Poincare group and quantum fields

Krzysztof Golec–Biernat

IFJ PAN

(25 października 2020)

Wersja robocza nie do dystrybucji

Kraków

2020

1 Poincare group 4

1.1 Definition . . . 4

1.2 Tensor notation . . . 5

1.3 Co and contravariant tensors . . . 7

1.4 Scrap . . . 7

1.5 Infinitesimal Lorentz transformations . . . 8

1.6 Finite Lorentz transformations . . . 9

1.7 Finite dimensional representations . . . 12

1.8 SL(2, C) and Lorentz group . . . . 13

1.9 Irreducible representations of SL(2, C) . . . . 15

2 UIR of the Poincare group 18 2.1 Unitary representations . . . 18

2.2 Generators . . . 19

2.3 Poincare group algebra . . . 20

2.4 Pauli-Lubanski vector . . . 22

2.5 Casimir operators . . . 25

2.6 Unitary irreducible representations . . . 26

2.7 Orbit and little group . . . 28

2.8 Wigner’s construction . . . 30

3 Massive particles 32 3.1 Standard momentum and little group . . . 32

3.2 Consistency condition . . . 33

3.3 Standard hermitian boosts . . . 34

3.4 Casimir operators . . . 37

3.5 Irreducible representations . . . 37 3

4 Massless particles 40 4.0.1 Standard momentum and little group . . . 40 4.0.2 Casimir operators . . . 43 4.0.3 Irreducible representations . . . 44

Poincare group

1.1 Definition

Let

g = (g_µν) =











1 0 0 0

0 −1 0 0

0 0 −1 0

0 0 0 −1











(1.1)

be the matrix of the metric tensor, where µ, ν = 0, 1, 2, 3. The proper Lorentz group SO(1, 3) is defined by a set of nonsingular, 4-dimensional real matrices Λ which obey the relation

Λ^Tg Λ = g , detΛ = 1 (1.2)

The Poincare group P is defined as a set of pairs

(Λ, a) , a ∈ R⁴ (1.3)

which obey the following composition law

(Λ₁, a1)(Λ₂, a2) = (Λ₁Λ₂, Λ1a2+ a₁) (1.4) The composition law is non-commutative since in general

(Λ₁Λ₂, Λ₁a₂+ a₁) 6= (Λ₂Λ₁, Λ₂a₁+ a₂) (1.5) The neutral element is given by (1, 0) since for any element (Λ, a) ∈ P

(1, 0)(Λ, a) = (Λ, a) (1.6)

while the inverse to this element is given by (Λ⁻¹, −Λ⁻¹a) since

(Λ⁻¹, −Λ⁻¹a)(Λ, a) = (1, 0) (1.7)

5

Notice that the set of elements (Λ, 0) ∈ P form a subgroup of P which is isomorphic to the Lorentz group since

(Λ₁, 0)(Λ2, 0) = (Λ1Λ₂, 0) (1.8) Considering the elements (1, a) ∈ P, we obtain the subgroup of P with the composition law

(1, a₁)(1, a₂) = (1, a₁+ a₂) (1.9) which is the abelian group of 4-dimensional translations.

The Poincare transformation is a composition of the Lorentz transformation and translation (in the indicated below order)

(Λ, a) = (1, a)(Λ, 0) (1.10)

1.2 Tensor notation

Let V be a real 4-dimensional vector space. Choosing the canonical base vectors e_µ, where µ = 0, 1, 2, 3, any vector x ∈ V can be written as a linear combination of the base vectors

x = x⁰e0+ x¹e1+ x²e2+ x³e3≡ x^µeµ (1.11) where we apply the Einstein summation convention in which the upper and lower identical indices are summed. The four numbers

x^µ= (x⁰, x¹, x², x³) (1.12) are coordinates of the vectors x in the base e_µ. The vector space V is equiped with a real symmetric two-form, i.e. for any x, y, z ∈ V and a, b ∈ R we have

g(ax + by, z) = ag(x, z) + bg(y, z) , g(x, y) = g(y, x) ∈ R (1.13) which is not positive definite, i.e. there exists vector x 6= 0 such that

g(x, x) < 0 (1.14)

This two form allows to define the scalar product of two vectors x, y ∈ V

x · y = g(x, y) (1.15)

