jM i
y k s
U r U r Y jM i jM i
We can now de ne a new op erator V r where
y
V r U r U r Y
U R for two particles
V r is also generated by the exchange algebra and
k s
V r jM i jM i for all jM i
This implies that V r is diagonal in the jM i basis An op erator
exp id r E
z
generated by E is diagonal in any basis and next we will show that V r is such
z
an op erator
In the jM i basis V r must commute with any op erator generated by E
z
y
V r exp id r E V r exp id r E
z z
This is equivalent to
y
exp id r fV r E V r g exp id r E
z z
which implies that
y
V r E V r E
z z
in the jM i basis To any element generated by the exchange algebra
exp ic r E
we can asso ciate a rotation matrix R jcj a rotation ab out the axis c by an angle
c
V
jcj Then using the rotation matrix R asso ciated to V r from we have
V
R z z
The z direction is invariant under the spatial rotation asso ciated to V r and con
sequently it must corresp ond to a rotation ab out the z axis
V r exp id r E
z
Chapter Quantum indistinguishability
The exchange sign is indep endent of U r
k
Using these results for V r we can see that the exchange sign is indep endent
of the particular form of U r From equation we know that the spin vectors
are null states of E
z
E jM i
z
As V r is generated by E this implies
z
V r jM i jM i
Comparing this to equation we see that
k s
k
The exchange sign is that required to pro duce the correct spin statistics con
nection for any op erator U r with the required prop erties The exchange sign is
top ological
Constructing U r
These results also suggest how to construct the op erator U r We saw previously
that any op erator generated by E do es not e ect the spin states jM i
z
exp i r E jM i jM i
z
If we de ne a new unitary op erator U r where
U r U r exp i r E
z
then
jM r i U r jM i U r jM i
The same p osition dep endent basis is generated by b oth U and U
U R for two particles
To determine U r it is su cient to determine the action of the asso ciated spatial
U
rotation R r on the z axis From the de nition of V r we have
U U
R r R r R z z
y
where R is a rotation by ab out the y axis the rotation asso ciated to the
y
op erator Y This condition on the rotations reduces to o
U U
R r z R r z
U
To simplify this further we can de ne e r R r z then
e r e r
or e r is o dd A simple solution with this prop erty is to take e r so
U
R r z r
So one example of the op erator U r which generates the p osition dep endent spin
basis is
U r exp i n E
where R z r
n
Smo othness of jM r i
The choice of U r in is smo oth everywhere except for the south p ole There
fore the p osition dep endent basis jM r i must also b e smo oth except p ossibly at the
south p ole We can cho ose a second op erator U r related to U r by a rotation
ab out the z axis as in equation U r will b e an alternative exchange rotation
rst from z to z then from z to r
U r exp i n E exp i E
y
U r is clearly smo oth near the south p ole so the p osition dep endent basis jM r i
is also smo oth there As b oth U r and U r generate the same p osition dep endent
spin basis the basis is smo oth everywhere
Chapter Quantum indistinguishability
Parallel transp ort of jM r i
The parallel transp ort condition is
y
hM r jrM r i hM jU r rU r jM i
U r and U r dr are in nitesimally di erent exchange rotations and so di er only
by a linear combination of the elements of the exchange algebra
y
U r rU r r E r E r E
z
Therefore
hM r jrM r i
r hM jE jM i r hM jE jM i r hM jE jM i
z
The physical spin states jM i of the Schwinger representation are null states of E
z
E jM i
z
So the term in vanishes As E and E op erators raise or lower the
total spin of one of the particles the and terms are inner pro ducts of a spin
state s s with a state s s These also vanish which demonstrates
that the p osition dep endent basis parallel transp orts spins
The Schwinger representation of SU n
We will see that the Schwinger representation of n spins is equivalent to the com
pletely symmetric representation of SU n The algebra of exchange op erators
generates the subgroup SU n while the n sets of spin op erators generate the group
n
SU
We b egin by de ning a set of matrices related to the spin op erators of the
j
Schwinger representation The spin op erators S corresp ond to a vector of n n
j
matrices S which are blo ck diagonal with the Pauli matrices in the j th
The Schwinger representation of SU n
blo ck The other terms in the matrix are zero
B C
B C
B C
B C
B C
B C
B C
B C
j
B C
S
B C
B C
B C
B C
B C
B C
B C
A
We can see immediately that these matrices have the commutation relations required
for the spin op erators of the n particles All the spin matrices commute with those of
a di erent spin while the three matrices of a single spin have the angular momentum
n
commutation relations These spin matrices generate the group SU n sets of
commuting matrices each generating the angular momentum group SU
As there exists a simple matrix analogue for spin op erators in the Schwinger
representation of spin we can exp ect such an analogy to continue for the exchange
ij
algebra The op erators E in the Schwinger scheme are related to a vector of n n
ij
matrices E The nonzero terms in these matrices are where the i th and j th
columns intersect the i th and j th rows
B C
B C
B C
B C
B C
B C
B C
B C
B C
B C
B C
B C
B C
B C
ij
B C
E
B C
B C
B C
B C
B C
B C
B C
B C
B C
B C
B C
B C
B C
A
These matrices form a matrix algebra with the same commutation relations as the
exchange op erators in the Schwinger representation We can see that the matrices
Chapter Quantum indistinguishability
span the space of Hermitian n n matrices and so the exchange algebra is the Lie
algebra su n The matrices de ned in generate the group of n n unitary
matrices with determinant one SU n So if we take u to b e
X
ij ij
u exp c E
ij
then u is in SU n In order to de ne matrices related to the exchange algebra
l
which have the same commutation relations with the matrices S as the op erators
m
ij
in the Schwinger representation we take the tensor pro duct of E with the
k
identity matrix I The elements u generated by the exchange algebra now p ermute
n
the blo cks in SU
ij l
By including the commutators of E I and S we generate a set of n n
matrices which span the space of Hermitian n n matrices These matrices de ne
the Lie algebra su n and generate the group SU n We see that the algebra
de ned by the Schwinger op erators is therefore also su n and the states j i are
vectors in a representation of SU n The existence of unphysical states where the
particles have di erent spins is a necessary consequence of using the larger algebra
of op erators which generate the p ermutations of the spins
The Schwinger representation of the spins is isomorphic to the symmetrised
tensor pro duct of ns copies of the n n de ning representation of SU n the
generators of which are the matrices we have just de ned To con rm this we can
check the highest weight of the Schwinger representation States in a representation
of SU n can b e lab elled using the eigenvalues of a maximal set of commuting
matrices the Cartan sub algebra To construct the Cartan sub algebra we will
ij
l
As with the Schwinger scheme select the diagonal generators E and S
z
z
j i je e e m m i
n n n
To de ne a highest weight we x the order the eigenvalues in the de nition of and
say that a weight vector of eigenvalues is p ositive if the rst non zero eigenvalue is
p ositive The highest weight state is then the state ji for which e is a maximum
followed by the other eigenvalues in order
U R for n particles
The highest weight state of the Schwinger representation will have all the quanta
in the oscillator a This makes e maximum and m maximum This highest weight
will b e sn j i where j i is the highest weight state of the Schwinger representation
with just one quanta
j i je e m i
n
This highest weight is also the highest weight of the n n de ning representation
of SU n whose matrix generators we de ned As a basis vector in the de ning
representation
B C
B C
B C
B C
j i
B C
B C
A
The representation of SU n which has a highest weight that is sn times the
highest weight of the de ning representation is lab elled by a Young tableau with a
single row of sn b oxes see section
...
This con rms that the Schwinger representation is a symmetrised tensor pro duct of
de ning representations as those are the symmetry conditions recorded by such a
tableau
U R for n particles
For n particles the op erator U R which generates the p osition dep endent spin basis
will have the form
n
X
ij
c R E U R exp i
ij
i j
The op erator must still generate a p osition dep endent basis which is smo oth parallel
transp orts and where
k sgn
j M R i jM R i
Chapter Quantum indistinguishability
which derives from the condition that a single state represents all p ermutations of
ij
particles We have just seen that the op erators E are naturally asso ciated with
the group SU n In the same way the op erator U R is connected to an element
u R of SU n
n
X
ij
c R E u R exp i
ij
i j
U R is the Schwinger representation of the element u R in the SU n subgroup
of SU n We can express this as U u R
We want to nd the e ect of the p ermutation condition on the elements
u R of SU n used in the construction To do this we must write the state j Mi
in terms of jMi As all p ermutations can b e written as a pro duct of two cycles we
will b egin by considering the exchange of the rst two particles As with the
n construction we can use the xed exchange rotation exp i E
y
s
j Mi exp i E jMi
y
s
U exp i E jMi
y
The element of SU n that pro duces the exchange is
B C
B C
exp i E
B C
y
A
I
where I is the n by n identity matrix This is an example of a phased
p ermutation matrix
i
1
e
B C
B C
i
2
B C e
B C
D
B C
B C
A
i
n
e
P
where exp i sgn and D is the n n de ning representation of the
j
p ermutation These phased p ermutation matrices are a subgroup of SU n and
the op erator U asso ciated with one of them will p ermute the n spins For a general
U R for n particles
p ermutation
k sgn i i
n
1
gD jMi j Mi U fe e
i i
n
1
g From where the diagonal matrix of phases has b een abbreviated fe e
the p ermutation condition
y k sgn
U u R u R j Mi jMi
y
For this to b e true for all jMi the element u R u R of SU n must also b e a
phased p ermutation matrix This gives us a condition on the map u R from the
con guration space to SU n
i i
n
1
gD u R u R fe e
Selecting a map u with this prop erty also de nes the op erators U R
We will now show that k s for any op erators U R with the required prop
erties This will involve the exchange of the rst two particles using the xed
The p ermutation condition can b e written exchange rotation exp i E
y
as
y k s
U R U R exp i E jMi jMi
y
The related element of SU n is then
y
u R u R exp i E
y
i
1
e
C B B C B C
C B B C B C
C B B C B C
A A A
i
n
I I
e
ij
This is a diagonal element of SU n and so must b e generated by the matrices E
z
X
y ij
exp i u R u R exp i E c E
ij
y z
ij
Therefore the eigenvalues of the state jMi for which we have the condition E jMi
z
are unity and consequently in the eigenvalue equation k is s
Chapter Quantum indistinguishability
We have yet to show that there actually exist maps u R from the con guration
space to SU n which have the prop erty and pro duce a smo oth p osition
dep endent basis jM R i when n greater than two This is the statement of the
problem reached in If such a map exists then using the Schwinger representation
of spin states
s
j M R i jM R i
The spin statistics connection then follows as in section
Maps from con guration space to SU n
The existence of such a map is an interesting geometrical problem In and
Atiyah constructs a map with the required prop erties however there are still
imp ortant unanswered questions which relate to an alternative more aesthetic con
struction
In the simplest but unproved construction u R is obtained by orthogonalising
an n n matrix w R where the j th column of w R w R is asso ciated with
j
particle j so the p ermutation condition for the columns of the matrix is sat
is ed
The direction of the particle i seen from j is t
ij
r r
i j
t
ij
jr r j
i j
Each of the n directions from the particle j can b e describ ed by its complex
stereographic co ordinates A unit sphere is centred at r and a line is drawn
i j
from the south p ole through the p oint where the line connecting particles j and i
intersects the sphere The line from the south p ole intersects the equatorial plane
of the sphere and the Cartesian co ordinates of the p oint where it meets the plane
provide the real and imaginary parts of The comp onents of w w are the
i j
k j
co e cients of z in the p olynomial
i
Y X
z
p
w P z z
j i
k j
k n k
i n
k n
U R for n particles
To orthogonalise the columns of w R the vectors w R must b e linearly indep en
j
dent for all con gurations R This is surprisingly di cult to prove and to date it
hasn t b een done although numerical results for det w R are encouraging It can
b e proved for n or and in some of the hard cases where it might b e exp ected
to fail for example when all the particles are in a line
To construct a sp eci c map which has the required prop erties Atiyah uses a
pro cedure which breaks the translational symmetry of the problem and xes an
origin The previous construction was indep endent of the origin used to de ne the
vectors r in R The construction follows a similar pro cedure to the more elegant
j
construction describ ed previously We also de ne p olynomials P z but now the
j
values of j are distinguished
a If jr j jr j then t r jr j
j i ij j j
b If jr j jr j then t is the second intersection of the line r r with the
j i ij i j
sphere of radius jr j see gure i
ri
rj o
t ij
Figure The de nition of t for jr j jr j
ij j i
The complex stereographic co ordinates of the directions t are again used to
ij
de ne the p olynomials The construction is still compatible with the p ermutation
condition as p ermuting the co ordinates r alters the lab els of the p oints but not the
j
geometry of the con guration which determines the p olynomials
In this construction of the map the p olynomials P can b e shown to b e linearly
j
indep endent The pro of is by induction on n Take r to b e the vector of greatest
n
Chapter Quantum indistinguishability
magnitude jr j jr j for all j Q Q are the p olynomials de ned by the
n j n
smaller con guration r r of degree n which we take to b e linearly
n
indep endent We cho ose the complex parameter on the sphere so that r lies at
n
in nity Then for j n the p olynomial of the n p oints is P Q While P
j j n
is a p olynomial of genuine degree n as none of its ro ots are in nite Therefore
the set of p olynomials P P is still linearly indep endent The induction then
n
starts from the trivial case n
With a map from the con guration space of the particles to SU n which has
the required p ermutation prop erties the Schwinger construction of the p osition
dep endent spin basis can pro ceed We have already seen that the exchange sign is
indep endent of the particular choice of map and will give the observed spin statistics
relation
Smo othness and parallel transp ort for n particles
The parallel transp ort of the p osition dep endent basis for two particles was demon
strated using the condition
E jM i
z
For the Schwinger representation of n spins we still have the condition
ij
E jMi
z
and the same argument can b e applied to show that the spin basis is parallel trans
p orted We see that this parallel transp ort condition is also indep endent of the
choice of map u R used in the construction
For the p osition dep endent basis to b e a smo oth function of R the map u R
from con guration space to SU n must also b e a smo oth function of R up to mul
tiplication on the right by a diagonal matrix In the rst construction the matrix
w is smo oth up to multiplication of the columns by phase factors and so would
generate a smo oth p osition dep endent basis In the second construction the change
Alternative constructions
b etween the two regimes as r moves across the sphere of radius jr j is not smo oth
j i
The construction is still however continuous and can b e made smo oth so this is
only a technical problem Given that we know a map u R exists which generates
a p osition dep endent basis with the required prop erties the exchange
sign of jM R i is determined by the Schwinger representation of the spins
Alternative constructions
Combining the Schwinger representation of spin with the map u R found by Atiyah
provides constructions of a p osition dep endent spin basis for n particles with all the
required prop erties so that in this framework the singlevaluedness of the wavefunc
tion requires that the system ob ey the spin statistics theorem In BR consider
alternative constructions with the prop erties intro duced in which don t pro duce
the physically correct spin statistics relation
The rst alternative construction involves a change in representation of SU n
The commutators for the creation and annihilation op erators in the Schwinger rep
resentation are replaced by anticommutators Anticommutators imply that a
j
with similar relations for the other op erators of the harmonic oscillators so the rep
resentation can only have a single quantum in each oscillator This means that the
representation is limited to states of spin
In this anticommuting Schwinger representation n spin particles are repre
sented by a distribution of n quanta b etween the oscillators with at most a single
quantum in each The highest weight state of this representation will b e the state
with the rst n oscillators a b a lled The weight of this state is the sum of
the rst n weights of the anti Schwinger representation with a single quantum With
only a single quantum the anticommutation relations can t a ect the available states
So the anticommuting Schwinger representation with a single quantum is isomor
phic to the de ning representation of SU n which is the commuting Schwinger
representation with a single quantum The sum of the rst n maximal weights of the
Chapter Quantum indistinguishability
de ning representation of SU n is the de nition of the n th fundamental weight
and consequently the n spin anti Schwinger representation is isomorphic to the
representation of SU n lab elled by a Young tableau with a single column of n
b oxes
.
.
