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Home , Coherent state

The Erwin Schrodinger International Boltzmanngasse

ESI Institute for Mathematical A Wien Austria

The Geometry of Coherent States

Timothy R Field

Lane P Hughston

Vienna Preprint ESI Septemb er

Supp orted by Federal Ministry of Science and Transp ort Austria

Available via httpwwwesiacat

The Geometry of Coherent States

Timothy R Field

DERA Malvern WR PS UK

Lane P Hughston

Merril l Lynch International Ropemaker Street London ECY LY UK

and Kings Col lege London The Strand London WCR LS UK

Abstract

We examine the geometry of the state space of a relativistic eld The mathematical to ols

used involve complex algebraic geometry and Hilb ert space theory We consider the Kahler geometry of

the state space of any quantum eld theory based on a linear classical eld equation The state space

is viewed as an innite dimensional complex pro jective space In the case of b oson elds a sp ecial

role is played by the coherent states the totality of which constitutes a nonlinear submanifold C of

the pro jective Fo ck space P F We derive the metric on C induced from the ambient FubiniStudy

metric on P F Arguments from dierential geometric algebraic and Kahlerian p oints of view are

presented leading to the result that the induced metric is at and that the intrinsic geometry of C is

Euclidean The co ordinates for the single particle Hilb ert space of solutions are shown to b e complex

Euclidean co ordinates for C A transversal intersection prop erty of complex pro jective lines in P F with

C is derived and it is shown that the intrinsic geo desic distance b etween any two coherent states is

strictly greater than the corresp onding geo desic distance in the ambient FubiniStudy geometry The

functional metric norm of a dierence eld is shown to give the intrinsic geo desic distance b etween two

coherent states and the metric overlap expression is shown to measure the angle subtended by two

coherent states at the vacuum which acts as a preferred origin in the Euclidean geometry of C Using

the atness of C we demonstrate the relationship b etween the manifold complex structure on P F and

the quantum complex structure viewed as an active transformation on the single particle Hilb ert space

These prop erties of C hold indep endently of the sp ecic details of the single particle Hilb ert space We

show how C arises as the ane part of its compactication obtained by setting the vacuum part of the

state vector to b e zero We discuss the relationship b etween unitary orbits and geo desics on C and on

P F We show that for a Fo ck space in which the exp ectation of the total numb er op erator is b ounded

ab ove the coherent state submanifold is Kahler and has nite conformal curvature

PACS Classication numb ers k Dr k

I Intro duction

The existence of complex structures in quantum theory and their p ossible role in a theory of quantum

gravity is a sub ject of much interest Such structures are essential in ordinary to

express the idea that the state vector evolves unitarily according to appropriate dynamical equations

One of the main applications of complex structures in physics recently has b een the sub ject of geometric

quantum mechanics an area which has b een develop ed by a numb er of authors Kibble

Anandan and Aharonov Cirelli et al Gibb ons Hughston Ashtekar and

Schilling Bro dy and Hughston

In this picture the natural geometry of the quantum is shown to b e characterized not

merely by a symplectic structure but also by a compatible complex structure and a Riemannian metric

from which one derives the concepts of quantum mechanical uncertainty and exp ectation

In this pap er we explore geometric asp ects of the complex structures on function spaces that arise

in quantum theory By a complex structure J we mean a linear op erator acting on the Hilb ert space

2

V of solutions to a linear eld equation satisfying J The op erator J is an endomorphism of

V and if we complexify V to dene the single particle Hilb ert space H then J has eigenspaces with

p

eigenvalues which we refer to as the p ositive and negative frequency single particle Hilb ert

spaces From this H we dene the Fo ck space F via tensor pro ducts of H with itself In addition to

the complex structure J we also have the quantum mechanical symplectic form and the real Hilb ert

space metric g all of which are mutually compatible Then the triple fJ g g constitutes a quantum

Kahler structure Such structures have received much attention in the literature and form the basis of

the relationship b etween classical and quantum eld theories On a classical level it is the J op eration

that bridges b etween the symplectic and metric structures and for real b oson elds enables one to

obtain a classical expression for the norms of elds Considerations of quantum eld theory and the

intro duction of Plancks constant then show that the squared norm is equal to the exp ectation of the

quantum numb er op erator in a corresp onding coherent state see eg Field

In classical Hamiltonian mechanics the symplectic structure taken together with a preferred

Hamiltonian function H on the phase space determines the system completely It is in the passage to

quantum theory that additional structure is required namely the quantum complex structure When

compatibility of the elements of the quantum Kahler structure is assumed any two of its elements

imply the other and thus can b e viewed as the addition of either J or g to the classical

n

system In terms of geometric quantum mechanics it is the complex pro jective space CP the quantum

mechanical state space that replaces the classical phase space This is a Kahler manifold endowed with

the FubiniStudy metric It is also a symplectic manifold and thus the geometry of Hamiltonian

carries over to the space In particular the equation governing the

evolution of a quantum mechanical state Schrodingers equation b ecomes Hamiltons equations of

classical mechanics In other words Schrodinger tra jectories are given by the orbits of a Hamiltonian

vector eld on the quantum state space The extra ingredient of quantum theory is the quantum metric

g which takes the form of a real Riemannian structure on the state manifold enabling one to calculate

quantum mechanical transition amplitudes and uncertainties

A further mo dication that quantum theory entails is that uncertainty terms app ear in the Hamil

tonian function of the system in addition to the function obtained by replacing classical phase space

co ordinates by the exp ectations of their asso ciated op erators The is an example of

