Representations of the Poincar´ e Group

The normal subgroup T(4) is abelian, so the fixed-point condition that defines the little groups is given by Equation 2.36 and the corresponding little groups are defined by Equation 2.37. As the normal subgroup is a translation group, the representations ξ are given by Equation 2.44 for elements aµ_{, µ= 0,1,2, ..., n}

(note thata0 is the time component). So for states|pilabelled by the four-vector

p≡pµ _{in the Hilbert space H}ξ_:

ξ(aµ)|pi=eipµaµ_{|pi. (2.95)}

If the Hamiltonian of the system is translationally invariant then the Lie alge- bra operators Pi_{, i = 1,2,3 are conserved quantities [26]. These operators have}

corresponding eigenvalues which are the character labels: Pi|pi=−i ∂ξ(ai₎ ∂ai ai₌₀ |pi =−i ∂ ∂aie ipiai ai₌₀ |pi =pi|pi. (2.96)

Similarly, the Hamiltonian operator P0 or H has eigenvalue given by e:

H|pi=e|pi, (2.97)

Note that, due to the choice of metric, the contravariant version of Equation 2.97 has opposite sign whereas the sign in Equation 2.96 remains negative. The explicit inclusion of a factor of _~ then allows the identification of the Pi operators with

momentum, so that:

Pi|pi=

˜

pi

~ |pi, where ˜pi :=~pi. Similarly, for ˜e:=~ce:

H|pi= e˜ ~c

|pi.

Elements k ≡ Λ of the homogeneous subgroup act on the representations of the normal subgroup in the following manner:

k·ξ =ξ(kak−1)

=ξ((Λ,0)(I, aµ)(−Λ,0)) =ξ((Λ,Λaµ)(−Λ,0)) =ξ(I,Λaµ)

=epµ(Λ·a)µ_.

Equation 2.36 is therefore satisfied whenever:

ξ(I,Λai) =ξ(I, aµ) ⇒pµ(Λ·a)µ=pµaµ ⇒pνΛνµa µ_=p µaµ ⇒pνΛνµ=pµ.

Stated another way,

Λµνpν =pµ

In matrix form this is given by:

    Λ00 Λ01 Λ02 Λ03 Λ0 1 Λ11 Λ12 Λ13 Λ0 2 Λ21 Λ22 Λ23 Λ03 Λ31 Λ32 Λ33         p0 p1 p2 p3     =     p0 p1 p2 p3     . (2.98)

Following the method of Wigner [81], there are four orbits, two of which have two sub-orbits each. The little groups can then be studied by choosing suitable representative vectorspµ. The orbits are defined by solutions to the “mass-shell”

condition:

pµpµ ≡p0p0−pipi =M2. (2.99)

These orbits are:

• Timelike, where pµpµ >0 and so M2 is positive. There are two sub-orbits,

depending upon the sign of p0. This orbit is discussed in Section 2.8.3.1;

• Spacelike, wherepµpµ <0 and soM2 is negative. This orbit is not discussed

further as it is non-physical.

• Massless or lightlike, wherepµpµ = 0 and soM is also equal to 0. There are

two sub-orbits, again depending upon the sign ofp0. This orbit is discussed in Section 2.8.3.2; and

• Null, where all components of the vector pµ = 0. This orbit is the trivial

Lorentz subgroup of the Poincar´e group and is not discussed further.

2.8.3

Representations of the Little Groups

2.8.3.1 Timelike Case

The timelike orbit is the orbit defined by pµpµ > 0. A representative vector is

b

pµ_{= (±M,0,0,0)}T_{, forn = 3 spatial dimensions, which has a length ofM. Then}

the orbitSO(1,3)·p_bµ _{is the set of images obtained through proper orthochronous}

Lorentz transformations acting on this representative vector. There are sub-orbits defined by the sign ofp_b0_{; if}

b

p0_{is positive then the vector and the orbit are said to be} future-pointing. If, on the other hand, p_b0 _{is negative then the orbit and the vector} are said to be past-pointing. The little group for both sub-orbits are the same; it is the set of all Lorentz transformations that leave this representative vector invariant. The length M of the vector can be used to label the representations of this orbit.