Calculations using the anti Schwinger scheme show
j M R i jM R i
The wavefunctions on this basis will b e symmetric under p ermutations This is
b osonic b ehaviour for spin particles not the correct spin statistics relation
This alternative construction is an example of the generalisation we will investigate
In this alternative construction the p osition dep endent basis has all the prop erties
required in We therefore deduce that the spin statistics connection dep ends on
the representation used for the spin states
A second alternative construction is for two spin zero particles In this case the
p osition dep endent spin basis is represented by the unit vector in the direction r
jM r i r jrj
where only one vector for example z corresp onds to the spin vector j i in the
normal representation of spin When r is substituted for r this p osition dep endent
basis changes sign despite the integer spin The wrong spin statistics connection
Both these alternative constructions of a spin statistics connection are unsatis
factory They each apply to a single value of spin and in the second case only to
two particles However they illustrate that the requirements intro duced in section
are insu cient to derive the spin statistics theorem on their own
Parastatistics
Parastatistics
One interesting generalisation of the condition that can b e made is to allow
a set of states lab elled by an additional quantum numb er to represent b oth the
initial and p ermuted states The p osition dep endent basis is then jM R i and we
can take the xed spin basis jM i to b e orthonormal
hM jM i
MM
The p ermutation condition is now
X
j M R i c R jM R i
and the parallel transp ort condition is
hM R jr M R i
Using these conditions we will derive the prop erties of the co e cients c R
From the orthogonality condition we can write
h M R j M R i
MM
Using the exchange condition this is
X
c R c R hM R jM R i
MM
Applying the orthogonality of the states for a second time this reduces to an equation
for the co e cients
X
c R c R
If we let C R denote a matrix with elements c R then equation is equiv
alent to the matrix equation
y
C R C R I
The matrix of co e cients C R is unitary
Chapter Quantum indistinguishability
Using the p ermutation condition we can write
h M R jrj M R i
P
c R c R hM R jrM R i
P
c R rc R hM R jM R i
Applying the parallel transp ort condition on b oth sides of the equation we
nd
y
C R rC R
As C R is unitary this implies
rC R
The matrix C R is a constant and so is indep endent of R
Taking the two p ermutations and we will expand the state obtained by
applying b oth p ermutations to jM R i
j M R i C j M R i
C C jM R i
From the action of the combined p ermutation is
j M R i C jM R i
We see that the matrices C de ne a representation of S
n
C C C
The representation acts on the additional quantum numb ers Any representation
C S can b e decomp osed into a direct sum of irreducible representations Taking
n
a spin state jM i in one of these irreducible representations then acting with U R
we obtain only other vectors in the irreducible representation Each irreducible
subspace is invariant under the action of U The p osition dep endent spin basis
decomp oses into subspaces that transform according to the irreducible representa
tions of S in C In this generalisation of the BR construction it is p ossible for the
n
The comp onents of the BR construction
p osition dep endent basis to transform according to any irreducible representation of
S Those subspaces transforming according to the higher dimensional representa
n
tions exhibit parastatistics
Changing condition in the construction allows the p osition dep endent basis
to exhibit parastatistics The same parastatistics will also b e present in the wave
functions de ned on this basis By noticing that parastatistics would b e consistent
with the alternate condition we do not in any way change the results for the
construction made using the Schwinger representation In the Schwinger construc
tion parastatistics is not present the spin states can only transform according to
a one dimensional representations of the p ermutation group which ever version of
condition we imp ose on the construction
The comp onents of the BR construction
To see how the construction can b e generalised it will b e useful to summarise the
essential ingredients that give the BR construction the prop erties required in section
This will also demonstrate why the anti Schwinger construction is viable Re
calling section the three requirements on the p osition dep endent basis are that
it is smo oth parallel transp orts and under p ermutations of the p ositions and spins
X
j M R i c R jM R i
This is the more general p ermutation condition that allows parastatistics In the
previous section by combining with the parallel transp ort condition we found
that states of the p osition dep endent spin basis must transform under p ermutations
according to an irreducible representation of S equation
n
We will consider the construction of the p osition dep endent basis using the ex
ij
change algebra of op erators E In order for the op erator U R to generate a basis
that exchanges spins along with p ositions we saw that the related map u R from the
Chapter Quantum indistinguishability
con guration space of the n particles to SU n must have a p ermutation prop erty
i i
n
1
gD u R u R fe e
Including parastatistics do es not change this requirement The map should also b e
smo oth up to multiplication by a diagonal matrix if the p osition dep endent basis is
to b e smo oth Given such a map we want to b e precise ab out the information used
to demonstrate that jM R i has the three required prop erties
For the p osition dep endent basis to b e smo oth we know already that the map
u R which de nes U R must b e smo oth up to multiplication by a diagonal ma
y
trix Given the p ermutation condition on the map multiplying U R U R
by any xed exchange rotation which p ermutes the spins pro duces an op erator
P
ij
R E We have the condition on the spin states exp i
z
ij
ij
ij
E jMi
z
This implies that
X
ij
exp i R E jMi jMi
ij
z
ij
and there can b e no extraneous phase intro duced the p osition dep endent basis is
smo oth
The argument for parallel transp ort is also based on the spin states jMi b eing
null states of the z comp onents of the exchange algebra As long as this is the case
the p osition dep endent basis parallel transp orts spins
The p ermutation condition on the spin states in the p osition dep endent basis is
satis ed given any map u R with the prop erty To demonstrate that the
exchange sign is top ological indep endent of the particular choice of u R we also
used the prop erty of the spin states We now see that the success of the
construction is derived from two basic prop erties Firstly the existence of a map
u R from con guration space to SU n with the required p ermutation prop erty
and smo oth up to multiplication by a diagonal matrix and secondly that the spin
Summary
vectors where the n spins are equal are null states of the Cartan sub algebra of the
exchange algebra
We can now see why the anti Schwinger scheme is also p ossible Changing the
commutation relations of the creation and annihilation op erators do es not e ect
either of these basic prop erties of the construction In fact we see that to pro duce a
p osition dep endent basis with the required prop erties we have not referred directly to
the Schwinger representation of the spin states at all The Schwinger scheme provides
the necessary representation of the op erators which form the su n algebra from
which the exchange sign is calculated but a di erent representation of su n could
still pro duce a satisfactory p osition dep endent basis A representation T SU n
must simply include states jMi where the total spins of the n particles are equal
and where
ij
T E jMi
z
Any map u R used to de ne a Schwinger construction will then apply equally well
to this alternative construction
Summary
In this chapter we have seen that in order to generate a p osition dep endent spin
basis which exchanges spins along with p ositions the algebra of spin op erators can
b e extended to included an algebra of exchange op erators Spin basis vectors are
now vectors in a representation of SU n The p ermutations of spin with p osition
are generated by a map from the con guration space of the particles to SU n the
group generated by the exchange algebra While explicit constructions of this map
exist due to Berry and Robbins for n and Atiyah in the general case the
prop erties of the p osition dep endent basis under p ermutations are determined by
the representation of spin used and are the same for all maps with the required
prop erties
Chapter Quantum indistinguishability
The p osition dep endent basis transforms under p ermutations of the particles
according to a representation of S For the Schwinger spin basis this is the trivial
n
representation for particles of integer spin and the alternating representation for
half integer spin The transformation prop erties of the p osition dep endent basis
determine how a wavefunction transforms under p ermutations Consequently for the
Schwinger scheme wavefunctions on the p osition dep endent spin basis are required
to ob ey the spin statistics theorem In order for the p osition dep endent spin basis to
b e smo oth parallel transp orted and have the correct p ermutation prop erty the spin
basis vectors must have zero weight with resp ect to the Cartan sub algebra of su n
Replacing the Schwinger representation with another representation of SU n will
pro duce an alternative construction of the p osition dep endent basis It will have
all the required prop erties but may transform under a di erent representation of
S leading to the wrong spin statistics relation as o ccurs with the anti Schwinger
n
construction So we are led to the question for a general representation of SU n
what is the relationship between spin and statistics
Chapter
Calculation of the exchange sign
for the irreducible
representations of SU
In this chapter we will generalise the Berry Robbins construction for two spinning
particles to nd the exchange sign for irreducible representations of SU This is
the simplest form of the construction not only as it has the least particles but the
p ermutation group has only two irreducible representations The elements of S are
I and b oth form their own class and the two irreducible representations are
the trivial representation and the alternating representation b oth one dimensional
In this straightforward case we exp ect to b e able to directly assemble the various
spin bases jM i that the representations can act on The p osition dep endent basis
will b e constructed in the same way as for the Schwinger representation by a unitary
transformation generated by the exchange algebra If this subspace transforms under
p ermutations of the particles as the trivial representation of S the particles are
b ehaving as b osons and the wavefunction is symmetric under exchange Conversely
spin vectors in a subspace transforming according to the alternating representation
of S are fermionic their wavefunctions are antisymmetric under exchange
Chapter Calculation of the exchange sign for the irreducible representations of
SU
The group SU
SU is the group of unitary matrices with determinant one The generators
of the group can b e de ned in terms of the generators of the two spins and of the
exchange algebra as in equations and
i
A A
S S
i i
i
I iI
A A
p p
E E
y x
I iI
I
A
p
E
z
I
are the Pauli matrices and I is the identity matrix The commutators of
i
these matrices provide the remaining six generators in the algebra A representation
of the group determines a representation of the generators Let V denote the carrier
space of the representation the spin vectors jM i b elong to V
The spin subspace
In our discussion of the Schwinger representation in chapter we saw that V is
comp osed of states
y y y y
n n n n
a a
b b
2 1
2 1
j i b b a j i a
where the spins of the two particles may b e di erent Similarly for a general rep
resentation only a subspace W of V contains spin states that can b e used in the
construction Spin vectors jM i in W have two prop erties
with the same value of total spin s for b oth and S jM i is an eigenvector of S
particles
E jM i that is the spin vectors are zero weight states of the exchange alge
z
bra This ensures the p osition dep endent basis is smo oth parallel transp orted
Preparing the subspace W
and that the exchange sign is top ological
These two conditions de ne the subspace W to which the Berry Robbins construc
tion can b e applied As we showed in chapter the prop erties of the p osition
dep endent basis dep end on the commutation relations of the generators which are
a prop erty of the algebra and the conditions on the spin vectors jM i The problem
now breaks down into two parts nding W for a general representation of SU
sections and and determining the exchange sign for a vector in W section
and following
Preparing the subspace W
An irreducible representation of SU is lab elled by three integers
f f f f
These are the lengths of the rows of the corresp onding Young tableau and we can
take f f f A vector in an irreducible representation of SU is constructed
by taking a tensor pro duct of states of the de ning representation of SU and
applying a characteristic unit of the symmetric group generated by the tableau f
We will construct vectors in the carrier space of f which b elong to W
Basis vectors of the tensor pro duct representation
As in the discussion of su in section the basis vectors of the de ning repre
sentation are eigenvectors of the Cartan sub algebra
B B C C
B B C C
B B C C
B B C C
p
S E
z z
B B C C
B B C C
A A
Chapter Calculation of the exchange sign for the irreducible representations of
SU
B C
B C
B C
B C
S
z
B C
B C
A
The eigenvectors
B C B C B C B C
B C B C B C B C
B C B C B C B C
B C B C B C B C
x x x x
B C B C B C B C
B C B C B C B C
A A A A
are lab elled by weights e s s
z z z
p p
p p
j
The weight lab els the vector x
j
A basis vector of the tensor pro duct of de ning representations is a tensor pro d
uct of basis vectors from
x x x x
j
l k
We will use the notation N x for the numb er of basis vectors x used in x and
j j
N x for the total numb er of terms in the tensor pro duct
Basis vectors of irreducible representations of SU
To prepare a vector in the carrier space of f the vectors x in are assigned to
j
b oxes of the tableau f The tensor pro duct is symmetrised with resp ect to the terms
in the rows of the tableau and then antisymmetrised with resp ect to the columns
This is the application of a primitive characteristic unit of S to the tensor pro d
N x
uct
Preparing the subspace W
For example if we take the representation of SU we assemble vectors in
V from the tensor pro duct of four basis vectors Let
x x x x x
We assign these vectors to the tableau
1 4 2
4
Symmetrising with resp ect to the rows of the tableau pro duces
1 4 2 1 2 4 4 1 2 4 4 4 SSS
4 2 1 2 1 4 2 4 1 4 4 4
SS S
where the ket symb ols remind us that the diagrams corresp ond to tensor pro d
ucts The kets are lab elled with an S to distinguish vectors to which have b een
symmetrised with resp ect to the symb ols in the rows of the tableau Finally we
antisymmetrise with resp ect to the vectors in the columns
1 4 2 4 4 2 1 2 4 4 2 4 4 1 4 1 AS AS AS AS
4 1 2 4 1 2 4 2 1 4 2 1 4 4 4 4 AS AS AS AS
2 1 4 4 1 4 2 4 1 4 4 1 4 2 4 2
AS AS AS AS
The lab el A denotes that the vector has now b een antisymmetrised with resp ect to
the columns The vectors in the central row cancel leaving
1 4 2 4 4 2 1 2 4 4 2 4 4 1 4 1 AS AS AS AS
2 1 4 4 1 4 2 4 1 4 4 1 4 2 4 2
AS AS AS AS
This is a basis vector of the representation of SU Alternatively the vector
could b e written as a linear combination of tensor pro ducts of the x s where each
j
term is a p ermutation of By recording the vector using tableau we avoid sp ec
ifying the order of the terms in the tensor pro duct Each b ox in the Young tableau
Chapter Calculation of the exchange sign for the irreducible representations of
SU
corresp onds to a particular term in the tensor pro duct but as this corresp ondence
is arbitrary it is convenient to b e able to suppress it
Basis vectors of a representation of SU SU
The SU SU subgroup is generated by S and S A representation of
SU SU is de ned by two representations of SU each of which can b e
lab elled by a Young tableau Vectors in a representation of SU SU are the
tensor pro duct of vectors of the two representations of SU The basis vectors of
the de ning representation of S are x and x while x and x are a basis of S
To record a vector in a representation of SU SU using tableau we assign the
vectors x or x to the rst tableau and x or x to the second For example
1 1 2 3 4
2
In this case b oth representations of SU are spin one and the vector has s
z
s To construct vectors in an irreducible representation of SU SU
z
we apply the SU symmetry conditions for b oth SU tableau to a vector in the
tensor pro duct representation
s s multiplets in representations of SU
The unitary groups have b een applied successfully to problems in particle physics
concerning the decomp osition of a representation into irreducible comp onents of
a subgroup of U n In Itzykson and Nauenb erg use Young tableau to de
comp ose an irreducible representation of SU m n into representations of the
SU m SU n subgroup This problem is related to nding the subspace W de
ned previously Applying their results to SU SU SU gives the numb er
of s s multiplets in a representation of SU The multiplets where the spin
eigenvalues are b oth equal form the subspace where the spins are the same de ned
in This subspace contains W
Preparing the subspace W
Representations of SU SU where b oth comp onents have spin s are la
b elled by two SU tableau b oth of which have s columns of length one
α 2s β 2s
α β
Columns of two b oxes don t a ect the representation of SU lab elled by the
tableau see section Representations of SU which when restricted to the
SU SU subgroup contain this representation are those whose tableau app ear
when the two tableau lab elling the representation of SU SU are multiplied
Multiplying tableau see section for the rules do esn t change the total numb er
of b oxes so the numb er of b oxes in the representation of SU is s This
is even and so only representations of SU lab elled by tableau f where jf j is even