2 2 2 2

this phenomenon for which the quantum Hamiltonian function is equal to x p x p It

is in this context that we are naturally led to consider the coherent states Glaub er Klauder

and Skagerstam cf also Schrodinger which are in some resp ects in the closest sense classi

cal in b ehaviour These states saturate the quantum mechanical uncertainty inequality xp h

and their quantum mechanical evolution corresp onds closely to classical evolution in the absence of

interactions

The pap er addresses various asp ects of the quantum Kahler structure with emphasis on the ge

ometric features of Fo ck space Our main results are as follows We b egin by developing geometric

quantum mechanics for relativistic quantum eld theory thus extending existing results in the liter

ature to pro jective Fo ck space P F We see how this applies to coherent states and derive geometric

prop erties of the coherent state submanifold C In particular in Theorem we show that the intrinsic

geometry of C is complex Euclidean and that the single particle state vectors serve as complex Eu

clidean co ordinates for this submanifold This enables one to deduce other prop erties of C and provides

a relationship b etween the complex structure on the single particle Hilb ert space H and that on the

ambient state manifold We discuss the geo desic and unitary orbits of C observing that C and the

state space P F share no geo desics and that the geo desic orbits of C are nonunitary We discuss the

corresp onding situation for truncated Fo ck spaces given by the condition that the exp ectation of the

total numb er op erator is b ounded ab ove In particular we deduce that the asso ciated submanifold of

coherent states has nite conformal curvature

The pap er concludes with a discussion of some physical issues We discuss the relationship b etween

the theory of p ositive op erator measures POMs and coherent states in the context of our results and

in this connection remark on prosp ects for a mo del of sto chastic state vector reduction

In the generation of a quantum Fo ck space we consider the Hilb ert spaces of real and complex

solutions to a set of linear classical eld equations In the former case there is only one available

complex structure since this maps real solutions of the eld equations to themselves In terms of the

Fourier transform k of the eld multiplication by a complex numb er a ib is replaced by

the action of the op erator a J b on where J acts linearly and multiplies the p ositive and negative

frequency parts of the eld by i and i resp ectively We say that J is the quantum complex structure

acting on the Hilb ert space In the case of a complex valued eld there are two p ossible complex

structures namely a the one given ab ove and b straightforward multiplication of by the complex

numb er However it is only the former choice that leads to p ositive values for the exp ectation of

energy Gero ch a

I I The quantum Kahler structure

Let V denote the Hilb ert space of normalizable real solutions to a classical linear eld equation and

n n

dene the single particle Hilb ert space H V C We adopt the notation H H for the n

fold tensor pro duct of H with itself where denotes the symmetric tensor pro duct for b osons and

n

antisymmetric tensor pro duct for fermions Then H is said to b e the nparticle Hilb ert space and we

2 n

dene Fo ck space as the Hilb ert space F C H H H see eg Gero ch for

various technical details of this construction which are not so relevant for the present discussion We

2

intro duce the quantum complex structure J a linear map J V V such that J Then if we

extend V to V C the map J admits eigenspaces with eigenvalues i and i that we call the spaces of

p ositive and negative frequency solutions denoted by H and H resp ectively so H H H The

+ +

spaces H give rise naturally to the p ositive and negative frequency Fo ck spaces F according to the

2 n

scheme F C H H H We use an abstract index notation Penrose and Rindler

a

for elements of H so for a typical element of H we write See Gero ch ab Wald

for more details of the abstract index notation in a Hilb ert space context We shall take H to have a

countably innite basis so that the index on can b e thought of as running over the natural numb ers

There are some technicalities asso ciated with the fact that in physics one deals typically with an innite

dimensional Fo ck space which is itself built up from an innite dimensional single particle Hilb ert space

1

H However in practice it is reasonable to assume that the underlying single particle Hilb ert space is

separable Streater and Wightman so any vector can b e decomp osed along countably many basis

states as for example o ccurs in the Fourier series analysis of an oscillator with b oundary conditions

In the case of a state not built up in this way one usually argues via continuity in the relevant function

space for further discussion of these issues see Schilling

We now develop in further detail the abstract index notation For a p ositive frequency eld we

1

use an unprimed Greek index so for example H for a negative frequency eld we use a primed

+

Greek index Then we can regard the italic index a as comp osite by writing a where as

0

ab ove and have eigenvalues i and i under the action of J resp ectively so J i

0 0 0

0

and J i A general complex eld decomp oses into indep endent p ositive and negative

0 0 0

a

frequency parts We shall refer to and resp ectively as the p ositive and

a

negative frequency parts of the complex eld For all the action of complex conjugation will b e

0

0

denoted and and thus it interchanges index typ e For a real eld we have

0

a

thus indicating that in this case the p ositive and negative frequency parts are complex

conjugates of each other

We can also represent the metric the symplectic structure and the complex structure in matrix

form according to this scheme For the metric we have

0

g

g a

ab

0

g

0

is Hermitian or real in a sense explained b elow Note that as in the case in which the tensor g

of the algebra of comp onent spinors a primed index commutes with any unprimed index so that

0 0 0 0

T Now complex conjugation interchanges for any multivalence tensor we have T

0 0 0 0 0 0

index typ e according to the scheme T T Again as in the case of comp onent

spinors a tensor can only b e Hermitian if it has equal numb ers of primed and unprimed indices in

0 0 0 0 0 0 0

T which case the Hermitian condition is T For the metric g in particular

0 0 0 0

we have g g g g Similarly we write the symplectic structure as

0

ig

b

ab

0

ig

This is compatible with the complex structure in that J J The action of the metric

0 0

a b

0 0

is given by g g g g and thus g is real for all real elds and

ab

0 0

0 0

The action of the symplectic structure is given by ig ig We write

the quantum complex structure in matrix form as

i

a

J c

0

b

i

0

The Dirac or quantum mechanical scalar pro duct b etween the elds and is dened by F