In a given base e_µ, we have

g(x, y) = g(x^µe_µ, y^νe_ν) = x^µy^νg(e_µ, e_ν) (1.16) and introducing

g_µν= g(e_µ, e_ν) (1.17)

we may write

x · y = gµνx^µy^ν (1.18)

Thus, for the Minkowski space with g_µν given by (1.1), we have for the canonical base vectors e₀· e₀= −e₁· e₁= −e₂· e₂= −e₃· e₃= 1 (1.19) while for µ 6= ν

e_µ· e_ν = 0 (1.20)

In the active interpretation, the Poincare transformation (Λ, a) is the vector space trans- formation V → V such that for any x, x⁰, a ∈ V

x⁰≡ (Λ, a) x = Λx + a (1.21)

where Λ is a Lorentz operator which obeys the linearity conditions

Λ(αx + βy) = α Λx + β Λy (1.22)

for any x, y ∈ V and α, β ∈ R, and preserves the scalar product

g(x, y) = g(Λx, Λy) (1.23)

With such a definition, we find the composition law (1.5)

x⁰= (Λ₁, a₁)(Λ₂, a₂) x = (Λ₁, a₁)(Λ₂x + a₂) = Λ₁(Λ₂x + a₂) + a₁

= Λ₁Λ₂x + (Λ1a2+ a₁) = (Λ₁Λ₂, Λ1a2+ a₁)x (1.24) In a given canonical base e_µ, the transformation (1.21) reads

x⁰= x^0µeµ= Λ(x^νeν) + a^µeµ= x^ν(Λe_ν) + a^µeµ (1.25) Defining the matrix Λ^µ_ν of the Lorentz operator Λ through the equation

Λe_ν = e_µΛ^µ_ν (1.26)

we find the following coordinate form of the transformation (1.21)

x^0µ= Λ^µ_νx^ν+ a^µ (1.27)

which could be interpreted as matrix multiplication with the coordinates forming columns,









 y⁰ y¹ y² y³











=











Λ⁰₀ Λ⁰₁ Λ⁰₂ Λ⁰₃ Λ¹₀ Λ¹₁ Λ¹₂ Λ¹₃ Λ²₀ Λ²₁ Λ²₂ Λ²₃ Λ³₀ Λ³₁ Λ³₂ Λ³₃



















 x⁰ x¹ x² x³









 +









 a⁰ a¹ a² a³











(1.28)

The Lorentz transformation condition (1.23) can also be written in the coordinate form since g(e_α, e_β) = g(Λe_α, Λe_β) = g(e_µΛ^µ_α, e_νΛ^ν_β) = Λ^µ_αΛ^ν_βg(e_µ, e_ν) (1.29) which gives

gαβ = g_µνΛ^µ_αΛ^ν_β (1.30)

1.3 Co and contravariant tensors

Introducing the symmetric contravariant metric tensor

g⁻¹= (g^µν) =











1 0 0 0

0 −1 0 0

0 0 −1 0

0 0 0 −1











(1.31)

such that

g^αµg_αν = δ_ν^µ (1.32)

we can lower and raise tensor component indices, e.g.

x_µ= g_µνx^ν, x^µ= g^µνx_ν (1.33)

Thus, the scalar product (1.18) can also be written as follows

x · y = xµy^µ= x^µyµ (1.34)

Introducing the inverse Lorentz matrix (Λ⁻¹)^µ_ν such that

Λ^µ_α(Λ⁻¹)^α_ν= (Λ⁻¹)^µ_αΛ^α_ν= δ^µ_ν (1.35) we can write (1.29) in the following form

g_µρΛ^µ_α= g_αβ(Λ⁻¹)^β_ρ (1.36) Using symmetry of the metric tensors, we can lower the indices to obtain

Λ_ρα= (Λ⁻¹)_αρ (1.37)

and raising α we find

Λ_ρ^α= (Λ⁻¹)^α_ρ (1.38)

1.4 Scrap

In the passive interpretation, the Poincare transformation is due to the change of the base vectors and the point to which they are fixed to, i.e.