can contain s s multiplets This agrees with the known results for the Schwinger
representations which corresp ond to tableau with a single row of s b oxes as in
We see that when constructing the subspace W we need only consider represen
tations where jf j is even Representations with jf j o dd contain no multiplets where
the spins of the two particles are equal If the same tableau f app ears when two
pairs of tableau for di erent values of spin s and s are multiplied then f has b oth
s s and s s multiplets This is a common situation for example the tableau
app ears in the pro duct of b oth the tableau multiplications
Consequently the representation of SU contains s s multiplets for s
and s We can see that a general representation of SU is likely to b e more
complex than the Schwinger scheme selecting the representation of SU will not
in general x the value of spin and there may b e many multiplets with the same
value of spin
Chapter Calculation of the exchange sign for the irreducible representations of
SU
Spin vectors with zero weight with resp ect to the exchange
algebra
Condition used to de ne the subspace W requires vectors in W to b e eigenvec
tors of E with eigenvalue zero In the tensor pro duct representation the generator
z
E is represented by E
z
z
X
E I I E I I
z
z
all p ermutations
The sum is over all the p ossible p ositions of E in the tensor pro duct Using
z
with eigenvalue we see that x is an eigenvector of E
z
p
e N x N x N x N x
z
with eigenvalues and S Similarly x is also an eigenvector of S
z z
N x N x s
z
s N x N x
z
From a basis vector in an irreducible representation f of SU is a linear
combination of vectors x all of which have the same weight e s s Conse
z z z
quently such a basis vector of an irreducible representation of SU also has weights
and where N x is the numb er of vectors x used to prepare
j j
the vector in V
From x is a null vector of E if and only if
z
N x N x N x N x
This provides one condition on spin vectors jM i in W
s s multiplets with E eigenvalue zero
z
An irreducible representation SU can b e restricted to the subgroup SU
f
SU generated by the spin op erators This representation SU SU can
f
s s multiplets with E eigenvalue zero
z
then b e decomp osed into irreducible comp onents
M
SU SU SU SU
s s e
f
z
1 2
s s e
z
1 2
An s s multiplet is lab elled by a pair of tableau
α 2s 1 β 2s 2
α β
From section and the eigenvalue equation we see that the E eigenvalue
z
of such a multiplet is
p
s s e
z
Each s s multiplet has a de nite value of e The s s multiplets are distin
z
guished not only by the two spins but also by the eigenvalue of E
z
The numb er of SU SU multiplets in a representation of SU is the
frequency of the tableau f in the pro duct of the two tableau lab elling the multiplet
Combining the condition for the E eigenvalue of the multiplet to b e zero with
z
the condition that s s the multiplet is lab elled by two tableau where
The numb er of representations in the decomp osition of a representation of f
s s
is the numb er of tableau f in the pro duct of the two identical tableau which lab el
the multiplet
The tableau lab els a spin irreducible representation of SU So
vectors in an s s multiplet with spin and E eigenvalue zero can for example
z
b e lab elled by a pair of tableau
Multiplying these tableau pro duces tableau with six b oxes
2
Chapter Calculation of the exchange sign for the irreducible representations of
SU
Each irreducible representations of SU lab elled by one of these tableau contains
the representation
when restricted to the SU SU subgroup In the example ab ove the represen
tations and of SU are equivalent to those lab elled by the partitions
and resp ectively If a tableau app ears more than once in the pro duct then
this representation of the subgroup contains multiple copies of the representation
of when decomp osed into its irreducible comp onents In our example
s s
contains two s s multiplets with spin and e
z
Results for general multiplets
We will consider the general result of multiplying two identical tableau s with spin
s As the lengths of the rows of this tableau could easily b e confused with the
spins of the two particles we will distinguish the two cases with an extra lab el T
for tableau The partition s is then s s and the spin of the representation of
T T
SU lab elled by this partition is
s s s
T T
As s is a tableau we know that
s s
T T
The multiplication of two such tableau is written
s T1 s T1 a b
s T2 s T2
Following the usual rules for multiplying Young tableau we lab el the b oxes in the rst
row of the second tableau with the symb ol a and the second row of the tableau b
A general result of such a multiplication can b e written
s s multiplets with E eigenvalue zero z
f1 s T1 α(a) f2 s T2 β(a) δ(b)
f3 γ(a) ε(b)
Where the a s are in sections of length and
s
T
Similarly the b s are in the sections of length and
s
T
The rows of the resultant tableau have lengths f f and f resp ectively
s s f f f jf j
T T
jf j is the total numb er of b oxes in the tableau The general pro duct tableau that we
are considering has only three rows Multiplying two tableau of two rows it is p ossi
ble as we saw previously to pro duce tableau with columns of four b oxes However
tableau with columns of four b oxes lab el a representation of SU equivalent to
that lab elled by the tableau with the columns of four b oxes removed Considering
only the tableau with three rows is su cient to provide results for all representations
of SU
There are several rules used in the multiplication of Young tableau section
These pro duce conditions on the lengths of the sections of the resultant tableau
Firstly the result of the multiplication must b e a tableau its rows decreasing in
length
f f f
Then when the b oxes lab elled with a s are added to the rst tableau they can not
b e put on top of each other This implies that
s s
T T
s
T
As b s also can t b e placed in the same column we obtain a forth condition
f s
T
Chapter Calculation of the exchange sign for the irreducible representations of
SU
Counting right to left and top to b ottom the numb er of a s must always b e greater
than or equal to the numb er of b s This provides two further conditions on the
lengths of the sections in our general pro duct tableau
s
T
Supp ose that given tableau s and f there exists a set of variables
which satisfy conditions to and so de ne one of the tableau of shap e f
in the pro duct s s
f s
T
f s
T
f
We will de ne a pro cedure that changes the lab elling of the b oxes in f without
changing its shap e This will b e referred to as procedure A A moves a b ox lab elled
b from the third row up to the second and a b ox a down from the second to the
third row
Schematically this is
a
b
We see that not only do es A not change the length of the rows in f it also do esn t
alter the total numb er of a s or b s in f Applying A de nes a second lab elling of f
To determine if this alternate lab elling is a p ossible result of the tableau multipli
cation we must check if the new variables satisfy the conditions
to A and its inverse are the only such exchanges of b oxes which can pro duce
s s multiplets with E eigenvalue zero
z
an alternative lab elling of the b oxes of the tableau which is also a p ossible result of
the multiplication of the two tableau
To nd the total numb er of tableau of shap e f in the pro duct s s the idea is
to start from the tableau with the maximum numb er of b s in the third row and
then count how many times A can b e applied b efore the conditions for multiplying
tableau are violated This will give the numb er of s s multiplets in the represen
tation f of SU with e
z
is the numb er of b oxes lab elled b in the third row of the tableau First we
consider those conditions that restrict the maximum value of can not b e greater
than the length of the third row of f by de nition and it can contain at most all s
T
of the b s In order for to b e a maximum must b e a minimum and so must
also b e maximum Condition determines the maximum value of Collecting
these conditions
f by de nition
s maximum no of b s
T
s s f max from
T T
The maximum value of is equivalently the minimum value of the maximum
value of or the minimum value of There are ve conditions which limit the
minimum value of
by de nition
f s from
T
f f s from
T
s s f from
T T
f s f from
T
Every satisfying conditions and corresp onds to an s s multiplet
There are three p ossible maximum values of de ned by The smallest of
these maximums is Similarly the maximum of the ve lower b ounds on
max
is The numb er of satisfying the conditions is then
min max min
To evaluate this we consider the fteen combinations that result from combining
Chapter Calculation of the exchange sign for the irreducible representations of
SU
each of the three upp er b ounds with each of the ve lower b ounds These p ossible
values of are enumerated in b elow Four of the combinations
max min
are redundant as they are equivalent to other pairs The variable f has also b een
eliminated from the results using equation
s
T
f f
f s
T
f s
T
f s
T
s s
T T
f f s
T
s s f
T T
f f s s
T T
f f s s
T T
s s f f
T T
To nd the numb er of s s multiplets in the representation f we nd the minimum
of the eleven integers de ned in If this is negative or zero then there are no
such multiplets in f
It is interesting to note that in xing the representation of SU even though
we do not x the spin of the multiplets we do determine whether their spins are
integer or half integer Cho osing f de nes the numb er of b oxes jf j in the tableau
and the numb er of b oxes jsj jf j in the representations of the spins Changing
the spin s involves moving b oxes from the rst row of s to the second Moving a
single b ox in this manner changes the numb er of columns of one b ox by two which
changes the spin by an integer amount Hence the spins of all multiplets in the
representation are either integer or half integer
Example The s s multiplets of with e
z
s s multiplets with E eigenvalue zero
z
By cho osing a tableau with a relatively large numb er of b oxes and three rows we
exp ect to nd several s s multiplets As the numb er of b oxes in the tableau is even
the representation will contain s s multiplets with E eigenvalue zero To pro duce
z
the tableau the two tableau s that we will multiply must each contain seven
b oxes This translates to representations of SU with half integer values of spin
b etween and We can take each value of spin in turn and evaluate the
integers for these values of f f s s
T T
Starting with spin s s evaluating the integers we nd
T T
s
T
f f
f s
T
f s
T
f s
T
s s
T T
f f s
T
s s f
T T
f f s s
T T
f f s s
T T
s s f f
T T
The minimum of these integers is two so there are two spin multiplets with
e in the representation of SU Continuing this pro cedure we can ll
z
out a table gure showing the spin of the multiplets available for the construc
tion of the p osition dep endent basis
Figure The s s multiplets of the representation of SU with e
z
multiplet spin no of multiplets
Chapter Calculation of the exchange sign for the irreducible representations of
SU
The exchange sign
So far we have found the subspace of spin vectors W for which the construction
of the p osition dep endent spin basis is de ned As a representation of SU can
contain many multiplets with di erent values of spin we will need to consider how
the basis vectors transform under the exchange of the spins in each of these di erent
multiplets
The p osition dep endent basis is generated as in the Schwinger scheme by a
unitary op erator U r The op erator is de ned using the same map u r from the
con guration space to the exchange subgroup SU that was used for the Schwinger
representation In chapter we saw that the prop erties of the p osition dep endent
basis are indep endent of the particular form of this map which is chosen providing
that it is smo oth up to multiplication by a diagonal matrix of phases and has the
desired p ermutation prop erty
i
e
A A
u r u r
i
e
For a general representation f of SU
U r u r
f
As with the Schwinger representation the sign change of vectors in the p osition
dep endent basis generated by U r determines the sign of the wavefunction under
the exchange of the particles Consequently evaluating the exchange sign for spin
vectors in a multiplet determines the statistics of wavefunctions on the p osition
dep endent basis constructed from it
De ning a xed exchange rotation
In the Berry Robbins construction the exchange sign is indep endent of the choice
of u r This top ological prop erty of the p osition dep endent spin basis allows us
to evaluate the exchange sign using a xed exchange rotation We can use the
The exchange sign
same exchange rotation as was used to investigate the prop erties of the Schwinger
representation exp i E
y
B C
B C
B C
B C
exp i E
y
B C
B C
A
This is an element of the SU exchange subgroup of SU Given any irreducible
representation of SU the element exp i E de nes an exchange rotation
y
f
which p ermutes the two spins
k
M i exp i E jM i j
y
f
where jM i is a vector in the subspace W For two particles there is no p ossibility
of parastatistics as b oth irreducible representations of the p ermutation group S are
one dimensional To evaluate the exchange sign we want to calculate
k
h M j exp i E jM i
y
f
for spin vectors in W
In the de ning representation of SU the exchange rotation exp i E acts
y
on the basis vectors x transforming them accordingly
j
x x x x
x x x x
In the tensor pro duct representation the exchange rotation is generated by E
y
X
E I I E I I
y
y
all p ermutations
Using this generator our exchange rotation in the tensor pro duct representation is
exp i E exp i E exp i E exp i E
y y y
y
We can see how this exchange rotation acts on a basis vector x in the representa
tion The vectors x in x are replaced with x s and x s with x s and vice versa
Chapter Calculation of the exchange sign for the irreducible representations of
SU
N x N x
3 4
The op erator also intro duces a sign factor
Vectors in an irreducible representation f of SU are generated by apply
ing the symmetry conditions of the tableau f to a vector x in the tensor pro d
uct representation For vectors in the subspace W we have already seen that
N x N x N x N x where the total numb er of terms in the pro duct
is jf j Op erating on a vector with exp i E is equivalent to op erating on
y
f
the vector x in the tensor pro duct representation with exp i E then applying
y
the symmetry conditions of the tableau f to the result We see that vectors in the
jf j
subspace W acquire a sign
This sign factor do es not on its own determine the exchange sign For example
consider the vector
x x x x
jf j
however the sign factor is This is invariant under the op erator exp i E
y
The symmetry conditions recorded in the tableau f also play an imp ortant role
We can however separate the two contributions determining the a ect of exchanging
jf j
x x and x x then including the sign factor
jf j
The sign can b e put into a more familiar form using the conditions on
f for it to contain an s s multiplet with e In section we had condition
z
jf j s s s s
T T T
s is an integer but s can b e half integer therefore
T
jf j s
This phase factor is reminiscent of the correct spin statistics connection and forms
the rst comp onent in our evaluation of the exchange sign for spin vectors in a gen
eral representation of SU
The exchange sign
Selecting a vector jM i
We have simpli ed the problem by considering only a single exchange rotation
exp i E To simplify it further we will sp ecify a particular vector jM i in each
y
multiplet for which we will determine the exchange sign
Any spin vector jM i can b e written as a series of spin lowering op erators S and
S acting on the highest weight state of the representation of SU SU The
e ect of the exchange rotation on these op erators can b e determined entirely from
the commutation relations of the op erators For simplicity if we use the de ning
representation of su we know that
I
A
exp i E
y
I
and the spin lowering op erators are
S
A A
S S
S
where all the matrices have b een written in blo cks Conjugating S by
exp i E we obtain the result
y
exp i E S fexp i E g S
y y
This is as exp ected changing the z comp onent of spin of one of the particles do esn t
change the exchange sign The exchange rotation simply swaps the particle that the
spin lowering op erator acts on As our choice of z comp onent of spin for the two
particles do esn t e ect the exchange sign we will cho ose the spins of b oth particles
to have maximum z comp onent s This spin vector is the highest weight state of
the representation of SU SU and so is unique By cho osing this state to
evaluate we know that
M i jM i j
which further simpli es the calculation
Chapter Calculation of the exchange sign for the irreducible representations of
SU
Constructing the highest weight state of a multiplet using
Young tableau
In sections and following we constructed basis vectors in irreducible represen
tations of SU and SU SU using Young tableau To construct the highest
weight state of an irreducible representation of SU SU we take a pair of
tableau lab elling the representation of SU SU Then basis vectors x of the
de ning representation are assigned to the rst row of the rst tableau x to the
second row of the rst tableau x to the rst row of the second tableau and x to
the second row
1 3
2 4
The tableau is symmetrised with resp ect to the vectors in the same row which is
trivial in this case then antisymmetrised with resp ect to vectors in the same col
umn The tableau represents the highest weight vector as it contains the maximum
numb er of vectors x and x antisymmetrising columns containing the same symb ol
would pro duce zero
For example the highest weight state of a spin multiplet with e can b e
z written
1 1 3 3
2 4
Applying the symmetry conditions of the two SU tableau we have the linear
combination of tensor pro ducts written
1 1 3 3 2 1 3 3 2 4 1 4 AS AS 1 1 4 3 2 1 4 3 2 3 1 3
AS AS
Each b ox in the pair of tableau corresp onds to a p osition in the tensor pro duct of
six basis vectors of the de ning representation
The exchange sign
We want to nd the highest weight state of an s s multiplet with e
z
in a representation f of SU This involves applying the symmetry conditions f
asso ciated with the multiplet to the highest weight vector of the representation of
SU SU we de ned previously The symmetry conditions asso ciated with
the multiplet are found by multiplying the pairs of tableau in the highest weight