If p_bµ _{= (±M,0,0,0)}T_{, then Equation 2.98 shows that Λ}0

i = Λi0 = 0, for

i = 1,2,3, and Λ0

0 = 1. The little group then, is the maximal set of remaining elements. In matrix form, it is the set of all proper orthochronous Lorentz matrices Λξ _{∈ SO(1,3) of the following form:}

Λξ =     1 0 0 0 0 Λ1 1 Λ12 Λ13 0 Λ21 Λ22 Λ23 0 Λ3 1 Λ32 Λ33    

This is a 4×4 matrix realisation of the group SO(3) - the group of all rotations in 3-dimensions.

The stabiliser group is:

Gξ=SO(3)_nT(4),

and the representations of the elements of the stabiliser group are given by Equa- tion 2.38, as:

τξ(g) = σ(R)⊗ξ(a), (2.100) for g ∈ Gξ _{and where R ≡ Λ}ξ _{are elements of SO(3) - rotations. Define}

X to be the quotient space of G/Gξ_{. That is,}

X is the three dimensional hyperbolic space H3. Then Θ is the coset representative function that takes elements of x∈H3 to

P(1,3):

Θ :x∈_H3 _{→ P(1,3).}

These cosets will be labelled by a subscriptB, which indicates a pure boost trans- formation. Then:

xB =g(B,0)Gξ.

The function Θ takes this coset to an element of the group P(1,3) and so: Θ(xB) =g(B,0).

To determine all the terms in Equation 2.41 requires the result of the product of a group element (actually, the inverse of that element) and the coset. By separating the Lorentz transformation in the group element into a pure boost and a pure rotation, this yields:

gxB0 = (Λ, a)(B0,0)Gξ

= (BB0,0)(R, a)Gξ

= (BB0,0)Gξ_;

∴Θ(g−1x) = xB−1_B0.

All the terms in Equation 2.41 can now be easily expressed and the representations of the group can be written in terms of Equation 2.42. Then, for Λ :=BR:

hxB0|τ|ψi=hx_B−1_B0|τξ[g(B0−1,0)g(Λ, a)g(B−1B0,0)]|ψi

=τξ(B0−1BRB−1B0,−B0a)ψ(xB−1_B0)

=τξ(R◦,−B0a)ψ(xB−1_B0),

where R◦ =B0−1BRB−1B0 is a pure rotation. Then, from Equation 2.100:

hxB0|τ|ψi=σ(R◦)⊗ξ(−B0a)ψ(x_B−1_B0)

=σ(R◦)e−icµ(B0a)µ_ψ(x

where cµ are character labels. The representations are guaranteed by the Mackey

induction method to be irreducible, and they will be unitary so long as the repre- sentationsσ(R◦) are unitary. The unitary irreducible representations of the group

SO(3) are well known.

So far, the representations discussed have been the representations of the pre- viously defined Inhomogeneous Lorentz Group. To change to representations of the (proper orthochronous) Poincar´e group, all that changes is the representations

σ of the little group. It is well known that the universal covering group of SO(3) is SU(2). The extension, therefore, to the Poincar´e group simply involves the addition of space inversion, time reversal and the combination of space inversion and time reversal - in other words, the operations summarised in Table 2.7.

The Hilbert space of the stabiliser group is given by Equation 2.39 as Hσξ⊗_C. The Hilbert space of the whole group is therefore given by:

Hτ ∼=Hσξ ⊗C⊗ L2(X,C, µ) =L2₍

H3,Hσ

ξ

, µ),

where µ is a measure on _H3. The Hilbert space _Hσξ is the Hilbert space of the little group - in this case it is the group SU(2).