0

1

0

g i g F for all real elds and is Hermitian in the sense that F

2

0

0 0 0 0

and We can use g to lower indices by writing g g The complex structure

a c a

satises J J The compatibility of the set fg J g can b e represented by the following

c b

b

conditions

b

a g J

ac ab c

a b

b g J J g

ab c d ad

a b

c J J

ab c d cd

ap bq pq

d g g

ab

Condition a is the familiar expression for the p ositive denite quantum metric in terms of

the symplectic form and complex structure ie g J A straightforward consequence is

that g J for any Condition b states that the metric g is compatible with the complex

structure J ie that the metric g is Hermitian Kobayashi and Nomizu so g g J J

We note that the term Hermitian is used conventionally in two related but rather dierent ways

First it can b e used to indicate a reality condition on noncomp osite tensors of equally mixed rank

n n

as indicated ab ove ie for tensors b elonging to the subspace H H On the other hand we say

+

q

s r 2n p

J T T The J J that a tensor of general even rank b elonging to H is Hermitian if J

pq r s abcd

c a

d

b

context will usually indicate in exactly which sense the term is b eing applied Thus c states that

is Hermitian in the latter sense Some authors use a convention for the symplectic tensor that

ab

diers by a factor of two from the one used here eg Wo o dhouse We adopt conventions such

ab a

that the tensor with its indices raised according to d ab ove is the inverse to so

ab bc

c

We refer to the compatible set fg J g as the quantum Kahler structure cf Ashtekar and Magnon

Ashtekar Observe that condition a ab ove implies that any two of fg J g are sucient to

determine the quantum Kahler structure

We now pro ceed briey to examine various op erations on Fo ck space in this spirit A state vector in

1 2

Fo ck space F can b e written j i where H H and so on The evaluation

2

of the squared Hilb ert space norm of a vector j i in F is given by jj jj For

1

any H we dene the annihilation and creation op erators A and C according to the prescription

p p

A j i A j i a

p p

( ) ( )

b C j i C j i

These ob ey the commutation relations C C A A and A C or

equivalently A C I These op erators are adjoints of each other in the sense that for

any vector F we have hC i h A i

1 2

C A We then dene the numb er op erator asso ciated with H by N jj jj

2 ( )

from which it follows that N jj jj Summing over an orthonor

mal basis f g using the identity we obtain the total numb er op erator N given by

N

N The numb er op erators satisfy the commutation relations N C

N C C N A N A A and N N For further details of these

and other relations see Gero ch ab

I I I Coherent states

There are various characterizations of coherent states in quantum eld theory First we give a denition

1 a 1

via exp onentiation of the single particle Hilb ert space H We b egin with a vector H and construct

a

from this a unique element of F obtained by exp onentiating denoted j i Explicitly we have

c

p p

a a a a b a b d

E n j i F

c

p

n has n indices cf the denition of Perelomovs generalized coherent where the term containing

states in pro of b of Theorem b elow The element j i is said to b e a coherent state vector Now

c

we intro duce the pro jection op erator P from the Fo ck space F down to the pro jective Fo ck space P F

a a a

It is imp ortant to note that if then P E P E if and only if Note

a b 1

however that although dene dierent vectors in H for they dene the same single

1

particle state for all nonzero values of b ecause the single particle states are elements of P H not

1

H Observe therefore that changing the phase or scale of a single particle state vector changes the

asso ciated coherent state We can consider the universal bund le U over the single particle state space

1

with pro jection U P H dened so the bre ab ove any p oint or state is the ray in the Hilb ert

space that it represents The map E U F dened in is a map from this bundle to Fo ck space

1 1

and this is nonconstant along the bres s for all s P H

The action of the creation and annihilation op erators on coherent states is as follows Let b e

1

an element of H and j i a coherent state vector dened as b efore Then from a we have

c

+

A j i j i a

c c

or equivalently A j i j i from which it follows that coherent states are eigenstates of the

c c

a 1

annihilation op erator A for any vector H On the other hand from b it follows that the

1

action of the creation op erator on a coherent state vector is given by dierentiation with resp ect to H

so

dj i

c

C j i b

c

d



For convenience we set Then h j i e so we have to divide by this factor to calculate the

c c

exp ectation of any op erator from its matrix element with a coherent state vector For the exp ectation

2

of the numb er op erator N in a coherent state P j i we obtain hN i jj jj and for

c

a 1

the total numb er op erator N we have hN i More generally we observe that for any vector H

with norm the for the total numb er of particles in the coherent state

a n 

asso ciated with is Prob N n e n the with mean and variance

By a resolution of the identity we mean an expansion of the unit op erator of the form

Z

p jc ihc j

t t t

tA

where t is an element of some generally multidimensional index set A endowed with a notion of

continuity and p is p ositive In the case of coherent states the choice A H determines p uniquely see

t t

Klauder and Skagerstam This uniqueness prop erty holds even though the coherent states are not

mutually orthogonal and form an over complete basis in our Hilb ert space notation h j i exp

c c

which is never zero The resolution of the identity leads to an imp ortant concept in quantum

measurement theory and quantum known as a positive measure POM We return to

a discussion of POMs in Sec VI

In the case of the quantum harmonic oscillator the space of coherent states evolves into itself under

the unitary evolution asso ciated with the Hamiltonian op erator H The orbits in classical phase space

are identical to the quantum mechanical orbits in the exp ectation phase space co ordinatized by the

exp ectation values of the p osition and momentum op erators hQ i and hP i In this sense the coherent

state description of the quantum harmonic oscillator is very closely classical Moreover the uncertainty

relation Q P h is saturated by the coherent states and Q P remain constant under

unitary Hamiltonian time evolution These prop erties also hold in the more general case of a system of

coupled harmonic oscillators see Glaub er

Now consider the bre bundle V dened as F where is the classical phase space of elementary

Hamiltonian mechanics and the bre F ab ove any p oint x p is the set of quantum mechanical

states P j i such that h jX P j i x p In geometric quantum mechanics the evolution can b e

viewed as taking place in V The quantum Hamiltonian op erator H is obtained by promoting x

