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WIGNER ROTATION IN DIRAC FIELDS

HOR WEI HANN (B. Sc. (Hons.), NUS)

A THESIS SUBMITTED

FOR THE DEGREE OF MASTER OF SCIENCE

DEPARTMENT OF PHYSICS

NATIONAL UNIVERSITY OF SINGAPORE

2006 Acknowledgements

This project has been going on for several years and I have had the fortune to work on it with the help of following people.

First of all, I am grateful to my supervisors Dr. Kuldip Singh and Prof. Oh Choo Hiap for offering this project. Dr. Kuldip has given a lot of advice and firm directions on this project, and I would not have been able to complete it without his guidance. It was my pleasure to be able to work with someone like him who is extremely knowledgeable but yet friendly with his students.

Secondly, I wish to thank some of my friends Mr. Lim Kim Yong, Mr. Liang Yeong Cherng and Mr. Tey Meng Khoon for their assistance towards the completion of this thesis. I learned a great deal about MATHEMATICATM, WinEdtTM and several other useful programs through these people.

I feel indebted to my family for sticking through with me and providing the necessary advice and guidance all the time. I have always found the strength to accomplish whichever goals through their encouragements.

Last but not least, I would like to mention a very special friend of mine, Ms. Emma Ooi Chi Jin, who has been a gem. I am appreciative of her companionship and the endless supply of warm-hearted yet cheeky comments about this project. The satisfaction that I derived from the completion of this project is as much hers as mine.

i Contents

Acknowledgements i

Summary iv

List of Symbols vii

1 Basic Quantum Field Theory 1

1.1 Introduction ...... 1

1.1.1 Lorentz and Poincar´eGroups ...... 2

1.1.2 Multiple-Particle Schro¨edingerEquation ...... 5

1.2 Classical Field Theory ...... 6

1.2.1 Conservation Laws in Classical Fields ...... 8

1.2.2 Field Quantization for Dirac Field ...... 12

2 Representations of Poincar´eGroup 22

2.1 Conventional Approach ...... 22

2.1.1 Evaluating the Wigner Matrix ...... 26

2.2 The Action of the Wigner Transformation on the Dirac Spinors ...... 31

ii 2.2.1 Comparing Wigner Transformation and its Effect on Dirac Spinors ... 35

3 Spin Measurements in Dirac Fields 39

3.1 Spin Projection for Single Particle with Arbitrary Momentum ...... 39

3.1.1 The Effect of Lorentz Transformation on the Dirac Spin Operator .... 45

3.2 Single Particle State with Fixed Momentum ...... 47

3.2.1 Bell Correlations for Spin Singlet States ...... 51

4 Spin in Poincar´eGroup 54

4.1 Defining Spin Observables in Relativistic Settings ...... 54

4.1.1 Spin Projection for Dirac Spinors ...... 54

4.1.2 Lie Algebra of the Poincar´eGroup ...... 58

4.1.3 Dirac Spin Operator and Pauli-Lubanski Vector ...... 61

4.2 Unitary Transformations on the Dirac Spinors ...... 62

5 Conclusion 68

iii Summary

Spin as defined in non-relativistic quantum mechanics is the two-dimensional irreducible representation of angular momentum of a quantum state. The Einstein, Podolsky and Rosen (EPR) correlations [1] and Bell-CHSH inequality [2, 3] are important in the fields of quantum information and entanglement, in which the spins or helicities of the quantum systems are manipulated to produce the necessary entangled states. However, a Minkowskian space-time picture is required to describe a free relativistic particle with spin. E. Wigner in his seminal 1939 paper [4] explored the irreducible unitary representations of the Poincar´egroup, which would describe all possible physical transformations of a quantum state. Thus, the spin generator can be realized as a linear combination of the Poincar´egenerators, as discussed in Bogolubov [5] and Terno [6], and it is named as the Pauli-Lubanski vector.

Other researchers like Czachor [7, 8], Doyeol [9], Terashima [10] have utilized different forms of the Pauli-Lubanski vector. However, the spin observable as defined by its generator, has to satisfy certain commutation relations in which the above-mentioned papers have not addressed. It is important to note that a judicial choice for the spin observable, constructed from the Poincar´egenerators, is to apply an appropriate Lorentz boost to the Pauli-Lubanski vector as explained by Bogolubov [5] and Terno [6]. One can thus define the spin states (or the helicities) of any quantum system with respect to this well-defined spin generator.

Another approach to addressing relativistic states is to consider quantum fields associated with underlying particles. For instance describes particles with the Klein-Gordon field for spin

1 0 particles or the Dirac field theory that describes particles of spin 2 . The operators attributed to the physical observables, for example, energy, momentum and angular momentum, can be constructed accordingly which would yield the conventional observables as in a non-relativistic scenario. The angular momentum operator in a field-theoretic formalization, as shown in this thesis, produces the same exact measurements as compared to the spin generator defined from

iv the Poincar´egenerators. The motivation for this thesis is to show the logical consistency be- tween the field-theoretic approach and the group description of relativistic quantum mechanics.

In this thesis, the group of transformations that leaves the particle momentum invariant is examined. The elements of this group is named as the Wigner elements, and it can be defined in terms of the Poincar´egenerators as shown by Soo and Lin [11]. Chapter one introduces the basics of quantum field theory which covers the field-theoretic realization of some physical quantities like energy and angular momentum. It is shown by Noether’s theorem that one can derive these observables for physical symmetries based on the concept of conserved currents.

For chapter two, the Wigner elements for the Poincar´egroup is introduced by way of induced representations. The equivalence between the induced representation of the Wigner elements for Dirac particles and the Lorentz transformation on Dirac spinors is then explored.

In chapter three, the field-theoretic realization of spin for Dirac particles is elucidated. The spin operator can then be used in the context of spin measurements and also entanglement, especially during the discussions on Bell correlations and Clauser-Horne-Shimony-Holt (CHSH) inequality. It is demonstrated that this spin operator exhibits similar measurement results that agree with the works of Terno [6], Czachor [7, 8], Doyeol [9] and Terashima [10] whose spin operator is based on the Pauli-Lubanski vector.

In the fourth chapter, the Dirac spin operator derived from the field-theoretic approach is shown to be equivalent to the induced representation of the Pauli-Lubanski vector of the Poincar´egroup, under appropriate coordinate representation. Finally, it is also shown that the Dirac spin operator is equivalent to the Wigner spin operator up to a proportional factor. These two spin operators agree exactly when the measurement axis is taken as along the direction of the particle momentum.

v List of Figures

3.1 Rotation of Bloch vector ...... 50

3.2 Bell Observable for Dirac Spin Operator ...... 52

3.3 Bell Observable for Dirac Spin Operator with momentum on xy-plane ...... 53

4.1 Geometrical Picture of the Unitary Rotation on Dirac Spinors ...... 64

4.2 Three-Dimensional Plot of the Cosine of Rotation Angle ϕ1 ...... 65

4.3 Two-Dimensional Plot of the Cosine of Rotation Angle ϕ1 ...... 66

vi List of Symbols

The Greek indices (µ, ν, α, β) etc. used in this report are normally reserved for space-time coor- dinate labels 0, 1, 2, 3, with x0 the time coordinate. They are also applicable to the labeling 1, 2, 3, 4 that denotes the spinorial components in the Dirac equations.

The Latin indices (i, j, k) are used for three-dimensional spatial coordinates 1, 2, 3 that represent the x, y and z coordinates respectively.

The letters in bold denote the three-dimensional vector of that quantity. For special cases like the three-dimensional vector of an operator, an arrow is placed over it. An example is σ~ = (σ1, σ2, σ3) where σi are the standard Pauli matrices.

A hat over any vector indicates the corresponding unit vector. However, a widehat over any symbol denotes its status as a group generator.

The space-time metric ηµν is diagonal and its metric signature is given by {1, −1, −1, −1}.

Repetition of the indices denotes a summation over the index , unless otherwise stated.

A dot over any quantity denotes the time-derivative of that quantity.

Dirac matrices γµ are defined so that γµγν + γνγµ = 2ηµν.

The complex conjugate, transpose and Hermitian adjoint of a matrix or vector M are written as M ∗, M T and M † respectively.

The element in any matrix can be labeled by specifying its row-column position, for example the

i i-th row, j-th column of matrix M is given by M j.

The symbol I is reserved for a square identity matrix.

For any non-singular square matrix M, the notation kMk denotes its eigenvalues.

In general, the Planck’s constant ~ and speed of light c are taken as one unless otherwise stated.

vii Chapter 1

Basic Quantum Field Theory

Quantum field theory has been visualized as a marriage between the theory of special relativity and quantum mechanics. By studying the proper Lorentz group and the Poincar´e group, it is possible to incorporate the Minkowskian space-time structure in quantum field theory by way of induced group representations. Their group elements represent the Lorentz transformations and space-time translations associated with the particle states. The Poincar´e representations for the angular momentum associated with Fermi particles are also discussed in this chapter.

1.1 Introduction

In non-relativistic systems, time is assumed to be universal and frame-independent. The Galilean group is taken as the space-time symmetry for such systems. However, in relativistic settings the space-time structure is Minkowskian in nature since one has to accommodate the principle that the speed of light is constant in all reference frames.

An event is characterized by its time of occurence, x0 = ct and its position in terms of its spatial coordinates, (xi ; i = 1, 2, 3); it is described in terms of a contravariant vector xµ =

(ct, x). In addition, the Minkowskian metric can be defined as [12] ηµν = diag(1, −1, −1, −1) with which a covariant vector can be given as

ν xµ = ηµν x = (ct, −x). (1.1)

µ 0 2 P k 2 2 2 2 The “norm” of the vector xµx = (x ) − k(x ) = c t − |x| is a Lorentz invariant term.

The Lorentz transformation is defined as an element in the set of transformations that acts

1 Sec. 1.1 Introduction 2

on the space-time coordinates:

µ 0µ µ ν x −→ x = Λ ν x (1.2)

which reflects the principle of covariance:

µ ν ηµν Λ α Λ β = ηαβ. (1.3)

0 The proper orthochronous Lorentz transformation, in which Λ 0 ≥ 1, is important in the study of quantum fields since this property also allows a consistent definition of the identity

0 transformation. For Λ 0 ≤ −1, this is associated with the class of time-reversing Lorentz transformations which has no physical relevance, but mathematically it is still admissible since the condition given in equation (1.3) is preserved.

1.1.1 Lorentz and Poincar´eGroups

There are two types of Lorentz transformation, namely the rotation in the three spatial dimensions and the Lorentz boost along any given axis. The rotation can be expressed as the

4 × 4 matrix:    1 0 0 0       0  R =    j   0 [R(nˆ, θ )]   i  0

j in which the three-dimensional matrix [R(nˆ, θ)] i denotes a rotation of angle θ about the axis nˆ. The set of all rotations is encompassed in the special orthogonal group of dimension three, denoted as SO(3), which possesses three distinct elements that can be represented by the Euler angles [12]. The other special transformation involves pure Lorentz boost in which the time coordinate is mixed with the spatial coordinates, take for example a boost along the x-axis:    cosh ξ sinh ξ 0 0       sinh ξ cosh ξ 0 0  [L(nˆ = x , ξ )] =   . 1    0 0 1 0    0 0 0 1

The parameter ξ is obtained via v = c tanh ξ, and v is defined as the velocity along the x-axis between two inertial frames. Sec. 1.1 Introduction 3

e The set of all proper orthochronous Lorentz transformations forms the Lorentz group L+, which can be characterized by the three representations obtained individually from the spatial e rotations and the Lorentz boosts. Note that the rotation group SO(3) is a subgroup of L+.

A translation on the space-time coordinates can be written in the following manner:

xµ −→ x0µ = xµ + aµ. (1.4)

The set of transformations involving both the Lorentz transformation and the translation, {T (Λ, aµ)} is described by an inhomogeneous Lorentz group, or simply a Poincar´egroup. Note that the Poincar´egroup encompasses the Lorentz group as a subgroup. The group multiplica- tion for the Poincar´egroup is defined by:

T (Λ1, a1) · T (Λ2, a2) ≡ T (Λ1 · Λ2, Λ1 a2 + a1) (1.5)

It is possible to generate the transformations in the Poincar´egroup through ten generators, which can be derived by considering infinitesimal transformations associated with the Lorentz transformation and translation. The interested reader can refer to Appendix A for the details.

The contravariant generators for the space-time translations {aµ ; µ = 0, 1, 2, 3} are defined by {Pbµ}, in which the time translation, Pb0 and spatial translations, Pbi are the Hamiltonian and the momentum operators respectively of a free particle. The finite translation acting on the space-time coordinates are well-defined (with no Lorentz transformation):

µ µ b T ( I4×4, a ) = exp(−i a Pµ) (1.6)

b bν with the corresponding covariant generators Pµ = ηµνP . The Lorentz transformation Λ can be described by an anti-symmetric rank two tensor {ωµν ; µ, ν = 0, 1, 2, 3} which is defined by the parameters in Λ:

i 0i i0 ij ξ = ω = −ω , θk = ²ijk ω . (1.7)

i Here ξ is the Lorentz boost along the i-th axis, and the θk are the parameters involved in the rotation along the axial vector. The generators for these Lorentz transformations are given by c {Mµν ; µ, ν = 0, 1, 2, 3} have been explicitly derived in Appendix A. Here the generators for the b i b Lorentz boost K and spatial rotation Ji are given by 1 Jbi = ²ijkMc , Kb = Mc (1.8) 2 jk i i0 Sec. 1.1 Introduction 4

The Lorentz transformation can be described by its generators (without translation) µ ¶ i Λ(ω) = exp − ωµν Mc . (1.9) 2 µν

In general, the elements of the Poincar´egroup are given as µ ¶ i T (Λ, aµ) = exp(−i aµPb ) exp − ωµν Mc (1.10) µ 2 µν

and it produces the following space-time coordinate transformation:

µ 0µ µ ν µ x −→ x = Λ ν x + a . (1.11)

Similarly, the momentum of a free particle also transforms accordingly:

µ 0µ µ ν µ p −→ p = Λ ν p +p ¯ (1.12)

in whichp ¯µ is the momentum of the particle in the new coordinate frame.

In this thesis, a momentum contravariant four-vector is defined as pµ ≡ (E/c, p) with

ν its associated covariant form pµ = ηµνp = (E/c, −p). The “norm” of the four-momentum

µ 2 2 2 2 pµp = (E/c) − |p| = m c is another Lorentz invariant term as well. In this context m is defined as the rest mass of the particle.

The generators in the Poincar´egroup can also be used to define the unitary operations acting on the quantum states. This is accomplished by considering the infinitesimal Lorentz transformation which arises from a Taylor’s series expansion of equation (1.10):

i U( I + ω, ²µ) ≡ I − ωµνMc − i ²µPb + ... (1.13) 2 µν µ

in which the infinitesimal Lorentz transformations are described by the corresponding Lorentz transformation Λ = I + ω and space-time translation ²µ. When the unitarity condition is imposed, both set of generators must be Hermitian:

c c † b b † Mµν = (Mµν) , Pµ = (Pµ) . (1.14)

Consider the following unitary operations defined for some arbitrary inhomogeneous Lorentz transformation (Λ and a): ³ ´ U(Λ, a) U( I + ω, ²) U −1(Λ, a) ≡ U Λ(I + ω)Λ−1, Λ² − ΛωΛ−1a (1.15) Sec. 1.1 Introduction 5

in which the group multiplication property from equation (1.5) has been used. If the first order terms of ω and ² are taken and equated respectively, the following transformation rules from the coefficients can be derived for the Poincar´egenerators: ³ ´ c −1 σ ρ c b b U(Λ, a) Mµν U (Λ, a) = Λ µ Λ ν − Mσρ − aσPρ + aρPσ , (1.16a) £ ¤ b −1 −1 ν b U(Λ, a) Pµ U (Λ, a) = Λ µ Pν . (1.16b)

These transformation rules are used to yield the unitary representations of the quantum states. Thus, it is possible to obtain all quantum states by performing the necessary Poincar´etrans- formations on the quantum state defined from a chosen reference frame. As such, one-particle states are being attributed as the irreducible unitary representations of the Poincar´egroup and this fact can be incorporated into the transformation properties of the quantum fields under some arbitrary Lorentz operations. It is important to note that the particle picture can be interpreted in the framework of quantum field theory.

1.1.2 Multiple-Particle Schro¨edingerEquation

A quantum state is obtained from the wavefunction, which in non-relativistic scenario is derived from the famous Schro¨edingerequation: " # ∂ ~2 Xn i ~ ψ(x ,..., x ; t) = − ∇2 + V (x ,..., x ) ψ(x ,..., x ; t) . ∂t 1 n 2m i 1 n 1 n i=1 The energy of the non-relativistic multi-particle system is given by Xn pˆ2 Hb ≡ i + V (x ,..., x ), where 2m 1 n i=1 ∂ Hb = i ~ , pˆ = −i ~∇ (1.17) ∂t i i are the Hamiltonian and momentum operators of the system respectively.

For a system of point masses characterized by its individual i-th coordinates qi(t), the wavefunction has to be modified into fields with continuous generic variables. The field picture Ψ(x, t) attributes the dynamical quantities of the system, ϕ(x, t) at each space-time coordi- nates, instead of the discrete set of qi(t) which describe the particles individually.

A quantum field can be obtained by either quantizing a classical field through the Lagrangian or second quantizing a first quantized system. In the first approach, one starts with a classical Sec. 1.2 Classical Field Theory 6

system described by fields obtained via the Lagrangian. Using the variational principle, the equations of motion can be written in terms of Poisson brackets and then replaced by quantum commutation relations. In the second approach, one first obtains the quantum wavefunction in a one particle system using the Schro¨edingeror Dirac equation. The mode functions (charac- terized by the momentum of the particle) in the wavefunction are then elevated to the status of an operator and this will in turn lead to the commutation relation.

In this thesis, the first approach is considered in which classical fields are formulated accord- ing to the usual Lagrangian or Hamiltonian dynamics. In addition, the conserved quantities in the relativistic quantum mechanics can be deduced via the familiar Noether’s theorem.

1.2 Classical Field Theory

According to classical mechanics, the Lagrangian, L(t) of a system with the dynamical variables ϕ(x, t) can be defined in terms of the Lagrangian density, L in which it is given as: Z L(t) = d3x L [ϕ(x, t), ∇ϕ, ϕ,˙ x, t] . (1.18)

Note that the Lagrangian has been modified which was originally defined by the n-particle

coordinates {qi(t); i = 1, . . . , n} into a continuous function parameterized by ϕ(x, t). The independent variables are now given by the space-time coordinates (x, t). Similarly the following functional is defined as the action: Z ZZ S def= dt L(t) = dt d3x L [ϕ(x, t), ∇ϕ, ϕ,˙ x, t] . (1.19)

The variation of the action is given by ZZ δ S[ϕ] = δ dt d3x L [ϕ(x, t), ∇ϕ, ϕ,˙ x, t] ZZ · ¸ ∂L ∂L ∂L = dt d3x δϕ + δ(∇ϕ) + δϕ˙ . (1.20) ∂ϕ ∂(∇ϕ) ∂ϕ˙

The dynamical variable ϕ(x, t) is assumed to be smooth, infinitely differentiable and fast van- ishing at the boundary limits for both the time and spatial coordinates (typically 0 ≤ t < ∞ and −∞ < x < ∞). Sec. 1.2 Classical Field Theory 7

∂ By considering δ∇ϕ = ∇δϕ and δϕ˙ = ∂t (δϕ), equation (1.20) reduces to: ZZ · µ ¶ µ ¶¸ ∂L ∂L ∂ ∂L δ S[ϕ] = dt d3x − ∇ − δϕ. (1.21) ∂ϕ ∂(∇ϕ) ∂t ∂ϕ˙

The boundary conditions whereby ϕ(x, t) vanishes quickly at the limits of integration have been applied into the calculations. According to the principle of least action, or Hamilton’s principle, the variation on the action equals to zero, or δS = 0. Since the field variables ϕ(x, t) are arbitrary, based on the Hamilton’s principle, the following condition (also called the Euler-Lagrange equation) have been derived for the Lagrangian density L: µ ¶ µ ¶ ∂L ∂L ∂ ∂L − ∇ − = 0 ∂ϕ ∂(∇ϕ) ∂t ∂ϕ˙ µ ¶ ∂L ∂ ∂L ∴ − µ = 0 (1.22) ∂ϕ ∂x ∂(∂µϕ)

µ 0 µ in which the contravariant notation x = (x , x) = (ct, x) and ∂µϕ = ∂ϕ/∂x is employed.