As any standard quantum mechanics text book will describe6_{, the timelike} states can be expressed as simultaneous eigenstates of the angular momentum operator in a given direction - by convention the z or 3 direction - as well as the square of the vector J = (J1, J2, J3) of angular momentum operators:

J2 =J12+J22 +J32.

This operator commutes with each component of angular momentum, and has eigenvalues such that:

J2|p_bi=s(s+ 1)|p_bi.

The number s is called the “spin quantum number”, and it takes the values:

s = 0,1₂,1,3₂, .... The half-integral values arise due to the fact that the universal covering group SU(2) of the timelike little group is the double cover of SO(3). The J3 operator has eigenvalues ms:

J3|pbi=ms|pbi.

This quantum number is called the “secondary spin quantum number” and takes the values: ms = −s,−s + 1, ..., s− 1, s. Since the timelike states can be si-

multaneously diagonalised in both of these operators, the states can therefore be labelled by the quantum numbers s and ms. The states still transform under

translations like Equation 2.95, and so the label _bp will be retained. Therefore the two quantum numbers along with the momentum p_bcan be used to label the timelike representations as the 2s+ 1 basis vectors |p_b;smsi. The corresponding

Poincar´e unitary irreducible representations for the timelike case can be denoted

by τ(M,s) _{where s is the spin quantum number and M is the (positive) length of} the momentum four-vector. Since there are an infinite number of pfor any length

M, the representations are infinite dimensional [26].

In timelike representations the value of pµpµ is given by M2. By replacing M

with a quantity that has dimensions of mass: ˜

M := ~

cM,

it follows from the form ofPµ_P

µ=P02−

P

iPi2, using the respective Lie algebra

eigenvalues, that: ˜ M2_c2 ~2 = ˜ e2 ~2c2 − X i ˜ p2 i ~2 ⇒e˜2 = ˜M2c4+c2p˜2,

which is the relativistic energy-momentum equation.

2.8.3.2 Massless Case

The massless orbit is the orbit defined by pµpµ = 0, where pµ = (06 ,0,0,0)T. A

representative vector for this orbit is p_bµ_{= (±M,0,0,±M)}T _{where both the signs}

must be the same. For the forward-pointing sub-orbit both components must be positive. Following Wigner [81] (and cf. [79]), the determination of the little group for the massless case is best done by representing the Lorentz transformations as in Section 2.7.2. This has SL(2,_C) matrices A obeying the transformation rule of Equation 2.82 with matrices X defined in Equation 2.80. Substituting the forward-pointing representative vector p_bµ _{into Equation 2.82 gives:}

a b c d 2M 0 0 0 a∗ b∗ c∗ d∗ = 2M 0 0 0 ⇒ 2M aa∗ 0 2M ca∗ 0 = 2M 0 0 0 .

From this, it follows that:

c= 0,

|a|2 = 1.

Recalling thatad−bc= 1, takea=e−iφ/2_{, andd=e}iφ/2_{, asad−bc= 1 ⇒d= 1/a.} Still following the argument of [81], the most general element Λξ _{of the little group}

is given by: Λξ= e−iφ/2 (x+iy)eiφ/2 0 eiφ/2 .

The element bof the general form of theSL(2,_C) matrix is chosen in this manner so that the little group matrices Λξ _{are easily decomposed into two subgroups of}

matrices, B(x, y)R(θ): B(x, y) = 1 x+iy 0 1 ;R(φ) = e−iφ/2 0 0 eiφ/2 . (2.101)

R(θ) is a rotation matrix of an angle φ around the 3 axis, and B(x, y) is a boost of magnitudexin the 1 direction, andyin the 2 direction. Clearly, these matrices satisfy the relations:

B(x1, y1)B(x2, y2) = B(x1+x2, y1+y2)

R(φ1)R(φ2) = R(φ1+φ2).