(Λ, a) : {0, e_µ} → {O⁰, e_µ⁰} (1.39) Under the Lorentz transformation any vector x ∈ V remains the same while its coordinates are different due to the change of the basis,

x = x^µeµ= x^ν0e_ν⁰ (1.40)

Assuming (1.29) for the change of the base vectors

e_ν⁰= e_µΛ^µ_ν0 (1.41)

we find the following transformation of the coordinates

x^µ= Λ^µ_ν0x^ν0 (1.42)

or introducing the inverse matrix Λ⁻¹

x^µ0 = (Λ⁻¹)^µ_ν⁰x^ν (1.43)

This should be compared to (1.27) for a^µ= 0,

x^0µ= Λ^µ_νx^ν, (1.44)

assuming that numerically

Λ^µ_ν = Λ^µ_ν0, (Λ⁻¹)^µ_ν= (Λ⁻¹)^µ_ν⁰ (1.45) With such an assumption x^ν0 are coordinates of the vector x in the new base e_µ⁰, while x^0µ are coordinates of the vector x⁰= Λx in the base e_µ.

The change of the reference points by the vector a = −−−→

OO⁰ amounts to the vector shift x → x⁰ such that

x⁰= x + a (1.46)

which leads to the following transformation of the coordinates in the basis e_µ

x^0µ= x^µ+ a^µ= (Λ⁻¹)^µ_ν(x^ν− a^µ) (1.47) The composition law (1.5) is also obeyed since in the matrix interpretation, we find

x^0µ= (Λ₁, a₁)(Λ₂, a₂) x = (Λ₁, a₁)(Λ₂x + a₂) = Λ₁(Λ₂x + a₂) + a₁

= Λ₁Λ₂x + (Λ1a2+ a₁) = (Λ₁Λ₂, Λ1a2+ a₁)x (1.48)

1.5 Infinitesimal Lorentz transformations

Let us consider infinitesimal Lorentz transformation

Λ^µ_ν = δ_ν^µ+ ^µ_ν, ^µ_ν 1 (1.49) where from (1.29)

g_µν(δ_α^µ+ ^µ_α)(δ_β^ν+ ^ν_β) = g_αβ (1.50) Up to linear terms in , we obtain

gµβ^µ_α+ g_αν^ν_β= 0 (1.51)

which can be written as

_αβ= −_βα (1.52)

where we lowered the first index. Thus, we found six infinitesimal parameters of the proper Lorentz transformation: _0iand _ij where i, j = 1, 2, 3 and i < j.

1.6 Finite Lorentz transformations

Let us write the proper Lorentz transformation in the form

Λ = (Λ^µ_ν) = e^Ω, (1.53)

gdzie the matrix Ω = (Ω^µ_ν) will be found from the Lorentz transformation condition

Λ^Tg Λ = g (1.54)

Multiplying from the right by the inverse Lorentz transformation matrix

Λ⁻¹= e^−Ω (1.55)

we find

e^ΩTg = g e^−Ω. (1.56)

Expanding this expression in powers of Ω



1 + Ω^T + 1

2!(Ω^T)² + ...

 g = g



1 − Ω + 1

2! Ω² + ... ,



, (1.57)

and comparing the same order terms on both sides, we find that

Ω^Tg = −g Ω . (1.58)

Introducing the matrix

ω = g Ω , ↔ ω_µν= g_µαΩ^α_ν (1.59)

we find that ω is antisymmetric

ω^T = (gΩ)^T = Ω^Tg = −g Ω = −ω . (1.60) Hence the six finite parameters of the Lorentz transformation

β = (β₁, β₂, β₃) , φ = (φ₁, φ₂, φ₃) (1.61) which can be organized in the following way

ω = (ω_µν) =











0 β1 β2 β3

−β₁ 0 φ₃ −φ₂

−β₂ −φ₃ 0 φ₁

−β₃ φ2 −φ₁ 0











(1.62)

and

β_i= ω_0i, φ_i=¹₂_ijkω_jk (1.63)

The matrix Ω for these parameters is given by

Ω = g⁻¹ω ↔ Ω^µ_ν= g^µαωαν (1.64)

and we find

Ω = (Ω^µ_ν) =











0 β1 β2 β3

β₁ 0 −φ₃ φ₂ β2 φ3 0 −φ₁ β3 −φ₂ φ1 0











(1.65)

Let us write Ω in the form

Ω = iβ · K − iφ · J (1.66)

which defines six generators of the Lorentz group,

K = (K₁, K₂, K₃) , J = (J₁, J₂, J₃) (1.67) By the comparison with (1.65), we find their matrix representation