For a Lagrangian that depends on multiple independent fields ϕm(x, t), m = 1 ...N, equation (1.22) can be generalized into: µ ¶ ∂L ∂ ∂L − µ = 0. (1.23) ∂ϕm ∂x ∂(∂µϕm)

Hamiltonian and Poisson Brackets

According to classical mechanics, Legendre’s transformation [13] can be applied onto the Lagrangian density so that the Hamiltonian density, H is then given as:

def H = π ϕ˙ − L [ϕ(x, t), ∂iϕ, ϕ,˙ x, t] (1.24)

R with the canonical conjugate field π = ∂L/∂ϕ˙. In addition, H(t) = d3x H is the Hamiltonian

of the system. It can be shown that the Hamiltonian density is a function of ϕ, πm and ∂iϕ only [14]:

H = H [ϕ, π, ∂iϕ] . (1.25)

The following functional derivative is introduced here: · ¸ δ def ∂ ∂ ∂ = − i (1.26) δϕ ∂ϕ ∂x ∂(∂iϕ) Sec. 1.2 Classical Field Theory 8

so that given any two functionals U[ϕ, πm, ∂iϕ, ∂iπm] and V [ϕ, πm, ∂iϕ, ∂iπm], the operation of the Poisson bracket can be defined as: Z µ ¶ δU δV δU δV { U,V } def= d3x − (1.27) PB δϕ δπ δπ δϕ R R where U(t) = d3x U and V (t) = d3x V. It can be shown [14] that the time evolution of a functional U obeys Z µ ¶ dU δU δH δU δH = d3x − ≡ { U,H} (1.28) dt δϕ δπ δπ δϕ PB

provided U has no explicit time dependence.

1.2.1 Conservation Laws in Classical Fields

Conservation laws are used to describe those dynamical quantities that are independent of time. These conserved entities are manifested in terms of field variables such as energy, momentum, angular momentum and also internal properties like charge, isospin etc. These variables can be constructed by way of generators that came from the corresponding symme- try transformations on the system. It has been established in Noether’s Theorem that for each symmetry transformation there always exists a corresponding conservation law for certain dynamical variables in the system.

Consider an infinitesimal transformation in the space-time coordinates given by:

xµ −→ x0µ = xµ + δxµ (1.29)

and the corresponding change in the field ϕm(x) is:

0 0 ϕm(x) −→ ϕm(x ) = ϕm(x) + δϕm(x), (1.30)

which then induces the following change in the Lagrangian density

L(x) −→ L0(x0) = L(x) + δL(x). (1.31)

It should be noted that the variation in the Lagrangian density comes in two forms, one in which there is a transformation in the space-time coordinates while the other consists of a Sec. 1.2 Classical Field Theory 9

metamorphosis in the fields ϕm under the new coordinates. The coordinate transformation given in equation (1.29) is assumed to have preserved Hamilton’s principle, in which Z Z δS = d4x0 L0(x0) − d4x L(x) def= 0 Z Z Z = d4x0 δL(x) + d4x0 L(x) − d4x L(x). (1.32)

Under the coordinate transformation x0µ = xµ +δxµ, the volume measure d4x0 can be computed from the original measure d4x using the Jacobi determinant: ¯ ¯ ¯ 0 0 ¯ ¯ 1 + ∂(δx ) ∂(δx ) ...... ¯ ¯ ∂(x0) ∂(x1) ¯ ¯ ¯ ¯ 1 1 ¯ 0µ ¯ ∂(δx ) ∂(δx ) ¯ ¯∂(x )¯ ¯ ∂(x0) 1 + ∂(x1) ...... ¯ d4x0 = ¯ ¯ d4x = ¯ ¯ d4x ¯ ∂(xν) ¯ ¯ . . . ¯ ¯ ...... ¯ ¯ ¯ ¯ . . . ∂(δx3) ¯ ¯ . . . 1 + ∂(x3) ¯ µ ¶ to first order ∂(δxµ) ∼= 1 + d4x. (1.33) ∂xµ

After some lengthy computations, it can be shown that [14] the action integral reduces to Z · µ ¶ ¸ ∂ ∂L ∂L δS = d4x δϕ − ∂ ϕ − δµ L δxν = 0 (1.34) ∂xµ ∂(∂ ϕ ) m ∂(∂ ϕ ) ν m ν | µ m {zµ m } f µ which is in effect an equation of continuity for the fields. Here µ ¶ µ def ∂L ∂L µ ν f = δϕm − ∂νϕm − δ νL δx (1.35) ∂(∂µϕm) ∂(∂µϕm)

denotes the four-dimensional “current” density. It is postulated that the “current” density f µ is conserved over time since Z Z Z 3 µ 3 0 3 i 0 = d x ∂µf = d x ∂0f + d x ∂if Z Z 3 0 3 0 0 ∴ d x ∂0f = ∂0 d xf = ∂0Ω = 0 (1.36) R in which Ω0 = d3xf 0 is the conserved quantity associated with the symmetry operation. Here it is assumed that the surface integral Z Z I 3 i 3 d x ∂if = d x ∇·f = f · dA = 0 Surface vanishes rapidly at infinity since the fields and their derivatives would have behaved in a similar manner. Sec. 1.2 Classical Field Theory 10

Canonical Energy-Momentum Tensor

The following term has been isolated from the conserved current in equation (1.35), which is dependent on the variation in the coordinate, δxν:

µ ∂L µ Θ ν = ∂νϕm − δ ν L (1.37) ∂(∂µϕm) to which it is labeled as the canonical energy-momentum tensor.

An infinitesimal translation on the space-time coordinates can be written as

x0µ = xµ + αµ =⇒ δxµ = αµ . (1.38)

Presently the fields have been assumed to preserve its structure such that any local variation

has vanished, where δϕm = 0:

0 0 ϕm(x ) = ϕm(x) . (1.39)

Making use of the variation in the coordinate, δxµ given by equation (1.38), the conserved current is given by µ ¶ ∂L f µ = − ∂ ϕ − δµ L δxν = −Θµ αν ∂(∂ ϕ ) ν m ν ν µ m µ ¶ 0 0 ν ∂L 0 ν ∴ f = −Θ να = − ∂νϕm − δ νL α . (1.40) ∂(∂0ϕm)

Note that the canonical conjugate momentum field for field index m is given by πm = ∂L/∂(∂0ϕm). R 0 3 0 ν As a result, the conserved quantity can be given by Ω = d x f = −Pν ² in which Pν repre- sents the field-theoretic realization of the momentum generator: Z Z def 3 0 3 ¡ 0 ¢ Pν = d x Θ ν = d x πm ∂νϕm − δ νL . (1.41a)

When ν = 0, the Hamiltonian P0 can be shown to be Z Z 3 ¡ 0 ¢ 3 P0 = d x πm ∂0ϕm − δ 0L = d xH = H. (1.41b)

For the spatial components ν = i, the field realizations of the momentum Pi are given as Z 3 Pi = d x (πm ∂iϕm) . (1.41c)

Note that the Hamiltonian and momentum operators have been constructed explicitly based on the conserved current given by equation (1.35). As long as the Lagrangian is well-defined, these field operators can be constructed with due ease. Sec. 1.2 Classical Field Theory 11

For an infinitesimal Lorentz transformation, the coordinate transformation can be parame- terized using the second-rank tensor from equation (1.7):

0µ µ µ ν µ µ ν x = x + ω νx =⇒ δx = ω ν x .

In the presence of the Dirac fields, the structure of the spin has to be incorporated into the Lorentz transformation since the operation of rotation can “mix up” the field indices. The

transformed field is linearly dependent on the rotation angle and ϕm(x) up to the first order: 1 ϕ0 (x0) = ϕ (x) + ωµν [Σ ] ϕ (x) m m 2 µν mn n 1 ⇒ δϕ = ϕ0 (x0) − ϕ (x) = ωµν [Σ ] ϕ (x) (1.42) m m m 2 µν mn n

µ µλ in which ω ν = ηλν ω and the quantities [Σµν]mn are the matrix representations of the gener- ators associated with the infinitesimal spin transformations. Note that the spin generator Σµν is anti-symmetric since the Lorentz transformation tensor ωµν possess the same property. The parameters associated with the Lorentz transformation are given by

0 i i Infinitesimal Lorentz boost ω i = ω 0 = δξ (1.43)

i ip k Infinitesimal spatial rotation ω j = η ²pjk δθ (1.44)

The conserved quantity can be given as (with reference to the conserved current from equation (1.35)): Z ½ µ ¶ ¾ ∂L ∂L Ω0 = d3x δϕ − ∂ ϕ − δ0 L δxµ ∂(∂ ϕ ) m ∂(∂ ϕ ) µ m µ Z ½ 0 m 0 m ¾ 1 1h i = d3x π (x) [Σ ] ϕ (x) − Θ0 x − Θ0 x ωµν (1.45) 2 m µν mn n 2 µ ν ν µ in which the anti-symmetric property of ωµν has been used in simplifying the workings. Here ¡ ¢ 0 0 Θ µ = πm ∂µϕm − δ µL and πm = ∂L/∂(∂0ϕm). Here the covariant field generator of the Lorentz transformation, which can also be referred to as the covariant total angular momentum tensor, can be identified as Z ½ ¾ 1 1h i M = d3x π (x) [Σ ] ϕ (x) − Θ0 x − Θ0 x = L + S . (1.46a) µν 2 m µν mn n 2 µ ν ν µ µν µν in which the orbital angular momentum generator is given as Z h i 3 0 0 Lµν = − d x Θ µ xν − Θ ν xµ Z n o 3 ¡ 0 0 ¢ = d x πm(x)[xµ ∂ν − xν ∂µ] ϕm(x) + δ µ xν − δ ν xµ L (1.46b) Sec. 1.2 Classical Field Theory 12

and the spin angular momentum generator is defined as Z ½ ¾ 1 S = d3x π (x) [Σ ] ϕ (x) . (1.46c) µν 2 m µν mn n

The contraction rules for the covariant total angular momentum tensor are given by Mij = bk b b b ²ijk J and M0i = −Mi0 = Ki, in which Jk and Ki are the generators for the angular momenta and the Lorentz boosts of the system respectively.

The equations (1.41a, 1.46a) provide a realization of Poincar´esymmetry generators through the classical fields. The symmetries associated with these transformations can be shown through explicit calculations of the Poisson brackets to obey the Lie algebra of the Poincar´egroup as shown in Appendix B. For example, h i ³ ´ c b b b {Mµν , Pσ}PB = ηνσPµ − ηµσPν −→ Mµν, Pσ = −i ηνσPµ − ηµσPν to which a factor of (−i) has been appended in the calculations. The classical fields have been realized as operators in the continuous quantum systems and this can be termed as the “second” quantization in quantum theory. The classical fields have allowed one to derive the conserved currents easily, so that the conserved quantities for the quantum systems can be deduced equivalently and unambiguously.

In covariant notation, the field generators can be re-written in terms of the Lie algebra of the Poincar´egroup: h i ³ ´ c c c c c c Mαβ, Mµν = −i ηµαMνβ − ηναMµβ + ηνβMµα − ηµβMνα (1.47a) − h i ³ ´ c b b b Mµν, Pλ = −i ηνλPµ − ηµλPν (1.47b) − h i b b Pµ, Pλ = 0. (1.47c) −

1.2.2 Field Quantization for Dirac Field

In this section, the equation for a massive spin-1/2 field is derived from the assumption that the relativistic momentum of the isolated system can be expressed as (~ = c = 1)

µ 2 2 2 2 pµ p = m =⇒ p0 = |p| + m , (1.48) Sec. 1.2 Classical Field Theory 13

in which the energy and momentum terms are replaced by their corresponding operators p0 → i∂0 , p → −i∇. Dirac showed that equation (1.48) can be written in linear form for both the space-derivatives and time-derivatives, which is written here as [14]: h i µ i γ ∂µ − m Ψ(x, t) = 0 . (1.49)

The terms γµ, µ = 0, 1, 2, 3 given in the Dirac equation are labeled as Dirac matrices, which satisfy

µ ν ν µ µν γ γ + γ γ = 2 η I4×4 . (1.50)

The 4 × 4 Dirac matrices imply that the wavefunction Ψ(x, t) has four components and trans- forms as a bi-spinor function.

At this moment, the Lagrangian density with real components defined for the Dirac field [14] is introduced as ½ ¾ 1 h i 1 h i † L def= Ψ(x, t) iγµ∂ − m Ψ(x, t) + Ψ(x, t) iγµ∂ − m Ψ(x, t) 2 µ 2 µ ½ ¾ i ³ ´ ³ ´ = Ψγµ ∂ Ψ − ∂ Ψ (γµ)† Ψ − m ΨΨ (1.51) 2 µ µ

in which the adjoint spinor Ψ = Ψ†γ0 is introduced. According to the classical field theory, the field Ψ(x, t), its space-time derivatives Ψ˙ , ∇Ψ and their hermitian conjugate fields are mutually independent of each other. Since matrices are involved, it is understood that any product between the elements in the Lagrangian density includes a summation over the field components. The canonical conjugate momentum fields for both independent fields Ψ , Ψ† are given respectively:

∂L i † ∂L i † ΠΨ = = Ψ , ΠΨ† = = − Ψ = Π . (1.52) ∂Ψ˙ 2 ∂Ψ˙ † 2 Ψ

According to the Euler-Lagrange equation, this Lagrangian is a valid choice since µ ¶ h i ∂L ∂L ¡ † 0¢ µ ¡ † 0¢ − ∂ν = − i ∂µ Ψ γ γ + m Ψ γ = 0 (1.53a) ∂Ψ ∂(∂νΨ) µ ¶ h i ∂L ∂L 0 µ † − ∂ν † = γ i γ ∂µ − m Ψ = 0 (1.53b) ∂Ψ ∂(∂νΨ )

in which the Dirac equation has been utilized in its Hermitian conjugate form, which is similar Sec. 1.2 Classical Field Theory 14

to equation (1.53a). The energy-momentum tensor from equation (1.37) can be worked out as

∂L ∂L Θ µ = ∂ Ψ + ∂ Ψ† − δµ L ν ∂(∂ Ψ) ν ν ∂(∂ Ψ†) ν ½ µ µ ¾ i ³ ´ ³ ´¡ ¢ = Ψγµ ∂ Ψ − ∂ Ψ γµ †Ψ − δµ L (1.54) 2 ν ν ν

with the following conserved four-momentum vector Z Z ½ ¾ i h ¡ ¢ ¡ ¢ i P def= d3x Θ 0 = d3x m ΨΨ − Ψ γ ·∇Ψ − ∇Ψ · γ Ψ (1.55a) 0 0 2 Z Z ½ ¾ i ¡ ¢ ¡ ¢ P def= d3x Θ 0 = d3x Ψ† ∂ Ψ − ∂ Ψ† Ψ . (1.55b) k k 2 k k

To determine the total angular momentum, the spin generator is identified as [Σµν] for the spin-1/2 particle. The infinitesimal Lorentz transformation Λδ,

0 µ £ ¤µ ν £ µ µ ¤ ν x = Λδ ν x = δ ν + ω ν x (1.56)

when applied to the coordinates in the Dirac wavefunction, is transformed accordingly as gov- erned by equation (1.42):

0 α 0 α 1 µν α β Ψ (x ) = Ψ (x) + ω [Σµν] β Ψ (x) · 2 ¸ h iα α 1 µν α β b β = δ β + ω [Σµν] β Ψ (x) ≡ S(Λδ) Ψ (x) (1.57) 2 β

in which α, β = 1, 2, 3, 4 are the spinorial indices of the Dirac wavefunction. Note that the

representation for the infinitesimal Lorentz transformation Λδ acting on the Dirac spinors is b b given by S(Λδ). The inverse for the operator S(Λδ) is known to exist because of the presence of an inverse Lorentz transformation, and this gives rise to the following property:

1 Sb−1(Λ ) = Sb(Λ−1) ≡ I − ωµν [Σ ] . (1.58) δ δ 4×4 2 µν

The Dirac equation is then transformed under the coordinate transformation as follows: · ¸ ∂ i γα − m Ψ(x) = 0 ∂xα · ¸ ³ £ ¤ ´ ∂ =⇒ i Sb(Λ ) γα Sb−1(Λ ) Λ β − m Ψ0(x0) = 0 . (1.59) δ δ δ α ∂x0 β

In addition, the Dirac equation has to obey the property of form invariance [15] or covariance under Lorentz transformation. The Dirac matrices have to transform as £ ¤ b α b−1 −1 α β S(Λδ) γ S (Λδ) = Λδ βγ (1.60) Sec. 1.2 Classical Field Theory 15

which is due to the invariance of physical laws in all inertial frames as governed by the principle of relativity. This fundamental relation according to equation (1.60) can be extended for some arbitrary, finite Lorentz transformation, helping to preserve the covariance nature of the Dirac equation. In standard representation, the Dirac matrices can be expressed as

γµγν + γνγµ = 2ηµν where µ, ν = 0, 1, 2, 3 , (1.61a)     i I2×2 0 0 σ γ0 =   , γi =   i = 1, 2, 3 (1.61b) i 0 −I2×2 −σ 0       0 1 0 −i 1 0 and σ1 =   , σ2 =   , σ3 =   . (1.61c) 1 0 i 0 0 −1

Combining the equations (1.56, 1.58, 1.60) and taking only the terms involving first-order power of ωµν, the commutation relation for the spin generator is derived as: · ¸ · ¸ 1 1 ωµν Σ γα − γα Σ = ωµν ηα γ − ηα γ 2 µν µν 2 ν µ µ ν or equivalently, since ωµν is arbitrary, h i α α α Σµν , γ = η ν γµ − η µ γν . (1.62) − A natural choice for the Lorentz generator, as required by the Lorentz transformation properties of the Dirac fields from equation (1.62), is the antisymmetric matrix labeled by the indices µ, ν: 1h i Σµν = γµ, γν (1.63) 4 − which can be shown easily that it obeys the commutation relation from equation (1.62).