state of the representation of SU SU We will demonstrate the pro cedure
by continuing the example ab ove discussing more generally what is going on
In the multiplication of two tableau we nd the representation
app ears twice in the result with the two alternative lab elling
a1 a1
a2 b1
b1 a2
Each lab elling corresp onds to a di erent multiplet If we take the rst lab elling
and apply those symmetry conditions to the highest weight state of the spin
representation of SU SU we nd
1 1 3 2 1 3 1 1 4 2 1 4 2 3 1 3 2 3 1 3
4 4 3 3
The symmetry conditions of the tableau lab elling the irreducible represen
tation of SU can now b e applied to this vector First the rows in each tableau
are symmetrised
Chapter Calculation of the exchange sign for the irreducible representations of
SU
1 1 3 2 1 3 1 1 4 2 1 4 2 2 3 1 3 2 2 3 1 3 4 4 3 3 S S S S
1 1 3 2 3 1 41 1 2 4 1 2 3 2 3 1 2 2 3 1 3 4 4 3 3 S S S S
31 1 1 2 3 1 4 1 4 2 1 2 2 3 3 1 2 2 3 1 3 4 4 3 3 S S S S
1 3 1 1 3 2 41 2 2 3 2 3 1 1 3 4 4 3 S S S 3 2 1 1 3 4 S
31 2 1 3 4
S
States are multiplied by a factor of two when two di erent p ermutations of the sym
b ols in the same row lead to the same lab elling of the tableau To simplify the result
all the tableau with two identical symb ols in the same column which would vanish
when the columns are antisymmetrised have b een omitted
We could now antisymmetrise the tableau with resp ect to the symb ols in the
same column however it would pro duce pages of tableau which can b e avoided If
two of the tableau ab ove contain the same symb ols in their columns then they will
b oth pro duce the same set of tableau when antisymmetrised However the two sets
of tableau could di er by a sign To determine this sign we can compare the two
tableau and to see whether an even or o dd p ermutation must b e used to rearrange
one tableau into the other For example the two tableau
1 1 3 2 1 3 2 3 1 3
4 4
b oth pro duce the same set of tableau when antisymmetrised but with opp osite signs
as a single exchange of the rst two symb ols in the rst column transforms the
second tableau into the rst Using this we can collect tableau that are related by
The exchange sign
column p ermutations without antisymmetrising and writing the results out in full
In our example collecting tableau we are left with
1 1 3 3 1 1 1 1 3 3 2 3 3 2 3 3 3 2 4 4 4 S SS
2 4 1 41 2 21 4 3 1 3 3 1 3 3 1 3 3 3 3
SS S
where the full vector is still found by antisymmetrising the columns Leaving the
vector in this form is su cient to compare pairs of vectors as we will do later We
have constructed a highest weight vector of the s s multiplets with spin in the
representation of SU
With this picture of the pro cedure the prop erties of a general construction of
the highest weight vector of a multiplet are clearer A state created by applying
rst the s s symmetry conditions of the SU SU multiplet followed by the
symmetry conditions f of SU is a vector in the representation f of SU as it
is a linear combination of basis vectors of the representation f It is also a highest
weight state of the SU SU subgroup as applying a spin raising op erator S
or S will pro duce zero consider applying the raising op erator to the state b efore
the symmetry conditions f are applied The multiplication of the two tableau gives
a set of linearly indep endent ways of combining the symmetry conditions
The e ect of the symmetry conditions on the exchange sign
changes vectors x to x x to x and vice The exchange rotation exp i E
y
s
versa It also multiplies a vector jM i by the sign factor For the highest
weight vector of an s s multiplet with e exchanging the spins do esn t change
z
the vector So from the exchange sign is
k
hM j exp i E jM i
y
f
Chapter Calculation of the exchange sign for the irreducible representations of
SU
To determine the e ect of the symmetry conditions on the exchange sign we will
compare the phase of the highest weight state of an appropriate multiplet with the
state where the vectors x are replaced with x x with x and vice versa
To make this clear we can return to our previous example We will write the
tableau recording the highest weight state of the s s multiplet as a column on the
left The tableau on the right are those where the vectors have b een exchanged
x x x x
1 1 3 33 1 2 3 4 1 4 2 SS
3 1 1 1 3 3 2 3 4 1 4 2 S S 1 1 3 3 3 1 3 2 1 4 4 2 S S 2 4 1 42 3 1 3 3 1 3 1 S S 41 2 23 4 1 3 3 1 3 1 S S 21 4 43 2 1 3 3 1 3 1
S S
The arrows connect tableau related by column p ermutations of the symb ols We
can see that the two columns record the same vector as the column p ermutations
that relate the connected tableau are all even If the p ermutation required in each
case was o dd the symmetry conditions would contribute an extra factor of to
the exchange sign
It is this principle of comparing the highest weight state of the multiplet to
the state where the symb ols have b een exchanged that we will use to evaluate this
second contribution to the exchange sign from the symmetry conditions of the Young
tableau
The exchange sign
Evaluating the e ect of the symmetry conditions for a general
highest weight state
A general highest weight state is generated by a general tableau which is itself
obtained by lling in the result of the pro duct of two tableau s with the symb ols
to From section such a general tableau is
f1 s T1 (1) α (3) β δ f2 s T2 (2) (3) (4)
f3 γ (3) ε (4)
To this tableau the symmetry conditions of the s s tableau are applied p ermuting
s with s s with s Then the symmetry conditions of the whole tableau f are
applied to form the highest weight vector
The highest weight vectors generated in this manner span the space of highest
weight vectors with e for the given spin s We will show that each vector
z
contains the state lab elled by the tableau from which the vector is generated
This sounds obvious but will prove useful later The identity p ermutation is con
tained in b oth sets of symmetry conditions so the question is could the same tableau
app ear with the opp osite sign as the result of a di erent set of p ermutations This is
not p ossible as a sign change intro duced in the s s symmetry conditions must b e
undone with a second antisymmetric p ermutation to return to the original tableau
The s s symmetry conditions always p ermute symb ols in di erent rows
If we consider our previous example applying the s s symmetry conditions
1 1 3 2 1 3 2 3 ... 1 3 ... 4 4
AS
Then applying the symmetry conditions of the tableau we return to the
original tableau with no change in sign
2 1 3 1 1 3 1 3 ... 2 3 ... 4 4
AS AS
Chapter Calculation of the exchange sign for the irreducible representations of
SU
The highest weight vector contains the state lab elled by the tableau used to generate
the vector
The vector generated by also includes the state lab elled by the exchanged
tableau where the symb ols and have b een swapp ed with and
s T1 (3) α (1) β δ s T2 (4) (1) (2) γ (1) ε (2)
AS
To show that this is a p ossible result of the application of the two sets of symmetry
conditions we must show there exists a set of p ermutations which applied to
will pro duce the tableau We can also nd the sign of the exchanged tableau
in the vector by counting the column p ermutations used
Starting from the tableau
s T1 (1) α (3)
s T2 (2) β (3) δ (4)
γ (3) ε (4)
we will assume that the identity p ermutation is chosen from the symmetry condi
tions of s s We now split the argument for the symmetry conditions of the tableau
f into two parts
If the sections and don t overlap f s the p ermutations necessary to
T
pro duce are straightforward Starting with the row p ermutations the s
in the rst row must b e swapp ed with s from the rst row this is a symmetric
p ermutation so there is no change in sign The s in the second row are also
exchanged symmetrically with s in the second row All the other terms must b e
exchanged antisymmetrically using column p ermutations The s in the second
row are swapp ed with s in the top row a sign change of Similarly the s
are also swapp ed antisymmetrically a sign change of and the s are swapp ed
with s in the second row a sign change of Combining these contributions
the sign of the state in the vector generated by the tableau is
The exchange sign
If the sections and overlap the exchange of the symb ols is more complex
as some symb ols must b e moved twice In the overlapping section we start with
columns of
1 3
4
The must b e swapp ed with a but there are no s in the same row or column To
accomplish the exchange we pair this column with one of the columns of three b oxes
that is not in the overlapping section This can always b e done as the maximum
length of is s the length of the section containing s To exchange the symb ols
T
in this pair of columns we pro ceed as follows
1 ..... 1 1..... 1 2 ..... 3 3..... 2
3 ..... 4 4 ..... 3
First the p ositions of the two columns are exchanged Swapping symb ols in the
same row do es not intro duce a sign change Then using the antisymmetric column
p ermutations
1 ..... 1 3..... 3 3 ..... 2 4 ..... 1
4 ..... 3 1 ..... 2
The combined p ermutation of b oth columns is even so again there is no sign change
The remaining symb ols not involved in these pairs of columns are exchanged as
in the rst case As the overlapping region has length f s the sign of the state
T
in the vector generated by the tableau is
f s
3
T 2
If we consider the alternative vector constructed from the tableau with one less
in the third row we can compare the sign of the states lab elled by the exchanged
tableau in the two cases To b e completely transparent we are comparing the signs of
the two di erent states lab elled by the two di erent exchanged tableau each de ned
from the tableau used to construct the resp ective vectors Earlier we referred to the
Chapter Calculation of the exchange sign for the irreducible representations of
SU
pro cedure relating the tableau used to construct two such vectors as pro cedure A
where A sends
Applying A adds one to and subtracts one from and so
For b oth cases and the sign of the vectors lab elled by the exchanged
tableau alternates b etween vectors generated by tableau related by A
Let us pro ceed for the moment as if the vectors generated by such tableau were
eigenvectors of the exchange op eration exp i E as we will discuss b elow in
y
general they are not Given a vector jM i generated by symmetry conditions s s
and f we have
s T1 (1) α (3) s T1 (3) α (1) πΕ ) β δ 2 s β δ exp(-i y s T2 (2) (3) (4) (-1) s T2 (4) (1) (2)
γ (3) ε (4) γ (1) ε (2)
where the vector is lab elled by the tableau to which the symmetry conditions will
b e applied to generate the vector For an eigenvector of the exchange op eration our
previous results show that either
s T1 (1) α (3) s T1 (3) α (1) β δ ( β + γ + ε ) β δ s T2 (2) (3) (4) (-1) s T2 (4) (1) (2)
γ (3) ε (4) γ (1) ε (2)
if f s or
T
s T1 (1) α (3) s T1 (3) α (1) ( β + γ + ε ) β δ - 3 ( f 3 - s T2 ) β δ s T2 (2) (3) (4) (-1) s T2 (4) (1) (2)
γ (3) ε (4) γ (1) ε (2)
for f s Substituting these results into we can evaluate equation
T
to obtain the exchange sign
s
for f s
T
k
s f s
3
T 2
for f s
T
The exchange sign
for eigenvectors of the exchange op erator
Using the relationship the exchange sign alternates b etween vectors
generated by tableau related by A Consequently
N N N N even
e o e o
jN N j N N o dd
e o e o
where N is the numb er of multiplets with exchange sign of and N is the numb er
e o
with exchange sign Half of the spin multiplets transform with each exchange
sign
This would amount to a derivation of the exchange sign if the vectors generated
by the tableau were eigenvectors of the exchange op erator this is not neces
sarily the case We can de ne a set of eigenvectors of the exchange op erator that will
have the correct exchange signs However the problem is then to show that these
vectors are linearly indep endent If they are linearly indep endent we have found the
dimension of the subspaces of W with each exchange sign
Let us start instead with the vector generated by the exchanged tableau
s T1 (3) α (1) β δ s T2 (4) (1) (2)
γ (1) ε (2)
then applying the same argument that we used on the vector generated by
we see that the vector contains a state lab elled by the original tableau and
the sign of this state is the same as the sign we calculated in the reversed situation
If the sign of the vector lab elled by the exchanged tableau is then we can take
the linear combination of vectors generated by the two tableau
s T1 (1) α (3) s T1 (3) α (1) β δ β δ s T2 (2) (3) (4) s T2 (4) (1) (2)
γ (3) ε (4) γ (1) ε (2)
The vector generated by acting on this combination of tableau with b oth sets of sym
metry conditions s s and f is not zero as it must include vectors lab elled by b oth
the original and exchanged tableau It is also clearly an eigenvector of exp i E
y
Chapter Calculation of the exchange sign for the irreducible representations of
SU
s
with eigenvalue If there are N multiplets then this pro cedure de nes
N highest weight vectors half with each exchange sign However these vectors could
b e linearly dep endent
One way to prove that the exchange signs of the multiplets are determined by
the signs calculated previously would b e to show that these vectors generated by
are linearly indep endent If there is only a single multiplet with spin s then
there is nothing to prove and we have determined the exchange sign of the multi
plet For two multiplets with spin s the two highest weight vectors generated by
tableau will have di erent exchange signs so they are linearly indep endent
Consequently all pairs of s s multiplets with e will consist of one with each
z
exchange sign
Tableau where f can contain at most one multiplet with each spin s to
see this consider applying A to such a tableau So we have found the exchange
signs of multiplets in a representation of SU lab elled by a tableau with two rows
For tableau with three row if f s we can show that the vectors generated by
T
are linearly indep endent Take the vector after applying the symmetry
conditions s s and f it must contain a tensor pro duct lab elled by the original
tableau which has columns
1 2
4
It is imp ossible for tensor pro ducts in the state to b e lab elled by tableau with more
columns of To see this consider the diagram f s
T
f1 s T1 (1) α (3) f2 s T2 (2) β(3) δ (4)
f3 γ (3) ε (4)
Using the s s symmetry conditions the s can only b e exchanged with s in a
higher row which will reduce the numb er of s in the third row Applying the sym
metry conditions f s in the rst or second row must undergo a row p ermutation
with a or to increase the numb er of columns of three b oxes containing s This
Numerical calculation of the exchange sign
prohibits the column created also containing b oth a and along with the
The same argument can b e applied to the vectors generated by the tableau where
s and s and s and s have b een exchanged to show that again the maximum
numb er of columns is We can now see how to show the vectors are
linearly indep endent The vector generated by tableau with the maximum numb er
of s in the third row is linearly indep endent of the other vectors as other
max
vectors can not contain as many columns of The pro cedure can now b e it
erated to show that the vector generated by tableau with m columns
max
of is indep endent of the vectors generated by tableau where k for
max
k m This proves the vectors are linearly indep endent for f s and con
T
sequently the exchange signs for these multiplets ob ey the relations and
As yet I have b een unable to verify that the vectors are linearly inde
p endent for tableau with f s For these vectors it is p ossible to increase the
T
numb er of columns of but only to a limited extent It therefore seems likely
that these vectors are linearly indep endent despite the di culty in proving it
Numerical calculation of the exchange sign
To verify our results for the numb er of s s multiplets with e and the
z
exchange sign of the multiplets and we can calculate these prop erties
numerically for the irreducible representations of SU of low dimension To do this
we construct pro jectors onto a subspace of the representation f For a diagonalisable
matrix A with eigenvalues the pro jector onto the subspace with eigenvalue is
j i
P
i
Y
A I
i
P
i
i j
j i
We must now consider the distinguishing prop erties of the subspace we will pro ject
onto
Chapter Calculation of the exchange sign for the irreducible representations of
SU
From the subspace must have equal spin eigenvalues for the two spins
s s s and the condition restricts to a subspace with e Of the
z
vectors in W we will pro ject onto a subspace of highest weight vectors of the rep
resentation of SU SU where the z comp onents of the spins are maximal
s s s This subspace is spanned by eigenvectors of the exchange rotation
z z
exp i E with eigenvalues By pro jecting onto the subspace with one of these
y
eigenvalues we can determine the numb er of multiplets with each exchange sign in
the representation of SU
The trace of a pro duct of pro jection matrices is the dimension of the subspace
they pro ject onto Using this we multiply the matrices de ned by the symmetry
conditions of a Young tableau which pro ject onto a subspace with those symmetry
conditions and the pro jectors onto a subspace where
e
z
s s s
z z
s s s s
Taking the trace of this matrix we nd the numb er of s s multiplets with e eigen
z
value zero This pro duct of pro jectors is multiplied by the pro jector of the exchange
rotation onto the subspace with exchange sign Taking the trace again we nd
the numb er of multiplets with this exchange sign By comparing to the results using
the pro jector of the exchange rotation onto the subspace with exchange sign
we can verify that the remaining spin multiplets do have the alternate sign under
exchange
The results of the numerical calculations made by following this scheme using
MATLAB are displayed in gures and The representation of SU is la
b elled with the appropriate tableau and the spins of the physical multiplets in the
representation are given along with their exchange sign
Before discussing the results it is worth commenting on some of the technical
problems that limit such calculations Although the representations of SU ap
jf j
p ear simple the size of the matrices in the tensor pro duct representation is To
Numerical calculation of the exchange sign
Young Tableau Multiplet Spin Exchange Sign