K₁=











0 −i 0 0

−i 0 0 0

0 0 0 0

0 0 0 0











K₂=











0 0 −i 0

0 0 0 0

−i 0 0 0

0 0 0 0











K₃=











0 0 0 −i

0 0 0 0

0 0 0 0

−i 0 0 0











(1.68)

and

J₁=











0 0 0 0

0 0 0 0

0 0 0 −i

0 0 i 0











J₂=











0 0 0 0

0 0 0 i

0 0 0 0

0 −i 0 0











J₃=











0 0 0 0

0 0 −i 0

0 i 0 0

0 0 0 0











(1.69)

Notice that with such definition, the parameters β and φ are real while the generators K are antihermitian while J are hermitian matrices

K^†= −K , J^†= J (1.70)

The generators K cannot be hermitian since the Lorentz group is not compact and cannot be represented by finite size unitary matrices, which would be the case if K were hermitian. It can be easily checked that these generators obey the commutation relations which define the Lie algebra of the Lorentz group SO(1, 3)

[ J_i, Jj] = i _ijkJ_k (1.71)

[ J_i, Kj] = i _ijkK_k (1.72)

[ K_i, K_j] = −i _ijkJ_k, (1.73) where _ijk is totally antisymmetric symbol with ₁₂₃= 1.

We see that J are generators of the rotation group in three dimensional position space, SO(3). Computing the Casimir operator

J²= (J₁)²+ (J₂)²+ (J₃)²=











0 0 0 0 0 2 0 0 0 0 2 0 0 0 0 2











(1.74)

we find that in the position space

J²= j(j + 1) = 2 (1.75)

which means that the representation (1.53) of the Lorentz group corresponds to spin j = 1.

The direction of the vector φ gives the rotation axis, while its length, φ = |φ|, the rotation angle. For rotation along the axis ˆ1, when φ = (φ, 0, 0), from the relation J²₁= 1 we find the following matrix

Λ(φ) = e^−iφJ1 = cos φ − iJ₁sin φ =











1 0 0 0

0 1 0 0

0 0 cos φ − sin φ 0 0 sin φ cos φ











(1.76)

which corresponds to anticlockwise rotation in the (2, 3) plane for φ > 0.

The generators K are vectorial operators and generate Lorentz boosts in direction given by vector β. For example, for the boost along the axis ˆ1, when β = (β, 0, 0), we find from the relation K²₁= −1 the following matrix

Λ(β) = e^iβK1 = cosh β + iK₁sinh β =











cosh β sinh β 0 0 sinh β cosh β 0 0

0 0 1 0

0 0 0 1











(1.77)

which correspond to the boost along the axis ˆ1 in the negative direction for β > 0.

Organizing the six generators K and J in the antisymmetric matrix

Jµν = −J_νµ=











0 K₁ K₂ K₃

−K₁ 0 J₃ −J₂

−K₂ −J₃ 0 J₁

−K₃ J₂ −J₁ 0











(1.78)

such that

K_i= J_0i, J_k= ¹₂_klmJ_lm (1.79)

and the six parameters β and φ in the tensor

ω^µν=











0 −β₁ −β₂ −β₃ β1 0 φ3 −φ₂ β2 −φ₃ 0 φ1

β₃ φ₂ −φ₁ 0











(1.80)

we find the Lorentz transformation in the following form

Λ = expⁿ−₂ⁱω^µνJ_µν^o= exp{iβ · K − iφ · J} (1.81)

1.7 Finite dimensional representations

Let us define the following operators which are built out the SO(1, 3) generators J and K J^±=1

2(J ± iK) (1.82)

In this way, we have two sets of operators

J⁺= (J₁⁺, J₂⁺, J₃⁺) , J⁻= (J₁⁻, J₂⁻, J₃⁻) (1.83) which from (1.71)-(1.73) obey

hJ_i⁺, J_j⁺ⁱ= i _ijkJ_k⁺ hJ_i⁻, J_j⁻ⁱ= i _ijkJ_k⁻ h

J_i⁺, J_j⁻ⁱ= 0 (1.84)

Notice that J^± are hermitian operators, (J^±)^†=1

2(J ± iK)^†=1

2(J^†∓ iK^†) =1

2(J ± iK) = J^± (1.85) Thus, we constructed two independent generators of group SU (2). Therefore, we obtain the following isomorphism for Lie algebras of the groups below

