Group Theory - QMII 2017

There are many references about the subject. Here are three of them:

1. The Quantum Theory of Fields Vol. I,S.Weinberg.

3. Group Theory in Physics, W. K. Tung.

1 The proper Lorentz group and poincare

1.1 Reminder

Recall that at the end of the day spacial relativity is a theory of transformation rules. The allowed set of transformations are those which leaves the interval invariant. The interval which is measured to be the same in all inertial frames of reference is

2 µ ⌫ ds = ⌘µ⌫dx dx . (1)

We found that under the general transformation dxµ ⇤µ dx⌫,theallowed ! ⌫ ⇤’s are those who satisﬁes the following condition:

⌘ =⇤T ⌘⇤ . (2)

We separate them to two classes - rotations and boosts. Note that in both cases the determinant is det(⇤) = 1. This is related to the fact that this continuous transformations are connected to the identity, i.e. ⇤(0, 0) = 1.

1.2 Discrete symmetries

The group O(1, 3) contains also transformations with ⇤ = 1. This is | | related to discrete symmetries. On top of the continuous rotations and boosts there are two additional discrete transformations - time reversal (T )and

1 parity (P )

1000 10 0 0 0 01001 00 10 01 T = ,P= . (3) B 0010C B00 10C B C B C B 0001C B00 0 1C B C B C @ A @ A All together we get the complete Lorentz group which has four discon- nected pieces

P L+" L" = P L+" proper-orthochronous ! improper orthochronous·

T T

l L+# = T L+" L#l= P T L+" proper nonorthochronous· P! improper nonorthochronous· · From now on we will consider only the proper-orthochronous Lorentz group

L+" .

1.3 Poincar´e

So far we consider just linear transformations of dxµ.Thereareadditional four transformations which leaves ds2 invariant. Those are translations of µ µ µ µ µ space time x x + a ,withsomeconstantfourvectora .Clearlydx0 = ! dxµ.IncludingthistransformationsaswellisknownasthePoincar´egroup.

2 2 Generatros and algebra

We would like to ﬁnd the generators and the algebra of the Lorentz and Poincare groups. We will start from the generators of translations which will give us an example of representations in quantum mechanics.

2.1 Poincar´e

For simplicity we will consider a one-dimensional problem. Consider the group of spatial translation in 1D x x + a.Wewouldliketoﬁndthe ! action of this group on the Hilbert space. To this end consider a basis of eigenvectors of the position operator Xˆ.Theactionofthetranslationgroup on this space is just

T (a) x = x + a . (4) | i | i

This is clearly a representation since

1 T (a)T (b)=T (a + b) ,T(0) = 1 ,T (a)=T ( a) . (5)

As we saw before, group elements are exponentiation of the generators, namely the operator T (a)canbewrittenas

T (x)=eiPxˆ , (6) where by deﬁnition Pˆ is the generator of translations. This is a hermitian operator so it has real eigenvalues, which are given by

Pˆ p = p p T (x) p = eipx p . (7) | i | i) | i | i

3 What can we learn about this p?Forthisendletusexaminethewave function (x)= x . h | i

1 1 T (a) = T (a) x (x)dx = x + a (x)d(x + a) | i | i | i Z1 Z1 1 = x (x a)dx T (a) (x)= (x a) . (8) | i )D Z1 ⇥ ⇤ Now we can ﬁnd the representation of the algebra acting on the space of wave functions

T (a) (x)=ei⇡(Pˆ)a (x) . (9) D ⇥ ⇤ Recalling our deﬁnition of the generators we ﬁnd

⇡(Pˆ) (x)= i@ (ei⇡(Pˆ)a (x)) = i@ (x a) a |a=0 a |a=0 = i@ (x) ⇡(Pˆ)=i@ . (10) x ) x

This is indeed the momentum operator that we are familiar with from quantum mechanics. Following similar steps we can show that

• The generator of time translation is the Hamiltonian (or the energy), so as we actually know (t) = eiHt (0) . | i | i • The generator of rotations is the angular momentum operator, so for a i✓nˆ J~ rotation of some angle ✓ around then ˆ axis we have (✓) = e · (0) . | i | i Having the generators of spatial and time translation it is easy to generalize it to be covariant. One can ﬁnd that the generator of space-time translation xµ xµ + aµ is the four momentum pµ,suchthat !

⌫ xµ + aµ = eia⌫ p xµ . (11) | i | i

To make all this statements more precise we will have to use Noether theorem. We will get to that at some later time of the course.