Figure Numerical results for representations of SU lab elled by Young tableau
with four b oxes
continue the calculation to include the representations lab elled by tableau with eight
b oxes would requires the manipulation of matrices of dimension There is also
an approximately factorial increase in the numb er of p ermutation matrices that are
required to de ne the symmetry conditions the order of the p ermutation group on
jf j symb ols is jf j but not all p ermutations are required for each set of symmetry
conditions The combination of these problems limited calculations to jf j or
To obtain further results the programs could b e improved or computing p ower
increased In particular an e cient algorithm for calculating the p ermutation ma
trices is likely to make a signi cant impact However even with these improvements
the di culty still grows rapidly with jf j and obtaining results for tableau of higher
dimensions is di cult
These results for representations of low dimension all agree with the analytic
results Both the numb er and exchange sign of the multiplets are as predicted The
most interesting case is probably the representation
Chapter Calculation of the exchange sign for the irreducible representations of
SU
Young Tableau Multiplet Spin Exchange Sign
Figure Numerical results for representations of SU lab elled by Young tableau
with six b oxes
Numerical calculation of the exchange sign
which is the lowest dimensional representation to contain two multiplets with the
same spin We see that the numerics con rm the prediction that the multiplets will
have di erent exchange signs despite b elonging to the same representation
Chapter
Character decomp osition of
SU n
The subspace of spin vectors jMi that can b e used to construct the p osition
dep endent basis corresp onds to vectors in particular representations of a subgroup
H of SU n The representation of H from which a vector jMi is selected deter
mines the spin s of the particles and how the vector transforms under p ermutations
In order to assemble these physical representations to b e used in the Berry Robbins
construction we rst consider the group of inner automorphisms of the maximal
torus of the exchange rotations the Weyl group and determine its action on spin
vectors jMi Both the Weyl group and H are formulated in terms of the semidirect
pro duct which is not necessary but provides a general p ersp ective on the calculation
To nd the numb er of subspaces which corresp ond to physical representations of the
subgroup H we use the character orthogonality relations
Character decomp ositions
A representation of a group G can b e restricted to elements of a subgroup H This
representation of H will in general b e reducible We de ne to b e the character
H
of an irreducible representation of H and X the character of the representation of
G
G Then the numb er of irreducible representations corresp onding to the character
Character decomp ositions
in the representation of G is given by
H
X
X
N X h h
G H
H
where parameterises the classes of H and h is an element of H in the class
is the order of the class and the order of H This decomp osition follows from
H
the character orthogonality relations which are discussed in more detail in section
The character orthogonality describ ed in equation assumes that the group
is nite However we will see that the subgroup H is continuous For a compact Lie
group the sum b ecomes an integral and the character decomp osition lo oks like
Z
X
X h h h dh N
G H
H
where h is the Haar measure on H
We have seen previously that to b e used in the Berry Robbins construction
vectors jMi in the carrier space of the representation of SU n must have certain
prop erties
jMi must b e an eigenvector of S with eigenvalue s for all n spins
j
ij
jMi must b e a null state of the z comp onents of the exchange algebra E
z
Only certain representations of H will corresp ond to vectors with these prop erties
we will refer to these as the physical representations of H In order to use the
character orthogonality relations to nd the numb er of physical representations of
H in SU n we must nd the classes and characters of H We will also require the
character of the representation of SU n for elements in the subgroup H With
these comp onents we can integrate the pro duct of characters over the subgroup to
nd the numb er of physical representations of H that an irreducible representation
of SU n contains
Chapter Character decomp osition of SU n
The semidirect pro duct
For SU n the action of the Weyl group can b e conveniently describ ed using the
semidirect pro duct If we take two groups G and H where G acts on H as a group
of automorphisms
H H h h
g g
Then the semidirect pro duct G n H is the group with elements g h and the
multiplication law
g h g h g g hh
g
The identity element of G n H is I I From the multiplication law we nd
G H
that
g h g h
1
g
Classes of G n H
All elements in the class of an element g h of G n H can b e found by conjugating
g h by an element g h of the group
g h g h g h g g g hh h g h
1
g
g g g
Elements g h in the class of g h are de ned by an element g of G in the
same class of G as g The elements h that can b e obtained by conjugation are
h restricted to those of the form hh
g
g
Irreducible representations of G n H
First we determine how G acts on irreducible representations of H Let f D W g
denote a complete set of inequivalent unitary irreducible representations D H
acting on vector spaces W Given D W let
D h D h
g
g
The semidirect pro duct
It is easy to check that D h is a representation of H it is irreducible and acts
g
W is equivalent to one of the irreducible on the space W Consequently D
g
representations D W and to denote this instead of we will use Given g
g
there exists a unitary transformation
g W W
g
such that
g
h g D h g D
g
Therefore G acts on the set of irreducible representations of H f D W g by
sending to The stabiliser of the representation is a subgroup of G denoted
g
G
G fx G g
x
For x G x is a unitary transformation on W and
D h x D h x
x
We now turn to consider an irreducible representation U of G n H acting on
the carrier space V We can restrict the representation U to the subgroup H and
decomp ose V into orthogonal subspaces which transform according to an irreducible
representation of H lab elled by
M
V V
If we select a particular subspace V which carries the th representation of H with
multiplicity r then we can cho ose an orthonormal basis for V
jj i
where j runs from to d the dimension of the representation and runs from
to r In this basis
h jk i U h jj i D
k j
Chapter Character decomp osition of SU n
so U h is blo ck diagonal in this basis and each blo ck is D h an irreducible repre
sentation of H
For x G the representation U x leaves V invariant
U x jj i x jk i
k j
The co e cients x factorise To see this we start by using the group multi
k j
plication law
U x U h U h U x
x
Applying this to the state jj i we obtain
h x jl i h jl i D x D
x
k j l k
l k k j
Equating co e cients and using equation we get
x D h D h x x
l k k j
k j l a ab bk
To simplify this relation we can think of the co e cients as de ning a d
k j
dimensional matrix parameterised by the indices and
x x A
k j
k j
so A x is a matrix Writing as a matrix equation we have
A x D h x D h x A x
Solving this we obtain
x A x D h D h x A x
Schur s lemma implies that the matrix in the square brackets is a multiple of the
identity
x A x c x I
Returning to the index notation
x x c x
j k
j k
The semidirect pro duct
This is in e ect the de nition of the tensor pro duct
C
where C is the matrix with comp onents c This shows that the co e cients
factorise
We are now interested in the prop erties of and C As U de nes a representation
of G we have
U x U x U x x
The matrices also ob ey this multiplication law
x x x x
Factorising using equation we have the condition
x x C x C x x x C x x
which implies that
x x x x x x
C x C x x x C x x
where x x is a phase factor The equations and resemble the def
initions of a representation of G they are called pro jective representations The
phase factor x x is the factor system of the pro jective representation The
equations show that b oth and C are pro jective representations of G where the
factor systems are the inverses of each other We will see later that the app earance
of these pro jective representations is not signi cant For the representations of H in
which we are interested the pro jective representations and C will turn out to b e
representations in the usual sense x x
The group G is an invariant subgroup of G so we can form the quotient group
G G as in section Elements of this quotient group are cosets g G We can
select a set of coset representatives
g G g G g G
m
Chapter Character decomp osition of SU n
so that the coset g G is lab elled by a Any g G can b e expressed uniquely as a
a
pro duct of a coset representative g and an element x in G g g x The pro duct
a a
of two coset representatives is
g g g x
a
b ab ab
This is determined by the multiplication law of G G For cosets of a quotient group
g G g G g g G g G
a a
b b ab
Using the coset representatives we can de ne new basis vectors
jaj i U g jj i
a
We will show that the space spanned by the vectors jaj i is invariant under U
If we take a general element of the representation U g h and act on a basis
vector we get
U g h jbk i U h U g U g jk i
b
U g U g U h 1 jk i
1
b
g g
b
h U g g jl i D
1
1
b
l k
g g
b
Using the coset representatives g g g x for a unique x in G consequently
b d
U g U x jl i U g h jbk i D h
1
1
d
l k
g g
b
x c x U g jj i h 1 D
1
d
j l l k
g g
b
x c x jdj i h D
1
1
j l l k
g g
b
The vector U g h jbk i is a linear combination of the vectors jdj i so the space
spanned by these vectors is invariant under the action of the of U As U is an
irreducible representation the representation C of G must b e irreducible The irre
ducible representations of G n H are lab elled by an irreducible representation of H
and a pro jective irreducible representation C of G with a factor system conjugate
to
The Weyl group
Equation de nes the representation U of G n H To simplify the structure
of the equation we can write the basis vectors as tensor pro ducts
jbk i jbi jk i j i
Using matrix notation we obtain
U h jbi jk i j i jbi D h 1 jk i j i
g
b
U g jbi jk i j i jdi x jk i C x j i
where g g g x These equations also de ne the representation U
b d
The Weyl group
We will rst de ne the Weyl group of the exchange p ermutations SU n and then
see how this group acts on the spin vectors jMi which are used to construct the
p osition dep endent spin basis
De nition of the Weyl group
For a Lie group G the maximal torus T is the group generated by the Cartan sub
algebra The normaliser of G is de ned to b e
Norm T fx G xtx T for all t T g
G
T is clearly an invariant subgroup of Norm T We de ne the Weyl group of G as
G
W Norm T T
G G
Elements of the quotient group are cosets xT where x is an element Norm T
G
There is a natural homomorphism g from W to the automorphisms of T
G
g xT t xtx
As x is an element of Norm T conjugating by x is an automorphism of T The
G
Weyl group acts on T as automorphisms of T
Chapter Character decomp osition of SU n
The Weyl group for SU n
For the exchange p ermutations SU n the maximal torus is the group of n n
diagonal matrices
i
1
e
C B
C B
t
C B
A
i
n
e
P
i n
j
where e This is the n dimensional torus T
Applying the de nition of the normaliser to SU n pro duces conditions
n
on the elements x of Norm T
SU n
xt t x
Writing this in comp onent form for the element x we get
j k
i
i
k
j
e x e x
j k j k
i
i
k
j
e For a given j the second case can This implies that either x or e
j k
only hold for one k the other entries in the row must all b e zero This implies that
x is a phased p ermutation matrix
i
1
e
C B
C B
D x
C B
A
i
n
e
where D is the n n de ning representation of S
n
if j k
D
j k
otherwise
For the de ning representation of the symmetric group
det D sgn
so D is not an element of SU n In order for x to have determinant unity we
P
i
j
require e sgn
The Weyl group
n
We can see that T is indeed a subgroup of these matrices obtained by
taking to b e the identity The inverse of x is
i
1
e
C B
C B
y T
x x D
C B
A
i
n
e
so these phased p ermutations x are unitary The phased p ermutation matrices
n
form the group Norm T We will call this group of matrices to indi
n
SU n
cate the similarity with S The Weyl group of the exchange angular momentum is
n
n
W T so W is isomorphic to S
n n
SU n SU n
The group
n
We will see later that plays an imp ortant role in de ning H and in preparation
n
we will investigate the structure of this group more closely We can parameterise
the elements x of by the n angles and the element of S An
n n n
element x is then de ned by If we lo ok at the multiplication of two elements
x and x of using the factorisation we nd that
n
This is the multiplication law of semidirect pro duct de ned in
n
S n T
n n
n
where S acts on T by p ermuting the angles
n
Classes of
n
We can now identify the classes of First we note that the inverse of an element
n
x is
Chapter Character decomp osition of SU n
Now if we take an element and conjugate with any element we obtain
the other elements in the class of
The conjugate element of S is in the same class of S as So one lab el for a
n n
classes of is a class of S
n n
We must also see how the phases a ect the class If we lo ok at the class of
I we see that
I
can b e any p ermutation so all other elements in the class of I can b e obtained
from a p ermutation of the phases These classes are lab elled by unordered sets of
n phases f g
If we consider the general vector of phases
and are arbitrary Applying the p ermutation to b oth sides
As is arbitrary let us de ne a new arbitrary vector of phases
In this general case it is clear that unlike the classes of the identity element of S
n
the phases are not constants of the conjugation However for each cycle of
there is a constant formed from a sum of the phases Assume contains the m
cycle ij k then taking the sum of the the i j k th phases on the right hand
side of equation we nd
X
i i j j j i
k k l
l in ij k
The Weyl group
We will de ne the sum of the phases in a cycle to b e
X
ij k l
l in ij k
Applying any p ermutation the vector of phases will still contain m phases
whose sum is Setting the values of these constants determines the class of
ij k
for a given A class of is determined by r angles where r is the numb er
n
of cycles in
The classes of S are lab elled by a partition where j j n In a partition
n
the integers corresp ond to the lengths of the cycles of elements
r j
of S in the class We lab el the classes of with a partition and the vector of
n n
P
i
j
The condition on the phases e sgn transfers angles
r
1
to a condition on the parameters
j
P
r
i
j =1
j
e sgn
sgn is constant for any p ermutation in the class as the numb er and length of
the cycles is the same for all elements in Using the condition the numb er
of parameters can b e reduced by one A class of is therefore lab elled by a
n
j
partition of n into r integers and r angles for each partition
j
Representations of
n
n
is the semidirect pro duct S n T so applying the results of section the
n n
n
irreducible representations of are lab elled by irreducible representations of T
n
and pro jective irreducible representations of the stabiliser of S with resp ect to the
n
n
representation of T
n
T is the group of diagonal matrices de ned in The group is Ab elian so
the irreducible representations are one dimensional The irreducible representations
Q of T are lab elled by an integer m where
m im
1 1 1
Q e
Chapter Character decomp osition of SU n
n
An irreducible representation of T is formed from a pro duct of these monomial
representations
P
m i m
j j
Q e
The irreducible representation Q is lab elled by the vector of n integers m
n
m m As Q is a representation of T we exp ect one of these integers
n
P
n i
j
to b e redundant For the group T we have the condition e If we de ne
a vector of integers m from m so that
m m m m m m m
n n n n
m m
The vector m lab els then we see that the representation Q is equivalent to Q
n
irreducible representations of T
n
In the semidirect pro duct the group S of automorphisms of T de nes a map
n
n
b etween irreducible representations of T For in S
n
P
i m
m m
1
j
(j )
Q e Q
The stabiliser of S with resp ect to the representation m is de ned as the group
n
m m
S f S Q Q g
n m n
This is the subgroup of p ermutations b etween equal integers in m If we divide the
integers m into sets fi j k g where m m m then S is the direct
j i j n m
k
pro duct of the symmetric groups on these sets of symb ols A representation of this
direct pro duct is de ned by cho osing a representation for each of the subgroups of
p ermutations amongst the sets fi j k g Representations of de ned using
n
pro jective representations of the stabiliser are not considered as they will not b e
needed later
n
To make this clear let us take the representation of T with n lab elled by
the vector of integers
m
The Weyl group
Then the pro duct of the symmetric groups S of p ermutations of f g and S of
f g form the group S An irreducible representation is lab elled by two parti
n m
tions of three and of two
n
A representation of is lab elled by a representation m of T and r partitions
n
one for each set of equal integers j j is the numb er of equal integers in one set
j j
Using the results in section we can construct an irreducible representation for
any choice of m and
r
The action of the Weyl group on spin vectors jMi
We will now consider the prop erties that make the Weyl group signi cant in the
construction of the p osition dep endent spin basis A representation SU n acts
on a complex vector space V We have seen that spin vectors jMi used in the
construction must b e in a subspace W of V From vectors in W have zero
weight with resp ect to the exchange angular momentum su n If we take x to b e an
n
automorphism of the maximal torus T of SU n the representation SU n
can b e restricted to elements x of For jMi in W
n
x jMi xt jMi
n
where t is any element of T This comes from the relation for zero weight vectors
t jMi jMi As a consequence the representation xt acting on the subspace
W is the same for any element of the coset xT and the representation is also
n
a representation of the Weyl group T of SU n
n n
We see that on the spin subspace W restricting the representation to the
automorphisms of SU n is equivalent to de ning a representation of W We
SU n
