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Before we begin our analysis of relativistic quantum information it will be helpful to have a clear picture of the current standing of the field of relativistic quantum

3.2 Relativistic Quantum Information circa 2009 19

information, with which we will draw many comparisons. Specifically we will focus on the treatment of fermion and photon one particle states which is based on an alternative representation to the one which will be explored in this thesis known as the Wigner representation. The outline of this section is therefore as follows, we will begin with a brief summary of the Wigner representation for a massive spin-1₂ system and a massless spin-1 system. We will then review several interesting results that have been found in the relativistic quantum information community. We will continually return to these results throughout this part of the thesis to make various comparisons.

3.2.1

The Wigner representation

During the 1930’s, Eugene Wigner asked the important question whether there exists a unitary representation of the Poincar´e group, the symmetry group for flat spacetime which is comprised of Lorentz boosts, rotations and translations. The need for a unitary representation should be clear: if one would like to unify relativity and quantum mechanics then one would hope that a unitary representation of the symmetry group of flat spacetime can be found. However it is (and was) a well known fact that there are no finite dimensional faithful unitary representations of the Poincar´e group, which stems fundamentally from the fact that neither the Poincar´e group nor the Lorentz group are compact.2 This question, in a seminal paper [22] was with the use of infinite dimensional representations, answered affirmatively and is now commonly known as the Wigner representation. We will now outline how to construct the Wigner representation for both massive spin-1₂ and massless spin-1 systems.

One-particle states

The goal is to construct a unitary representation of the Poincar´e group acting on an infinite dimensional Hilbert space H. First lets begin by considering the Lie algebra of the Poincar´e group defined by the commutation relations

iJαβ_{, J}γδ_{ = η}βγ_Jαδ_{− η}αγ_Jβδ _{− η}αδ_Jγβ_{+ η}βδ_Jγα _(3.8)

iPα, Jβγ = ηαβPγ− ηαγPβ (3.9)

Pα_{, P}β_{ = 0 (3.10)}

where Pα is identified as the generators of traslations and Jαβ as the generators of the Lorentz group. The generators can further be decomposed into generators of rotations Jmn _{and generators of Lorentz boosts J}0n_{. Note that for simplicity we will occasionally}

keep indices implicit.

In order to quantize we promote the generators Pα _{and J}αβ _{to operators ˆP}α _and

ˆ

Jαβ _{which act on states in an infinite dimensional Hilbert space. Following Weinberg’s}

2_{We will see that we can recover unitarity for finite dimensional systems by introducing an ap-}

propriate inner product, however by doing so we will no longer have a representation in the strict mathematical sense as boosts will map states from one hilbert space to another.

treatment, we identify our Hilbert space by the set of states which satisfies the following ˆ

Pα|p, σi = pα|p, σi , (3.11) i.e. by the set of momentum eigenstates |p, σi, where σ can be any discrete label, but here we will take it to represent the spin of the fermion or the polarization of the photon. Under an arbitrary translation |p, σi → U (1, a) |p, σi = eip·a_{|p, σi. We are left}

to identify the action of the Lorentz group on the states |p, σi. To do this first consider an arbitrary Lorentz transformation Λ which is represented by the unitary operator U (Λ, 0) ≡ U (Λ). The action of U (Λ) on |p, σi must be to produce an eigenstate of ˆPα

with eigenvalue Λp. Therefore the action of U (Λ) must result in a transformation of the form

U (Λ) |p, σi =X

σ0

Cσσ0|Λp, σ0i (3.12)

In order to determining the coefficients Cσσ0 the standard procedure is to first choose a

representative four-momentum kα, e.g. for a massive fermion a suitable choice may be the rest frame momentum kα = (1, 0, 0, 0), however any other choice is equally valid. We then define the action of a set of privileged Lorentz transformation Lα

β(p) as

|p, σi ≡ N (p)U (L(p)) |k, σi (3.13) where pα _{= L}α

β(p)kβ and N (p) is some appropriately chosen normalisation. For the

massive case Lα_β(p) would correspond to a pure boost. We see that this definition of the Lorentz transformation L(p) acting on |k, σi simply leaves σ invariant. Considering again an arbitrary Lorentz transformation acting on an arbitrary momentum eigenstate we have