The covariant total angular momentum tensor is given by Z ½ h i h i¾ def 3 1 † † 1 0 0 M = d x Π [Σ ] Ψ + Ψ [Σ ] Π † − Θ x − Θ x (1.64a) µν 2 Ψ µν µν Ψ 2 µ ν ν µ Z 1 n o L def= − d3x Θ0 x − Θ0 x µν 2 µ ν ν µ Z ( · ¸ i ³ ¡ ¢ ¡ ¢´ ³ ¡ ¢ ¡ ¢´ = d3x Ψ† x ∂ Ψ − x ∂ Ψ − x ∂ Ψ† − x ∂ Ψ† Ψ 4 µ ν ν µ µ ν ν µ ) ³ ´ 0 0 + δ µxν − δ νxµ L (1.64b) Z ½ ¾ Z ½ ³ ´ ¾ def 1 3 † † i 3 † £ ¤ £ ¤† S = d x Π [Σ ] Ψ + Ψ [Σ ] Π † = d x Ψ γ , γ − γ , γ Ψ µν 2 Ψ µν µν Ψ 8 µ ν − µ ν − (1.64c) Sec. 1.2 Classical Field Theory 16

where the orbital and spin angular momentum are identified in their respective covariant form. In terms of its spatial index, the three-dimensional orbital and spin vectors are given by (letting µ = i, ν = j): h i k def 1 ijk 1 ijk M = ² Mij = ² Lij + Sij (1.65a) 2 2 Z ½ ¾ 1 i ³ ¡ ¢ ¡ ¢´ ³ ¡ ¢ ¡ ¢´ ∴ Lk = ²ijk L = ²ijk d3x Ψ† x ∂ Ψ − x ∂ Ψ − x ∂ Ψ† − x ∂ Ψ† Ψ 2 ij 8 i j j i i j j i Z ½ ¾ i ³ ¡ ¢´k ³ ¡ ¢´k = d3x Ψ† x × ∇Ψ − x × ∇Ψ† Ψ (1.65b) 2 Z ½ ¾ 1 i ³£ ¤ £ ¤ ´ ∴ Sk = ²ijk S = ²ijk d3x Ψ† γ , γ − γ , γ † Ψ 2 ij 16 i j − i j − Z ½ ¾ = d3x Ψ† Ξk Ψ . (1.65c)

The standard representation of the Dirac matrices gives rise to a diagonalized form in the spin generator, where it is given by

i £ ¤ i Ξk = ²ijk γ , γ = ²ijk Σ 8 i j − 2 ij   1 σk 0 =   . (1.66) 2 0 σk

It is interesting to note that the temporal index in spin operator, (for example, letting µ = 0, ν = j) vanishes: Z ½ ¾ i ³£ ¤ £ ¤ ´ S = d3x Ψ† γ , γ − γ , γ † Ψ = 0 . (1.67) 0j 8 0 j − 0 j −

This implies that the total angular momentum of the Dirac field is dependent on the orbital component only. This result seems reasonable since a Dirac field can be interpreted as an infinitely large distribution of particles occupying each space-time position and the total spin will disappear due to the isotropy of space. To study the effects of the Lorentz transformation on the spin of a single particle, the particle picture has to be invoked. This will be elucidated subsequently.

Canonical Quantization

The classical field theory has been derived without any constraints imposed based on the requirements of a quantum system. However, the spin-1/2 particles, like the electrons, have been Sec. 1.2 Classical Field Theory 17

observed to satisfy the Pauli exclusion principle, which states that no two or more fermions in a quantum system can have identical quantum numbers. To incorporate the fermionic structure into the classical field, it is postulated that the anti-commutation rules have to be introduced into the quantization procedure. The canonical quantization involving the independent fields Ψ(x, t) and Ψ†(x0, t0) can be surmised to be given as:

£ ¤ £¡ ¢ ¡ ¢ ¤ b α b β 0 0 b † α b † β 0 0 Ψ (x, t), Ψ (x , t ) + = Ψ (x, t), Ψ (x , t ) + = 0 (1.68a) £ ¡ ¢ ¤ £¡ ¢ ¤ b α b † β 0 b † β 0 b α 3 0 Ψ (x, t), Ψ (x , t) + = Ψ (x , t), Ψ (x, t) + = δαβ δ (x − x ) (1.68b)

Note that these anti-commutation relations are postulated for different spinorial components of the Dirac field, where α, β = 1, 2, 3, 4. The fields have been imbued with the character of an operator and the anti-commutators between them represent the algebra of the Dirac field. The current choice of these anti-commutators are going to reflect that these particles (or called field quanta) satisfy the Fermi-Dirac statistics.

The simple solutions for the free spin-1/2 particle are described by the stationary states derived from the Dirac equation (for zero momentum particles)       I 0 ∂ Ψ Ψ 0 2×2 0 p0 p0 iγ ∂0Ψ(0, t) = mΨ(0, t) =⇒ i     = m   . (1.69)

0 −I2×2 ∂0Ψ−p0 Ψ−p0

The independent solutions to the above first-order differential equation corresponding to the positive and negative energies from the mass-shell condition given by equation (1.48), and they are given as         c c 0 0 (1) −imt  1  (2) −imt  2  (3) imt   (4) imt   Ψp0 = e , Ψp0 = e , Ψ−p0 = e , Ψ−p0 = e 0 0 c1 c2 (1.70a)     1 0 c1 =   , c2 =   . (1.70b) 0 1

Note that the entries in the spinor vector are governed by the number of eigenstates for the spin measurements, which in general are the (2j +1)-states for a spin-j particle. The positive energy p 2 2 (1) (2) is taken as p0 = |p| + m , in which Ψ and Ψ are the positive energy states with the corresponding spin 1/2 and spin −1/2 measurements with respect to some pre-determined axis, Sec. 1.2 Classical Field Theory 18

typically the z-axis. Similarly the wavefunctions Ψ(3) and Ψ(4) are the spin 1/2 and spin -1/2 states for the negative energy respectively. Then the plane-wave solutions can be formulated as

µ α −ipµx Ψp0 (x, t) = Np e u(p, si) , i = 1, 2 (1.71a)

µ α ipµx Ψ−p0 (x, t) = Np e v(p, sj) , j = 1, 2 (1.71b)

where N|p| is the proper normalization constant. Note that the plane-wave solutions are dis-

tinguished by the spinor functions u(p, sα) and v(p, sα), which are charge conjugate to each other. The former describes the positive energy states of a particle while the latter is reserved for the anti-particle. The spinors are going to satisfy the Dirac equation individually h i h i µ µ iγ ∂µ − m u(p, si) = γ Pµ − m u(p, si) = 0 (1.72a) h i h i µ µ iγ ∂µ − m v(p, sj) = γ Pµ + m v(p, sj) = 0 . (1.72b)

As a result, the general field can be expanded in terms of its plane-wave functions:

2 Z 3 ½ ¾ 1 X d p µ µ ˆ −ipµx † ipµx Ψ(x, t) = p a(p, si) u(p, si) e + b (p, si) v(p, si) e (1.73) 3 2p (2π) i=1 0

as a Fourier expansion over the plane-wave solutions. Different mode coefficients a(p, si) and

† b (p, si) exist for each positive and negative energy solution arise from the fact that the energy solutions are independent of each other. In addition the integration is achieved over the Lorentz

3 invariant measure d p/2p0. The conjugate momentum field is also given by

i Πˆ (x, t) def= Ψ†(x, t) (1.74) Ψ 2 2 Z 3 ½ ¾ X µ µ i d p 0 † † ipµx † −ipµx = p N a (p, si) u (p, si) e + b(p, si) v (p, si) e . 3 2p p 2 (2π) i=1 0

Presently the spinor functions u(p, si), v(p, si) have not been properly defined, so they are first assumed to obey the following orthogonality conditions:

† def u (p, si) u(p, sj) = 2p0 δij (1.75a)

† def v (p, si) v(p, sj) = 2p0 δij (1.75b)

† † def u (p, si) v(−p, sj) = v (p, si) u(−p, sj) = 0 . (1.75c) Sec. 1.2 Classical Field Theory 19

Substituting the equations (1.73, 1.74) into the anti-commutation relations (1.68a, 1.68b), while making use of the orthogonality conditions of the spinor functions (1.75a, 1.75b, 1.75c), the following anti-commutation relations for the mode coefficients can be derived:

£ † 0 ¤ (3) 0 a(p, si), a (p , sj) + = 2p0 δ (p − p ) δij (1.76a)

£ † 0 ¤ (3) 0 b(p, si), b (p , sj) + = 2p0 δ (p − p ) δij (1.76b)

£ 0 ¤ £ 0 ¤ a(p, si), a(p , sj) + = b(p, si), b(p , sj) + = 0 (1.76c)

£ 0 ¤ £ † 0 ¤ a(p, si), b(p , sj) + = a(p, si), b (p , sj) + = 0 . (1.76d)

Hamiltonian and Momentum Operators

The Hamilton operator can be formulated in terms of these mode coefficients by making use of the constructed fields: Z Z ½ ¾ i h ¡ ¢ ¡ ¢ i Hb = d3x Θ 0 = d3x m ΨΨ − Ψ γ ·∇Ψ − ∇Ψ · γ Ψ 0 2 Z ½ ¾ 1 X2 = d3p a†(p, s ) a(p, s ) − b(p, s ) b†(p, s ) (1.77) 2 i i i i i=1

Z ½ ¾ Z 1 X2 X2 ∴ Hb = d3p a†(p, s ) a(p, s ) + b†(p, s ) b(p, s ) + d3p p . (1.78) 2 i i i i 0 i=1 |i=1 {z } vacuum energy This expression may differ from standard texts ([14], [16]) because of the normalization condi- tion employed for the mode coefficients. The Hamiltonian has to be normal-ordered because of the presence of an infinitely divergent vacuum energy, which means that if normal-ordering is applied:

† † :: b(p, si) b (p, si) :: = −b (p, si) b(p, si) (1.79)

the energy of the Dirac field becomes finite. As a result, the modified, normal-ordered Hamil- tonian is changed to: Z ½ ¾ 1 X2 :: Hb :: = d3p a†(p, s ) a(p, s ) + b†(p, s ) b(p, s ) . (1.80) 2 i i i i i=1

At this juncture it is necessary to interpret the physical meaning of the mode coefficients. A vacuum state is defined to be a state in which no particles nor anti-particles exist and also Sec. 1.2 Classical Field Theory 20

to which all operations acting on the state are going to leave it invariant. Due to the existence of such a vacuum state, it is possible for to define the number operators for both the particles and anti-particles in the following manner:

† def b a (p, si) a(p, si) = Np0 (p, si) (1.81a)

† def b b (p, si) b(p, si) = N−p0 (p, si) (1.81b) in which the number operator acting on the vacuum state yield:

† def a (p, si) a(p, si)|0i = 0 =⇒ a(p, si)|0i = 0 (1.82a)

† def b (p, si) b(p, si)|0i = 0 =⇒ b(p, si)|0i = 0 . (1.82b)

† Here the operators a(p, si) and a (p, si) are interpreted as the annihilation and creation oper- ators for the particle of momentum p and spin si along z-axis. Similarly the operators b(p, si)

† and b (p, si) are defined as the annihilation and creation operators for the anti-particle of momentum p and spin si along z-axis as well. Note that both the particles and anti-particles possess positive energies, but one is the charge-conjugate state of the other. Once the number operator is well-defined, a Fock state for fermionic particles can be defined to also obey the Pauli exclusion principle. The generalized Fock state for a fermionic system can be described as

|Ψi = |n1, n2 . . . , ni,...i ∀ i ∈ N and ni = 0, 1 (1.83) where the occupation number ni describes the number of particles in the i-th energy eigenstate, in addition to the fact that ni has to be either 0 or 1 due to the exclusion principle.

For pedagogical purposes, the spatial momentum operator can be shown in terms of the creation and annihilation operators: Z ½ ¾ i ¡ ¢ ¡ ¢ P def= d3x Ψ† ∂ Ψ − ∂ Ψ† Ψ k 2 k k Z ½ ¾ X2 d3p = p a†(p, s ) a(p, s ) + b†(p, s ) b(p, s ) (1.84) 2p k i i i i i=1 0 where normal-ordering has been applied to the anti-particle creation/annihilation terms. The orbital angular momentum operator possesses position-coordinate terms which cannot be sim- plified into a diagonal form. Sec. 1.2 Classical Field Theory 21

Special interest in the spin leads the Dirac spin operator which can be used to compute the spins of the particles under the influence of arbitrary Lorentz transformation. Bearing in mind the definition of the spin operator given in the equation (1.65c), the Dirac spin operator can be simplified into a diagonal form which is simpler and easier for calculation purposes. The derivation of the diagonalized spin operator shall be explained in further detail in the later chapters.

A localized state can be constructed from the field operator by

| x, ti = Ψˆ †(x)| 0i (1.85) and the one-particle wavefunction is derived as

ψ(x) = h x, t| ψi = h0|Ψ(ˆ x)|ψi

2 Z 3 ½ ¾ X µ µ 1 d p −ipµx † ipµx = p h0|a(p, si)|ψi u(p, si)e + h0| b (p, si)|ψi v(p, si)e 3 2p (2π) i=1 0 2 Z 3 ½ ¾ X µ 1 d p −ipµx = p hp, si| ψi u(p, si) e . (1.86) 3 2p (2π) i=1 0

† Here the action of b (p, si) acting on the vacuum state has vanished, and the one-particle wavefunction with momentum p and spin si is given by hp, si| ψi. Note that the single particle wavefunction satisfies the Dirac equation since the field operator is defined accordingly.

Chapter Summary

In this chapter we have given a basic introduction to quantum field theory, and most im- portantly the angular momentum operators as given in equations (1.65a, 1.65b, 1.65c) for the Dirac field. These operators will be derived using the method of induced group representations in the next chapter. The Wigner transformation [12, 16] will be calculated explicitly and it will be shown to be consistent with the spin measurements based on the angular momentum operator given in equation (1.65c). Chapter 2

Representations of Poincar´eGroup

The Lorentz transformations associated with the Poincar´egroup can be constructed in which the quantum state differs by only a mixture of the internal spin indices but possess the same physical observables. Here this operation is termed as the Wigner transformation [12, 14] and there have been many successful attempts [9, 10, 11] to derive it. In this chapter these methods are explicitly expounded and shown to be equivalent to one another. In particular, the Wigner transformation can be written as a linear combination of the Poincar´egenerators, as shown by Soo and Lin [11]. In the subsequent section another method to derive the Wigner transformation using the Dirac spinors is shown to have the same results as the one by Soo and Lin [11]. Lastly, it is noted that the Wigner transformation is only unique up to some similarity transformation.

2.1 Conventional Approach

The effects of the inhomogeneous Lorentz transformations acting on the Dirac fields can be elucidated by considering the particle states obtained from the irreducible unitary representa- tions of the Poincar´egroup. The unitary operation representing the Lorentz transformations acting on the Poincar´egenerators are given by the equations (1.16a, 1.16b). Since the momen- tum four-vector commutes among each other according to equation (1.47c), the particle states can be characterized by the four-momentum, pµ, together with additional internal degrees of freedom, σ. The internal degree of freedom pertains to the spin vector, which can be affected by transformations in the space-time coordinates. Thus, the one-particle state is an eigenvector

22 Sec. 2.1 Conventional Approach 23

of the momentum operator: Pbµ | p, σi = pµ | p, σi (2.1)

and the state transforms accordingly for a space-time translation:

µ b µ U( I, a) | p, σi ≡ exp(−i a Pµ) | p, σi = exp(−i a pµ) | p, σi . (2.2)

Given a homogeneous Lorentz transformation Λ, the momentum p is changed to Λp. According to equation (2.1), the one-particle state must possess an eigenvalue of Λp as well: h i ³ ´ bµ −1 bµ µ bν P U(Λ, 0) | p, σi = U(Λ) U (Λ) P U(Λ) | p, σi ≡ U(Λ) Λ νP | p, σi

= (Λp)µ U(Λ) | p, σi (2.3)

in which equation (1.16b) has been employed for the Lorentz transformed momentum generator. It is obvious that the U(Λ) | p, σi is a linear combination of the states | Λp, σ0i, where X (jn) 0 U(Λ) | p, σi = Dσ0σ (Λ, p) | Λp, σ i . (2.4) σ0

(jn) in which Dσσ0 (Λ, p) is termed the Wigner coefficient and it depends on the irreducible rep- b bµ resentations of the Poincar´egroup. As a special example, the Casimir operator PµP cannot change the value of the momentum:

b bµ µ 2 PµP | p, σi = pµp | p, σi = m | p, σi (2.5)

and this leaves the momenta of all particle states invariant. Hence to distinguish each state, the standard four-momentum given by kµ = (m, 0, 0, 0) is chosen, from which all momenta can be achieved by way of a pure Lorentz boost p = L(p) k. Note that the standard four-momentum is non-unique and it also depends on the characteristics of the particle, for example whether it is a massive or a massless particle. Notice that a simple three-dimensional rotation, W which is an element of the Poincar´egroup, will render the standard four-momentum invariant:

µ ν µ W ν k = k (2.6)

which implies that there exist a subgroup of elements consisting of some arbitrary Wigner ro- tations, W , and this subgroup is called the little group and labeled as H(k). It is important to distinguish that this little group is not unique in the Lorentz group but it is actually iso- morphic to other subgroups under a similarity transformation. The reason is because there Sec. 2.1 Conventional Approach 24

are no well-defined frame for the standard momentum, kµ, due to the equivalence principle in special relativity. The definition of this little group is dependent on the choice of the standard momentum as well as the Lorentz transformation, Λ.

The Wigner coefficient for the standard four-momentum can be evaluated as X X (jn) 0 0 U(W ) | p, σi = Dσ0σ (W, k) | k, σ i ≡ Dσ0σ(W ) | k, σ i . (2.7) σ0 σ0 Note that the Wigner coefficients form a representation of the little group, which means that for any little group elements W, W 0, the group multiplication property holds:

U(W ) U(W 0) ≡ U(WW 0) X 0 0 0 0 ∴ U(WW ) | k, si = Ds0s (WW ) | k, s i ≡ U(W ) U(W ) | k, si s0 X 0 0 = Dσs (W ) Ds0σ (W ) | k, s i σ,s0 X 0 0 =⇒ Ds0s (WW ) = Ds0σ(W )Dσs(W ) . (2.8) σ However, the choice for the normalization condition leads to a problem whereby the subse- quent transformation equations relating to the particle states involves some tedious momentum- dependent constants. Instead, the following normalization condition is chosen throughout this report:

0 0 def 0 3 0 hp , σ | p, σi = (2p ) δ (p − p )δσ0σ (2.9)

This is a natural choice from previous normalization conditions imposed on the creation and annihilation operators in equations (1.76a, 1.76b). This implies that the Wigner coefficients cannot sum up to unity, but instead up to a phase factor that depends on the momentum of the particle, p and the Lorentz transformation, Λ: X † 0 ∗ 0 h p, s| U (Λ)U(Λ) | p, s i = Dσs(Λ, p)Dσ0s0 (Λ, p) hΛp, σ | Λp, σ i σ,σ0 X 0 0 ∗ 0 h p, s| p, s i = 2p δss0 = Dσs(Λ, p) Dσ0s0 (Λ, p) (2Λp) δσσ0 σ,σ0 X 0 ∗ p =⇒ D 0 (Λ, p) D (Λ, p) = δ 0 σs σs (Λp)0 ss σ p0 ∴ D†(Λ, p) D(Λ, p) = I (2.10) (Λp)0 2×2 Sec. 2.1 Conventional Approach 25

Note that equation (2.4) is modified such that the normalization factor is appended as an extra term and this allows the Wigner coefficients to be unitary. However this is actually unnecessary since there are no fundamental changes in its behavior. The transformations mentioned above whereby the representations of the Poincar´egroup is being derived from the little group is termed as the method of induced representation.