2 2

and p to the corresp onding p osition and momentum op erators X and P so that H X P in

appropriate physical units However the quantum Hamiltonian function h h jH j i is equal to

2 2 2 2

Let us lab el the two bracketed terms h h resp ectively The term h P X p x

0  

comes from the quantum metric g of Sec I and is the essential extra ingredient in quantum theory

a ab

We can separate the Hamiltonian vector eld X r h uniquely into its horizontal and vertical

b

h

parts X X resp ectively in the bundle V For the quantum harmonic oscillator the pro jection to

0 

of any tra jectory in V is always the classical orbit in corresp onding to X The coherent states

0

are characterized by the fact that the Heisenb erg inequality xp h is saturated and this

xes the values of x p The coherent state evolution has X and thus the tra jectory is purely



horizontal in V cf Schilling

In the case of the quantum electro dynamics of a free eld the quantum eld is describ ed

in essentially the same way as an innite collection of harmonic oscillators see Dirac and the

ab ove prop erties of coherent states apply Thus a classical state of the quantum electromagnetic eld

can b e represented by a coherent state in P F where the Fo ck space F is built up in the standard

way from H the space of square integrable real solutions of Maxwells equations The creation and

M

annihilation op erators are based on the electromagnetic p otential A and the eld op erator is dened

y

as A A The exp ectation of this op erator in a coherent state j i is the corresp onding ancestor

c

a

classical solution of Maxwells equations H cf the discussion of Sec VI

M

IV The FubiniStudy geometry

n

The pro jective form of the FubiniStudy metric on CP that one most usually encounters in quantum

theory see eg Kobayashi and Nomizu Page Hughston can b e written in the elegant

form

Z dZ Z dZ

[ ]

2

ds a

2

Z k Z

n

where Z are homogeneous co ordinates for CP with n and k is the holomorphic sectional

curvature which in subsequent calculations we shall take to equal one In application to coherent

0

states it will b e useful also to have this metric expressed in nonhomogeneous co ordinates since if Z

is the co ordinate of the vacuum part of a coherent state then it is necessarily nonvanishing Thus

the coherent states form a submanifold of the ane part of the pro jective Fo ck space the latter

a ab

consisting of elements of the form f g A C C and whose compactication is

a ab

fP g B CP ie states for which the probability of no quanta b eing present is

zero Observe that the image of any state vector j i under the creation op erator C lies within the

compactication ie C j i B j i F H From now on we deal with coherent states

+ + +

viewed as forming a submanifold of A and we shall make use of the following result Kobayashi and

Nomizu

n

Lemma The equivalent nonpro jective form of the FubiniStudy metric on CP is given by

d d d d

2

ds b

2

0

where Z Z n are inhomogeneous co ordinates

a 0 a a 0 0 2

Proof We have d Z dZ Z dZ Z together with its complex conjugate and thus

a 0 a a 0 0 a a 0 0 2 a

d d Z Z dZ dZ Z Z dZ dZ Z Z dZ dZ Z Z dZ dZ Z Z Also d

a 0 a 0 a a 0 a 0 0 a

1

0 a a 0

Z Z dZ Z Z dZ and similarly for its complex conjugate Straightforward algebra then

a a

0 2



Z (Z )

0

a 0 a

shows that the numerator of the right hand side of b is equal to Z Z dZ dZ Z Z dZ dZ

a 0 a

2 0 0 a b

Z Further manipulation shows that the bracket in the Z Z Z Z dZ dZ Z dZ dZ

0 a 0 b

[ ]

ab ove expression is equal to Z Z dZ dZ Z dZ dZ Z Z dZ Z dZ and dividing

a 2

through by completes the pro of

a

We shall use this lemma to provide a dierential geometric pro of of Theorem b elow which

concerns the intrinsic geometry of the submanifold of coherent states Before stating this result we make

some general remarks on the curvature tensor of the FubiniStudy geometry and of Kahler manifolds

in general

Consider again the FubiniStudy line element a This provides a oneparameter family of

n

FubiniStudy metrics on CP The Riemann tensor derived from the asso ciated metric connection is

1

given by Kobayashi and Nomizu Page Hughston R k g g

abcd

a[c bjd] a[c bjd]

2

1

the Ricci tensor by R k n g and the Ricci scalar by R k nn We adopt

ab cd ab ab

2

the convention that k and in this case for n the FubiniStudy metric b ecomes the intrinsic

distance measure on the sphere of unit radius For n the line element is the distance measure on

p

the sphere of radius k

Kahler geometries have a sp ecial curvature prop erty that relates to Theorem b elow First we

give the denition of a Kahler manifold cf Morrow and Ko daira

Denition A complex manifold M is said to b e Kahler if it comes equipp ed with a Hermitian

0 0

2

0 0 0

metric h with ds h dz dz such that the asso ciated real form ih dz dz is

closed Then is said to b e a Kahler form for M

This is equivalent Morrow and Ko daira to the existence on M of a real valued function K the

0 0

K Consequently a complex Kahler scalar function such that i K or equivalently h

submanifold N of a Kahler manifold M is itself Kahler since the restriction of the function K to N

p

In the case of the FubiniStudy provides the intrinsic Kahler form by applying the op erator

n 1 1 2 2 2 n 2

metric on CP the Kahler scalar function takes the form K k log k j j j j j j

n

where are inhomogeneous co ordinates on CP Now we have the following result

Prop osition Let b e a p ositive form on a complex manifold M Then is a Kahler

1 n

form for M if and only if for all x M there exist holomorphic Euclidean co ordinates z z around

0

0

2

0 0 0

x such that ih dz dz and h O jz j at x and thus the Kahler metric osculates

0 0

the at Euclidean metric to second order

Proof The implication towards b eing a Kahler form is clear To prove the reverse impli