now want to investigate the prop erties of the Weyl group An element of W
SU n
n
is a coset xT Using the decomp osition an element of the coset can b e
Chapter Character decomp osition of SU n
written
i i
1 1
e e
C B C B
C B C B
D
C B C B
A A
i i
n n
e e
i
1 1
(1)
e
B C
B C
D
B C
A
i
n
1
(n)
e
P
i
j
The only condition on the phases is that e so the coset consists of all
elements x of which are constructed from the same element of S As is the
n n n
semidirect pro duct of S and T the multiplication law for the cosets is just that
n n
of S It follows that the Weyl group W is isomorphic to the p ermutation
n
SU n
group S
n
This is just the structure that we require in order to construct the p osition de
p endent basis In the group SU n we need a subgroup which p ermutes the n spins
However if the spin states jMi are to transform according to an irreducible rep
n
resentation of S then a representation of should also provide a representation
n n
of S in the spin subspace W As any representation of on W descends to a
n n
representation of W spin vectors transform according to a representation of S
n
SU n
The physical subgroup H
The n spin subgroup
The exchange p ermutations which pro duce W are not the only signi cant
SU n
subgroup used to de ne the subspace of spins W From and spin vectors
jMi in W have zero weight with resp ect to su n and identical spins s with resp ect
n n
to the n spin subgroup SU Matrices U in SU are constructed from n
The physical subgroup H
matrices u in SU
j
u
C B
C B
U
C B
A
u
n
These matrices clearly form a subgroup of SU n Irreducible representations of
n
SU are constructed by taking the tensor pro duct of n irreducible representa
tions of SU
n
R U R u R u R u
n
j
where R u is an irreducible representation of SU An irreducible representa
n
tion R SU where the n spins are the same is a tensor pro duct of n identical
n
representations of SU R SU R SU
n
If we restrict a representation of SU n to the SU subgroup we can de
n
comp ose this reducible representation of SU into irreducible comp onents and
nd the numb er of these representations where the n spins are identical The spin
vectors jMi used in the construction must b e chosen from the subspaces spanned
by these representations
De nition of H
We want to restrict a representation of SU n to a subgroup generated by the
n
p ermutation op erations and the spin subgroup SU We will call this the
n
subgroup of physical transformations of the spin vectors H The vectors jMi used
in the construction of the transp orted spin basis will b elong to particular irreducible
representations of this subgroup
Given a vector jMi in the subspace W applying one of these transformations
will pro duce another vector in W with the same spin s To form elements in H we
Chapter Character decomp osition of SU n
n
take the pro duct of an element of SU with an element of
n
i
1
u e I
B C C B
B C C B
D I
B C C B
A A
i
n
u I e
n
i
1
e u
C B
C B
D I
C B
A
i
n
u e
n
The elements of de ned in equation are made into a subgroup of SU n
n
by taking the tensor pro duct with the identity matrix as with the generators
of the exchange angular momentum in section Elements of H are parameterised
by u u u and As with we can write down the multiplication law
n n
for elements of H constructed using equation
u u u u
This is also the multiplication law of a semidirect pro duct
n
H n SU
n
n
where acts on SU by p ermuting the elements of SU u u for
n
n
Classes of H
The classes of are lab elled by partitions of n into r parts and a vector of an
n
gles of length r for each partition In section we noted that classes of a
semidirect pro duct are lab elled rstly by classes of the automorphism group The
n
automorphisms of SU are provided by so and will also distinguish
n
classes of H
If we conjugate an element u of H with all elements v we obtain
The physical subgroup H
the class of u
u v u v
v u u
We are interested in the extent to which u is determined by u To investigate this
we will follow the pro cedure used to determine the classes of If we apply the
n
n
p ermutation to u and de ne a new arbitrary element of SU from v
w u
Then from we nd the relation
u w u w
If we consider the element u of SU it is clear that provided is not xed by
we can obtain any element of SU as the rst term in u If the new rst
element is to b e w then let the element w b e u The class of individual elements
u in SU is not in general maintained by conjugation
j
Let contain the m cycle ij k l then the pro duct of the elements of u
in this cycle is
w u w u u u w u w w u w
i i j j
l l
i j l
i j
k
w u u u w
i i j
l
i
The result is in the same class of SU as the element u u u To obtain u
i j
k
we apply the p ermutation This changes the order of the elements of SU
however there will still b e m terms whose pro duct is in the same class of SU as
u u u The class of SU of the pro duct of the elements of u in the same cycle
i j
k
of is a constant of conjugation
The class of an element u of SU is lab elled by a complex numb er of mo dulus
see section To distinguish the unity where the two eigenvalues of u are and
classes of H we require one eigenvalue for each cycle in If the class of is
lab elled by a partition of n into r parts the classes of H for this choice of are
Chapter Character decomp osition of SU n
distinguished by a vector of eigenvalues of SU
r
As H is a semidirect pro duct the classes of H are lab elled by classes of and
n
the constants of conjugation A class of H is lab elled by a partition of n into
r parts and for each a vector of angles and a vector of eigenvalues of SU
b oth of length r
The volume element of H
The group H has an unusual structure it consists of a nite numb er of continuous
parts one for each element of S To sum a pro duct of characters over the group H
n
involves a sum over the elements of S and an integral over the continuous param
n
eters of the classes for each element of S For two elements and in the same
n
class of S we have found that the continuous parameters lab elling their classes are
n
the same Therefore the sum over the elements of S can b e reduced as for a nite
n
group to a sum over the classes of S where each class is weighted by its order
n
For each class of S there still remains an integral over the eigenvalues of
n
SU and the angles These integrals require a volume element for the region of
H that is to b e integrated over The angles lab el elements of the torus T each set
lab els a single element which are all weighted equally So to integrate with resp ect
to the angle we use the in nitesimal angle d The condition on the angles
j j
P
i
j
e sgn reduces the numb er of angles that we need to integrate over by
one So if their are r cycles in the class of S there will b e r integrals
n
The parameters are eigenvalues of SU and for each choice of there are
j
many matrices in SU with the required eigenvalue The eigenvalues have mo dulus
i
j
one so we can write e where runs from zero to The in nitesimal volume
j j
element for SU is d where
j j j
j j
j
The physical subgroup H
see section Alternatively we can write
j j j j j j
We see the two eigenvalues of SU app ear symmetrically in the volume element
We can now write the integral of a pro duct of two characters over the group
H To corresp ond to the character decomp ositions in and we will take
the character X to b e a reducible character of a representation of SU n
SU n
restricted to elements of the subgroup H and to b e an irreducible character of
H
H then
X
X
N A
S
n
S
n
where
Z Z
A d d d d
r
1 r 1
H
X
H
SU n
and
r r
This is the form of the character orthogonality relations we will use later to de
comp ose the irreducible representations of SU n The volume is found by
H
integrating the in nitesimal volume elements as with the unitary group in theorem
Z Z
d d d d
r
H
1 r 1
Each of the r angle integrals with resp ect to contribute a factor of An
integral with resp ect to pro duces a factor of which when evaluated turns
SU
out to b e
r r
H
Using these results we will b e able to determine the decomp osition of a representa
tion of SU n given the characters of the representations of SU n and H
Chapter Character decomp osition of SU n
Representations of H
As H is a semidirect pro duct we can apply the general theory develop ed in section
to nd the irreducible representations of H
n
H n SU
n
n
is also a semidirect pro duct S n T and it is the group S that
n n n n
n n
de nes the automorphisms of b oth T and SU From this we can re ne the
de nition of H
n n
H S n T SU
n
n
Irreducible representations of H are lab elled by irreducible representations of T
n
SU and pro jective representations of the stabiliser of the representation of
n n
T SU in S
n
n
An irreducible representation R of SU was de ned in
n
R u R u R u R u
n
j
R u is an irreducible representation of SU The irreducible representations Q
n
of T are lab elled by the vector of integers m From
P
m i m
j j
Q e
m n n
Together Q and R u de ne an irreducible representation of SU T
The group of automorphisms S de nes maps b etween irreducible representations
n
For an element of S
n
R u R u
Similarly
m m
Q Q
n
These two relations de ne maps b etween the irreducible representations of SU
n
T
The physical subgroup H
n n
The stabiliser under S of the representation of SU T lab elled by R
n
and m consists of the elements of S which map the representation R m into itself
n
S f S R R and m mg
n R m n
j
If all the R in R are di erent then clearly the stabiliser contains only the identity
element This applies equally if all the m in m are di erent
j
i j k
We can divide the n symb ols into sets fi j k g where R R R
and m m m Then if is an element of S which only p ermutes sym
i j n
k
b ols in the same set we know that R R and m m so is in the stabiliser of
R m The p ermutation group on a set of symb ols fi j k g is a subgroup of S
n
We de ne S to b e the direct pro duct of the p ermutation groups on all such sets
n R m
of symb ols fi j k g Then S is the stabiliser of the representation R m of
n R m
n n
SU T As S is formed from subgroups of S we can see that it is
n R m n
also a subgroup of S
n
To clarify this let us take an example for n If we cho ose a representation of
n n
SU T where
R R R R R R
m m m m m
Then the stabiliser is the pro duct of the symmetric groups on the symb ols f g
and f g Any in this subgroup will map the representation R m back into
itself
If there are q sets of symb ols fi j k g then an irreducible representation of
q
S is lab elled by q partitions where each de nes an irreducible repre
n R m
sentation of the p ermutation group on one set fi j k g We can use this to de ne
an irreducible representation of H The representation is lab elled by a representa
n n q
tion R of SU a representation m of T and a representation of
S These representations will b e su cient for our problem without considering
n R m
pro jective representations of the stabiliser
Chapter Character decomp osition of SU n
We can use the general results for the semidirect pro duct in section to
de ne how such a representation of H acts on basis vectors of the carrier space of
1 r
m
S the representation Let S b e the stabiliser of R u Q
n R m n R m
is an irreducible representation of the stabiliser A vector in the representation of
H can b e written
jhi jbi c jv i j i
where c is a complex numb er of mo dulus one the space acted on by the one dimen
m
sional representation Q jv i is a basis vector of the carrier space of R so it can b e
written as a tensor pro duct of n basis vectors of the representations R
j
jv i jv i jv i jv i
n
1 q
S and The basis vector j i is in the carrier space of the representation
n R m
jbi is de ned by the coset representatives of S S
n n R m
b
From equations and an irreducible representation of H is de ned
by
P
i m
j
(j )
b
n
I I jhi jbi e c jv i j i
S
SU
n
I u jhi jbi c R u jv i j i I
n 1
S
b
T
n
1 q
I n jhi jdi c j v i j i I
n 1
SU
T
where with S
n R m
b d
Physical representations of H
Using the semidirect pro duct we have classi ed the irreducible representations of
H The vectors of SU n eligible for use in the construction can b elong only to
particular representations of the subgroup H These are the representations that we
will now determine
From vectors used in the construction must have equal spins s with resp ect
to the n spin subgroup Restricting an irreducible representation of H lab elled by R
The physical subgroup H
q n n
m and to the SU subgroup pro duces the representation R SU
j
see equation The n spins are identical if the representations R SU used
to construct R are the same
n
R R R
This is the rst condition on the physical irreducible representations of H which can
b e used to generate a p osition dep endent spin basis
Spin vectors used in the construction are also zero weight vectors of the exchange
angular momentum condition This ensures that the Weyl group acts as the
p ermutation group on the n spins and that a representation of descends to the
n
Weyl group For this to b e the case we saw that a representation of the group
n
of automorphisms of the exchange angular momentum can not dep end on the phases
From equation
jMi I jMi
If we restrict a physical representation of H to the subgroup then the represen
n
tation of we obtain should ob ey the condition From we see that
n
for this to b e the case the representation of H must b e constructed from the trivial
n
representation of T
m m m
n
From the two conditions and on the representations R and m we
can determine the stabiliser S for these physical representations of H
n R m
S f S R R and m mg S
n R m n n
All elements of S map a representation R with n identical spins back into itself and
n
n
map the trivial representation of T to the trivial representation As irreducible
representations of H are lab elled by representations of S the physical repre
n R m
sentations are lab elled by an irreducible representation of S A representation of
n
S is lab elled by a single partition Selecting the representation of S can b e
n n
seen as a choice of representation of the Weyl group which p ermutes the spins in
Chapter Character decomp osition of SU n
the construction The physical representations of H are lab elled by a choice of spin
n
s which de nes the R SU in R SU and a representation of S
j n
Rewriting equations to for physical representations of H we obtain
I I n jv i j i jv i j i
S
SU
n
I u jv i j i R u jv i j i I
n 1
S
T
n
n
I n 1 I jv i j i j v i j i
SU
T
In de ning the vectors in the carrier space of the representation the constant c
n
has b een removed As we are considering only the trivial representation of T its
inclusion would serve no purp ose For the physical representations of H the quotient
group S S is S S This is a group of one element and so there is only a
n n R m n n
single vector jbi which is also omitted As the stabiliser S is now the whole of
n R m
S we do not need to de ne a separate element of S Equation de nes
n n R m
a representation P of the symmetric group where
P jv i jv i jv i jv i jv i jv i
1 1 1
n
n
P is a representation which p ermutes the tensor pro duct
Equations and de ne the physical irreducible representations
of H In order to investigate further the prop erties of these representations it will
s
b e useful to construct them explicitly A physical representation B H is
s s s
B u R u R u P
n
s
It is the numb er and typ e of these representations B in a representation of SU n
restricted to H that we want to determine A spin statistics connection is a rela
tionship b etween the spin and the representation of S for particular choices of
n
representations of SU n The representation of S determines how spin vectors
n
transform under p ermutations of the spins If is the trivial representation spin
vectors are symmetric when p ermuted a prop erty which then transfers to the p o
sition dep endent spin basis The alternating representation of the symmetric group
provides antisymmetric spin vectors and other representations of S corresp ond to
n
The physical subgroup H
spin vectors which pro duce parastatistics in the p osition dep endent basis
s
Characters of B H
We intend to use the character orthogonality relations to decomp ose a representa
tion SU n restricted to H To apply the character orthogonality relations we
must know the characters of the irreducible representations of H whose presence
we want to determine in these are the characters of the physical representations
s
B H
s
The representation B H de ned in is the tensor pro duct of a represen
s s
tation S and the representation R u R u P of H The char
n n
acter of a representation is the trace of the representation and the trace of a tensor
pro duct of two matrices is the pro duct of the traces of the two matrices Therefore
s
to nd the character of B H it is su cient to know the characters of the repre
s s
sentations S and R u R u P and take their pro duct is
n n
an irreducible representation of S for which the characters are well known see sec
n
s s
tion It remains to determine the trace of matrices R u R u P
n
As a preliminary we will state a lemma for the trace of a tensor pro duct multiplied
by P
n
Lemma For any n m m matrices U U
Y
n i j l
T r U U P T r U U U
cycles of
where P permutes the tensor product of n vectors and ij k l is a cycle in
j
j
Pro of Let U b e a matrix with elements U P acting to the left p ermutes
p q
j j
n
columns in U U so
n n
U U P U U
p p q q
p q n n p q
1 1
n
1
1 1
(n) (1)
Setting p q and summing
i i
X
n n
T r U U P U U
q q q q
n
1
1 1
(1) (n)
q q
n
1
Chapter Character decomp osition of SU n
Now let ij k l b e a cycle in so we know that i j Taking the terms in
with these lab els
X
l i j l i j
U T r U U U U U
q q q q q q
i i j j
l k
q q q
i j
l
s s
Using lemma the characters of R u R u P are the pro duct
n
s
on the cycles of of the characters of R u where
ij k l
u u u u u
i j
ij k l k l
s
The representation R SU can also b e lab elled by the integer s this is the
numb er of b oxes in the Young tableau of one row which distinguish irreducible
representations of SU The character of such a representation of SU is given
by the Weyl character formula section
s
s
s s
R u
s
are the eigenvalues of u They lab el the classes of SU and
s
If we combine these results we can write the character of the representations
H
s s
R u R u P of H
n
s
s
Y
s
u
H
cycles of
where and are the eigenvalues of u It follows that there is one eigenvalue for
ij l
each cycle in As the character is a class function this agrees with the de nition