According to standard literature [12, 16], the explicit form for W (Λ, p) is

W (Λ, p) = L−1(Λp)ΛL(p) (2.11)

k − L(p) → p − Λ → (Λp)

and it can be shown that this is an element of the little group associated with the standard vector kµ. Note that L(Λp) is the Lorentz boost of the momentum (Λp)µ from standard vector kµ. Then the Wigner transformation corresponding to the present choice of kµ is just a simple SO(3) rotation. Furthermore, the Lorentz boost L(p) can be computed from the boost generator b c c Ki = M0i [16], where L(p) = exp(−iηpˆ · K). The explicit form for L(p) is given as   cosh η pˆj sinh η L(p) =   . (2.12) i i j pˆ sinh η δij + (cosh η − 1)ˆp pˆ p ± with tanh η = (p0)2 − m2 p0 and the components of the particle momentum are given by pˆi, i = 1, 2, 3 . The Lorentz boost L(p) transforms the standard momentum kµ to momentum

µ p . In addition, the general Lorentz transformation ΛL for pure boost only is written as:   cosh ξ nˆj sinh ξ ΛL =   (2.13) i i j nˆ sinh ξ δij + (cosh ξ − 1)ˆn nˆ

where nˆ is the unit vector along the direction of the boost. This transforms the momentum pµ to a Lorentz-transformed momentum (Λp)µ. However, the general Lorentz transformation for

a simple spatial rotation, ΛR is given by:    1 0 0 0     0  ν   (ΛR) µ =   (2.14)  i   0 [R(nˆ, θ )]   j  0 Sec. 2.1 Conventional Approach 26

in which the three-dimensional spatial rotations, R(nˆ, θ ) are elements in the simple orthogonal group SO(3). However, this is not relevant for evaluating the Wigner coefficients since it is trivial to show that both the Wigner transformation and Lorentz rotation, ΛR belong to the same little group H(Λ, k).

The conventional way of characterizing the Lorentz transformation, Λ, according to equation b c c bi 1 ijk c (1.9) is described by the generators for boosts Ki = M0i = −Mi0 and rotations J = 2 ² Mjk, that is µ ¶ i Λ = exp − ωµνMc . 2 µν Note that the parameters associated with the Lorentz transformation are given by the anti- symmetric tensor ωµν. In addition, the matrix representations of the Lorentz generators in the four-vector coordinate are given by:     0 iδ 0 0 b bi bi Ki ≡   , J ≡   (2.15) iab iδai 0 0 −i² where the indices a, b represent the rows and columns respectively.

2.1.1 Evaluating the Wigner Matrix

The work of Soo et.al. [11] has considered the Taylor’s expansion of the Wigner transfor- mation in equation (2.11) as well as the explicit form of L(p) in equation (2.12) for an arbitrary Lorentz transformation, Λ:

W (Λ, p) = L−1(Λp)Λ L(p) µ ¶ ³ ´ i ≡ exp iη0pˆ ·Kc exp − ωµνMc exp(−iηpˆ · Kc) (2.16) Λ 2 µν in which the Lorentz boosts, L(p) and L(Λp) and the arbitrary Lorentz transformation, Λ are parameterized in terms of its generators. In addition, the spatial component of the Lorentz- transformed momentum is given by pΛ. Note that the corresponding Lorentz boost for the Lorentz transformed momentum starting from the standard vector, kµ, is given by tanh η0 = . ¡ ¢ 0 |pΛ| (Λp) , in which L Λp k = Λp. Consider the infinitesimal Lorentz transformation being Sec. 2.1 Conventional Approach 27

parameterized by the anti-symmetric tensor ωµν (expressed in first order terms of ω):

i Λ(ω) = I − ωµνMc + ... (2.17a) 2 µν h iα α ∼ α i µν c β α α β =⇒ (Λp) = p − ω Mµν p ≡ p + [ω] βp . (2.17b) 2 β

A Taylor series expansion of the Wigner transformation is considered as follows: ¯ ¯ ¯ ∂W (Λ, p)¯ W (Λ, p) = W (Λ, p)¯ + ωµν ¯ + ... ¯ ∂ωµν ¯ ω=0 ω¯=0 ∂ h i¯ = I + ωµν L−1(Λp)Λ L(p) ¯ + ... ∂ωµν ¯ · ¸¯ ω=0 · ¸¯ ∂L−1(Λp) ¯ ∂Λ(ω) ¯ ∴ W (Λ, p) ∼= I + ωµν ΛL(p) ¯ + ωµν L−1(Λp) L(p) ¯ ∂ωµν ¯ ∂ωµν ¯ · ¸¯ ω=0 · µ ¶¸¯ ω=0 ∂L−1(Λp) ¯ i ¯ = I + ωµν ¯ L(p) + ωµν L−1(Λp) − Mc ¯ L(p) . (2.18) µν ¯ µν ¯ ∂ω ω=0 2 ω=0 In the second term of equation (2.18), the expression can be re-written as · ¸¯ ∂L−1(Λp) ¯ ωµν ¯ L(p) ∂ωµν ¯  ω=0 ¯   p00 p0i ¯ p0 pk ∂ − ¯ = ωµν  m ³ m ´ ¯  m ³ m ´  (2.19) µν 0j 00 i j ¯ j 0 ∂ω p p ˆ0 ˆ0 ¯ p p j k − m δij + m − 1 p p m δjk + m − 1 pˆ pˆ ω=0 in which the substitution Λp = p0 has been made. Making use of the following results: · ¸¯ · ¸ ∂(p0)α ¯ i α ωµν ¯ = ωµν − Mc pβ = [ω]α pβ (2.20a) µν ¯ µν β ∂ω ω=0 2 β i h i i h i [ω] p = − ω0iMc + ωi0Mc + ωijMc p = − 2ξiKb + 2θkJbk p 2 0i i0 ij 2 i     0 ξj p0 =     ξi −²ijkθk pj     |p| (pˆ · ξ) |p| (pˆ · ξ) =   ≡   (2.20b) p0ξi − |p| (pˆ × θ)i p0ξi − |p|~mi " #¯ ( ) ∂(pˆ0)i ¯ 1 ¡ ¢ X ¡ ¢ ωµν ¯ = p0ξi − |p|~mi − pˆi pˆm p0ξn − |p|~mn . (2.20c) ∂ωµν ¯ |p| ω=0 m,n

P m 0 n n At this juncture the sum m,n pˆ (p ξ − |p|~m ) describes the relation between the components of the momentum and its Lorentz boost. Assuming that any sum over the cross-terms is zero for (m 6= n), that is X ¡ ¢ pˆm p0ξn − |p|~mn = p0(pˆ · ξ) (2.20d) m,n Sec. 2.1 Conventional Approach 28

in which the axial and boost vectors are given by θ = (θ1, θ2, θ3)† and ξ = (ξ1, ξ2, ξ3)† respec- tively. These are the parameter vectors associated with the infinitesimal Lorentz transforma- tion Λ. In addition, the vector normal to both the axial and momentum vectors is given by m~ = pˆ × θ. Thus equation (2.19) can be simplified by using the Lorentz boosts from equations (2.12, 2.13), and this can be worked out as: · ¸¯ ∂L−1(Λp) ¯ ωµν ¯ L(p) ∂ωµν ¯  ω=0  0 |p| (pˆ · ξ) − p ξj + |p| ~mj =  m ³ ´n m m ³ ´ o  p0 i |p| i p0 i j β j i β p0−m i j − m ξ + m ~m m − 1 pˆ [ω] β p +p ˆ [ω] β p − |p| (pˆ · ξ)p ˆ pˆ   p0 pk  m ³ m ´  pj p0 j k m δjk + m − 1 pˆ pˆ  ³ ´  p0 k |p| k p0 k  0 m − 1 (pˆ · ξ)ˆp + m (m) − m ξ  =  ³ ´ ³ ´n o  p0 i |p| i p0 i p0 p0 n n ikn m − 1 (pˆ · ξ)ˆp + m (m) − m ξ m − 1 |p| (pˆ × ξ) + 2 (m × pˆ) ² µ µ ¶ ¶ µ ¶· ¸ p0 p0 |p| p0 p0 ≡ i ξ − − 1 (pˆ · ξ)pˆ − m · Kc + i − 1 (pˆ × ξ) − 2 (pˆ × m) · Jb . m m m m |p| (2.21a)

The third term of equation (2.18) can also be simplified in terms of its matrix elements ³ ´ i µν c (note that the expression − 2 ω Mµν can be re-written as the matrix [ω] ): h i h i L−1(p) [ω] L(p)       0 j 0 l p − p 0 ξk p p =  m ³ m ´     m ³ m ´  pi p0 i j j jkm m pk p0 k l − m δij + m − 1 pˆ pˆ ξ −² θ m δkl + m − 1 pˆ pˆ  ³ ´  p0 l p0−m l |p| l  0 m ξ − m (pˆ · ξ)ˆp − m (m)  =  ³ ´ ³ ´ ³ ´³ ´  p0 i p0−m i |p| i i pl pi l ilm m p0−m i l i l m ξ − m (pˆ · ξ)ˆp − m (m) ξ m − m ξ − ² θ + m pˆ mˆ − mˆ pˆ  ³ ´  p0 l p0−m l |p| l  0 m ξ − m (pˆ · ξ)ˆp − m (m)  =  ³ ´ n¡ ¢ ³ ´ o  p0 i p0−m i |p| i p m m p0 m ilm m ξ − m (pˆ · ξ)ˆp − m (m) ξ × m − θ + m − 1 (pˆ × m) ² µ µ ¶ ¶ p0 p0 |p| ≡ −i ξ − − 1 (pˆ · ξ)pˆ − m · Kc m m 2m · µ ¶ ¸ |p| p0 + i − (pˆ × ξ) − θ + − 1 (pˆ × m) · Jb . (2.22a) m m Sec. 2.1 Conventional Approach 29

in which equation (2.20b) has been used for the matrix representation of the arbitrary infini- tesimal Lorentz transformation Λ(ω).

After some careful re-grouping of the terms from all the matrix elements given in equations (2.21a , 2.22a), the boost terms in equation (2.18) has canceled out completely and leaving behind the rotation terms:   h i 0 0 W (Λ, p) = I +  n o  (p0−m) n n ijn 0 − |p| (pˆ × ξ) − θ ² · ¸ (p0 − m) ∴ W (Λ, p) ≡ I − i (p × ξ) + θ · Jb |p|2 · ¸ 1 ¡ ¢ = I − i piξj − pjξi − ²ijnθ Mc (p0 + m) n ij · ¸ 1 ¡ ¢ = I + i ωij − piω0j − pjω0i Mc (p0 + m) ij ³ ´ b = I + i Ωp · J (2.23a)

k ij 1 i 0j j 0i bk 1 ijk c where Ωp ≡ −²ijk[ω − (p0+m) (p ω − p ω )] is the Wigner angle and J = 2 ² Mij is the rotation generator for the Poincar´egroup. The Wigner angle can be re-written as a sum of the contributions from the rotation and the boost:

p × ξ |p|ξ Ω ≡ θ − ≡ θ + (nˆ × pˆ) . (2.24) p (p0 + m) p0 + m

ij Here the angles of rotation are represented by the Euler angles θk = ²ijkω and the boost parameter ξi = ξ(ˆn)i = ω0i. The finite Wigner transformation is given as

h ³ ³ ω ´ ´iN W (Λ(ω), p) = lim W Λ , p N→∞ N ³ ´ b = exp iΩp · J . (2.25)

This has been verified by Soo and Lin [11] and also for some special cases by Terashima [10].

(j) The representation matrix Dσs (W (Λ, p)) can be constructed from the angular momentum generators Ji explicitly, depending on the angular momentum of the particle. In this paper,

1 σ spin- 2 particles will be considered and the appropriate generators are given by J = 2 according to the isomorphism between the proper Lorentz group and the SU(2) ⊗ SU(2) algebra. The Sec. 2.1 Conventional Approach 30

necessary details can be found in Appendix A. As a result, the representation matrix becomes: s p0 ³ σ ´ D(1/2)(W (Λ), p) = exp iΩ ¦ (Λp)0 P 2 s · ¸ p0 1 ³ σ ´ 1 ³ σ ´2 = I + iΩ ¦ + iΩ ¦ + ... (Λp)0 2×2 1! P 2 2! P 2 s ½· ¸ · ¸ ¾ p0 1 |Ω |2 |Ω | 1 |Ω |3 ³ ´ = 1 − P + ... I + P − P + ... iΩb ¦ σ (Λp)0 2! 22 2×2 2 3! 23 P

s · µ ¶ µ ¶ ¸ p0 |Ω | |Ω | ³ ´ ∴ D(1/2)(W (Λ), p) = cos P I + i sin P Ωb ¦ σ (2.26) (Λp)0 2 2×2 2 P by using the relation (σ ¦ a)(σ ¦ b) = (a ¦ b) + iσ ¦ (a × b) . (2.27)

For the sake of convenience, I2×2 is assigned as the 2 × 2 identity matrix, and σ as the “vector” that comprise of the Pauli matrices (σ1, σ2, σ3). Note that the Wigner angle |Ωp| is dependent on both the rotation and boost parameters for an arbitrary Lorentz transformation Λ. In q p0 addition, notice that an additional normalization factor of (Λp)0 has been appended to the matrix so as to maintain the condition given in equation (2.10).

In the absence of rotation, the arbitrary Lorentz boost can be parameterized according to equation (2.13). In fact, Doyeol et.al. [9] attempted to construct the matrix by the associative property of the representation matrix:

D(1/2)(W (Λ, p)) = D(1/2)(L−1(Λp)ΛL(p))

≡ D(1/2)(L−1(Λp))D(1/2)(Λ)D(1/2)(L(p)). (2.28)

The representation matrix D(1/2)(W (Λ, p)), in the absence of rotation, is thus given as p · µ ¶ 0 0 (1/2) p /(Λp) 0 ξ D (W (Λ, p)) = (p + m) cosh I2×2 + {(p0 + m) [(Λp)0 + m]}1/2 2 µ ¶ µ ¶ ¸ ξ ξ (p · nˆ) sinh I −i sinh σ · (p × nˆ) (2.29a) 2 2×2 2 s · µ ¶ µ ¶ ¸ p0 Ω Ω = cos p I + i sin p (σ ¦ mˆ ) (2.29b) (Λp)0 2 2×2 2 in which

µ ¶ ¡ ξ ¢ ¡ η ¢ ¡ ξ ¢ ¡ η ¢ Ωp cosh cosh + sinh sinh (nˆ ¦ pˆ) cos = £ 2 2 2 2 ¤ (2.30a) 2 1 1 1 1/2 2 + 2 cosh (ξ) cosh (η) + 2 sinh (ξ) sinh (η)(nˆ ¦ pˆ) Sec. 2.2 The Action of the Wigner Transformation on the Dirac Spinors 31

and µ ¶ ¡ ξ ¢ ¡ η ¢ Ωp sinh sinh (nˆ × pˆ) sin mˆ = £ 2 2 ¤ (2.30b) 2 1 1 1 1/2 2 + 2 cosh (ξ) cosh (η) + 2 sinh (ξ) sinh (η)(nˆ ¦ pˆ) p0 where cosh η = m and m = nˆ × pˆ represents the axis of rotation of the equivalent Wigner transformation on the Dirac spinors. Here the parameters ξ and nˆ are defined in the same manner as the general Lorentz boost as given in equation (2.13). Soo’s results (equation (2.26)) can be shown to agree with Doyeol’s result (equation (2.28)) in the situation when there is an absence of rotation in the Lorentz transformation Λ. The explicit derivation can be found in Appendix D. These results have been widely quoted by the following current papers ([7, 9, 10, 11, 17, 18, 19]), although their results differ slightly due to their choices for the simplified form of the Lorentz boost Λ.

2.2 The Action of the Wigner Transformation on the Dirac Spinors

The construction of the Wigner transformation given in equation (2.16) requires an explicit dependence of some fixed reference frame, in this case the rest frame of the particle kµ = (m, 0). According to standard literature [12, 16], the following transformation rules have been derived for the annihilation and creation operators: X −1 ∗ (jn) U(Λ) a(p, si, n) U (Λ) = Dji (W (Λ, p)) a(pΛ, sj, n) (2.31a) Xσ † −1 (jn) † U(Λ) a (p, si, n) U (Λ) = Dji (W (Λ, p)) a (pΛ, sj, n) (2.31b) Xσ −1 ∗ (jn) U(Λ) b(p, si, n) U (Λ) = Dji (W (Λ, p)) b(pΛ, sj, n) (2.31c) Xσ † −1 (jn) † U(Λ) b (p, si, n) U (Λ) = Dji (W (Λ, p)) b (pΛ, sj, n) (2.31d) σ

where jn is the spin of particles of species n, and pΛ is the spatial part of Λp. For massive particles, these transformation relations can be implemented in the derivation of the Dirac fields. As such, the Lorentz transformation for the Dirac fields can be re-expressed in terms of Sec. 2.2 The Action of the Wigner Transformation on the Dirac Spinors 32

the Dirac spinors and they are given as: s X ¡ ¢ p0 X h iα ∗(jn) α ˆ −1 β Dji W (Λ, p) u (p, si) = S(Λ ) u (Λp, sj) (2.32a) (Λp)0 β i β s X ¡ ¢ p0 X h iα (jn) α ˆ −1 β Dji W (Λ, p) v (p, si) = S(Λ ) v (Λp, sj) . (2.32b) (Λp)0 β i β

in which Sˆ(Λ) is the representation of the proper Lorentz transformation Λ acting on the Dirac spinors u(p, s, n) and v(p, s, n). Those who are interested in the derivation of the above- mentioned equations can refer to Appendix D for the details.

The positive energy solutions from equation (2.32a) with positive-definite masses are the subject of discussion here. The Dirac spinors in this case are given as:      1   0          p  0  p  1  u(p, ↑) ≡ p0 + m   u(p, ↓) ≡ p0 + m   (2.33)  p3   p1−ip2   0   0   p +m   p +m  p1+ip2 −p3 p0+m p0+m

P ∗ α 0 0 0 in which α uα(p, s) u (p , s ) = 2p0 δ(p − p ) δss0 is the proper normalization condition. Here the spins are defined with respect to the z-axis, with ↑ denoting spin up and ↓ spin down. The explicit form for the representation matrix D(Λ) acting on the Dirac spinors corresponding to a Lorentz transformation Λ can be given as [14]: µ ¶ i Sˆ(Λ) = exp − ωµνΣ (2.34a) 4 µν

in which   ω00 = 0  ωµν = ωi0 = −ω0i = −ξi (2.34b)    ij ji ijk ω = −ω = ² θk 1h i Σµν = γµ, γν , (2.34c) 4 −      I2×2 O2×2     γ0 =    O2×2 −I2×2 O2×2 σi γµ =   , αi =   (2.34d)  O σ σ O   2×2 i  i 2×2  γi = γ0αi =  −σi O2×2 Sec. 2.2 The Action of the Wigner Transformation on the Dirac Spinors 33

Note that the group parameters are encoded in the covariant tensor ωµν, with six variables that represents the boost and the components of the axial rotation vector θ. In this instance, ξ ≡ |ξ|

ij is the boost parameter (tanh ξ = β) and θk = ²ijkω are the covariant angular variables.