This little group is isomorphic to the group E(2) = O(2)_n T(2) - the two dimensional Euclidean group, or inhomogeneous rotation group. The generators of this group can be constructed from the Pauli-Lubanski vector [76], which is defined as:

Wλ := 1₂λµνσ_J

µνPσ. (2.102)

In the timelike case the momentum operatorPσ has only two non-zero components

(both of which have the value of M for the forward-pointing sub-orbit), and so the generators of E(2) are:

W0 =W3 =M J12 =M J3

W1 =M(J20+J23) = M(−K2 +J1)

W2 =M(J01+J31) = M(K1+J2).

These generators obey the following set of commutation relations:

J3, W1 =iW2 J3, W2 =−iW1 W1, W2= 0.

A general Lorentz transformation can be broken up into two parts:

Λ =XL, (2.103)

whereL=B(x, y)R(φ) involves only transformations of the kind defined in Equa- tion 2.101 - the little group portion of the Lorentz transformation. TheX portion of the Lorentz transformation therefore involves a boostB(z) along the 3 axis and a rotationR(θ, γ) which brings the orientation of the representative vectorp_bµ_into

the direction of a general vectorpµ_{. This means that any vectorp}µ _{can be written}

as:

pµ=R(θ, φ)B(z)p_bµ

The cosets x∈ G/Gξ _{can now be written as:}

x= (R(θ, γ)B(z),0)Gξ

= (X,0)Gξ_.

For brevity, these cosets will be denotedxX. In the notation of Equation 2.103, an

inverse group element of the Poincar´e group is, in general, (L−1X−1,−L−1X−1a). Therefore: g−1xX0 = (L−1X−1,−L−1X−1a)(X0,0)Gξ = (X−1X0,0)(L−1,−L−1a)Gξ = (X−1X0,0)Gξ =xX−1_X0 and Θ(g−1xX0) = (X−1X0,0).

All the terms in Equation 2.41 can now be easily expressed and the representations of the group can be written in terms of Equation 2.42:

hxX0|τ|ψi=hx_X0|τ(Θ−1(x_X0)gΘ(g−1x)|ψi =hxX−1_X0|τξ h (X0−1,0)(XL, a)(X−1X0,0)i|ψi =τξh(X0−1XL,−X0a)(X−1X0,0)iψX−1_X0 =τξ((X0−1XLX−1X0,−X0a))ψX−1_X0 =τξ((L◦,−X0a))ψX−1_X0,

where L◦ is a transformation of the little group. From Equation 2.38:

hxX0|τ|ψi=σ(L◦)⊗ξ(−X0a)

=σ(L◦)e−icµ(X0a)µ_ψ

X−1_X0,

where cµ are character labels and it should be recalled that X and X0 are trans-

formations of the type seen in Equation 2.104.

The Hilbert space of the stabiliser group is given in Equation 2.39 as Hσξ ⊗C. The Hilbert space of the whole group in the massless case is therefore:

Hτ ∼=Hσξ ⊗_C⊗ L2₍

X,C, µ) =L2₍

X,Hσξ, µ),

whereµis a measure on_Xand Hσξ is the Hilbert space of the little group - in this case, the group E(2).

As in Section 2.8.3.1 for timelike states, the massless states can be expressed as simultaneous eigenstates of the J3 operator as well as the square of the vector

W:= (W1_{, W}2_):

This operator commutes with all three generators of the E(2) algebra. In a basis labelled by the representative vector |p_bi, the W2 operator has eigenvalues w:

W2|p_bi=w|p_bi

and theJ3 generator has eigenvalues λ:

J3|pbi=λ|pbi.

Therefore massless states can be labelled by these numbers. Retaining the p_b

label in the ket, the basis vectors will be labelled as |p_b;wλi. However, there has never been a particle discovered with a non-zero w and zero M [76]. In the case of massless particles, the λ label, which corresponds to ms in timelike particles,

relates to the helicity of the particle.