4 2.2 Lorentz

To get the generators of the Lorentz group we di↵erentiate with respect to the continuous parameters of the group elements. Recall that boosts and rotations takes the form of

. ch⌘ sh⌘ .. ⇤ 0sh⌘ ch⌘ 1 ,R0 c s 1 . (12) ⇠ ⇠ ✓ ✓ .. B .C B s✓ c✓C B C B C @ A @ A Then we ﬁnd six independent generators, three Ki for boosts and three Ji for rotations

0100 0010 0001 010001 000001 000001 K = i ,K = i ,K = i , 1 2 3 B0000C B1000C B0000C B C B C B C B0000C B0000C B1000C B C B C B C @00 0 0A @000 0A @0000A 000 0 01 0000 11 000101 J = i ,J = i ,J = i . 1 2 3 B00 0 1C B000 0C B0 100C B C B C B C B00 10C B010 0C B0000C B C B C B C @ A @ A @ A Note that the representation is not Hermitian. The commutation relations are

[J ,J ]=i✏ J , [K ,K ]= i✏ J , [J ,K ]=i✏ K . (13) i j ijk k i j ijk k i j ijk k

There is a compact way of organizing the generators in an antisymmetric matrix Jµ⌫,where

J = ✏ J ,J = K ,J = J . (14) ij ijk k 0i i µ⌫ ⌫µ

Their elements can be written as

(J )⇢ = i(⌘⇢ ⌘ ⌘⇢ ⌘ ) . (15) µ⌫ µ ⌫ ⌫ µ

5 The commutation relations are

[J ,J ]=i (⌘ J ⌘ J ⌘ J + ⌘ J ) . (16) µ⌫ ⇢ µ ⌫⇢ µ⇢ ⌫ ⌫ µ⇢ ⌫⇢ µ

Now we can get group elements by exponentiating the algebra,

i !µ⌫ J ⇤=e 2 µ⌫ , (17) where !µ⌫ = !⌫µ and the 1/2istheretoavoidover-counting.

3 Irreducible representations

We are now at a stage to study the irreducible representations of the Lorentz group. Nevertheless there is a crucial di↵erence between the Lorentz group and the groups that we study so far - the Lorentz group is not compact. As a result the irreducible representations can be classiﬁed in one of two possibilities

• Finite dimensional but non-unitary representation.

• Inﬁnite dimensional but unitary representation.

This is a deep statement and has far reaching consequences in the study of relativistic quantum mechanics. In quantum field theories the field operators transform under certain finite dimensional representations, while for the consistency of quantum mechanics, the states must transform under certain unitary representation.

3.1 Finite dimensional irreducible representation

The easiest way to study the representations is by complexiﬁng the algebra1. The Lorentz algebra has a particularly simple structure. Deﬁne a new basis

1Roughly speaking it means that we are now allowing ourself to take ✓ C instead of 2 ✓ R. 2

6 for so(1, 3)C,

1 1 M = (J + iK ) ,N = (J iK ) ,i=1, 2, 3 . (18) i 2 i i i 2 i i

The commutators are

[Mi,Mj]=i✏ijkMk , [Ni,Nj]=i✏ijkNk , [Mi,Nj]=0. (19)

That means that the algebra is isomorphic to that of the group SU(2) ⌦ SU(2). Each irrep of Lorentz can therefore be denoted as (jM ,jN ), where 2 2 jM,N are the eigenvalues of the two quadratic Casimir operators M ,N. Here are some examples for the most common representations:

• (0, 0): Trivial (Lorentz scalar)

1 • ( 2 , 0): Right Weyl spinor

1 • (0, 2 ): Left Weyl spinor

• ( 1 , 0) (0, 1 ): Dirac spinor 2 2 1 1 • ( 2 , 2 ): Fundamental (4-vector)

• (1, 0) (0, 1): Antisymmetric 2-tensor, e.g. Fµ⌫ 1 1 To see how does it work lets take ( 2 , 2 ) as an example. First note that if we µ i deﬁne (12 2, ), there is a one-to-one mapping ⌘ ⇥ 1 xµ X = xµ xµ = Tr[µX] . (20) ! µ , 2

The transformation of X is given by

i↵ M i i N i i↵ i i i X e i Xe i = e i Xe i . (21) !

Recall that we complexiﬁed the algebra, so now we take ↵, C,andin 2 order to go back to our original basis (i.e. being consistent with Eq. (18)) we

7 must impose Re[↵]=Re[],Im[↵]= Im[], so

i✓ i i✓ i X e i Xe i⇤ AXA† (22) ! ⌘

I’ll leave it as an exercise to show that

µ 1 µ " • General Lorentz (L+) transformation is given by ⇤ ⌫(A)= 2 Tr[ A⌫A†].