1 n 1 n

cation b egin with holomorphic co ordinates z z such that dz dz give an orthonormal ba

0

0

sis of T dz dz the dual tangent space to M at x This implies that ih where

0

M x

0

0

2

0 0 0 0 0

z a a z h O jz j That is real implies a a The Kahler

0 0 0 0

1 n

0 0 0

a 0 To complete the pro of we dene z at x implies that a z h condition h

0

( )

0

1

0

holomorphic Euclidean co ordinates z as z z z z a

2

V The geometry of coherent states

We b egin our discussion with a result concerning the nonlinear geometry of the coherent state sub

manifold

Lemma Given a pair of distinct coherent states the complex pro jective line L P F joining

them intersects C exactly twice at the coherent states themselves

Proof For supp ose that L intersects C in three or more distinct p oints This implies a relation of the

form j i j i j i for ji j i j i wlog normalized with In the harmonic oscillator

c c c

n

1

n=0

p

case we expand this equation according to j i jni for energy eigenstates jni exp

c

2

n!

Glaub er Taking the Dirac pro duct with hmj for all m gives innitely many linear equations for

the two unknowns and whose solution space is empty for distinct

A similar argument applies if we generalize the coherent states according to their algebraic char

acterization via the exp onential map E of ie we not not sp ecialize to the case of the harmonic

a a a a a a

oscillator Then supp ose E E E with nonzero distinct elements of

1 n

H This relation implies innitely many equations to b e satised by The square of the H part

2 n 2 2 n n 2 2 n 2 2 2

of the relation gives y If all of are nonzero

and not all equal to unity then these equations are overdetermined and have no solution there are

2 2 2

innitley many equations for one unknown So for a solution in to exist we must have all

a a a 2 n 2

equal to unity Thus are unit vectors and y b ecomes n This

a a

has no solution for nonzero unless in which case are the same unit vector

A similar prop erty holds in the continuous case ie any normalized coherent state vector j i is

c

decomp osable as a continuous integral over the states of C via the resolution of unity Provided

an analyticity assumption holds this decomp osition is unique see xIV of Glaub er for a pro of of

this result

We now state our main result concerning the geometry of C the result to follow rst app ears in

Field and later in Field

Theorem a The metric induced on the coherent state submanifold from the ambient Fubini

a

Study metric on the quantum mechanical state space is intrinsical ly at The co ordinates on the

1

single particle Hilb ert space H are complex Euclidean co ordinates for the coherent state submanifold

If instead we b egin with the coherent state submanifold and decide a priori to place on it the

complex Euclidean metric giving us the manifold with metric C then we have the following equivalent

E

result

Theorem b The Euclidean coherent state submanifold has an isometric emb edding into the

i

FubiniStudy state manifold C F where i is the inclusion map in this case an isometry

E F S

We remark that the theorem stated in the latter form relates to the work of Rawnsley

which discusses the geometric quantization of a Kahler manifold K and how the resulting coherent

states give a map E from K into the quantum mechanical pro jective state space In particular following

Coroll it is remarked that by Ko dairas theorem on Ho dge manifolds Griths and Harris

Morrow and Ko daira there exists a holomorphic line bundle connection over K for which the

map E is an emb edding This result is indep endent of the curvature of K which is determined by

the Poisson bracket of the classical eld theory b eing quantized In our theorem we have eectively

adopted complex Cartesian co ordinates for K so that K is complex Euclidean space and so trivially a

Kahler manifold cf Corollary b elow and also Lemma to follow on truncated Fo ck spaces Our

approach diers from that of Rawnsley in that we have fo cussed attention on the description in terms

of geometric quantum mechanics as opp osed to geometric quantization It is nevertheless noteworthy

that the two approaches yield descriptions of the coherent states with essentially the same underlying

geometry With regard to the choice of Cartesian co ordinates we should also compare the remarks

made on p of Dirac For further background to the material covered in Rawnsley

see Wo o dhouse Observe that our theorem is indep endent of the details of the single particle

1

Hilb ert space H

Proof We give three indep endent pro ofs of this result from the p oints of view of dierential

geometry op erator algebra and the Kahler form

a Dierential geometry From the way that we dened coherent state vectors we can regard

a 1

H as complex co ordinate functions for the coherent state submanifold It will b e helpful to

p

(n) ( ) (n)

intro duce some further notation We dene n so that is the tensor contribution

n

to the coherent state vector j i which lies in H there b eing n factors in the symmetrized tensor

c

(n) n

pro duct ab ove similarly we dene Then setting we nd n Restricted

(n) (n)

down to the coherent state submanifold C we can calculate the tangent vector to a coherent state

n

induced by an element d of T H The comp onent in H of the dual tangent vector is given by

p

(n) ( )

d nd n and similarly for the complex conjugate To calculate the FubiniStudy

line element we nd the co ordinate inner pro duct of a tangent vector with itself The contribution to

n

this of any pair of vectors lying in distinct H vanishes as follows from our expression for the Hilb ert

space norm given earlier Thus when evaluating the inner pro duct of two Fo ck space vectors in the

abstract index notation we contract over vectors and their conjugates with the same numb er of indices

(n)

1

(n) n1 n2 2

Hence d d d d n j d j for all n We also need the

(m)

(m) (n1)!

  

(n) (n)

( )

1 n

( ) n1 (n)

p p

d d co ordinate inner pro duct d

(m)

(m) (n1)! (m)

n! n!