SO(1, 3) ∼= SU (2) ⊗ SU (2) (1.86)

This identification means that we can construct finite dimension representations of SO(1, 3) as a tensor product of finite dimension irreducible representations of SU (2), which are eumera- ted by spin j = 0,¹₂, 1, . . .. The SO(1, 3) representations will be enumerated by the pairs (j⁺, j⁻) and the resulting representations will have dimension equal to N = (2j⁺+ 1)(2j⁻+ 1). In ge- neral, it will be reducible representations. The subsequent representations in ascending order of their dimensionality are shown in Table 1.7.

The one dimensional representation (0, 0) is given by 1 and its generators J_i^±= 0. The two dimensional representations are constructed from generators which are proportional to Pauli matrices σ_i. For (¹₂, 0) we have

J_i⁺=¹₂σi, J_i⁻= 0 (1.87)

and the generators

J_i= (J_i⁺+ J_i⁻) =1

2σ_i, K_i= −i(J_i⁺− J_i⁻) = −i

2σ_i (1.88)

(j₁, j2) N = (2j1+ 1)(2j₂+ 1)

(0, 0) 1

(¹₂, 0) (0,¹₂) 2

(1, 0) (0, 1) 3

(¹₂,¹₂) (³₂, 0) (0,³₂) 4

(2, 0) (0, 2) 5

(1,¹₂) (¹₂, 1) 6

Tablica 1.1: Finite dimension representations of SO(1, 3)

give the representation

A^(1/2,0)= exp{ −¹₂β · σ +₂ⁱφ · σ} (1.89) Similarly, for (0,¹₂) we have

J_i⁺= 0, , J_i⁻=¹₂σi (1.90)

which gives

Ji= (J_i⁺+ J_i⁻) = 1

2σi, Ki= −i(J_i⁺− J_i⁻) = i

2σi (1.91)

and the representation

A^(0,1/2)= exp{¹₂β · σ +₂ⁱφ · σ} (1.92) These are the spinor representations which act in complex two dimensional space of vectors, called spinors. Notice that from

det M = exp{Tr ln M } (1.93)

we have

det A^(1/2,0)= det A^(0,1/2)= 1 (1.94) since Pauli matrices are traceless. Therefore, the two spinor representations are SL(2, C) ma- trices which form group. Their tensor product gives vector Lorentz group representation

A^(1/2,0)⊗ A^(0,1/2)= Λ (1.95)

1.8 SL(2, C) and Lorentz group

The explicit relation between the Lorentz group SO(1, 3) and the group SL(2, C) of 2 × 2 complex matrices with determinant equal one is the following. For any momentum p^µ, define the hermitean matrix

P = p_µσ^µ= p₀+ p₃ p₁− ip₂ p₁+ ip₂ p₀− p₃

!

(1.96) with the determinant

det(P ) = p²₀− p²₁− p²₂− p²₃= p² (1.97)

where σ^µ= (1, σ) and σ = (σ₁, σ₂, σ₃) denotes three Pauli matrices. Consider the transformation

P⁰= AP A^† (1.98)

where A ∈ SL(2, C). The matrix P⁰ is hemitian and has the same determinant as P

det(P⁰) = det(A) det(P ) det(A^†) = det(P ) (1.99) Thus, the transformation (1.98) induces Lorentz transformation: p⁰_µ= Λ_µ^νp_ν. Both A and −A give the same Λ since SL(2, C) is simple connected double covering of the Lorentz group.

Nevertheless, SL(2, C) has six independent parameters like the Lorentz group.

Introducing

σ¯^µ= (1, −σ) (1.100)

which obeys

Tr(σ^νσ¯_µ) = 2 δ^ν_µ (1.101)

and computing

Tr^A(p_νσ^ν)A^†σ¯^µ= Tr(p_ν⁰σ^νσ¯_µ) = 2p⁰_µ= 2Λ_µ^νp_ν (1.102) we obtain the inverse relation

Λ_µ^ν =¹₂Tr(¯σµA σ^νA^†) (1.103) We can also write the relation (1.98) with the help of ¯σ^µ using the relations