The Dirac spinors can be re-written such that for a rest state kµ = (m, 0), the momentum p arises from a Lorentz boost L(p) that can be derived from equation (2.34a):

X h iα uα(p, s, n) = Sˆ(L(p)) uβ(0, s, n) β β · µ ¶¸ X i ¡ ¢ α = exp − ω0jσ + ωj0σ uβ(0, s, n) 4 0j j0 β β ³η ´ √ ∴ u(p, s) = exp (pˆ · α) 2m b (2.35) 2 s

where         c1 c2 1 0 b1 =   , b2 =   for c1 =   , c2 =   (2.36) 0 0 0 1

and  ¡ ¢ ¡ ¢  ³ ´ cosh η I sinh η (pˆ · σ) η  2 2×2 2  exp (pˆ · α) = ¡ ¢ ¡ ¢ . (2.37) 2 η η sinh 2 (pˆ · σ) cosh 2 I2×2

Equation (2.35) can be used to evaluate the effect of the Lorentz boost L(Λp) as well: µ ¶ η0 √ u (p , s) = exp (pˆ · α) 2mb Λ 2 Λ s  ³ ´ ³ ´  η0 η0 √  cosh 2 I2×2 sinh 2 (pˆΛ · σ)  = 2m  ³ ´ ³ ´  bs (2.38) η0 η0 sinh 2 (pˆΛ · σ) cosh 2 I2×2

where tanh η0 = √ |pΛ| . As a result, equations (2.35) and (2.38) can be used in the evalua- 2 2 |pΛ| +m (1/2) tion of Dσs (W (Λ, p)) from equation (2.32a): s p0 X h iα X X ∗ ˆ −1 β ∗(jn) ∗ α uα(p, sk) S(Λ ) u (pΛ, sj) = Dji (W (Λ, p)) uα(p, sk)u (p, si) (Λp)0 β α,β σ α X ∗(1/2) = Dji (W (Λ, p)) (2p)0 δki. σ

P ∗ α 0 0 (3) Here the normalization condition for the Dirac spinors α uα(p, s)u (p , s ) = (2p)0 δ (p − 0 p ) δss0 is used. The Lorentz transformation of concern is merely just a boost along the nˆ-axis. Sec. 2.2 The Action of the Wigner Transformation on the Dirac Spinors 34

s 1 p0 h i ∴ D∗ (1/2)(W (Λ, p)) = u†(p, s ) Sˆ(Λ−1) u(p , s ) (2.39a) jk (2p)0 (Λp)0 k Λ j s · µ ¶¸· ¸ h i 0 0 η † η 2m p (pˆ·α) i µν (pˆΛ·α) = e( 2 )b exp ω σ e 2 b (2p)0 (Λp)0 k 4 µν j s   1 p0 C η I2×2 S η (pˆΛ · σ) †  2 2  = 0 bk Cη (Λp) S η (pˆΛ · σ) C η I2×2 2 2     C ξ I2×2 −S ξ (nˆ · σ) C η0 I2×2 S η0 (pˆΛ · σ) 2 2  2 2      bj −S ξ (nˆ · σ) C ξ I2×2 S η0 (pˆΛ · σ) C η0 I2×2 2 2 2 2 s   0 AB 1 p †   = 0 bk bj (2.39b) Cη (Λp) BA in which

A ≡ C η C ξ C η0 I2×2 − S η S ξ C η (pˆ · σ)(nˆ · σ) − C η S ξ S η0 (nˆ · σ)(pˆΛ · σ) 2 2 2 2 2 2 2 2 2

+ S η C ξ S η0 (pˆ · σ)(pˆΛ · σ) (2.39c) 2 2 2

B ≡ S η C ξ C η (pˆ · σ) − C η S ξ C η (nˆ · σ) + C η C ξ S η0 (pˆΛ · σ) 2 2 2 2 2 2 2 2 2

− S η S ξ S η (pˆ · σ)(pˆΛ · σ)(nˆ · σ) (2.39d) 2 2 2

η0 η0 ξ ξ η 0 0 η η and the symbols C η = cosh 2 , S η = sinh 2 ,C ξ = cosh 2 , S ξ = sinh 2 ,C = cosh 2 , S = 2 2 2 2 2 2 η 0 sinh 2 are shorthand notations for the hyperbolic functions of rapidity η , ξ, η respectively. The four-momentum and its Lorentz-transformed momentum of the particle are given by:

p0 = m cosh(η), p = m sinh(η) pˆ (2.40a)

∴ (Λp)0 = m {cosh(η) cosh(ξ) + (nˆ · pˆ) sinh(η) sinh(ξ)}

pΛ = m {[cosh(η) sinh(ξ) + (nˆ · pˆ) sinh(η) cosh(ξ)] nˆ

+ sinh(η)[pˆ − (nˆ · pˆ) nˆ]} (2.40b)

Note that the Wigner transformation matrix is given by: s 0 h †(1/2) 1 p D (W (Λ, p)) = C η C ξ C η0 I2×2 − S η S ξ C η0 (pˆ · σ)(nˆ · σ) − 0 2 2 Cη (Λp) 2 2 2 2 i C η S ξ S η0 (nˆ · σ)(pˆΛ · σ) + S η C ξ S η0 (pˆ · σ)(pˆΛ · σ) (2.41) 2 2 2 2 2 2

since the vectors bs0 , bs only span the two-dimensional square matrix A at the top left corner of the 4 × 4 matrix in equation (2.39b). Sec. 2.2 The Action of the Wigner Transformation on the Dirac Spinors 35

After some lengthy calculations, the Wigner transformation matrix is given by:  s ¡ ¢ ¡ ¢ ¡ ¢ ¡ ¢ 0  ξ η ξ η †(1/2) p cosh 2 cosh 2 + sinh 2 sinh 2 (nˆ · pˆ) D (W (Λ, p)) = q I2×2 + (Λp)0  1 2 [1 + cosh ξ cosh η + sinh ξ sinh η (nˆ · pˆ)]  ¡ ¢ ¡ ¢ sinh ξ sinh η  q 2 2 i (nˆ × pˆ) · σ~ . (2.42) 1  2 [1 + cosh ξ cosh η + sinh ξ sinh η (nˆ · pˆ)]

It can be seen that there is perfect agreement between Doyeol’s results from equations (2.30a, 2.30b) and equation (2.42), even though one is the complex transpose of the other. It can be shown easily that the Wigner matrices agree completely if equation (2.32b) has been chosen for computational purposes. This indicates that in the absence of rotation, the two methods involved in obtaining the Wigner transformation matrix are equivalent.

2.2.1 Comparing Wigner Transformation and its Effect on Dirac Spinors

The equivalence between the two results indicates that the formulation of the Wigner ro- tation is consistent with the pedagogy of the quantum field theory, that is, one can describe the effect of the Lorentz transformation on quantum fields given the Wigner transformation matrix. This allows some straightforward calculations on the observables, for example the spin operator, which has been constructed based on the equation (1.65c). However, the Wigner transformation matrix cannot be established as a unique representation of the arbitrary Lorentz operation, since it is dependent on the reference frame of the particle/state. To quote Weinberg [16], the arbitrary Lorentz boost can be constructed as a series of simple rotations and boost:

−1 L(p) = Rpˆ (ψ)Bz(|p|)R (ψ) . (2.43) ⊥ pˆ⊥

Here Rpˆ⊥ (ψ) is the active rotation applied to the momentum of the particle such that the momentum points along the z-axis, in which it has been rotated about an axis pˆ⊥ that is perpendicular to its momentum, by an amount of angle ψ in the counter-clockwise direction. In

† addition, Bz(|p|) is a pure boost along the z−axis from rest to the momentum vector (0, 0, |p|) . It transpires that the definition of the Wigner transformation from equation (2.16) is equivalent Sec. 2.2 The Action of the Wigner Transformation on the Dirac Spinors 36

to another simple rotation based on equation (2.25), that is: ³ ´ −1 def b W (Λ, p) = L (Λp)Λ L(p) = exp iΩp · J (2.44a) p × ξ |p|ξ Ω ≡ θ − ≡ θ + m (2.44b) p (p0 + m) p0 + m

ij i i 0i θk = ²ijkω , ξ = ξ(ˆn) = ω , m = nˆ × pˆ . (2.44c)

However, this is merely a sufficient but not necessary condition for the Wigner transformation, meaning that it changes if a different pure Lorentz boost is being used in the calculations. Assume that there exist another Lorentz boost which differs from the previous one in equation (2.43) by another pure rotation:

0 −1 L (p) ≡ L(p)Rpˆ(φ) = Rpˆ (ψ)Bz(|p|)R (ψ)Rpˆ(φ) (2.45) ⊥ pˆ⊥

in which Rpˆ(φ) is a simple rotation about the momentum pˆ−axis by an angle φ along the © ª counter-clockwise direction, and it can be written in terms of its generators, J i, i = 1, 2, 3 , for an element in the group SO(3): ³ ´ b Rpˆ(φ) = exp iφpˆ · J . (2.46)

Notice that the rotation axis has to be chosen as parallel to the momentum vector since it is imperative that the Lorentz boosts, L0(p) and L(p), are equal to the identity transformation when the particle is at rest (p = 0).

However, this also implies that the matrix representation of the Wigner transformation, D(1/2)(W (Λ, p)), for any arbitrary Lorentz transformation is non-unique. Each matrix repre- sentation depends on the particular form of the Lorentz boost, L(p), and they are connected to one another through a similarity transformation. This can be deduced easily based on the premise that all Lorentz boosts are related by some rotation matrix, as given in equation (2.45). Thereafter the Wigner transformation matrix based on the Lorentz boost given in Sec. 2.2 The Action of the Wigner Transformation on the Dirac Spinors 37

equation (2.45) can be evaluated as:

h i−1 h i 0 0−1 0 W (Λ, p) ≡ L (Λp)Λ L (p) = L(Λp)Rpˆ(φ) Λ L(p)Rpˆ(φ)

−1 −1 = R (φ) L (Λp)Λ L(p)Rpˆ(φ) pˆΛ

−1 = Rpˆ (φ) W (Λ, p)Rpˆ(φ) Λ ³ ´ ³ ´ ³ ´ b b b = exp −iφpˆΛ · J exp iΩp · J exp iφpˆ · J (2.47) ³ ´ b 6= exp iΩp · J

in which the symbol pˆΛ represents the spatial part of the Lorentz-transformed momentum Λp.

This poses a serious challenge for the calculation of the Wigner matrix elements associated with the arbitrary Lorentz transformation, Λ. Since there are non-unique representations of the boost, L(p), which produce different Wigner transformations as shown in equation (2.47), there is no judicious choice for the Wigner matrix. As a result, it can concluded there is no faithful representation of the Lorentz transformation Λ, since the Lorentz boost L(p) has to be chosen in advance before defining the Wigner rotation as W = L−1(Λp)ΛL(p).

This particular group property can be found to hold in the Dirac fields as well, especially pertaining to the fact that the group describing the spinors is given by SU(2)⊗SU(2). Since the Dirac spinors are eigenvector solutions to the Dirac equation (from equations (1.72a, 1.72b): h i µ γ pµ − m u(p, si) = 0 i = 1, 2 (2.48a) h i µ γ pµ + m v(p, sj) = 0 j = 1, 2 . (2.48b)

This illustrates that even if the equations (2.32a, 2.32b) uniquely describe the matrix represen- tation of the Wigner transformation, D(1/2)(W (Λ, p)), the spinors inherent to the Dirac field are generic to their respective Dirac equations. As such, the matrix D(1/2)(W (Λ, p)) associated to an arbitrary Lorentz transformation, Λ, can be found to be dependent on the choice for the Dirac spinors. As an example, assume that there exist a 2 × 2 matrix, M ∈ SU(2), such that: X2 0α α u (p, si) = Mim u (p, sm) i = 1, 2 (2.49a) m=1 X2 ∗ δii0 = Mmi Mmi0 (2.49b) m=1 h i h i µ 0 µ 0 iγ ∂µ − m u (p, si) = γ Pµ − m u (p, si) = 0 i = 1, 2 . (2.49c) Sec. 2.2 The Action of the Wigner Transformation on the Dirac Spinors 38

To quote equation (2.39a), the matrix D(1/2)(W (Λ, p)) can be evaluated for the new Dirac spinor in the following manner: s ³ ´ 1 p0 h i D∗ (1/2) W (Λ, p) = u†(p, s ) Sˆ(Λ−1) u(p , s ) ji (2p)0 (Λp)0 i Λ j s 2 2 0 X h iα X 1 p 0∗ ˆ −1 ∗ 0β = Mimuα (p, sm) S(Λ ) Mnju (pΛ, sn) (2p)0 (Λp)0 β m=1 s n=1 X2 1 p0 h i = M M∗ u0†(p, s ) Sˆ(Λ−1) u0(p , s ) im nj (2p)0 (Λp)0 m Λ n m,n=1 X2 ³ ´ ∗ ∗ (1/2)0 = MimMnj Dnm W (Λ, p) m,n=1 ³ ´ ³ ´ ³ ´ ∴ D(1/2) W (Λ, p) = M D0(1/2) W (Λ, p) M† 6= D(1/2) W (Λ, p) ³ ´ ³ ´ =⇒ D0(1/2) W (Λ, p) = M† D(1/2) W (Λ, p) M . (2.50)

It is interesting to note that the similarity transformations acting on the physical four- dimensional space-time vector belongs to the group SO(3), while the analogous similarity trans- formations for the Dirac spinors belong to the group SU(2), and it is a well-known fact that these two groups are homomorphic to each other. Note that the group elements of SU(2) form a double-valued representation for each member in SO(3). However, the crucial question at this stage is to evaluate without ambiguity the true representation of the general Lorentz transformation for the Dirac particles or even the quantum fields.

Chapter Summary

In this chapter, the Wigner transformation has been derived in terms of the Poincar´egen- erators, as shown in equation (2.25). This result implies that the little group of the Wigner transformations is simply a three-dimensional rotation group. In addition, this result is demon- strated as being equivalent to the Wigner transformation involving the Dirac fields, as given in equation (2.42). Lastly, the elements of the little group for the Wigner transformation are equivalent to each other, up to some similarity transformations as proven in equations (2.47, 2.50). For the next chapter, the relation between the effects of Wigner rotation and spin projection on a Dirac field will be explored. Chapter 3

Spin Measurements in Dirac Fields

In this chapter the spin projection on a Dirac field will be explored in which the spin angular momentum operator (1.65c) from chapter one will be derived explicitly using the mode coefficients. Making use of this spin operator, the effects of the Lorentz transformation acting on the quantum state is then calculated. Lastly, the operator is also used to show that the violation of the Bell correlations as shown in Czachor [7] can be reproduced here.

3.1 Spin Projection for Single Particle with Arbitrary Momentum

The operator for spin projection in arbitrary momentum defined for Dirac fields can be quoted from equation (1.65c): Z rˆ · S~ = d3x Ψ†(x) rˆ · Ξ~ Ψ(x) . (3.1)

In this definition, the Dirac fields can be derived as an expansion of its plane wave solutions as given in equation (1.73):

2 Z 3 ½ ¾ 1 X d p µ µ Ψ(ˆ x, t) = a(p, s ) u(p, s ) e−ipµx + b†(p, s ) v(p, s ) eipµx (3.2a) (2π)3/2 2p i i i i i=1 0 2 Z 3 ½ ¾ 1 X d p µ µ Ψˆ †(x, t) = a†(p, s ) u†(p, s ) eipµx + b(p, s ) v†(p, s ) e−ipµx (3.2b) (2π)3/2 2p j j j j j=1 0

In addition, the spin projector in rest frame is given as (from equation (1.66)):   k 1 σ 02×2 Ξk =   . (3.3) 2 k 02×2 σ

39 Sec. 3.1 Spin Projection for Single Particle with Arbitrary Momentum 40

2 Z ZZ 3 3 0 ½ ¾ 1 X d p d p µ µ ∴ rˆ · S~ = d3x a†(p, s ) u†(p, s )eipµx + b(p, s ) v†(p, s )e−ipµx 2(2π)3 2p 2p0 j j j j i,j=1 0 0   3 k ½ ¾ X σ 0 0 µ 0 µ k 0 0 −ipµx † 0 0 ipµx rˆ   a(p , si) u(p , si) e + b (p , si) v(p , si) e . (3.4) k k=1 0 σ

In fact, equation (3.4) can be split into terms relevant to its product of creation and anni- hilation operators for particle and anti-particle. The expanded term consists of:   Z ZZ X 3 3 0 σk 0 1 3 −i(p−p0)·x d p d p † 0 † k  0 3 d x e 0 a (p, sj)a(p , si)u (p, sj)ˆr u(p , si) 2(2π) 2p0 2p k i,j,k 0 0 σ

(3.5a)   Z ZZ X 3 3 0 σk 0 1 3 i(p+p0)·x d p d p † † 0 † k  0 3 d x e 0 a (p, sj)b (p , si)u (p, sj)ˆr v(p , si) 2(2π) 2p0 2p k i,j,k 0 0 σ

(3.5b)   Z ZZ X 3 3 0 σk 0 1 3 −i(p+p0)·x d p d p 0 † k  0 3 d x e 0 b(p, sj)a(p , si)v (p, sj)ˆr u(p , si) 2(2π) 2p0 2p k i,j,k 0 0 σ

(3.5c)   Z ZZ X 3 3 0 σk 0 1 3 −i(p0−p)·x d p d p † 0 † k  0 3 d x e 0 b(p, sj)b (p , si)v (p, sj)ˆr v(p , si) 2(2π) 2p0 2p k i,j,k 0 0 σ

(3.5d)

Consider the well-known Dirac delta function which is very useful for such simplifications: Z Z d3x e−i(p−p0)·x ≡ (2π)3δ3(p − p0) = d3x e−i(p−p0)·x. (3.6)

As such, equation (3.5a) can be evaluated in the following manner:   ZZ X 3 3 0 σk 0 1 d p d p 3 3 0 † 0 † k  0 3 0 (2π) δ (p − p ) a (p, sj)a(p , si)u (p, sj)r ˆ u(p , si) 2(2π) 2p0 2p k i,j,k 0 0 σ   Z X 3 σk 0 d p † 1 † k  = 0 a (p, sj)a(p, si) 0 u (p, sj)r ˆ u(p, si). (3.7) 2(2p ) 2p k i,j,k 0 σ

In addition, the positive energy solutions u(p, si) can be readily obtained by applying a Lorentz Sec. 3.1 Spin Projection for Single Particle with Arbitrary Momentum 41

boost to its rest frame solutions:

√ h ³η ´ ³η ´ i u(p, s ) = 2m cosh I + sinh pˆ · α~ b i 2 4×4 2 i √ = 2m D(p, η) bi (3.8) in which the rapidity tanh η = |v|/c = |p|/p0 is taken with respect to the velocity of the particle in the Lorentz-boosted frame. The matrices form for α~ and bi are written as:   0 σk α~ = (α1, α2, α3) , αk =   σk 0         c1 c2 0 0 b1 =   , b2 =   , b3 =   , b4 =   0 0 c1 c2     1 0 c1 =   , c2 =   . (3.9) 0 1

The Dirac spinor terms from equation (3.7) can be further simplified:     X σk 0 rˆ · σ~ 0 1 † k  m † †   0 u (p, sj)r ˆ u(p, si) = 0 bj D (p, η) D(p, η) bi 2p k p k 0 σ 0 rˆ · σ~

def aa = θji for i, j = 1, 2. (3.10)

The square matrices can be evaluated as   rˆ · σ~ 0 D†(p, η) D(p, η) 0 rˆ · σ~  †   ¡ η ¢ ¡ η ¢ ¡ η ¢ ¡ η ¢ cosh I2×2 sinh pˆ · σ~ rˆ · σ~ 0 cosh I2×2 sinh pˆ · σ~ =  2 2    2 2  ¡ η ¢ ¡ η ¢ ¡ η ¢ ¡ η ¢ sinh 2 pˆ · σ~ cosh 2 I2×2 0 rˆ · σ~ sinh 2 pˆ · σ~ cosh 2 I2×2   2 ¡ η ¢ rˆ · σ~ + 2 sinh (rˆ · pˆ) pˆ · σ~ sinh(η)(rˆ · pˆ) I2×2 =  2  2 ¡ η ¢ sinh(η)(rˆ · pˆ) I2×2 rˆ · σ~ + 2 sinh 2 (rˆ · pˆ) pˆ · σ~   X11 X12 ≡   (3.11) X21 X22 for which the property for the Pauli spin matrices has been extensively applied:

i j k σ σ = δij I2×2 + i²rijk σ . (3.12) Sec. 3.1 Spin Projection for Single Particle with Arbitrary Momentum 42

From equation (3.10), since only the product of positive energy solutions u(p, si) are con- sidered, it can be deduced that:     h i m X11 X12 ci m θaa = c† , 0†     = c† X c ji p0 j p0 j 11 i X21 X22 0 m h ³η ´ i = c† rˆ · σ~ + 2 sinh2 (rˆ · pˆ) pˆ · σ~ c p0 j 2 i · µ ¶ ¸ m m = c† rˆ · σ~ + 1 − (rˆ · pˆ) pˆ · σ~ c . (3.13) j p0 p0 i

Note that only the diagonal matrix inside the square matrix of equation (3.11) is considered.