2.8.4

Poincar´e Group Field Equations

2.8.4.1 Casimir Operators of the Poincar´e Algebra

The first Casimir operator of the Poincar´e Lie algebra is the momentum operator squared:

C1 :=PµPµ. (2.106)

This operator is invariant under Lorentz transformations since the square of any four-vector is always invariant under Lorentz transformations. It is clearly invari- ant under space-time translations. In states |pi that transform under translations like Equation 2.95, the eigenvalues of the Lie algebra generators are given in Equa- tions 2.96 and 2.97 as:

Pµ|pi=pµ|pi,

where pµ = (e,p)T. If the length of the four-vector pµ is given by M, then

M2 =−e2+p2 and:

C1|pi:=PµPµ|pi

= (−p02+p12+p22+p32)|pi =M2|pi.

In other words, the Casimir operatorC1 is defined to be:

C1 =M2I, (2.107)

whereI is the identity operator. In the timelike representation τ(M,s) _{the Casimir} operator is the mass squared c2

~2

˜

M2_{. In the massless representation τ}(0,s) _the Casimir operator has a value of zero, although this does not mean that the com- ponents ofpµ are all zero (this would correspond instead to the null case).

The second Casimir operator is constructed from the Pauli-Lubanski operator

Wµ. This operator is defined in Equation 2.102 and has the following properties [76]: WµPµ = 0 Wλ, Pµ = 0 Wλ, Jµν =i(Wµηλν−Wνηµλ) Wλ, Wσ = 1₂µνλσ_J µνPλPσ.

The second Casimir operator is then defined to be:

C2 :=WµWµ (2.108)

In the timelike representation τ(M,s) _{and with a basis state |p;sm}

si where p =

(M,0,0,0)T _{the momentum Lie algebra generators do the following:}

Pi|p;smsi= 0, P0|p;smsi=M|p;smsi.

Therefore, the components of the Pauli-Lubanski operator do the following:

W0|p;smsi= 0,

W1|p;smsi= (1₂1230J23P0+1₂1320J32P0)|p;smsi

=M J1|p;smsi.

Similarly, W2|p;smsi=M J2|p;smsiand W3|p;smsi=M J3|p;smsi(cf. Section

2.7.1 for the notation of Jµν and Ji). Therefore, for timelike representations, the

value ofC2 is given by:

C2|p;smsi: =WµWµ|p;smsi

=M2J2|p;smsi

=M2s(s+ 1)|p;smsi,

where the fact that J2 is the Casimir operator of SO(3) with eigenvalues(s+ 1) has been used. This allows the second Casimir operator of the Poincar´e group to be defined as:

C2 :=M2s(s+ 1)I

for timelike particles. The second Casimir operator in the massless case, however, has already been defined in Equation 2.105. For physical massless particles, this Casimir operator has eigenvaluew= 0.

2.8.4.2 Measure

An arbitrary state vector |ψi can be expanded in terms of the basis vectors |pλi

as follows:

|ψi=X

λ

Z

where ˜dp is the integration measure. This measure is given by Tung (in [76]) as: ˜ dp= 1 N d3_p 2p0 = 1 (2π)3 d3_p 2p0, (2.109)

with N = (2π)3 _{by convention. The value of p}0 _{is given by the “mass-shell”} condition - Equation 2.99 - and the first Casimir. This condition states that:

M2 −pµpµ=M2+p2−(p0)2 = 0,

so that:

p0 =±pp2_+M2_.

Only the positive root applies for forward-pointing representations. Therefore Equation 2.109 can be modified to:

˜

dp= 1

(2π)3

d3p

2(p2_+M2₎. (2.110)

2.8.4.3 The Klein-Gordon Equation

The Klein-Gordon Equation can be derived directly from the eigenvalue equation of the first Casimir operator for timelike unitary irreducible representations. This equation is: PµPµ|p;smsi=M2|p;smsi ⇒(−P02 +P2−M2)|p;smsi= 0 ⇒(1 c2 ∂2 ∂t2 − ∇ 2 + c 2 ~2m˜ 2 )|p;smsi= 0.

Similarly to this derivation the Dirac equation, the Weyl equation and Maxwell’s equations can all be derived from the eigenvalue equations of the representations

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