(n)

1

n1 (n)

d for all n In b ab ove a vector is given and similarly d

(m)

(n1)! (m)

(n)

by the collection f g for all values of n and thus to evaluate the line element induced on the

coherent state submanifold C we must sum over all m n in the ab ove identities This

(n)  (n)  2

d e d together with its complex d d e d d jd yields j and

(m) (m)

mn mn

(n) 

conjugate The denominator in the FubiniStudy line element equals e and thus the

(m)

mn

2

induced line element reduces to ds d d as required

b Operator algebra Here we shall assume only the canonical commutation relations CCR

for the creation and annihilation op erators A C together with ab for these prop erties

characterize the coherent states completely Klauder and Skagerstam For a Lagrangian including

interaction terms there is a corresp onding change in A and C but the fundamental algebraic relation

CCR b etween them and the prop erties still hold Sub ject to freedom of choice in the vacuum

state this gives the generalized coherent states as intro duced by Perelomov and b ecomes

C ji We shall adopt the Dirac notation for state vectors according to Z j i Z j i exp

c

h j In this notation the FubiniStudy line element b ecomes

hd jd i h jd ihd j i

2

ds f g

F S

2

h j i h j i

1

We abbreviate so that j i F denotes the state vector coherent to H Then by ab we

have jd i C j id and hd j d h jA Using the CCR A C we calculate hd jd i

d d h jA C j i Rewriting the op erator in the matrix element in terms of a commutator gives

d d h jA C C A j i which by the CCR and equals d d d d h j i

Similarly h jd i h jC j id h j j id h j i d Thus the line element induced on C

2 2

d d as required The pro of is illustrated b elow reduces to ds

F S

Fubini-Study metric on projective state space

differential geometric commutator

dψ|dψ termψ|dψ 2 term

quantum cancel commutator ψ ψ ψα 2 ψ |()α d |

ψ ψα ψ β ψ Complex Euclidean metric

d β d | [A , Cα ]|

n

c Kahler form Recall the Kahler scalar function for CP where now we take n to b e countable

(1) 2 (j ) 2

innity For A F dened in Sec IV we have K log j j j j ad inf where

0

(j ) 1

1 j

as b efore For C we have the coherent state vector asso ciated with H given by

p

j

(j ) (j ) 2 j

S

Summing over j to innity to sum to j and thus j j j with

innity is in fact necessary for atness as we discuss b elow we obtain the remarkably simple relation

K j

C

0 0 0

as required Then the induced metric on C is given by h

The theorem has the immediate following consequence

Corollary The coherent state submanifold is a Kahler manifold

We see therefore that the theorem is a global geometric prop erty of the coherent state submanifold

in itself Kahler which from Prop osition is a sp ecial case of a second order atness prop erty that

applies lo cally to any Kahler manifold

We have seen in Lemma that the coherent state submanifold C is nonlinear in the sense that the

complex pro jective line joining two distinct coherent states lies in the complement of C except at its two

intersection p oints This is an algebraic result whose pro of relies up on the uniqueness of decomp osition

of any given state into coherent states It suggests the following geometric prop erty of the coherent

state submanifold

Prop osition Given any two distinct coherent states the complex pro jective line joining them

intersects C transversal ly ie the line joining the two coherent states do es not lie in the tangent space

to C at either intersection p oint

Proof Since C is homogeneous we can assume that one of the coherent states is the vacuum state

that is P ji where ji is the element of Fo ck space which is the exp onential of the origin in the vector

1

space H Then from Theorem the intrinsic geo desic distance s from P ji to P j i is given

c

12

by s Recall eg Hughston that the geo desic distance b etween the two states in P F

with resp ect to the ambient FubiniStudy metric on P F is determined by the cross ratio

h jihj i

c c

cos

hjih j i

c c

where we take to b e the principal value determined by the ab ove equation so that the cross

1 

ratio ab ove xes Clearly hji hj i h ji and thus cos e It follows that

c c

 12  12

d d e and so d ds e Thus d ds is a monotone decreasing function

of b eginning at d ds where and decaying to zero as tends to innity Tangency at

P j i would require d ds for some and this is not p ossible given the form of the function

c

d ds

The metho d ab ove also proves another result that one exp ects intuitively from the nonlinear

geometric prop erty of C derived in Lemma

Corollary The geo desic distance along the pro jective line in P F joining two distinct coherent

states is strictly greater than the intrinsic geo desic distance within C

Theorem also proves the following simple geometric prop erties of C

Lemma The intrinsic C geo desic distance b etween two coherent states is given by the Hilb ert

1

space norm of the dierence eld of the two corresp onding vectors in H The corresp onding distance

of a coherent state from the vacuum is equal to its Hilb ert space norm

Lemma The overlap h j i of two normalized coherent state vectors is the cosine of the angle

c c

these states subtend at the vacuum state

These results illustrate the geometric character of two emergent linear structures in quantum theory

1

On the one hand addition of elements of the Hilb ert space H of solutions to some classical linear eld

equation yields a new classical eld As we have seen the intrinsic geometry of the asso ciated coherent

1

states is Euclidean with the elements of H serving as Euclidean co ordinates On the other hand any

two distinct coherent states can b e sup erp osed in the quantum mechanical sense of joining them with the

unique complex pro jective line in the ambient FubiniStudy geometry of the underlying state space P F

we have seen that this sup erp osition is noncoherent These two features are present in a linear theory

of gravity cf the discussion in Penrose The coherent states provide a natural preferred basis

together with a unique probability distribution for the asso ciated resolution of unity The state space

geometry illustrates how a quantum sup erp osition of distinct classical eld congurations is outside the classical domain

PF 1 PS CP

ψ P 2 ψ C P 1

PC

Fig. 1-1. Coherent state submanifold,

embedded inside projective .

jS i j i j i

1 2

a a a 1

jC i E u u u H

1 2 i

Theorem has an imp ortant consequence for the relationship b etween the quantum complex struc

ture J of Sec I and the manifold complex structure on the state space P F Our theorem shows that

in a suitable sense these two complex structures are identical Since the coherent state submanifold

is Euclidean and has as Euclidean co ordinates the single particle state vectors themselves the active

transformation J of induces a corresp onding transformation on the tangent space to C at the vac

uum state P ji This is b ecause the notion of nite displacement from some origin in any Euclidean

space is vectorial Thus to nd the action of the manifold complex structure of the state space on

any tangent vector V to C at the vacuum we regard this vector as a displacement in the Euclidean

space C and accordingly as a coherent state given by this displacement from the vacuum Then we