σ¯^µ= (σ₂σ^µσ2)^T, σ^µ= (σ₂σ¯^µσ2)^T (1.104) Then

p⁰_µ(σ₂σ¯^µσ2)^T = A(p_µσ2σ¯^µσ2)^TA^† (1.105) and taking the transposition of both sides, we obtain

p⁰_µ(σ₂σ¯^µσ₂) = A^∗(p_µσ₂σ¯^µσ₂)A^T (1.106) Multiplying from the left and right by σ₂, we find

p⁰_µσ¯^µ= (σ₂A^∗σ₂)(p_µσ¯^µ)(σ₂A^Tσ₂) (1.107) Thus, with

B = σ₂A^∗σ₂ (1.108)

we finally obtain

B(p_µσ¯^µ)B^†= p⁰_µσ¯^µ (1.109)

1.9 Irreducible representations of SL(2, C)

Since SL(2, C) is not compact group, its finite dimension, irreducible representations are not unitary. To construct them, let us introduce analytic functions of two complex variables f (z1, z2) and define the action of the representation D(A) of A ∈ SL(2, C) in the linear space H of such functions by

[D(A)f ](z₁, z₂) = f (z^TA) (1.110) where

z^TA = (z₁, z₂) α β γ δ

!

= (αz₁+ γz₂, βz₁+ δz₂) (1.111) The operators D(A) form the representation since

[D(A₂)D(A₁)f ](z₁, z₂) = ˜f (z^TA₂) = [D(A₁)f ](z^TA₂)

= f (z^TA₂A₁) = [D(A₂A₁)f ](z₁, z₂) ‘ (1.112) and

D(1) = 1 , D(A⁻¹) = D⁻¹(A) (1.113)

where D⁻¹ means the inverse to the operator D.

The finite dimension, irreducible representations act in the invariant subspaces H^j⊂ H of the functions f (z₁, z₂) which are homogenous of degree 2j,

f (λz1, λz2) = λ^2jf (z1, z2) (1.114) where

j = 0,¹₂, 1, . . . (1.115)

Indeed, H^j is invariant under the action of the representation element

[D^j(A)f ](z₁, z2) = ¯f (z1, z2) = f (αz₁+ γz₂, βz1+ δz₂) (1.116) since ¯f is homogeneous function of degree 2j

f (λz¯ 1, λz2) = f (λ(αz₁+ γz₂), λ(βz₁+ δz₂))

= λ^2jf (αz1+ γz₂, βz1+ δz₂) = λ^2jf (λz¯ 1, λz2) (1.117) From analyticity, any homogeneous function of degree 2j can be written as

f (z1, z2) =

j

X

m=−j

c^j_m z^j+m₁ z^j−m₂ p(j + m)!(j − m)! ≡

j

X

m=−j

c^j_mφ^j_m(z₁, z2) (1.118)

where φ^j_m are base functions in H^j. In particular, for j = 1/2 and m = ±, we find

φ+(z₁, z2) = z₁, φ−(z₁, z2) = z₂ (1.119)

while for j = 1 and m = −1, 0, 1, we obtain φ+(z₁, z2) = z₁²

√2, φ0(z₁, z2) = z₁z2, φ−(z₁, z2) = z₂²

√2 (1.120)

The matrix of the representation D^j(A) is defined from the action on the base functions [D^j(A) φ^j_m](z₁, z₂) = φ^j_m(z^TA) =

j

X

m⁰=−j

φ^j_m0(z₁, z₂) [ ˜D^j(A)]_m⁰_m (1.121)

In particular, for j = 1/2 we obtain

[D^1/2(A) φ₊](z₁, z₂) = φ₊(z^TA) = αz₁+ γz₂= φ₊α + φ−γ

= φ₊D˜+++ φ₋D˜−+

[D^1/2(A) φ−](z₁, z2) = φ−(z^TA) = βz1+ δz₂= φ₊β + φ−δ (1.122)

= φ₊D˜₊₋+ φ₋D˜−−

and the matrix representation

D˜^1/2(A) = D˜++ D˜+−

D˜−+ D˜−−

!

= α β

γ δ

!

= A (1.123)

Similarly, it can be shown that for j = 1 the matrix representation reads

D˜¹(A) =







α² √

2αβ β²

√

2αγ αδ + γβ √ 2βδ

γ² √

2γδ δ²





 (1.124)

Obviously these are not unitary matrices.