† The product of u (p, si) and v(p, si), the positive and negative energy solutions respectively, can be calculated from equation (3.5b). The integral can be derived as follows:   ZZ X 3 3 0 σk 0 1 d p d p 3 3 0 † † 0 † k  0 3 0 (2π) δ (p + p ) a (p, sj)b (p , si)u (p, sj)r ˆ v(p , si) 2(2π) 2p0 2p k i,j,k 0 0 σ   Z X 3 σk 0 d p † † 1 † k  = 0 a (p, sj)b (−p, si) 0 u (p, sj)r ˆ v(−p, si). (3.14) 2(2p ) 2p k i,j,k 0 σ

The Dirac spinors can be simplified into the following equation:     rˆ · σ~ 0 rˆ · σ~ 0 1 †   m † †   0 u (p, sj) v(−p, si) = 0 bj D (p, η) D(−p, η) bi 2p 0 rˆ · σ~ p 0 rˆ · σ~

def ab = θji for j = 1, 2 and i = 3, 4. (3.15)

The square matrices are given by   rˆ · σ~ 0 D†(p, η) D(−p, η) 0 rˆ · σ~  †   ¡ η ¢ ¡ η ¢ ¡ η ¢ ¡ η ¢ cosh I2×2 sinh pˆ · σ~ rˆ · σ~ 0 cosh I2×2 − sinh pˆ · σ~ =  2 2   2 2  ¡ η ¢ ¡ η ¢ ¡ η ¢ ¡ η ¢ sinh 2 pˆ · σ~ cosh 2 I2×2 0 rˆ · σ~ − sinh 2 pˆ · σ~ cosh 2 I2×2  ¡ ¢  cosh(η)rˆ · σ~ − 2 sinh2 η (rˆ · pˆ) pˆ · σ~ i sinh(η)(pˆ × rˆ) · σ~ =  2  2 ¡ η ¢ i sinh(η)(pˆ × rˆ) · σ~ cosh(η)rˆ · σ~ − 2 sinh 2 (rˆ · pˆ) pˆ · σ~   Y11 Y12 =   (3.16) Y21 Y22 Sec. 3.1 Spin Projection for Single Particle with Arbitrary Momentum 43

ab The matrix element θji can be calculated as     h i m Y11 Y12 0 m θab = c† , 0†     = c† Y c ji p0 j p0 j 12 i Y21 Y22 ci |p| h i = i c† (pˆ × rˆ) · σ~ c (3.17) p0 j i

† Similarly, the product of u(p, si) and v (p, si), can be calculated from equation (3.5c) and the integral derived as follows:   ZZ X 3 3 0 σk 0 1 d p d p 3 3 0 0 † k  0 3 0 (2π) δ (p + p ) b(p, si)a(p , sj)v (p, sj)r ˆ u(p , si) 2(2π) 2p0 2p k i,j,k 0 0 σ   Z X 3 σk 0 d p 1 † k  = 0 b(−p, si)a(p, sj) 0 v (−p, sj)r ˆ u(p, si). (3.18) 2(2p ) 2p k i,j,k 0 σ

The Dirac spinors can be simplified into the following equation:     rˆ · σ~ 0 rˆ · σ~ 0 1 †   m † †   0 v (−p, sj) u(p, si) = 0 bj D (−p, η) D(p, η) bi 2p 0 rˆ · σ~ p 0 rˆ · σ~

def ba = θji for j = 3, 4 and i = 1, 2. (3.19)

The square matrices are given by   rˆ · σ~ 0 D†(−p, η) D(p, η) 0 rˆ · σ~  ¡ ¢  cosh(η)rˆ · σ~ − 2 sinh2 η (rˆ · pˆ) pˆ · σ~ i sinh(η)(pˆ × rˆ) · σ~ =  2  2 ¡ η ¢ i sinh(η)(pˆ × rˆ) · σ~ cosh(η)rˆ · σ~ − 2 sinh 2 (rˆ · pˆ) pˆ · σ~   Y11 Y12 =   Y21 Y22

ba Therefore the matrix element θji can be calculated as     h i m Y11 Y12 ci m θba = 0†, c†     = c† Y c ji p0 j p0 j 21 i Y21 Y22 0 |p| h i = i c† (pˆ × rˆ) · σ~ c ≡ θab . (3.20) p0 j i ji Sec. 3.1 Spin Projection for Single Particle with Arbitrary Momentum 44

ba ab It is noteworthy that the matrix elements θji and θji , defined for the creation/annihilation † † operators a (p, sj)b (−p, si) and b(−p, si)a(p, sj) respectively, are equal.

Last but not least, the term involving products of the negative energy solutions v(p, si) is considered:   ZZ 1 X4 d3p d3p0 rˆ · σ~ 0 (2π)3δ3(p − p0) b(p, s )b†(p0, s )v†(p, s ) v(p0, s ) 2(2π)3 2p 2p0 j i j i i,j=3 0 0 0 rˆ · σ~   Z X4 d3p rˆ · σ~ 0 = b(p, s )b†(p, s )v†(p, s )  v(p, s ). (3.21) 2(2p )2 j i j i i,j=3 0 0 rˆ · σ~

  rˆ · σ~ 0 1 †   def bb Let 0 v (p, sj) v(p, si) = θji for i, j = 1, 2 2p 0 rˆ · σ~     h i m X11 X12 0 = 0†, c†     p0 j X21 X22 ci m = c† X c p0 j 22 i · µ ¶ ¸ m m =⇒ θbb = θaa ≡ c† rˆ · σ~ + 1 − (rˆ · pˆ) pˆ · σ~ c . ji ji j p0 p0 i

† It is shown here that the coefficients of the creation/annihilation operators a (p, sj)a(p, si) and

† b(p, sj)b (p, si) are the same. Thus, the spin operator can be simplified into the following equation: Z ( X2 d3p h i rˆ · S~ = θaa a†(p, s )a(p, s ) + b(p, s )b†(p, s ) 2(2p0) ij i j i j i,j=1 ) h i ab † † + θij a (p, si)b (−p, sj) + b(−p, si)a(p, sj) . (3.22)

Notice that the matrix θaa is Hermitian since θaa = θaa†, while the matrix θab is anti-Hermitian because θab = −θab†. In addition, the helicity states are defined as the spin projection along the direction of the particle momentum (rˆ = pˆ), and the coefficients from equation (3.22) can be simplified as: · µ ¶ ¸ m m h i θaa = c† pˆ · σ~ + 1 − (pˆ · pˆ) pˆ · σ~ c = c† pˆ · σ~ c , (3.23a) ij i p0 p0 j i j |p| h i θab = i c† (pˆ × pˆ) · σ~ c = 0 . (3.23b) ij p0 j i Sec. 3.1 Spin Projection for Single Particle with Arbitrary Momentum 45

This implies that the spin operator for Dirac fields has no cross-terms involving the particle and anti-particle mode functions and it is given by Z ( ) X2 d3p h i pˆ · S~ = c† pˆ · σ~ c a†(p, s )a(p, s ) + b(p, s )b†(p, s ) . (3.24) 2(2p0) i j i j i j i,j=1

† † In fact, due to the presence of either both creation operators a (p, sj)b (−p, si) or both annihi-

lation operators b(−p, si)a(p, sj), it is expected that these operators would disappear through

aa normal ordering. As a result, only the matrix element θij can be viewed as a characterization of the internal angular momentum or the spin of the Dirac field. The normal-ordered Dirac spin operator (where :: denotes normal-ordering) is thus written as Z X2 d3p h i :: rˆ · S~ :: = θaa a†(p, s )a(p, s ) − b†(p, s )b(p, s ) . (3.25) 2(2p0) ij i j i j i,j=1

3.1.1 The Effect of Lorentz Transformation on the Dirac Spin Op- erator

The effect of Lorentz transformations on the spin operator is explored here in which:

¯ ® X ³ ´¯ ® ¯ (1/2) ¯ |p, sii −→ U(Λ) p, si = Dji W (Λ, p) pΛ, sj (3.26a) j ³ ´ :: rˆ · S~ ::−→ U(Λ) :: rˆ · S~ :: U †(Λ) (3.26b)

where pΛ is the spatial part of the Lorentz transformed momentum, Λp. The transformation properties of the creation/annihilation operators given in equations (2.31a, 2.31b, 2.31c, 2.31d,) are utilized and as a result, the spin operator given in equation (3.22) transforms as: Z ³ ´ X2 d3p h i U(Λ) :: rˆ · S~ :: U †(Λ) = U(Λ) θaa a†(p, s )a(p, s ) − b†(p, s )b(p, s ) U †(Λ) 2(2p0) ij i j i j i,j=1 Z · X d3p = θaa D (W )D∗ (W )a†(p , s )a(p , s ) 2(2p0) ij mi Λ,p nj Λ,p Λ m Λ n i,j,m,n ¸ ∗ † − Dmi(WΛ,p)Dnj(WΛ,p)b (pΛ, sm)b(pΛ, sn) Z · ¸ X d3p = Θaa a†(p , s )a(p , s ) − b†(p , s )b(p , s ) 2(2Λp)0 mn Λ m Λ n Λ m Λ n m,n (3.27) Sec. 3.1 Spin Projection for Single Particle with Arbitrary Momentum 46

in which the coefficients of their respective mode functions are given by

(Λp)0 X (Λp)0 £ ¤ Θaa def= θaa D (W ) D∗ (W ) ≡ D(W ) θaa D†(W ) . (3.28a) mn p0 ij mi Λ,p nj Λ,p p0 Λ,p Λ,p mn i,j

Making use of the Wigner matrix obtained from equation (2.25), and also the Wigner angle ³ ´ b σ~ Ωp from equation (2.24), the Wigner transformation matrix is given as J ≡ 2 : s · µ ¶¸ p0 i D(1/2)(W (Λ, p)) def= exp Ω ¦ σ~ (Λp)0 2 p s · µ ¶ µ ¶ ¸ p0 Ω Ω ³ ´ = cos p I + i sin p Ωb ¦ σ~ (3.29) (Λp)0 2 2×2 2 p

|p|ξ with Ωp ≡ θ + p0+m (nˆ × pˆ). Here a spatial rotation is determined by the axial vector θ while a Lorentz boost is represented by the boost rapidity ξ and boost axis nˆ. The magnitude of the

Wigner angle Ωp is given as: ¯ ¯ ¯ |p|ξ ¯ Ω = |Ω | = ¯θ + (nˆ × pˆ)¯ . (3.30) p p ¯ p0 + m ¯ ³ ´ ³ ´ Ωp Ωp In the absence of rotation (θ = 0), the algebraic expressions for cos 2 and sin 2 are given by (referring to equations (2.30a, 2.30b)):

µ ¶ ¡ ξ ¢ ¡ η ¢ ¡ ξ ¢ ¡ η ¢ Ωp cosh cosh + sinh sinh (nˆ ¦ pˆ) cos = £ 2 2 2 2 ¤ (3.31a) 2 1 + 1 cosh (ξ) cosh (η) + 1 sinh (ξ) sinh (η)(nˆ ¦ pˆ) 1/2 µ ¶ 2 2 ¡ ¢ 2 ¡ ¢ sinh ξ sinh η (nˆ × pˆ) Ωp b 2 2 sin Ωp = £ ¤ (3.31b) 2 1 1 1 1/2 2 + 2 cosh (ξ) cosh (η) + 2 sinh (ξ) sinh (η)(nˆ ¦ pˆ)

0 b in which cosh(η) = |p|/p and Ωp is the unit vector for nˆ × pˆ . Most importantly, the matrix given by θaa can be evaluated as: ½ µ ¶ ¾ m m m³ ´ θaa = rˆ + 1 − (rˆ · pˆ) pˆ · σ~ = rˆ + ² pˆ · σ~ (3.32) p0 p0 p0 r ³ ´ p0 aa in which ²r = m − 1 (rˆ · pˆ). Therefore, the coefficient Θmn expressed in equation (3.28a) can be evaluated as: · µ ¶ µ ¶ µ ¶¸ i m³ ´ i Θaa = c† exp Ω ¦ σ~ rˆ + ² pˆ · σ~ exp − Ω ¦ σ~ c mn m 2 p p0 r 2 p n · µ ¶ µ ¶ ¸ m Ω Ω ³ ´ ³ ´ = c† cos p I + i sin p Ωb ¦ σ~ rˆ + ² pˆ · σ~ p0 m 2 2×2 2 p r · µ ¶ µ ¶ ¸ Ω Ω ³ ´ cos p I − i sin p Ωb ¦ σ~ c 2 2×2 2 p n Sec. 3.2 Single Particle State with Fixed Momentum 47

·½ m h³ ´ i h i ∴ Θaa = c† cos Ω Ωb × (rˆ + ² pˆ) × Ωb − sin Ω Ωb × (rˆ + ² pˆ) mn p0 m p p r p p p r ¾ ¸ h ¡ ¢i b b + Ωp · rˆ + ²rpˆ Ωp ¦ σ~ cn m h i ≡ c† r · σ~ c (3.33a) p0 m Λ n h³ ´ i h i h ¡ ¢i b b b b b ⇒ rΛ = cos Ωp Ωp × (rˆ + ²rpˆ) × Ωp − sin Ωp Ωp × (rˆ + ²rpˆ) + Ωp · rˆ + ²rpˆ Ωp

(3.33b)

Note that the eigenvalues of the matrix [rΛ · σ~ ] are given by

° °2 ¯³ ´ ¯2 ¯ ¯2 ° ° 2 ¯ b b ¯ 2 ¯b ¯ °rΛ · σ~ ° = cos Ωp ¯ Ωp × (rˆ + ²rpˆ) × Ωp¯ + sin Ωp ¯Ωp × (rˆ + ²rpˆ)¯ h ¡ ¢i2 ¯ ¯2 b ¯b ¯ + Ωp · rˆ + ²rpˆ ¯Ωp¯

¯¡ ¢¯2 °¡ ¢ °2 = ¯ rˆ + ²rpˆ ¯ ≡ ° rˆ + ²rpˆ · σ~ ° . (3.33c) ¡ ¢ Both matrices rΛ · σ~ and rˆ + ²rpˆ · σ~ share the same pair of eigenvalues, and this can be attributed to the fact that the Wigner transformation matrix D(1/2)(W (Λ, p)) with proper normalization condition belongs to the SU(2) group. The interpretation one can derive from

the vector defined in equation (3.33b) is a rotation of the vector (rˆ + ²rpˆ) about the axis Ωp by a clockwise angle of Ωp ≡ |Ωp|.

Therefore, it is possible to evaluate the expectation value of the spin operator under different inertial frames since the spin operator has been shown to transform as given in equation (3.27) under a Lorentz transformation Λ. The next section shows that the Dirac spin operator can be used as a spin observable under suitable conditions.

3.2 Single Particle State with Fixed Momentum

1 Suppose the spin- 2 particle possesses a unique momentum, p¯, and a specific spin state, sk,

and the quantum state of the particle is given by |p¯, ski. The spin operator acting on this state can be considered as: Z ¯ ® X d3p h i :: rˆ · S~ :: ¯p¯, s = θaa a†(p, s )a(p, s ) − b†(p, s )b(p, s ) a†(p¯, s )|0i k 2(2p0) ij i j i j k i,j Z X d3p m h³ ´ i = c† rˆ + ² pˆ · σ~ c a†(p, s ) 2p0 δ(3)(p − p¯) δ |0i 2(2p0) p0 i r j i jk i,j Sec. 3.2 Single Particle State with Fixed Momentum 48

¯ ® m X h¡ ¢ i ¯ ® ∴ :: rˆ · S~ :: ¯p¯, s = c† rˆ +² ¯ p¯ˆ ¦ σ~ c ¯p¯, s (3.34) k 2¯p0 i r k i i in which µ ¶ p¯0 ¡ ¢ ²¯ = − 1 rˆ · p¯ˆ . (3.35) r m

and c1 = (1, 0) and c2 = (0, 1) . Note that equation (3.34) refers to the effect of the Lorentz

transformation on the spin vectors. Suppose the vector rˆ+¯²r p¯ˆ is parameterized in the following manner:   sin ψ cos φ ¯ ¯   ¯ ¯   rˆ +² ¯r p¯ˆ = rˆ +² ¯r p¯ˆ  sin ψ sin φ  . (3.36)   cos ψ

The eigenvalues of the Pauli matrix in the equation (3.34) for a unit vector are ±1 and its associated eigenvectors are   ψ cos 2 λ1 = +1 , vˆ1 =   (3.37a) ψ iφ sin 2 e   ψ −iφ − sin 2 e λ2 = −1 , vˆ2 =   (3.37b) ψ cos 2

The Pauli matrix from equation (3.34) can be diagonalized in the following manner:

   −1 ψ ψ −iφ ψ ψ −iφ ¡ ¢ ¯ ¯ cos 2 − sin 2 e 1 0 cos 2 − sin 2 e rˆ +² ¯r p¯ˆ ¦ σ~ = ¯rˆ +² ¯r p¯ˆ¯    ψ iφ ψ ψ iφ ψ sin 2 e cos 2 0 −1 sin 2 e cos 2 ¯ ¯ −1 = ¯rˆ +² ¯r p¯ˆ¯ PDP (3.37c)

where     ψ ψ −iφ cos 2 − sin 2 e 1 0 P =   , D =   = σz . (3.37d) ψ iφ ψ sin 2 e cos 2 0 −1

Making use of the relations given in equations (3.37c, 3.37d) and substitute them into equation (3.34), it is possible to obtain the following equations:

¯ ® m X h¡ ¢ i ¯ ® :: rˆ · S~ :: ¯p¯, s = c† rˆ +² ¯ p¯ˆ ¦ σ~ c ¯p¯, s k 2¯p0 i r k i i ¯ ¯ X h ii ¯ m −1 ® = ¯rˆ +² ¯r p¯ˆ¯ PDP ¯p¯, si 2¯p0 k i Sec. 3.2 Single Particle State with Fixed Momentum 49

X h ik ¯ ® ¯ ¯ X h ii X h ik ¯ ® ~ m −1 ⇒ :: rˆ · S :: P ¯p¯, sk = ¯rˆ +² ¯r p¯ˆ¯ PDP P ¯p¯, si m 2¯p0 k m k i k m ¯ ¯ X h ii h il ¯ ® = ¯rˆ +² ¯r p¯ˆ¯ P D ¯p¯, si 2¯p0 l m i,l ¯ ® ¯ ¯ X h il ¯ ® ~ ¯ 0 0 m ¯ ¯ ¯ 0 0 ∴ :: rˆ · S :: p¯ , sm = rˆ +² ¯r p¯ˆ D p¯ , sl 2¯p0 m ¯ l ¯ ¯ ® 1 ¯m m ¯ ¯ ® :: rˆ · S~ :: ¯p¯0, ±s0 = ± ¯ rˆ + ²¯ p¯ˆ¯ ¯p¯0, ±s0 (3.38a) 2 ¯p¯0 p¯0 r ¯ ¯ ® ¯ 0 0 in which the substitution involving the new spin basis p¯ , sm has been made:

¯ ® X h ik ¯ ® ¯ 0 0 ¯ p¯ , sm = P p¯, sk m k · µ ¶¸ X i ¡ ¢ k ¯ ® = exp − ψ wˆ · σ~ ¯p¯, s (3.38b) 2 k k m and  ¡ ¢  cos φ − π  2  def  ¡ ¢  wˆ =  sin φ − π  . (3.38c)  2  0

One can conclude from equation (3.38a) that the Dirac spin operator can be used to compute the dichotomous spin projections on any arbitrary fixed momentum states, provided the substi- tutions as mentioned in equations (3.38b, 3.38c) have been made. In fact, equation (3.38b) can be interpreted as a three-dimensional rotation of angle, ψ, in the counter-clockwise direction

about the axis, wˆ . The resultant vector, rˆ +² ¯rp¯ˆ, is thus rotated to be parallel to the z-axis.