1

act on the corresp onding H element with J and exp onentiate this according to to give a new

displacement vector in C In turn this is asso ciated with a unique tangent vector to C at P ji which

gives the required action of the manifold complex structure on the original vector V

It is protable to think in terms of the geometry of the emb edding of C into P F as an ane

subspace whose compactication is the set of states for which the vacuum entry is zero In this way

symmetry is broken in ane C via the sp ecication of the lo cation of the vacuum state The geo desics

in C which are straight lines pass through innity or more precisely they pass through the innite

dimensional complex pro jective plane P where f F g The only geo desics in the

ambient geometry that pass through a given p oint P P are circles that pass through an antip o dal

p oint for example the vacuum Observe however that C and P F share no geo desics in common since

any geo desic in the state space has length whereas Theorem implies that geo desics of C have

innite length with resp ect to the same distance measure Motion of a coherent state along such a

geo desic corresp onds to scaling the amplitude of its asso ciated single particle state vector As this

amplitude tends to innity the compactication of C is approached and the exp ectation of the total

numb er op erator approaches innity

The isometry group of C corresp onds to Killing vector elds of the induced Euclidean metric and

consists of rigid rotations and translations giving rise to orbits that are circles or straight lines inside

C Observe that the pro jective unitary group PSU is isomorphic to the isometry group of the

FubiniStudy metric Hence the Killing orbits of C will not in general b e unitary orbits of the ambient

state space

We examine to what extent the innite dimensionality of the situation describ ed ab ove aects the

result of Theorem For supp ose we truncate the Fo ck space at the N particle states and dene

1 N

F C H H for some nite p ositive integer N We dene analogues of coherent states as

T

(n)

previously where now the co ordinates for n N are all assumed to vanish The ab ove pro cedure for

calculating the induced metric on the resulting submanifold can b e followed closely and some interesting

features of the truncated coherent state submanifold C emerge We dene the real function S

T N

1

N n

with as ab ove In physical terms the truncated Fo ck space F is appropriate for a

T

n=1

n!

situation in which the exp ectation of the total numb er op erator N is b ounded ab ove by N For example

one might consider the quantum electro dynamics of a photon eld constrained within a nite spatial

volume with a saturation density For a coherent state vector j i we have hN i S S

c N 1 N

a a

Thus hN i N since S NS as is straightforward to verify In the limit

N 1 N

jj we nd hN i N and thus the distribution amongst the n particle states b ecomes more

strongly p eaked at the N particle states as the amplitude increases without b ound

Following the same argument as given in pro of a of the Theorem we obtain for the induced

metric on C

T

2

S

S S

N 1 N 2

N 1

2 a 2 a

ds f jd j g d d

a a

2

S S S

N N

N



The case of Theorem is given by setting N and then S e so that the ab ove line element

0

reduces to the at Euclidean line element It is natural to ask in the case of nite N whether the

submanifold C is Kahler and whether it p ossesses intrinsic curvature Regarding the Kahler geometry

T

we have the following result

Lemma The induced metric on C is Kahler with Kahler scalar function given by K

T N

log S

N

Proof There are two ways of pro ceeding Simplest is to use the same argument as in the Kahlerian

pro of c of Theorem summing only to nite N in the expression for K This incorp orates a pro of



of Theorem as a sp ecial case in the limit N since S e and thus K as in

0 0

Alternatively it suces to evaluate the form asso ciated with K and show that this is identical to

N

a

the line element up on replacement We have K S S d and thus

N 1 N a

a 2 2 b a

K S S d d S S S S d d which is of the

N 1 N a N N 2 b a

N 1 N

required form

0

a

Regarding the intrinsic curvature we adopt co ordinates on C and write the induced

T

0 0 0

K Here the prime op eration is dd and we adopt the index notation K metric as g

N N

0 0 0 0 0

To g g K K g as explained in Sec I I That g is Hermitian implies

N N

calculate the Christoel connection we need the inverse metric to g written as in a with all

ab

0 0 0

raised indices and by insp ection we nd g K H K where the function

N

N N

0 0

H is dened by H K K K The derivatives of are and

N N

N N N

0 0 0 0 0 0

0 0 0

so g K K and g K K Observe that

( ) ( )

N N N N

0 0 0

0 0 0

these relations imply the symmetries g g g g which in fact hold for an

( ) ( )

0 0 0

arbitrary Kahler manifold as follows directly from g K and that commute among one

1

ad a

g g g g and the calculation of another The Christoel symb ols are dened by

c bd b cd d bc bc

2

0

1

0

these pro ceeds using the ab ove identities We nd g g K

0

N

( ) )

N

(

K

N

0

0 0

where the function is dened by K H K H K The calculation of

N N N N

N N N

0 0

1

0 0

0

pro ceeds in the same way with the roles of and interchanged so that K

0

0

)

N

(

K

N

0 0

0 0 0 0

0 g 0 which vanishes g vanishes for we have The mixed symb ol

N

[ ]

0 0

0 0 0 0

vanish by the identities and The remaining Christoel symb ols Thus

0

0 0 0 0

are non zero This prop erty holds In summary only the symb ols and g g

for a general Kahler manifold since the ab ove symmetries of the derivatives of g hold in the general

case For a general Kahler manifold there exist simple identities for the Riemann curvature tensor in

0 0 0

3

0 0

g K terms of the Christoel connection We have Ko et al

0 0 0 0 0 0

3

0 0 0 0 0 0

It follows g K and

that for a general Kahler manifold the only nonvanishing comp onents of the Riemann tensor are