It can easily be shown that

D^j(A) , D^j((A^T)⁻¹) , D^j(A^∗) , D^j((A^†)⁻¹) (1.125) fulfil the representation conditions. However, for SL(2, C) there exists the matrix

C = iσ₂= 0 1

−1 0

!

, C⁻¹= −iσ₂= 0 −1

1 0

!

(1.126) such that for any A ∈ SL(2, C)

CA C⁻¹= (A^T)⁻¹, CA^∗C⁻¹= (A^†)⁻¹ (1.127) which can be easily proven starting from the SL(2, C) matrices

A = α β γ δ

!

, A⁻¹= δ −β

−γ α

!

(1.128)

where αδ − βγ = 1. Now, one can check that the first relation in (1.127) is fulfilled

0 1

−1 0

! α β γ δ

! 0 −1

1 0

!

= δ −γ

−β α

!

(1.129) while the second relation can be found by taking complex conjugation of the first one.

Therefore, the representations (1.125) are pairwise equivalent

D^j(A) ∼ D^j((A^T)⁻¹) , D^j(A^∗) ∼ D^j((A^†)⁻¹) (1.130) and there exists only two inequivalent representations, e.g.

D^(j,0)(A) ≡ D^j(A) , D^(0,j)(A) ≡ D^j((A^†)⁻¹) (1.131) In particular, for j = 1/2 we have

D˜^(1/2,0)(A) = A = α β γ δ

!

D˜^(0,1/2)(A) = (A^†)⁻¹= δ^∗ −γ^∗

−β^∗ α^∗

!

(1.132) Let us notice that for A ∈ SU (2) ⊂ SL(2, C)

A⁻¹= A^† => A = (A^†)⁻¹ (1.133)

and the two representations (1.131) are identical

D^(j,0)(A) = D^(0,j)(A) = D^(j)(A) (1.134)

Thus, D^(j)(A) in this case is unitary since SU (2) is compact group.

By definition, the representation D^(j,k) is a tensor product of irreducible representations D^(j,k)(A) = D^(j,0)(A) ⊗ D^(0,k)(A) (1.135) which matrix has double indices

D˜_aa^(j,k)0,bb⁰(A) = ˜D_ab^(j,0)(A) ˜D^(0,k)_a0b⁰ (A) (1.136) The tensor product representation is in general reducible. In particular

D^(1/2,0)⊗ D^(0,1/2)= D^(1,0)⊕ D^(0,0) (1.137) where D^(0,0)= 1 and D^(1,0) is given by (1.124). It can be shown that such representation is unitary equivalent to the vectorial Lorentz group representations.

UIR of the Poincare group

2.1 Unitary representations

Topologically, the Poincare group P is not compact. Thus, their unitary and irreducible re- presentations (UIR) are infinite dimensional. It means that they are realized in an infinite dimensional Hilbert space with the inner product h .|.i by linear unitary operators U

(Λ, a) → U (Λ, a) (2.1)

such that the following homeomorphism is valid

U (Λ1, a1) U (Λ₂, a2) = U (Λ₁Λ₂, Λ1a2+ a₁) (2.2) With this composition law, the unitary operators U form group with the neutral element

U (1, 0) = 1 (2.3)

and the inverse element for each U (Λ, a)

U (Λ⁻¹, −Λ⁻¹a)) = [U (Λ, a)]⁻¹ (2.4) Since we consider the unitary representations

[U (Λ, a)]⁻¹= [U (Λ, a)]^†≡ U^†(Λ, a) (2.5) and we have

U (Λ, a) U^†(Λ, a) = U^†(Λ, a) U (Λ, a) = 1 (2.6) From (1.10) we find

U (Λ, a) = U (1, a) U (Λ, 0) (2.7)

Introducing the shorthand notation

U (a) = U (1, a) , U (Λ) = U (Λ, 0) (2.8)

19

we obtain in the indicated order

U (Λ, a) = U (a) U (Λ) (2.9)

Thus, we may consider two subgroups. The subgroup of the Lorentz transformations for which U (Λ₁) U (Λ₂) = U (Λ₁Λ₂) (2.10) and the abelian subgroup of four dimensional translations for which

U (a) U (a⁰) = U (a⁰) U (a) = U (a + a⁰) (2.11)