of the classes of H A class of H is lab elled by a class of S with one angle and
n
one eigenvalue for each cycle in the class of S The characters of the physical
n
s
representations B H are then given by the pro duct of characters
s
B s
u u
H H S
n
Representations of SU n
The physical representations are indep endent of the angles so the angles do not
s
app ear in the characters of B H The characters are still functions of the eigen
values and the class of in S
n
Representations of SU n
The irreducible representations of SU n are lab elled by a vector of n integers
f f f The integers are the lengths of the rows of the Young tableau
n
asso ciated with the representation Classes of SU n are lab elled by n complex
numb ers of mo dulus one with the condition that their pro duct is
n
unity We will often think of this condition as a de nition of
n
n n
The complex numb ers are the eigenvalues of the elements of SU n and the classes
of an element u of SU n is determined by its diagonal matrix of eigenvalues
C B
C B
C B
C B
f g
n
C B
C B
A
n
The order of the terms in is therefore arbitrary
i
Chapter Character decomp osition of SU n
Characters of SU n
The characters of SU n are a function of the class Using the Weyl formula we
can write the irreducible characters as a ratio of determinants
f n f n f n
1 1 1
n
f n f n f n
2 2 2
n
f f f
2n 1 2n 1 2n 1
n
f
X
SU n
n n n
n
n
As these ratios are awkward to write we will abbreviate the Vandemonde determi
nants by writing only the rst row of the matrix So for example
f n f n
1 1
n
f n f n
2 2
n
f n
1
f n
1
j j
n
f f
n n
n
Using this notation the character of an irreducible representation of SU n is
f n f n
1 1
j j
f
n
X
SU n
n n
j j
n
Restricting a representation of SU n to H
To write down the character of SU n for an element h H we must nd the
eigenvalues of h From equation an element in H can b e factorised
i
1
e u
C B
C B
h D I
C B
A
i
n
u e
n
i i i i
n n
1 1
e e where When is the identity the eigenvalues are e e
n n j
is an eigenvalue of u
j
Representations of SU n
We take a vector v to b e an eigenvector of h with eigenvalue
i
1
e u
C B
C B
D I v v
C B
A
i
n
u e
n
We will treat v as the direct sum of n two dimensional vectors
v v v v
n
If is not the identity let j k l b e an m cycle in For this cycle we obtain
from the eigenvector equation m equations
i
j
e u v v
j j
k
i
l
u v v e
j
l l
These can b e combined in order to nd an eigenvector equation for v
k
i m
j
k l
e u u u v v
j
l k k k
Similar equations are obtained for the other v in the cycle In each case the elements
j
u in the pro duct have undergone a cyclic p ermutation If v is an eigenvector of
k
u u u which we will call u with eigenvalue then v will also satisfy
j
l k j l k k
with
1
ip m i m
j
l
m
e e
where p is an integer is an m th ro ot of unity multiplied by a phase determined
by the sum of the phases in the cycle and an m th ro ot of the eigenvalue of u
j k l
A cyclic p ermutation of the u s will have the same eigenvalue so each version of
equation will yield the same values of which is as we require By setting
vectors v not in the cycle equal to the zero vector we nd eigenvectors v whose
i
is also eigenvalues are given by the relation for each of the cycles in As
an eigenvalue of u and there are m ro ots of unity each m cycle contributes m
j k l
eigenvalues Summing over the cycles in we obtain the n eigenvalues of h The
characters of SU n restricted to the subgroup H are then found by plugging the
Chapter Character decomp osition of SU n
eigenvalues of h into the Weyl character formula The eigenvalues
and the sum of the phases in a cycle were the parameters
j
j k l k l
used in section to distinguish classes of H
The decomp osition of SU
The two spin example is the simplest character decomp osition and the calculations
in this case can b e done most explicitly The results can also b e compared to those
obtained using Young tableau
In this section we will only b e considering the case where n The p er
mutation group S contains two elements I and H contains two disjoint
comp onents dep ending on the element of S used to construct h in H and the
classes of H are lab elled by di erent parameters in these regions For the class I
of S two parameters are eigenvalues of SU and there is also an angle
The second angle used to construct elements of H is determined by the condition
i
1 2
e sgn I When is there is only one parameter the eigenvalue
i
of u u The angle is determined by the condition e sgn
To sum the pro duct of characters of SU and H over the classes of H we must
integrate over the two continuous regions and sum the results
An irreducible representation of SU is lab elled by f f f f and the cor
f
resp onding character is X From equations and for the character
SU
decomp osition of a reducible representation of H the numb er of physical represen
tations B H in a representation f of SU is given by
s
S
f
2
N A A
I
s
Z Z Z
d d d A
I
I
f
s
I I X
I I I
SU H
Z
f
s
X d A
SU H
The decomp osition of SU
The factors of are from the order of S is dep ending on the whether
S
2
the representation B H is constructed from the trivial or alternating representa
s
tion of S For example to nd the numb er of representations where the vectors are
antisymmetric under the p ermutation of the two spins A is subtracted from A and
I
f
the result is divided by two In equations and X is the
SU
character of the representation f of SU restricted to the classes of H lab elled by
The character of SU for elements of H
The eigenvalues of an element of H are given by equation The two classes
of S are one two cycle or two one cycles I For I the four eigenvalues are
i i i i
1 1 1 1
e e e e
and are eigenvalues of u and u resp ectively These eigenvalues can b e sub
stituted into the Weyl formula for the character of the representation f of
SU
i f i f i f f i
1 1 1 1 1 1 1 1
e e e e
f i f f i i f i
2 1 2 2 1 1 2 1
e e e e
f i f f i i f i
3 1 3 3 1 1 3 1
e e e e
f
X I
I I
SU
i i i i
1 1 1 1
e e e e
i i i i
1 1 1 1
e e e e
i i i i
1 1 1 1
e e e e
If we solve equation for the eigenvalues of H using the two cycle
we obtain four di erent eigenvalues Using the condition on the sum of the angles
i
12
e sgn to eliminate the angle the eigenvalues dep end only on
i i i i
Chapter Character decomp osition of SU n
Substituting these terms into the Weyl character formulae we obtain the character
of SU for elements of H generated using
f f f f
1 1 1 1
i i i i
f f f f
2 2 2 2
i i i i
f f f f
3 3 3 3
i i i i
f
X
SU
i i i i
i i i i
i i i i
Evaluation of A
I
s
The character was de ned in Substituting the character formulae into
H
the de nition of A we obtain the following integral
I
s s s s
Z Z Z
d d d A
I
I
i f i f i f f i
1 1 1 1 1 1 1 1
e e e e
f i f f i i f i
2 1 2 2 1 1 2 1
e e e e
f i f f i i f i
3 1 3 3 1 1 3 1
e e e e
i i i i
1 1 1 1
e e e e
i i i i
1 1 1 1
e e e e
i i i i
1 1 1 1
e e e e
where taking the complex conjugate of the character of SU in changes
the sign of the e ect on the eigenvalues is removed by p ermuting columns in
the determinants A change in the sign of the determinant in the numerator will b e
The decomp osition of SU
cancelled by the same sign change in the denominator
To evaluate this integral we will employ the Littlewo o d Richardson rule to ex
press the SU character as a sum of SU characters The same technique will b e
rep eated in the general case The co e cients found from the Littlewo o d Richardson
theorem give the multiplicity of irreducible of representations of U m U n in the
decomp osition of an irreducible representation U m n this is discussed in section
As these co e cients are the multiplicities of representations they can also b e
applied to the decomp osition of characters
X
f f
Y
U m n U m
U n
f is a vector of m n integers lab elling the irreducible representation of U m n
and similarly is a vector of m integers and a vector of n integers lab elling the
f
resp ective irreducible representations of U m and U n The co e cients Y can
b e evaluated using the rules for multiplying tableau
The character of U m n is a function of the m n eigenvalues to We
m n
can take the rst m eigenvalues to to b e the eigenvalues of the U m subgroup
m
and assign the remaining n eigenvalues to the U n subgroup By substituting the
Weyl character formulae for the characters of the unitary group into equation
we can rewrite the Littlewo o d Richardson decomp osition
f m n f m n
1 1
j j
m n
m n m n
j j
m n
n n
m m
1 1
1 1
X
j j
j j
m
m n
m
f
Y
m n m n
j j j j
m
m n
m
The determinants in the characters have b een abbreviated as de ned in We
notice that as we have dealt with the decomp osition of the unitary group we have
not needed to assume any relations b etween the eigenvalues We can regard equation
as a factorisation of a the m n determinants into a sum of terms involving
smaller determinants and we will refer to this as the Littlewo o d Richardson factori
sation We should note that while using this formulae seems to imply a signi cant
Chapter Character decomp osition of SU n
simpli cation in the evaluation of a large determinant the rules for evaluating the
f
are complex E ectively we are transferring some of the di culty co e cients Y
in evaluating the character into the evaluation of the co e cients
We will now apply the Littlewo o d Richardson factorisation to the equa
tion for A We factorise the ratio of the determinants of matrices into
I
a sum of pro ducts of ratios of the determinants of matrices
Z Z Z
X
f
A d d d Y
I
I
i i i i
1 1 1 1 1 1 1 1
j e e j j j e e
i i i i
1 1 1 1
j e e j j e e j
s s s s
j j j j
j j j j
Z Z Z
X
i j j j j f
1 1
d d d e Y
I
s s s s
1 1
1 1
j j j j j j j j
j j j j j j j j
j j is the sum of the integers in The three integrals now separate and can b e
solved in turn
The integral with resp ect to is zero unless
j j j j
If A is not zero then the phase integral gives which cancels with a factor of
I
in The integrals with resp ect to and are b oth integrals of the pro duct of
I
two irreducible characters of SU over the group SU Using the orthogonality
of irreducible characters we know that these integrals will b e zero unless the repre
sentations and are b oth the spin s representation of SU A representation
of SU is equivalent to the representation this can b e veri ed
by lo oking at the Weyl character formula therefore A is zero unless
I
s s
If we de ne a vector of integers s s s where s s s then combining the
The decomp osition of SU
conditions from the three integrals A is zero unless
I
s
If A is not zero the integrals with resp ect to and b oth pro duce factors of
I
which cancel the remaining terms in Only one term in the sum can have
I
SU
and which ob ey condition and we have the result
f
A Y
I
ss
f
Y is found by multiplying two identical tableau s and nding the numb er of tableau
ss
f in the result This is equivalent to the result determined directly from the Young
tableau in chapter for the numb er of spin s subspaces with zero weight with resp ect
to E
z
Evaluation of A
We will apply a similar pro cedure to evaluate A
Z
f f f f
1 1 1 1
j i i i j i
A
j i i i j i
s s
j j
d
j j
Taking the complex conjugate of the SU character p ermutes columns in the
matrices but any change of sign in the numerator is cancelled by a similar factor
from the denominator We will rearrange this equation into a form where we can
use character orthogonality to evaluate the integrals Let
i
e
Changing variables in the integral
Z
f f f f
1 1 1 1
j i i i i j
A
j i i i i j
s s
j j
d
j j
The integral with resp ect to is p erio dic with p erio d so the integral with
resp ect to is p erio dic with p erio d Using this the integral can b e returned
Chapter Character decomp osition of SU n
to an integral over SU Applying the Littlewo o d Richardson factorisation
and some algebra we obtain
Z
jf j
X
1 1 1 1
i j j j j
f
A Y
j j j j
s s s s
j j j j
d
j j j j
This is a sum of integrals over SU of the pro duct of three characters of SU
Solving such an integral is equivalent to decomp osing the tensor pro duct of two
irreducible representations of SU the solutions of which are the Clebsch Gordan
co e cients
Z
C d
SU
SU SU
The Clebsch Gordan co e cients for SU are given by
if j j
C
otherwise
To apply this to the integral for A we recall that the representation of SU is
equivalent to the representation and de ne the integer to b e
We also know from the Weyl character formula that the characters of SU are
real Solving for A
X
f j j jf j
Y C C A i
s
s
We have now evaluated the second integral in the formula using the Littlewo o d
Richardson and Clebsch Gordan co e cients
Before combining the results for A and A we can state another prop erty of
I
s
A An irreducible representation B H restricted to the subgroup of H con
s
nected to the identity is a tensor pro duct of the representations R of SU This
is an irreducible representation of SU SU If a representation of SU
contains no representation of this subgroup ie A then we also know it can
I
not contain the corresp onding representation of H Therefore A is zero if A is zero
I
The decomp osition of SU
f
The rules for multiplying Young tableau determine the co e cient Y The num
ss
b er of b oxes in a tableau f which is the pro duct of two tableau s is twice the numb er
of b oxes in s So if A is nonzero
I
jf j s s s s
where s is an integer but s can b e half integer Using this we can replace the phase
jf j s
i in with
The numb er of physical representations of H in SU
Substituting the expressions for A and A into equation for the numb er of
I
physical representations in the decomp osition of the representation f of SU we
nd that
s
X
f f j j f
N Y C C Y
s
s
ss S
s
2
This gives the numb er of physical irreducible representations B H when the rep
s
resentation of SU is restricted to H
is a partition of two and if we add the results for the two p ossible values of
which are we nd that
S
2
f f f
N N Y
ss
s s
The numb er of representations of H with spin s from which we can select spin vec
tors jM i to use in the construction is given by the numb er of copies of the tableau
f found when two identical tableau lab elling a representation of SU with spin s
are multiplied This agrees with the result for the numb er of multiplets with spin s
and zero weight with resp ect to E which we obtained directly in chapter
z
From the results in chapter we also exp ect that approximately half the mul
tiplets available to the construction will transform according to each irreducible
representation of S For this to agree with equation A should b e zero
when A is even and for A o dd While we have not b een able to establish
I I
this from these results we show that this is plausible If we consider a choice of
Chapter Character decomp osition of SU n
f
is not zero then changing and and for which Y
f
For example slightly do esn t change Y
f f
Y Y
1 2 1 2 1 2 1 2
j j
The sign multiplying these two terms in the sum is di erent Consequently
the terms cancel Unfortunately this argument can not b e applied to all shap es of
tableau Using this pro cedure we see that we exp ect most terms in A to cancel
and we will b e left with a small integer although restricting this to or has not
b een achieved
In section we evaluated the exchange signs of spin multiplets in the low
dimensional irreducible representations of SU numerically For these representa
tions lab elled by tableau with up to six b oxes we can also determine the numb er
of multiplets with each exchange sign using The analytic results from the
character decomp osition agree with those computed directly
The decomp osition of SU
Before tackling the general case it will b e useful to see how the techniques intro duced
to solve the integrals in the decomp osition of SU are mo di ed when the symmetric
group is less trivial If we take n to b e three we are dealing with representations of
the classical groups SU and S S is a group of six elements in three classes
the identity three two cycles and two three cycles The formula for the character
decomp osition will now involve a sum over these three classes Rewriting equations
and for the integral of the characters over H when n
f
A A A N
I
S S
s
3 3
The decomp osition of SU
Z Z
A d d d d d
I
I
f
s
I I X
I I I
SU H
Z Z
A d d d
f
s
X
SU H
Z
f
s
A X d
SU H
We will solve the integrals for the A s in turn factorising the determinants until we
reach pro ducts of SU characters
Evaluation of A
I
For elements of H connected to the identity the character of the representation of
SU is
f i f i f f i f i i f i
1 2 1 3 1 1 2 1 3 1 1 1
e e j e e j e e
i i i i i i
1 2 3 1 2 3
j e e e j e e e
This is found by substituting the eigenvalues from into the Weyl character
formula for SU We will apply the Littlewo o d Richardson factorisation
twice to the character of SU Splitting the character rst into the sum of the
pro ducts of characters of SU and SU Then factorising the character of SU
in the pro duct of two characters of SU Substituting the characters into equation
for A
I
Z Z
X
f
d d d d d Y A
I
I
1 1 2 1
1 1
j j j j j j
i j j j j j j
1 2 3
e
j j j j j j
s s s s s s
j j j j j j
j j j j j j
f
which app ear when the factorisation The Littlewo o d Richardson co e cients Y
is applied twice are the numb er of tableau f in the pro duct of the three tableau
and
Chapter Character decomp osition of SU n
P
i
j
To solve the phase integrals we use the condition e to eliminate
Then for A to b e nonzero
I
j j j j j j
The three integrals are each pro ducts of two irreducible characters of SU
integrated over SU From character orthogonality either b oth characters corre
sp ond to the same irreducible representation of SU or the integral is zero We
have the second condition
s
where is de ned as The volume is cancelled when the integral is
I
nonzero Combining the two conditions we nd that
f
A Y
I
sss
s is de ned as previously to b e a vector s s where s s s It is clear that
this result for A will generalise to any value of n A is the numb er of tableau f in
I I
the pro duct of n identical tableau with spin s
Evaluation of A
The character of SU in this case will b e similar to that used in the integral for A
I
i i
12 12
however the eigenvalues will b e replaced by p owers of ie ie
i
12
e and the equivalent terms with These eigenvalues are the solutions of
for a single two cycle We rst apply the Littlewo o d Richardson factorisation
once to separate the terms involving We rewrite as
Z Z
X
f
d d d Y A
1 1 1 1
j i i i i j
i j j j j
12 3
e
j i i i i j
s s s s
1 2
j j j j j j
j j j j j j