1 This is related to the choice of the Pauli matrices as the spin- 2 representations of the SU(2) 1 1 algebra, in which the spin up (ms = 2 ) and spin down (ms = − 2 ) correspond to the Bloch

vectors pointing along the zˆ and −zˆ directions respectively. The vector (rˆ+¯²rp¯ˆ) is now rotated to the z-axis.

In addition, if the measurement axis is taken as along the particle momentum, (rˆ = p¯ˆ), equation (3.38a) simplifies to ¯ µ ¶ ¯ ¯ ® 1 ¯m m ¯ ¯ ® :: pˆ · S~ :: ¯p¯0, ±s0 = ± ¯ pˆ + 1 − (pˆ · pˆ) pˆ¯ ¯p¯0, ±s0 2 ¯p¯0 p¯0 ¯ 1 ¯ ® = ± ¯p¯0, ±s0 . (3.39) 2 Sec. 3.2 Single Particle State with Fixed Momentum 50

Figure 3.1: Rotation of the Bloch vector rˆ +² ¯r p¯ˆ about the axis wˆ = (sin φ, − cos φ, 0) by a counter-clockwise angle ψ.

This illustrates that the eigenvalues and their respective eigenstates of the spin operator, rˆ · S~ , can be determined for for any arbitrary momentum p¯ by performing the necessary rotations as given in equations (3.38b, 3.38c). If the measurement axis is aligned with the momentum ˆ 1 (rˆ = p¯) one can derive the helicity states in which the helicities are given as ± 2 . However, it is also possible to define a dichotomous spin operator by normalization so that the measurement results are automatically ±1: ¯ ® p¯0 ¯ ® ~ ¯¯ def ¯ ¯ ˆ ~ ¯¯ :: λr,p¯ · S :: p, sk = ¯ ¯ :: r · S :: p, sk m rˆ +² ¯r p¯ˆ 0 ¯ ® p¯ ~ ¯ = q ¡ ¢ :: rˆ · S :: p¯, sk (3.40) m2 + rˆ · p¯ 2

in which the vector λr,p¯ depends on the measurement axis rˆ and the particle momentum p¯.

The expectation value of the Dirac spin operator can be evaluated for a single particle

quantum state with its fixed momentum p¯ and also spin sk : ­ ® ­ ¯ ¯ ® ~ def 1 ~ :: rˆ · S :: = p¯, sk¯ :: rˆ · S :: ¯p¯, sk hp¯, sk|p¯, ski m X h¡ ¢ i ­ ¯ ® = c† rˆ +² ¯ p¯ˆ · σ~ c p¯, s ¯p¯, s (2¯p0)2 i r k k i i m X h¡ ¢ ii = rˆ +² ¯r p¯ˆ · σ~ δki 2¯p0 k i m h¡ ¢ ik = rˆ +² ¯r p¯ˆ · σ~ 2¯p0 k (−1)k−1 m³ ´ = rˆ +² ¯r p¯ˆ (3.41) 2 p¯0 z Sec. 3.2 Single Particle State with Fixed Momentum 51

The spin measurement for a single particle with fixed momentum p¯ and spin sk can be described as a spin projection along the standard z-coordinates. This is shown to be consistent with the helicity basis obtained from equation (3.38b) in the rest frame (p = 0), in which the Bloch vector for spin s is rotated to the z-axis as given by matrix P. Note that the magnitude of the ³ ´ m ˆ 0 vector p¯0 rˆ +² ¯r p¯ changes with the particle energy p and it is a unit vector only either in the particle rest frame or in the helicity direction (rˆ = pˆ).

3.2.1 Bell Correlations for Spin Singlet States

1 Assume that there exists a Stern-Gerlach type experiment which is able to create two spin- 2 1 relativistic particles in a singlet state. In addition, one can define the state vectors |p, ± 2 i from the helicity basis, where the total spin and total momenta are zero. As an ansatz, let the singlet state be defined as (assuming they are well localized around momenta p): µ¯ À ¯ À ¯ À ¯ À ¶ ¯ ¯ ¯ ¯ +− 1 ¯ 1 ¯ 1 ¯ 1 ¯ 1 |ψ i12 = √ ¯p, ¯−p, − − ¯p, − ¯−p, (3.42) 2 2 1 2 2 2 1 2 2 in which the subscripts 1 and 2 denote the particles as two distinct, well-defined wave packets which can also be described as plane waves in the external space. In addition, the binary operators based on the Dirac spin operator can be defined as: ˆ def aˆ · Sp e def b · Sp ae = q¡ ¢ , b = q¡ ¢ (3.43a) 1 − β2 + β2(aˆ · pˆ)2 1 − β2 + β2(bˆ · pˆ)2

in which both operators ae and eb give rise to unit eigenvalues, ±1.

The EPR-Bohm-Bell observable or simply Bell observable from the CHSH inequality [3, 21] is given by ³ ´ 0 0 0 C aˆ, aˆ0, bˆ, bˆ = hψ|ae ⊗ eb|ψi + hψ|ae ⊗ eb |ψi + hψ|ae0 ⊗ eb|ψi − hψ|ae0 ⊗ eb |ψi (3.44)

0 in which ae, ae0, eb, eb are the binary operators that would yield unit eigenvalues ±1. Since quan- tum mechanics dictate that for spin singlet states, the Bell observable has to obey: ¯ ³ ´¯ ¯ 0 ¯ √ 2 < ¯ C aˆ, aˆ0, bˆ, bˆ ¯ ≤ 2 2 (3.45)

whereby the number 2 indicates the maximum value for equation (3.44) using local realistic theories [2]. Sec. 3.2 Single Particle State with Fixed Momentum 52

In non-relativistic quantum mechanics (β = 0), maximal violation of the Bell inequality 0 occurs when aˆ = ( √1 , √1 , 0) , aˆ0 = (− √1 , √1 , 0), bˆ = (0, 1, 0) and bˆ = (1, 0, 0). Making 2 2 2 2 use of the spin measurements from equation (3.41), the expectation value of the relativistic EPR-Bohm-Bell operator is given by D ¯ ~ ˆ ~ ¯ E 2 ˆ 2 ˆ ¯ aˆ · Sp b · Sp ¯ (1 − β )aˆ · b + β (aˆ · pˆ)(b · pˆ) ψ+−¯° ° ⊗ ° °¯ψ+− = −q q (3.46a) ° ~ ° °ˆ ~ ° 12 2 °aˆ · Sp° °b · Sp° 1 − β2 + β2 (aˆ · pˆ) 1 − β2 + β2(bˆ · pˆ)2 " # " ³ ´ 0 0 0 0 0 aˆ · bˆ aˆ · bˆ aˆ · bˆ aˆ · bˆ (aˆ · pˆ) (bˆ · pˆ) ∴ C aˆ, aˆ0, bˆ, bˆ = −(1 − β2) + + − − β2 λaλb λaλb0 λa0 λb λa0 λb0 λa λb 0 ¡ ¢ ¡ ¢ 0 # (aˆ · pˆ) (bˆ · pˆ) aˆ0 · pˆ (bˆ · pˆ) aˆ0 · pˆ (bˆ · pˆ) + + − . (3.46b) λa λb0 λa0 λb λa0 λb0 in which r³ ´ 0 2 2 2 0 ˆ ˆ λr = 1 − β + β (rˆ · pˆ) , rˆ = aˆ, aˆ , b, b . (3.46c)

Note that the particle momentum is taken as pˆ = (cos φ, sin φ, 0), which is co-planar to 0 the measurement axes aˆ, aˆ0, bˆ, bˆ . The Bell observable from the Dirac spin vector can thus be √ plotted into the following Figures 3.2 and 3.3. Note that there is a maximal violation of −2 2 in equation (3.46b) for the non-relativistic scenario (β = 0), but the Bell observable increases to -2 in the ultra-relativistic limit (β → 1).

Figure 3.2: Bell Observable (3.46b) for Dirac spin operator where pˆ = (cos φ, sin φ, 0).

These results have been observed in several papers [10, 7, 9, 20, 6] in which the Bell cor- relation decreases to the classical value 2 in the ultra-relativistic limit (β → 1). According to Sec. 3.2 Single Particle State with Fixed Momentum 53

Figure 3.3: Bell Observable for Dirac spin operator where φ = 0 or pˆ = (1, 0, 0).

Terashima [10], the spin projection along the particle momentum increases in magnitude for increasing relative velocity β. At the ultra-relativistic limit, the spin vector becomes either completely parallel or anti-parallel to the momentum.

Chapter Summary

In conclusion, the spin state of a relativistic particle can be evaluated using the Dirac spin operator, which has been derived from the field conserved currents. It is observed that the field theoretic approach for deriving the spin operator is consistent with the group representations in the Poincar´egroup, and the same Bell violation is observed as shown in Figure 3.2 and Figure 3.3.

The next chapter is devoted to exploring the Lie algebra of the Poincar´egroup, in addition to explaining the connection between the Dirac field operators and the Poincar´egenerators. The Poincar´egenerators are used to construct the Pauli-Lubanski vector and it is shown to be a relativistic extension of the standard spin representation in non-relativistic quantum theory. Chapter 4

Spin in Poincar´eGroup

The spin angular momentum is inherent to the Poincar´egroup since it is possible to define the spin “vector” as a linear combination of the Poincar´egenerators. In this chapter the properties of this spin “vector” are examined and its measurements are also compared to the Dirac spin operator, which has been obtained previously from field-theoretic considerations. Lastly the Dirac spin operator can be shown to be equivalent to the Wigner spin operator up to a proportionality factor.

4.1 Defining Spin Observables in Relativistic Settings

4.1.1 Spin Projection for Dirac Spinors

The spin or internal angular momentum of the Dirac particles was first formulated by Pauli [22] in 1927 by adding an interaction term to the Schr¨odingerequation. The interaction Hamiltonian for a non-relativistic electron in a magnetic field was written as:

e~ H(spin) = − σ~ · B (4.1) 2mc

where e and ~ represent the electron charge and reduced Planck’s constant respectively. Subse-

µ quently Dirac wrote down the Dirac equation iγ ∂µΨ = mΨ in his seminal paper [23] in which the solution Ψ was a multi-component wave-function. In the non-relativistic approximation, it was shown by others, for example Sakurai [24], that the positive energy solutions of the Dirac equation also satisfied the Pauli spin equation. The spin projection is defined as the matrix in

54 Sec. 4.1 Defining Spin Observables in Relativistic Settings 55

which its eigenvectors are the Dirac spinors, and is given as [14]:   02×2 I2×2 1 γ5 def= γ0γ1γ2γ3 = −i   ≡ ² γµγνγλγσ (4.2a) 4! µνλσ I2×2 02×2

γ5γµ + γµγ5 = 0

5 µ iγ (γµw ) u(p, ±s) = ± u(p, ±s) (4.2b)

5 µ iγ (γµw ) v(p, ±s) = ± v(p, ±s) . (4.2c)

µ Here ²µνλσ denotes the Levi-Civita rank-4 tensor where ²0123 = +1. The four-vector w is defined as a space-like vector whereby

µ 0 2 2 µ w wµ = (w ) − |w| = −1 , w pµ = 0 . (4.3)

Note that in the rest frame (p = 0), the four-vector wµ is given by w = (0, sˆ) in which sˆ represents the direction of wµ for the Dirac particle at rest. It is postulated that the construction

5 µ of such a matrix iγ (γµw ) allows one to define the spin “vector” for the Dirac spinors.

5 µ µ It is noted that the spin projector iγ (w γµ) commutes with γµp , and it is shown here: h i ³ ´ µ 5 ν 5 µ ν ν µ γµp , iγ (γνw ) = −iγ γµp γνw + γνw γµp −

5 µ = −γ (2pµw ) = 0 (4.4)

by making use of the relation γµγν + γνγµ = 2ηµν. This implies that it is possible to construct

µ 5 µ simultaneous eigenspinor for the 4 × 4-matrices γ pµ and iγ (w γµ) with eigenvalues m and

5 µ ±1 respectively. The eigenvalues λ of the spin projector iγ (w γµ) are given by: ° ° ° 5 µ ° λ = °iγ (w γµ) ° = ±1 (4.5)

The corresponding eigenvectors for the eigenvalue λ1 = −1:     − 3  w   (w − 1)       3   +  1  −(w + 1)  1  w  e11 = 0   , e12 = 0   (4.6) w   w  0   0   w      w0 0 Sec. 4.1 Defining Spin Observables in Relativistic Settings 56

and similarly the eigenvectors for the eigenvalue λ2 = 1     0  w   0         0  1  0  1  w  e21 = 0   , e22 = 0   (4.7) w  3  w  −   (w − 1)   w      w+ −(w3 + 1) in which w± = w1 ± iw2 and w1, w2 and w3 are the spatial components of the four-vector wµ. Note that the spin projector can also be written as   w · σ~ −w0I 5 µ 2×2 iγ (w γµ) =   . (4.8) 0 w I2×2 −w · σ~

In the momentum rest frame (p = 0), it is obvious that the spin projector reverts to the matrix given in equation (1.66):   sˆ · σ~ 0 5 µ  2×2  ~ 0 iγ (w γµ)(p=0) = ≡ 2sˆ · Ξγ . (4.9) 02×2 −sˆ · σ~ where sˆ represents the direction of wµ for the Dirac particle at rest. Based on the requirements imposed by the equations (4.2b, 4.2c, 4.3, 4.9), the components of the four-vector wµ are given by µ ¶ |p| p0 w = , pˆ . (4.10) m m

Making use of the explicit construction of the four vector wµ from equation (4.10), a useful

5 µ relation involving iγ (w γµ) can be derived in which:       0 0 p pˆ · σ~ − |p| I pˆ · σ~ 0 p I − |p| pˆ · σ~ 5 µ m m 2×2 m 2×2 m iγ (w γµ) =   =     |p| p0 |p| p0 m I2×2 − m pˆ · σ~ 0 pˆ · σ~ m pˆ · σ~ − m I2×2 µ ¶ p0 pˆ = 2pˆ · Ξ~ γ0 − · γ~ m m pµγ = 2pˆ · Ξ~ µ . (4.11) m

µ p γµ Note that Dirac spinors u(p, ±s) and v(p, ±s) are eigenspinors of the matrix m since they are the solutions to the Dirac equation:

pµγ (pµγ − m) u(p, ±s) = 0 ⇒ µ u(p, ±s) = u(p, ±s) (4.12a) µ m pµγ (pµγ + m) v(p, ±s) = 0 ⇒ µ v(p, ±s) = −v(p, ±s) (4.12b) µ m Sec. 4.1 Defining Spin Observables in Relativistic Settings 57

This implies that the Dirac spinors are simultaneous momentum eigenstates to the matrices

µ 5 µ γµp and iγ (w γµ) for a free particle. They are also characterized by the positive definite mass (m > 0) and the helicity (s = ±|s|). Note that an additonal γ0 matrix is present in equation (4.9) since the negative sign from the eigenvalue of v(p, s) in equation (4.12b) reverses the helicity. Indeed, the three-dimensional vector of spin angular momentum can be written down as given in equation (1.65c), Z pˆ · S~ = d3x Ψ†(x) pˆ · Ξ~ Ψ(x)

2 Z 3 ½ ¾ 1 X d p µ µ Ψ(ˆ x, t) = a(p, s ) u(p, s ) e−ipµx + b†(p, s ) v(p, s ) eipµx (2π)3/2 2p i i i i i=1 0 1 for the helicity direction (rˆ = pˆ). It is possible to obtain the spin- 2 representation in non-

relativistic quantum mechanics by looking at the positive-energy Dirac spinor u(p, si). In fact, making use of equation (4.2b) and letting p = 0, one can show that

5 µ iγ (γµw ) u(0, ±s) = ± u(0, ±s) (4.13)       sˆ · σ~ 02×2 ci ci ⇒     = ±   02×2 −sˆ · σ~ 0 0 µ ¶ σ~ 1 ⇒ sˆ · c0 = ± c0 (4.14) 2 i 2 i 0 where ci forms the eigenvector for the Pauli matrices sˆ · σ~ defined along the vector direction sˆ. This exemplifies the Pauli spin formalism clearly and most importantly, this is a natural exten-

5 µ sion for the manifestly covariant spin operator iγ (γµw ) in the rest frame for the momentum. To see that it is covariant under a Lorentz transformation Λ, one can show that:

5 µ −1 0 −1 1 −1 2 −1 3 α β ρ σ ¡ −1 µ α µ¢ iγ (γµw ) → i(Λ ) α(Λ ) β(Λ ) ρ(Λ ) σγ γ γ γ γµ(Λ ) α Λ µw h ih ih ih i ³ ´ b 0 b−1 b 1 b−1 b 2 b−1 b 3 b−1 µ = i S(Λ)γ S (Λ) S(Λ)γ S (Λ) S(Λ)γ S (Λ) S(Λ)γ S (Λ) γµw £ ¤ ³ ´ b 0 1 2 3 b−1 µ = iS(Λ) γ γ γ γ S (Λ) γµw ³ ´ b 5 b−1 µ = iS(Λ)γ S (Λ) γµw (4.15)

in which the matrix γ5 transforms according to the principle of form invariance, as given by

µ b µ b−1 −1 µ α µ µ ν γ → S(Λ)γ S (Λ) = (Λ ) αγ , x → Λ νx .

It should be noted that the covariant spin operator does not depend on the external space- time coordinates but is an observable for the internal space, which in this case is the particle Sec. 4.1 Defining Spin Observables in Relativistic Settings 58

spin. It is noted that the four-vector wµ is relevant to the classification of the Poincar´egroup, in which its generator Wcµ is shown to be a covariant generalization of the spin s. This will be covered in the subsequent section.