2.2 Generators

From (1.10), the general form of the unitary representation is given by

U (Λ, a) = U (1, a) U (Λ, 0) = exp{ ia^µPµ} exp{ −₂ⁱω^µνJµν} (2.12) where a^µ and ω^µν= −ω^νµ are ten parameters of the Poincare group while P_µ and J_µν are the corresponding hermitian generators

P_µ^†= P_µ, J_µν^† = J_µν= −J_νµ (2.13)

For the Poincare transformation with infinitesimal parameters a^µ= ^µand ω^µν, we have up to the linear order in these parameters

U (1 + ω, ) = 1 + i^µP_µ−¹₂iω^µνJ_µν (2.14) Let us find the transformation laws for these generators under the Poincare transformations.

To this end, let us compute using the composition law (2.2)

U (Λ, a) U (1 + ω, ) U^†(Λ, a) = U (Λ(1 + ω), Λ + a) U (Λ⁻¹, −Λ⁻¹a)

= U (1 + ΛωΛ⁻¹, Λ − ΛωΛ⁻¹a) (2.15) and from (2.14), we obtain

U (Λ, a) U (1 + ω, ) U^†(Λ, a) = 1 + i(Λ − ΛωΛ⁻¹a)^αP_α−¹₂i(ΛωΛ⁻¹)^αβJ_αβ

= 1 + i(Λ)^αPα−¹₂i(ΛωΛ⁻¹)^αβ(J_αβ+ 2P_αaβ) (2.16) On the other hand, from (2.14)

U (Λ, a) U (1 + ω, ) U^†(Λ, a) = 1 + i^µU (Λ, a)PµU^†(Λ, a) −¹₂iω^µνU (Λ, a)JµνU^†(Λ, a) (2.17) To compare the corresponding expressions on the rhs of (2.16) and (2.17), we compute

(Λ)^αP_α= Λ^α_µ^µP_α= ^µ(P_αΛ^α_µ)

(ΛωΛ⁻¹)^αβ= Λ^α_µω^µν(Λ⁻¹)_ν^β = Λ^α_µΛ^β_νω^µν=¹₂ω^µνΛ^α_µΛ^β_ν− Λ^α_νΛ^β_µ^ (2.18)

where we used antisymmetry of ω^µν. Therefore, we find

U (Λ, a) PµU^†(Λ, a) = P_αΛ^α_µ (2.19) and

U (Λ, a)J_µνU^†(Λ, a) = ¹₂(J_αβ+ 2P_αa_β)^Λ^α_µΛ^β_ν− Λ^α_νΛ^β_µ^

= (J_αβ+ P_αa_β− P_βaα) Λ^α_µΛ^β_ν (2.20) Thus, P_µ and J_µν are vectorial and tensorial operators, respectively. Raising and lowering indices, we obtain with the help of relation (1.38)

U (Λ, a) P^µU^†(Λ, a) = (Λ⁻¹)^µ_αP^α

U (Λ, a) J^µνU^†(Λ, a) = (Λ⁻¹)^µ_α(Λ⁻¹)^ν_β(J^αβ+ P^αa^β− P^βa^α) (2.21)

2.3 Poincare group algebra

The commutation relations between the generators P_µ and J_µν can be found with the help of the transformation laws (2.19) and (2.20). For pure translations Λ^µ_ν = δ^µ_ν and

U (a) PµU^†(a) = P_µ

U (a) JµνU^†(a) = J_µν+ P_µaν− P_νaµ (2.22) Thus, for infinitesimal translations U (a) we have up to linear terms in a^µ

U (a)P_µU^†(a) = (1 + ia^νP_ν)P_µ(1 − ia^νP_ν)

= P_µ+ ia^ν[P_ν, Pµ] = P_µ (2.23) and from the comparison of both sides of the last equality, we find

[P_µ, P_ν] = 0 (2.24)

We also consider up to linear terms in a^µ

U (a)J_µνU^†(a) = (1 + ia^αP_α)J_µν(1 − ia^αP_α) = J_µν+ ia^α[P_α, J_µν]

= J_µν+ P_µaν− P_νaµ

= J_µν+ a^α(g_ναPµ− g_µαPν) (2.25) and by the comparison of the last expressions in the two lines, we obtain

[J_µν, Pα] = i (g_ναPµ− g_µαPν) (2.26)