We can now evaluate the phase integral A is nonzero only if
j j j j
The decomp osition of SU
The integral with resp ect to is the integral of two irreducible characters of SU
from which we nd the condition
s
for A nonzero The integral for is the same as the integral for A in SU
where the representation f is replace by Combining these results we have deter
mined A
X X
j j f s
C C Y Y A
s
s
g
We see that the solution still involves two sums over the co e cients Y E ectively
the solution of A when n requires knowledge of all the solutions of A from
n
Evaluation of A
A is de ned in equation into which we substitute the Weyl characters of
SU and H The eigenvalues of the elements of H connected to a three cycle in
S are and multiplied by each of the cub e ro ots of unity We simplify the
formulae for A by changing the variable so that
i
e
By applying the the Littlewo o d Richardson factorisation twice we reduce the char
acter of SU into a pro duct
Z
X
1 1
2
j j
i f j j j j
3
d e Y A
j j
s s
1 1 1 1
j j j j j j
j j j j
j j
The expresion can now b e written as a pro duct of SU characters
Z
X
2
i f j j j j
3
A d e Y
s s s s s s
j j j j j j
j j j j j j
1 1 1 1 1 1
j j j j j j
j j j j j j
Chapter Character decomp osition of SU n
This has reduced the expression for A to a sum of three integrals over SU
of the pro duct of four irreducible characters of SU The integral of a pro duct
of four characters of SU is an extension of the Clebsch Gordan co e cients the
Racah co e cients see They are functions of the four integers which lab el the
irreducible representations of SU and in corresp ondence with the Clebsch Gordan
co e cients we lab el them C Using this notation the term A is
X
2
j j j j i f
3
A C C C e Y
s s s
This is a sum of generalised Littlewo o d Richardson co e cients multiplying gener
alised Clebsch Gordan co e cients
From the group theory we know that A should b e an integer but the expression
2
i j j j j
3
contains the phase e If a term in the sum
f
C C C Y
s s s
is nonzero for one choice of and then it will b e the same for any p ermutation
of and In particular exchanging and the term will have the conjugate
phase which ensures that the sum over all is an integer
Physical representations of H in SU
The solutions of equations to which de ne A A and A com
I
bined with the irreducible characters of S provide an analytic decomp osition of
SU into those representations physically relevant to the construction of a p osi
tion dep endent spin basis
To nd the multiplicity of any particular physical representation of H the so
lutions for A A and A which dep end on s are substituted into the sum over
I
the p ermutation group where the representation of the p ermutation group
enters
f
I A A A N
I
S S S
s
3 3 3
The decomp osition of SU
The characters of the irreducible representations of S which app ear in are
recorded in gure
I
S
3
S
3
S
3
Figure Irreducible characters of S
The total numb er of physical representations with spin s in the given represen
f
over the three irreducible representations tation of SU is found by summing N
s
of S
n
f
A A N
I
s
We see that in this case the simple solution for A do es not determine the total
I
numb er of physical representations as it did for representations of SU Unfor
tunately not only is the solution for SU more complex but the factors that are
more di cult to evaluate A and A are more signi cant
Example The representation of SU contains a spin
multiplet which exhibits parastatistics
With this example we can see how equation is used to evaluate the exchange
sign The calculation also provides an explicit case of parastatistics in a representa
tion of SU For three particles parastatistics corresp onds to vectors transforming
according to the two dimensional irreducible representation of S The character of
this representation is recorded in the second row of gure In this example b oth
f and are lab elled by the partition From
A A N
I
is zero we avoid needing to evaluate A As
S
3
Chapter Character decomp osition of SU n
which is the numb er of ways of multiplying three tableau consisting A Y
sss
I
of a single b ox for spin and obtaining the tableau This is the tableau
multiplication
x y
where the lab els x and y are used to distinguish the tableau From the rules for
tableau multiplication section there are two distinct results with shap e
x y
y x
Consequently A is
I
To evaluate A using equation we must consider all p ossible ways of
multiplying three tableau and obtaining As the representation is simple these
can b e written out in full
The dot denotes a tableau with no b oxes Except for the last case which we evaluated
are unity for each of these multiplication schemes In for A the co e cients Y
I
each case we must also evaluate the sum of co e cients
b
C C C C
s s s
for s This can b e done by applying the Clebsch Gordan formula twice
b
C
b
C
b
C
b
C
The decomp osition of SU
For the two non zero cases we must also consider the p ossible phase factors
2
i j j j j
3
e
that can o ccur For as the three tableau are identical there is only
a single contribution of this typ e and the phase factor is unity Using Y
the contribution to A from the tableau multiplication of this typ e is
b
C Y
For the tableau are all di erent and there are such multiplications
dep ending on which tableau is assigned to which particle In three cases the phase
2 2
i i
3 3
and in three it is e The contribution to A from the tableau multipli is e
cations of this form is
2 2
i i
b
3 3
e e C Y
Summing the two non zero contributions A
We can substitute the two results for A and A into equation to obtain
I
the nal result
N
The representation of SU contains a single spin subspace transforming
according to the two dimensional representation of S If we also use our results for
A and A in equation we see that this is the only physical representation
I
of H which is contained in this irreducible representation of SU Wavefunctions
on a p osition dep endent spin basis constructed from this representation will exhibit
parastatistics
While this result is interesting we should not b e surprised by it There is a
simpler line of reasoning which leads to the same conclusion If we notice that
the representation of SU is lab eled by the same tableau which lab els the
two dimensional representation of S we can see that vectors generated by these
symmetry conditions must necessarily transform according to the given irreducible
Chapter Character decomp osition of SU n
representation of the symmetric group While this can not b e applied to larger more
complex tableau it do es at least show that for spin there exists a representa
tion of SU n where the p osition dep endent basis transforms according to every
p ossible representation of S All typ es of parastatistics can b e exhibited in some
n
p osition dep endent spin basis
The decomp osition of SU n
Having tackled a more typical example in the decomp osition of SU we can ap
ply the same techniques to evaluate the co e cients A in the decomp osition of an
irreducible representation of SU n into physical representations of H
The sum over H of a pro duct of characters of H was de ned in equation
Inserting the characters of the physical representations of H and the irreducible
representation of SU n into this formula we have an equation for the numb er of
s
representations B H in the decomp osition of the representation of SU n
X
f
N A
S
n S
s
n
S
n
where
Z Z
A d d d d
r r
H
f
s
X
H SU n
is a partition of n lab elling a class of S and r is the numb er of cycles in the class
n
The irreducible characters of S are known as are the numb er of elements in
n
the classes of the symmetric group Therefore the problem is to determine A for a
f
can b e written as a sum of known co e cients general class With this N
s
The decomp osition of SU n
Evaluation of A
is a partition of n From equation the eigenvalues of
r
elements of H connected to an element of S in the class are of the form
n
j j
i p i i p i
j j j j j j
e e or e e
j p j p
j j
where p is an integer and an eigenvalue of SU There are eigenvalues for
j j
each cycle j
Using the symb ols and for the eigenvalues we can write the Weyl character
j p j p
formula for the representation of SU n
f n f n f n f n f n f n
1 1 1 1 1 1
j j
r
r
r
1 2
n n n n n n
j j
r
r
r
1 2
With the Littlewo o d Richardson theorem we split this ratio of n n determinants
into a sum of pro ducts of determinants of size one for each cycle in the
j j
class Each of these irreducible characters of SU is lab elled by a vector of
j
j
integers
j j
j j
1 1
Y X
j j
j j
j
f f
X Y
1 r
SU n
j j
j j
1 r
j j
j
i
j j
Using this factorisation we can rewrite equation The phase e can b e
j
taken out of the determinants lab elled by We obtain the integral
Z Z
j
P
j j
Y X
i
j
j
f
j
Y A d d A e
j
1 r
r
H
1 r
where A is an integral over one of the eigenvalues of SU de ned by
j j
Z
j j
i i
j j j j j
1 1
j e e j
j j
d A
j
j
j j
i i
j j j j j
j e e j
j j
s s
j j
j j
j j
j j
j
Each of the th ro ots of unity app ear in the character lab elled by The char
j
j
acter lab elled by is real as taking the complex conjugate p ermutes the columns
in the determinant but a sign change in the numerator would b e cancelled by the
Chapter Character decomp osition of SU n
corresp onding sign from the denominator
Following this factorisation we will rst solve the phase integrals in then
nd a general solution of integrals of the form of To solve the phase integrals
we use the condition
P
i
j
e sgn
As all elements of S in the same class have the same cycle structure sgn is a
n
function of the classes of in S We use this to eliminate
n r
if o dd
r r
Then solving the phase integrals either A is zero or the factors ob ey the condition
j r
j j j j
for all j
j r
r
If A is nonzero the r phase integrals pro duce a factor of which cancels
r
j j
r
the similar term in and a phase sgn
H
de ned in we will rst change To solve the integral over SU for A
j
the variable to simplify the notation
i
j
e
Substituting into the equation for A
j
j j
Z
i i
j
j j j j j
1 1
j j e e
A d
j
j
i i
j j j j j
e j e j
s s
j j
j j
j
j
j j
is the function of that results from substituting for in the volume element
The integral is p erio dic with p erio d and using this we return equation
j
to an integral over SU We apply the Littlewo o d Richardson factorisation
j
to separate the di erent ro ots of unity in the character
Z
P
j
2
X
l
j
l j j i
l=1
j
d e Y A
j
1
j
l l
s s
Y
j j
1 1
j j j j
A
j
j
j j j j
l
j
The decomp osition of SU n
l l l
lab el characters of SU Clearly this integral is very close to where
a pro duct of characters of SU however to reduce the integral to a sum of
j
j j
known co e cients we must factorise We cancel and j j and multiply
we nd that top and b ottom by Factorising
j
X
x x
j j j j
A
if o dd
j
x
Substituting where the sum is up to the integer part of and
j j
this into equation and incorp orating the extra p olynomial in into the
to a pro duct of character determined by s we have reduced the equation for A
j
characters of SU
Z
l l
P
j
2
Y X
l
1 1
j
j l j j j i
l=1
j
A
Y A d e
j
1
j
j j
l
j
j
s y s y
X
j j
j j
A
j j
y
This is the typ e of integral for which we can write down a solution using the gener
alised Clebsch Gordan co e cients
j
P
j
2
X X
l
j
l j j i
l=1
j
C e A Y
j
1
j
1
j
s y
j
y
y y
y
In this case as previously is de ned to b e
Results for the decomp osition of SU n
Collecting together the results for the decomp osition of SU n the numb er of phys
ical representations lab elled by s and contained in a representation f of SU n
is
X
f
A N
S
n
S
s
n
S
n
where lab els a class of S The co e cients A are
n
r
Y X
r
j j f
r
A A sgn Y
j
1 r
j
Chapter Character decomp osition of SU n
with the condition
j r
j j j j for all j
j r
and where A is
j
j
P
j
2
X X
l
j
i l j j
l=1
j
e A Y C
j
1
j
1
j
s y
j
y
These results give an analytic decomp osition of the representation of SU n into
physical representations of H in terms of the Littlewo o d Richardson and Clebsch
Gordan co e cients and the characters of the irreducible representations of S
n
f
by While this solution of the decomp osition problem is complete to evaluate N
s
hand we require b oth n and jf j to b e small This reduces the numb er of co e cients
Y and C required to a manageable quantity which can b e calculated For larger
values of n and jf j there do however exist algorithms for calculating the co e cients
involved in the decomp osition The decomp osition of any particular representation
could then b e computed using these results If we consider the direct numerical de
comp osition of representations of SU n from chapter the largest representations
for which the calculation was p ossible were for n and jf j While this could
have b een improved with more e cient algorithms the fundamental problem was
jf j
that the dimension of the matrices grew as n while the numb er of terms in the
symmetriser also grew as approximately jf j Therefore the analytic results provide
a signi cant computational advantage
As we saw previously for I A is small as most contributions cancel We
can give a qualitative argument to show that this is inconsistent with all spin s
multiplets transforming according to the same irreducible representation S Each
n
s
of the integrals A is indep endent of the representation of S used to de ne B
n
Taking the set of solutions of the integrals A we can regard them as a character
of the symmetric group which is then decomp osed into irreducible representations
With this picture we can consider what A would lo ok like if all multiplets with
spin s were to transform according to the representation of the symmetric group
The decomp osition of SU n
In this case we would require
A p
S
n
where p is an integer Let us consider the case when the dimension of the subspace
with spin s is large ie A is large In this case p will also b e large However for
I
I equation will not agree with the formulae and where
A is small We see that for typical representations of SU n multiplets with spin
s will transform according to di erent representations of S
n
Chapter
Conclusions
In chapter we tasked ourselves with discovering the relation b etween spin and the
statistics of particles in a p osition dep endent spin basis constructed using a general
representation of SU n When we lo oked at the case of two particles with spin in
chapter we saw the rst di culty with constructing a general spin statistics con
nection The subspace W of vectors available to construct the p osition dep endent
spin basis is formed from s s multiplets with multiple values of spin In the
Berry Robbins construction the spin statistics connection involves the natural as
so ciation b etween the completely symmetric representations of SU lab elled by
Young tableau with a single row and the unique s s multiplet that the represen
tation contains In the general case this typ e of relationship is not p ossible as xing
the representation of SU do es not x the value of spin We did however notice
that the representation of SU do es determine whether the spin of the multiplets
which make up W is integer or half integer
Turning to the exchange sign we saw that in representations which contain sev
eral multiplets with the same spin s half if the numb er of s multiplets is o dd
of the multiplets will transform according to each irreducible representation of S
For these representations of SU there can b e no clear spin statistics connection
as sp ecifying the spin along with the representation is still insu cient to determine
the exchange sign
The decomp osition of SU n using the characters of the physical representa
tions of H shows that quantum mechanics on a p osition dep endent basis generated
by a general representation of SU n admits parastatistics We saw that at least
for spin representations exist which display all typ es of parastatistics An op en
question is whether there exist representations containing all forms of parastatis
tics for any spin For any spin s and n particles p osition dep endent bases can b e
constructed which pro duce either b osonic or fermionic statistics For example the
representation of SU n lab elled by a tableau with one row and sn b oxes exhibits
b osonic or fermionic statistics dep ending on whether s integer or half integer If
n
instead we select the representation of SU n lab elled by a tableau s n
this will also contain a single spin s multiplet However the symmetry conditions
of the representation sp eci cally the antisymmetry intro duced in the rst column
will contribute an extra sign change under the exchange of any two particles In this
representation half integer spin particles will b e b osons and those with integer spin
fermions In general representations of SU n contain several multiplets with spin
s which transform according to di erent irreducible representations of S Typically
n
such a representation will also contain subspaces with many di erent spins which
can b e used to generate the p osition dep endent basis The full decomp osition of a
representation of SU n into the physical representations of H is given in terms of
generalised Littlewo o d Richardson and Clebsch Gordan co e cients
So what are the implications of these results for the spin statistics connection
To include such a relationship what we require is a rule for selecting a p osition
dep endent spin basis given a value of spin If the rule is to asso ciate a representa
tion of SU n to each spin then the chosen representation of SU n must contain
s
unique representations of B when restricted to H In order for a representation
f of SU n to x s there can only b e a single way of multiplying n tableau s and
obtaining f The simplest strategy for selecting such a set of representations is to
take the set of symmetric representations
We see that the original scheme of Berry and Robbins is the most natural sys
tematic approach to cho osing a set of representations with which to construct the
Chapter Conclusions
p osition dep endent spin bases that is a orded by the representations of SU n
This do es not amount to a spin statistics theorem as Berry and Robbins have
already p ointed out there still needs to b e a convincing physical justi cation for
insisting that spin vectors b ehave according only to these representations We have
however seen that extending the construction to general representations of SU n
do es not in general provide equally valid schemes for establishing a spin statistics
connection
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