U (Λ) |p, σi = N (p)U (Λ)U (L(p)) |k, σi

= N (p)U (L(Λp))U (L−1(Λp)ΛL(p)) |k, σi (3.14) The Lorentz transformation L−1(Λp)ΛL(p) can be viewed as the following sequence of operations k → p → Λp → k. This transformation belongs to a subgroup of the Poincar´e group which we refer to as Wigner’s little group and consists of the set of Lorentz transformations which leaves the standard momentum kα invariant, i.e. Wα

βkβ = kα. Explicitly they are given by the following set of Lorentz transformations

Wα_β = L−1(Λp)α_γΛγ_δL(p)δ_β

The action of a Lorentz transformation on an arbitrary state has therefore been split into a component which acts purely on the momentum subspace U (L(Λp)) and a com- ponent U (L−1(Λp)ΛL(p)) which acts purely on the discrete subspace σ. Note however that U (L−1(Λp)ΛL(p)) is not independent of the momentum subspace. We therefore have

U (Λ) |p, σi =X

σ0

Dσσ0(W (Λ, p)) |Λp, σ0i (3.15)

where we can identify Dσσ0(W (Λ, p)) = U (L−1(Λp)ΛL(p)). The matrices D_σσ0 furnish

3.2 Relativistic Quantum Information circa 2009 21

matrix. Dσσ0(W (Λ, p)) is the σ-representation of W , e.g. for a spin-1

2 system the Dσσ0

would correspond to the spin-1₂ representation of the Lorentz transformation W . The Hilbert space in the Wigner representation is given by the tensor product H = L2_{⊗ H}

σ and the momentum eigenstates can be decomposed as |p, σi = |pi ⊗ |σi.

Wigner rotations for massive spin-1₂ particles

For a massive spin-1₂ particle the action of a Lorentz transformation can be determined by first choosing a standard momentum kα_{. This we define as the rest frame momentum}

kα=     m 0 0 0    

The corresponding little group is therefore the group of spatial rotations SO(3) and the action of a Lorentz transformation simply rotates the spin. This is clearly true if we restrict Λ to rotations, since the boost L and the rotation Λ commute W (Λ, p) = Λ and D(12)(W ) is the corresponding SU (2) rotation. If we restrict Λ to a pure boosts in order

to determine W (Λ, p) and hence D(1₂)_{(W ) we will require the spin-}1

2 representation of

an arbitrary Lorentz boost Λ1 2(p) = r γ + 1 2 σ0+ r γ − 1 2β2 β i_σ i

where β is the boost parameter and γ the corresponding Lorentz factor. The Wigner rotation is D(1₂)_{(W (Λ, p)) = L}−1 1 2 (Λp)Λ1 2L 1

2(p) and for an infinitesimal Lorentz boost is

explicitly given by D(12)(W (Λ, p)) = ˆσ0− i 2 ijkδvipj p0_{+ m} σˆ k _(3.16)

where δviis the infinitesimal boost. Note that the Wigner rotation has broken manifest Lorentz covariance. This as we shall see in §4.3.4 is because the Wigner formalism is defined in a specific reference frame defined by the choice of the standard four- momentum kα.

Wigner rotations for massless spin-1 particles

The structure of Wigner’s little group for massless particles is more complicated. This stems from the fact that a null four-vector is orthogonal to itself, for the purpose of this thesis we will simply outline the basic result further details can be found in [24, 41]. The standard choice for kα _{is the null vector}

kα =     1 0 0 1    

There are two distinct transformations that leave kα _{invariant rotations in the x − y}

plane by an angle θ = θ(Λ, p) which we label R(θ) and transformations that have the structure of the Euclidean group of translations E(2) which we label as S(α, β). The transformations S(α, β) acting on a polarisation vector ψα _{corresponds to a gauge}

transformation of the form ψα _{→ ψ}0α _{= ψ}α _{+ λk}α_{, where λ is some arbitrary con-}

stant and hence does not change the physical state. The action of a general Lorentz transformation therefore has the form

W (Λ, k) = S(α, β)R(θ) (3.17) That is, a component R(θ) which changes the physical state of the polarization and a component S(α, β) which is pure gauge. Explicit forms of the Wigner rotation for photons can be found in [41].