• If ✓ is real (i.e. A is unitary) ⇤is a pure rotation.

• If ✓ is imaginary (i.e. A is hermitian) ⇤is a pure boost.

4 Unitary irreducible representations

2 µ The Poincare group has two quadratic Casimir operators - P = P Pµ,and 2 µ W = W Wµ,where

1 W µ = ✏µ⌫⇢J P (23) 2 ⌫⇢ is known as the Pauli-Lubinski vector. For massive particles at rest (pµ = (m, 0, 0, 0)) we ﬁnd

m W 0 =0,Wi = ✏ijkJ = MJi, (24) 2 jk so W 2 = M 2J 2,witheigenvalues s(s +1).SinceW 2 is a Lorentz scalar it / will be true in any frame of reference (in the case of massless particles there are interesting consequences which are byond our scope). Having the two Casimirs in our hand we can study all unitary irreducible representation (in the same way that we did for SU(2) SU(2)). We can ⌦ get an intuition why it should be infinite-dimensional from the fact that the i 2 surfaces area of a sphere dx dxi = dr is finite, while the surface area in our µ 2 case dxµdx = ds is infinite. In what follows we will consider only representations with P 2 = M 2 > 0, which are known as time-like representations.

8 4.1 Time-like

In order to build all states of this representation we choose a representative µ vector pt.l. (m, 0, 0, 0). This vector is invariant under spatial rotations, ⌘ µ i.e. SO(3). This is also the maximal subgroup of L+" which leaves pt.l. invariant and is called the little group. µ ˆµ The basis vectors with eigenvalues pt.l. of the operator P are denoted by m, s; 0, ,s.t. | i

Pˆµ m, s; 0, = pµ m, s; 0, , Pˆ2 m, s; 0, = m2 m, s; 0, , (25) | i t.l. | i | i | i Jˆ m, s; 0, = m, s; 0, , Wˆ 2 m, s; 0, = m2s(s +1) m, s; 0, . 3 | i | i | i | i

To proceed one have to notice that any L+" transformation can be uniquely decomposed into a rotation s.t. the momentum is to the direction of the z- axis, then to breform a boost to the z-direction, and then rotating back

1 ⇤=R(↵,, 0)L3(⇠)R (,✓, ) , (26) where the angles are Euler angles and ⇠ is a boost parameter2. Acting with this transformation on our state we see that the ﬁrst rotation do just nothing because of the little group. The action of the boost on our initial state changes the momentum,

m, s; pˆz , L (⇠) m, s; 0, . (27) | i⌘ 3 | i

Note that now it is clear that the representation is inﬁnite dimensional. Fi- nally we can preform the second rotation to get

m, s; p, R(↵,, 0) m, s; pˆz , H(p) m, s; 0, , (28) | i⌘ | i⌘ | i where we deﬁned

H(p)=R(↵,, 0)L3(⇠) . (29)

2Note that again 6-independent variables are needed.

9 4.1.1 Conclusion and summary

The state vectors m, s; p, that we have constructed are: {| i} 1. Span an invariant vector space under Poincare transformations.

2. The representation is unitary and irreducible.

To study the action of the group on the states we ﬁrst deﬁne

1 R(⇤,p) H (⇤p)⇤H(p) . (30) ⌘

To see that this is indeed a rotation note that

µ 1 µ 1 µ R(⇤,p) p = H (⇤p) ⇤ H(p) p = H (⇤p) ⇤ p · t.l. · · · t.l. · · 1 µ µ = H (p0) p0 = p , (31) · t.l.

µ namely leaves pt.l. invariant. Now we can check what is the action of general Lorentz transformation on a general state

1 ⇤ m, s; p, =⇤H(p) m, s; 0, = H(⇤p) H (⇤p)⇤H(p) m, s; 0, | i | i | i ~ s 0 = H(⇤p)R(⇤,p) m, s; 0, = ⇥H(p0) m, s; 0,0 ⇤ [R(⇤,p)] | i | iD s 0 = m, s; p0,0 [R(⇤,p)] . (32) | iD

In we introduced the representation s [R(⇤,p)]0 of SO(3) acting on a ~ D spin state s. Under translation we have

ˆµ µ T (a) m, s; p, = eiaµP m, s; p, = ... = m, s; p, eiaµp . (33) | i | i | i

Thus we showed that the space spanned by m, s; p, is invariant under {| i} Poincare, irreducible (we can get to all this states by the above transformations), and it is unitary since the generators are hermitian.