0 0 0 0 0 0 0 0

R R R R and these p ossess the Hermitian and symmetry prop erties R

0

0

( (

) )

0 0

R R R As a consequence the Ricci tensor is determined by a R

( )

0

and the Ricci scalar R is real For a general Kahler manifold we have Ko et al Hermitian R

0 0

the useful simplifying identity R so that in our case we nd

K K

N N

N N

a R R

) (

) ( ( )

K K K K

N N N N

Thus with indices in any p osition the Riemann curvature vanishes in the limit that N for then

0

K and K b oth vanish This is in accordance with Theorem It makes sense to discuss the

1

limit N since the underlying H serves to co ordinatize C for each N We calculate the Ricci

0 T

tensor of C by taking the trace on and in the expression for R from a We obtain

T

R K K n K K K K n K K

N N

N N N N N N N N

b

1

where n is the holomorphic dimension of H and this is Hermitian For the Ricci scalar R we nd

K K

N N

2

N N

R n nn n c

K K K K

N N N N

which is real Observe that b oth the Ricci tensor and Ricci scalar diverge as the holomorphic dimension

n approaches Nevertheless the Weyl curvature is well dened in this limit as we demonstrate The

0

Weyl tensor Penrose and Rindler of a nreal dimensional Kahler manifold is determined by

1 1

1

C R R R so that in our case as n dim H R

C

2n2 (2n1)(2n2)

approaches we obtain

0

K

N

N

C R

K K

N N

We observe that this quantity is nite which has the following consequences

Prop osition For the complete Fo ck space N the Weyl curvature of the coherent

0

1

state submanifold vanishes for all values of dim H including countable innity This is despite

0

1

the divergence of the Ricci tensor in the limit dim H Thus C is conformally at for all single

0

1

particle Hilb ert spaces H

Prop osition For the Fo ck space F N nite the submanifold C has nite conformal

T T

1

curvature for any single particle Hilb ert space H

By comparison the FubiniStudy geometry of the underlying state manifold P F has divergent Ricci

1

tensor and scalar for dim H and nite nonzero Riemann and Weyl curvatures prop ortional to

0

1

the holomorphic sectional curvature k for all H

VI Discussion

Theories of sto chastic state vector reduction have recently b een studied eg Percival Hugh

ston and in particular a prop osal of this kind has b een given in the context of geometric

quantum mechanics Hughston It would b e interesting to see whether such an approach is viable

in the innite dimensional case that arises in the context of quantum eld theory whose geometry we

have describ ed ab ove The submanifold of coherent states suggests itself as a natural geometric ob ject

to study in the context of a sto chastic mo del in which the coherent states provide the preferred basis

for state vector reduction cf Hughston where it is the energy eigenstates that feature in this

resp ect The physical motivation for such a mo del is that the coherent states are precisely those from

which a unique classical eld conguration can b e inferred within the quantum eld theory context

In this way the coherent states play an imp ortant role in many quantization pro cedures cf Rawnsley

Wo o dhouse Klauder We take the view therefore that coherent states are central in

understanding the results of quantum measurement Evidence for this exists in the theory of quantum

optics in which the empirically determined photon numb ers ob ey the Poisson statistics of coherent

states eg Klauder and Skagerstam The theory of p ositive op erator valued measures Peres

is imp ortant in this regard As remarked in Sec III a POM provides a resolution of the unit

op erator amongst nonorthogonal states and constitutes the sp ectrum for an asso ciated measurement

a consistent theory of POM measurements has b een develop ed in Peres Ivanovic

We have observed that the space of coherent states provides a natural POM and moreover that this

is unique The underlying geometry of the coherent state manifold relates to the notion of quantum

versus classical sup erp osition in a physically intuitive way as discussed in SecV We envisage a sto chas

tic pro cess on the state manifold with drift oriented towards the coherent state submanifold C and

vanishing diusion tensor on C An initial incoherent state is driven towards C via the sto chastic evo

lution combined with drift until C is reached when ordinary unitary evolution pro ceeds according to

the Hamiltonian ow Thus C b ecomes an invariant attractor for the sto chastic evolution regarding

the invariance of C cf Glaub er which discusses the preservation of coherent states under unitary

n

evolution for a wide class of Hamiltonians The sto chastic dierential geometry of pro cesses on CP

is well understo o d cf Hughston and app ears to generalize naturally to the innite dimensional

1

case provided H is separable A suggestion for a sto chastic dierential equation governing this typ e

of evolution is given in chapter of Field and is still under investigation

We have not addressed the issue of b oundedness of the eld op erators asso ciated with a given

1

H and their kinematical representation While the former can b e dealt with by intro ducing the

corresp onding Weyl form of the canonical commutation relations Reed and Simon the

latter amounts to the choice of vacuum state together with a representation of the quantum complex

structure J on some kinematical background eg spacetime For zero restmass elds on Minkowski

space and linearized gravity in vacuum various represenations of J and the asso ciated metric g are

given in Field

An op en problem is that of constructing a geometric phase space formulation of quantum eld

theory for which the state space is an arbitrary Kahler manifold and the corresp onding construction

of physical Our present view is that the study of sto chastic pro cesses on CP may itself

suggest a consistent measurement theory in which certain mo dications of the state space geometry

1

b ecome necessary We have seen that eld theory based on a linear H can b e formulated naturally in

geometric terms As o ccurs with spacetime in the passage from Maxwells theory to Einsteins general

relativity the quantum state space may require global departures from the standard complex pro jective

geometry and maximal isometry group in order to accomo date the quantum description of nonlinear

phenomena such as gravitation

Acknowledgements The authors are grateful to BD Bramson R Penrose H Urbantke and

NMJ Wo o dhouse for helpful discussions

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