4.1.2 Lie Algebra of the Poincar´eGroup

The four-vector wµ defined by equation (4.3) is a relativistic extension of the three-dimensional spin in the rest frame, even though it has been defined in the context of finding the eigenspinors

5 µ for iγ (w γµ). This has been quoted in several papers [6, 7, 9, 10, 20] in which the authors assumed the existence of this relativistic spin vector. The relativistic spin can be written in the form of the Pauli-Lubanski-Bargmann (also known as simply Pauli-Lubanski) vector [5]:

1 ³ ´ Wcν def= ²αβµνPb Mc = Wc0 , Wc (4.16a) 2 α βµ ³ ´ = Pb · Jb , Pb0 Jb − Pb × Kc (4.16b) Ã ! b b def 1 c c0 P SP = W − W m Pb0 + m ³ ´ Ã b b b ! 1 P · J P = Pb0 Jb − Pb × Kc − (4.16c) m Pb0 + m

where Pbα, Mcβµ are the four-momenta and Lorentz generators respectively in the Poincar´egroup. b b i b 1 cjk b i c0i In addition, the rotation Ji and boost generators K are given by Ji = 2 ²ijkM and K = M . There are some commutation relations for the Poincar´egenerators that one can quote from equations (1.47a, 1.47b, 1.47c). The commutation relations for the Poincar´egenerators are given below h i h i b b bk b b b0 Ji, Pj = i²ijkP , Ki, Pj = iδijP − − h i h i h i ˆ ˆ0 ˆ ˆ0 ˆ ˆ Ji, P = Pi, P = Pi, Pj = 0 . − − −

One can derive the following commutation relations for the Pauli-Lubanski vector: h i Wcν, Pbµ = 0 (4.17a) − h i ³ ´ Mcαβ, Wcµ = i ηβµWcα − ηαµWcβ (4.17b) − h i cµ cν µναβ c b W , W = −i² WαPβ (4.17c) − Sec. 4.1 Defining Spin Observables in Relativistic Settings 59

Since the operator Wcµ commutes with the four-momentum operator Pbν from equation (4.17a), most notably the Hamiltonian Pb0 = Hb, a linear combination of Wcµ is invariant under transla- tion. In addition, one can also show that

c bµ c0 b0 c b WµP = W P − W · P ³ ´ ³ ´ = Pb · Jb Pb0 − Pb0Jb − Pb × Kc · Pb = 0 . (4.18)

Here the norm of the Pauli-Lubanski vector, is given by

1 Wc Wcν = Mc Mcµβ Pb Pbα − Mc Mcβµ PbαPb (4.19) ν 2 µβ α αµ β

which is actually a Casimir invariant of the Poincar´ealgebra. This can be proven by taking c cν commutator between WνW and the generators in the Poincar´egroup which all vanish.

b If one uses the spin vector SP furnished by equation (4.16c), it is possible to derive the b relativistic spin generator in the Poincar´egroup. In fact, the spin vector SP can be obtained by defining a pure Lorentz boost L−1(p) on the Pauli-Lubanski vector Wcν such that:

1 £ ¤ Sbi = L−1(p) i Wcµ (4.20a) P m µ £ ¤ £ ¤ Pbj L−1(p) 0 = L−1(p) j = − (4.20b) j 0 m bj bk £ −1 ¤j P P L (p) = δjk + (4.20c) k m(Pb0 + m)

in which the particle momentum changes to the rest frame due to the Lorentz boost:

L−1(p)(p0, p) = (m, 0) . (4.20d)

b bj It is noted here that the spin vector SP allows one to define a triplet of spin operators SP for j = 1, 2, 3 such that they satisfy the three-vector commutation relation: " Ã !# h i bj bi bj bi 1 cj c0 P J , SP = J , W − W − m Pb0 + m − 1 h i 1 h i = Jbi , Wcj − ¡ ¢ Jbi , Wc0Pbj m − m Pb0 + m − c0 b i ijk c ijk W Pk ijk bk = − ² Wk + i² ¡ ¢ = i² SP (4.21) m m Pb0 + m Sec. 4.1 Defining Spin Observables in Relativistic Settings 60

and also obey the spin commutation relation: " Ã ! Ã !# h i bi bj bi bj 1 ci c0 P 1 cj c0 P SP , SP = W − W , W − W − m Pb0 + m m Pb0 + m ( " # " − # ) 1 h i Wc0Pbj Wc0Pbi = Wci , Wcj − Wci , − , Wcj m2 − Pb0 + m Pb0 + m ( − − ) bj bi 1 ijαβ c b i0αβ c b P P 0jαβ c b = −i² WαPβ + i² WαPβ + i ² WαPβ m2 Pb0 + m Pb0 + m ( ) h i h i c b 1 ijk c b ijk c b imn bj bi jmn WmPn = −i −² WkP0 + ² W0Pk + i −² P + P ² m2 Pb0 + m " # ck ijk 1 ck mW bk ijk bk = i² mW − P ≡ i² SP . (4.22) m2 Pb0 + m

b It is important that the spin operator SP must satisfy the spin commutation relation because it defines the spin transformation in the internal space of a quantum particle. This operator b SP has been quoted in several sources like Bogolubov [5], Terno [6], in which the spin operator is taken as the Lorentz-transformed Pauli-Lubanski vector Wcµ along the −p direction. This b implies that the triple of operators SP reduces in the rest frame to the non-relativistic angular c b momentum generator Wrest/m = J.

However, the papers by Czachor [7, 8], Doyeol [9], Terashima [10], quoted the spin operator b c b0 Srel = W /P as the relativistic spin operator defined with respect to the Pauli-Lubanski vector Wcµ. However, the spin operator does not obey the spin commutation relation, which can be shown as follows: h i h i ci b0 cj b0 −i ijαβ c b −i ijk c b ijk c b W /P , W /P = ² WαPβ = −² WkP0 + ² W0Pk − (Pb0)2 (Pb0)2 ¡ ¢ ijk c b ijk 1 b b b = i² Wk/P0 − i² P · J Pk (4.23) (Pb0)2

It is obvious that the commutator between the spin operators does not close up to the spin b commutation relation since there is a non-trivial term involving Pk on the right hand side of equation (4.23). This implies that the choice for the spin operator with Wc/Pb0 is invalid, even b b if it reverts to the similar non-relativistic expression J as SP . Sec. 4.2 Unitary Transformations on the Dirac Spinors 61

4.1.3 Dirac Spin Operator and Pauli-Lubanski Vector

It is noted that the space-like vector wµ defined by equation (4.3) allows one to define the spin

5 µ projection iγ (γµw ) for Dirac spinors. In the Dirac spinor representation (1/2, 0)⊕(0, 1/2), it

5 µ is shown that the Dirac spinors are the helicity bases for the spin projection iγ (γµw ), by way of equations (4.2b, 4.2c). To achieve this, first one has to choose the covariant Lorentz generators c Mµν by referring to [8, 5] for the momentum representation of the Poincar´egenerators:

Pb0 = γ0p · γ~ + mγ0 , Pbi = pi (4.24a) ∂ Jb = −ip × + ~s (4.24b) ∂p p ∂ p × ~s Kc = −ip0 − p (4.24c) ∂p p0 + m and the Pauli-Lubanski vector is shown to be

c0 b b W ≡ P · J = p · ~sp (4.24d) h i i Wc ≡ Pb0 Jb − Pb × Kc = γ0p · γ~ + mγ0 ~s − γ0p × γ~ (4.24e) p 2

Here the spin vector ~sp denotes the finite-dimensional angular momentum matrices correspond-

(jn) 1 ing to the (2jn +1)-dimensional representation D of the rotation group. For spin- 2 represen-

tation, the spin vector ~sp defined for bispinors is given by the components of the spin generator:   σk 0 k k 1   sp = Ξ = . (4.25) 2 0 σk

The explicit construction of the field-theoretic observables can be derived from equation (1.65c): Z n o Sk = d3x Ψ† Ξk Ψ (4.26)

and this has been explored in chapter 3. Therefore one can conclude that the Dirac spin operator is an extension of spin generators from the Poincar´egroup in the bispinor space. It is observed that if one has to define a spin operator that satisfies the spin commutation relation, b it is possible to determine the generator SP from the Pauli-Lubanski vector such that à ! b b 1 c c0 P SP = W − W . (4.27) m Pb0 + m b The relativistic spin operator obtained through the spin generator SP can be used to define a triplet of spin operators for the Dirac spinors that would satisfy the spin commutation relation. Sec. 4.2 Unitary Transformations on the Dirac Spinors 62

4.2 Unitary Transformations on the Dirac Spinors

The expectation value for the spin operator given by equation (3.41) indicates that the spin

measurement is dependent on the vector projection of (rˆ +² ¯r p¯ˆ) onto the z-axis. This is a direct consequence of taking the Pauli matrices the SU(2) generators , where one of them σ3 is taken as a diagonal matrix with diagonal elements ±1. This is a convenient choice for the

measurement since the spin operator gives the magnetic quantum number ±ms for any spin-s quantum state in any Stern-Gerlach type experiments. However, one can ask for another set

of basis vectors instead of the original basis, {ci = (1, 0) , (0, 1)} as given in equation (3.9), such that the helicity is defined for another axis other than the particle momentum. It turns out that this is possible by re-defining the spins of the Dirac spinors by referring to the SU(2)

1 covering group for spin- 2 representation.

The Dirac spinors are eigen-solutions to the Dirac equation, in which

¡ µ¢ γµp u(p, si) = m u(p, si) i = 1, 2 (4.28a)

¡ µ¢ γµp v(p, si) = −m v(p, si) i = 1, 2 . (4.28b)

µ Note that the matrix (γµp ) has degenerate eigenvalues of ±m, and this implies that the Dirac spinors for each of the eigenvalue is not uniquely determined. Thus, one can subject the bispinors up to an unitary transformation within the eigenspaces of u(p, sj) or v(p, sj), in which X 0 u (p, si) = Uij u(p, sj) (4.29a) j X 0 v (p, si) = Vij v(p, sj) (4.29b) j

for U, V ∈ SU(2). It is obvious that the new set of Dirac spinors given in equations (4.29a, 4.29b) also satisfy the Dirac equation. It is important to observe that the transformed bispinors

0 0 5 µ u (p, si) and v (p, si) are no longer eigenspinors of the spin projection iγ (γµw ). Since the matrices U, V are elements of SU(2), they can be represented by exponential maps of the Pauli Sec. 4.2 Unitary Transformations on the Dirac Spinors 63

matrices: µ ¶ i ³ϕ ´ ³ϕ ´ U = exp − ϕ eˆ · σ~ = cos 1 I − i sin 1 eˆ · σ~ (4.30a) 2 1 1 2 2×2 2 1 µ ¶ i ³ϕ ´ ³ϕ ´ V = exp − ϕ eˆ · σ~ = cos 1 I − i sin 1 eˆ · σ~ . (4.30b) 2 2 2 2 2×2 2 1

The unitary transformation U involved in changing the spin indices can be shown as

X2 0 u (p, si) = Uij u(p, sj) j=1  ¡ ¢ ¡ ¢    ¡ ¢ ¡ ¢   η η 0 X2 η η cosh 2 I2×2 sinh 2 pˆ · σ~ ci cosh 2 I2×2 sinh 2 pˆ · σ~ cj    = Uij   ¡ η ¢ ¡ η ¢ ¡ η ¢ ¡ η ¢ sinh 2 pˆ · σ~ cosh 2 I2×2 0 j=1 sinh 2 pˆ · σ~ cosh 2 I2×2 0   X U 0  j1  =⇒ cj = Uji ci ≡ . (4.31) j Uj2

By referring to equation (3.10), one can substitute in the new Dirac spinors given in equations (4.29a, 4.29b) since they are also eigen-solutions to the Dirac equation:   · µ ¶ ¸ 0 1 0 rˆ · σ~ 0 0 m m aa def †   0 † ˆ ~ ˆ ˆ ˆ ~ 0 θij = 0 u (p, si) u (p, sj) ≡ ci 0 r · σ + 1 − 0 (r · p) p · σ cj 2p 0 rˆ · σ~ p p   h i · µ ¶ ¸ m m Uj1 = U ∗ , U ∗ rˆ · σ~ + 1 − (rˆ · pˆ) pˆ · σ~   1i 2i p0 p0 Uj2 · µ ¶ ¸ m m = c† U † rˆ · σ~ + 1 − (rˆ · pˆ) pˆ · σ~ U c (4.32) i p0 p0 j

Note that certain matrices in equation (4.32) can be computed in the following manner:

¡ ¢ h ³ϕ ´ ³ϕ ´ i ¡ ¢ h ³ϕ ´ ³ϕ ´ i U † rˆ · σ~ U = cos 1 I + i sin 1 eˆ · σ~ rˆ · σ~ cos 1 I − i sin 1 eˆ · σ~ 2 2×2 2 1 2 2×2 2 1 ³¡ ¢ ´ ³ ´ ³ ´ = cos ϕ1 eˆ1 × rˆ × eˆ1 · σ~ − sin ϕ1 eˆ1 × rˆ · σ~ + (rˆ · eˆ1) eˆ1 · σ~

def = ρˆϕ1 · σ~ ³¡ ¢ ´ ³ ´ ⇒ ρˆϕ1 = cos ϕ1 eˆ1 × rˆ × eˆ1 − sin ϕ1 eˆ1 × rˆ + (rˆ · eˆ1)eˆ1 . (4.33a) ³ ´ ³ ´ ³ ´ ¡ ¢ † ¡ ¢ U pˆ · σ~ U = cos ϕ1 eˆ1 × pˆ × eˆ1 · σ~ − sin ϕ1 eˆ1 × pˆ · σ~ + (pˆ · eˆ1) eˆ1 · σ~

def = τˆϕ1 · σ~ ³¡ ¢ ´ ³ ´ ⇒ τˆϕ1 = cos ϕ1 eˆ1 × pˆ × eˆ1 − sin ϕ1 eˆ1 × pˆ + (pˆ · eˆ1)eˆ1 . (4.33b) Sec. 4.2 Unitary Transformations on the Dirac Spinors 64

in which the vectors ρˆϕ1, τˆϕ1 are rotated vectors of rˆ and pˆ respectively as defined by the rotation matrix U. The matrix θaa is re-written in terms of these rotated vectors, in which: µ µ ¶ ¶ 0 m m θ aa = ρˆ + 1 − (rˆ · pˆ) τˆ · σ~ p0 ϕ1 p0 ϕ1 ¯ µ ¶ ¯ q ¯m m ¡ ¢ ¯ ¡ ¢ def= ¯ rˆ + 1 − rˆ · pˆ pˆ¯ rˆ · σ~ = 1 + β2 (rˆ · pˆ)2 − 1 rˆ · σ~ (4.34a) ¯p0 p0 ¯ µ ¶ q m m ¡ ¢ ⇒ ρˆ + 1 − (rˆ · pˆ) τˆ = ± 1 + β2 (rˆ · pˆ)2 − 1 rˆ (4.34b) p0 ϕ1 p0 ϕ1 in which β = |p|/p0 is the relative velocity of the particle. Notice that there is double covering for the SU(2) matrices over the rotation group SO(3), which explains the ± sign on the right hand side of equation (4.34b). One can choose to define the rotation matrix U such that the the Bloch vector in equation (4.34b) reverts back to the original measurement axis rˆ (positive sign).

Note that the vector magnitudes on both sides are equal, that is ¯p ³ p ´ ¯ q ¡ ¢ ¯ 2 2 ¯ 2 2 ¯ 1 − β ρˆϕ1 + 1 − 1 − β (rˆ · pˆ) τˆϕ1¯ ≡ 1 + β (rˆ · pˆ) − 1 r ³¡ ¢ ´ q ¡ ¢ 2 2 2 2 1 + β ρˆϕ1 · τˆϕ1 − 1 = 1 + β (rˆ · pˆ) − 1 (4.35)

since the inner product rˆ · pˆ = ρˆϕ1 · τˆϕ1 is invariant under a three-dimensional rotation.

Figure 4.1: The geometrical picture of the operation of the SU(2) matrix U, defined by³ a rotation´ of m m angle ϕ1 about the axis eˆ1 in the anti-clockwise sense. The vector p0 rˆ + 1 − p0 (rˆ · pˆ)pˆ q ¡ ¢ 2 2 makes an angle of ϕ1 with 1 + β (rˆ · pˆ) − 1 rˆ. From the diagram, eˆ1 can be deduced to be perpendicular to both rˆ and pˆ.

The exponential map for the SU(2) matrix U is given by µ ¶ i ³ϕ ´ ³ϕ ´ U = exp − ϕ eˆ · σ~ = cos 1 I − i sin 1 eˆ · σ~ 2 1 1 2 2×2 2 1 Sec. 4.2 Unitary Transformations on the Dirac Spinors 65

which actually effects a three-dimensional rotation on any vector about the rotation axis eˆ1

by a counter-clockwise angle ϕ1. This can be seen clearly in Figure 4.1 in which the vector ³ ³ ´ ´ m m p0 rˆ + 1 − p0 (rˆ · pˆ)pˆ is rotated about the axis eˆ1 by a counter-clockwise angle ϕ1. The simplest choice for the rotation axis is the vector perpendicular to both rˆ and pˆ , provided they are not collinear (rˆ · pˆ) 6= ±1 , and it is given by

rˆ × pˆ eˆ = . (4.36a) 1 |rˆ × pˆ|

The rotation angle ϕ1 can be obtained by taking the dot product between the vectors ³ ³ ´ ´ m m p0 rˆ + 1 − p0 (rˆ · pˆ)pˆ and rˆ, and it is given by: h ³ ´ i p ³ p ´ m m 2 2 2 p0 rˆ + 1 − p0 (rˆ · pˆ)pˆ · rˆ 1 − β + 1 − 1 − β (rˆ · pˆ) cos ϕ1 = q ¡ ¢ = q ¡ ¢ (4.36b) 1 + β2 (rˆ · pˆ)2 − 1 1 + β2 (rˆ · pˆ)2 − 1

Figure 4.2: The 3D plot of cos ϕ1 versus β and (rˆ · pˆ). The x-axis corresponds to the relative velocity β, y-axis is the dot product (rˆ · pˆ) and the z-axis is cos ϕ1.

The function given in equation (4.36b) is plotted into Figure (4.2). In fact, if one chooses certain value for β, for example β = 0.9, equation (4.36b) can be plotted into the following figure: Sec. 4.2 Unitary Transformations on the Dirac Spinors 66

Figure 4.3: The 2D plot of cos ϕ1 versus (rˆ · pˆ) where the x-axis corresponds to the dot product (rˆ · pˆ) and the y-axis is cos ϕ1.

Notice that the two-dimensional plot given in Figure (4.3) is a sinusoidal wave with max- imum value one. It can observed that when (rˆ · pˆ) = 0, ±1, the rotation angle given by nπ, where n is an integer. Notice that in the ultra-relativistic limit (β → 1), the cosine function from equation (4.36b) is given by cos ϕ1 = (rˆ · pˆ), and one can consider merely rotating the momentum vector p since it overwhelms the magnitude of the measurement axis rˆ.

This implies that given any value for (rˆ · pˆ), it is always possible to transform the Dirac ³ ³ ´ ´ m m spinors in terms of their spin indices, such that the vector p0 rˆ + 1 − p0 (rˆ · pˆ)pˆ is rotated to be parallel to rˆ by the unitary matrix U. However, this also leads to the conclusion that

0aa matrix element θij in the integral (3.25) can be written as

2 Z q X d3p ¡ ¢ h i h i ~ 2 2 † † :: rˆ · S :: = 1 + β (rˆ · pˆ) − 1 rˆ · σ~ a (p, si)a(p, sj) − b (p, si)b(p, sj) 2(2p0) ij i,j=1 s · ¸ · ¸ X h i Z 3 2³ ´ d p |p| 2 † † = rˆ · σ~ 1 + (rˆ · pˆ) − 1 a (p, si)a(p, sj) − b (p, si)b(p, sj) . ij 2(2p0) p0 i,j (4.37)

The spin operator obtained from equation (4.37) is similar to the Wigner spin operator quoted by Terno [6] in which X h i Z h i ~ 1 † † :: rˆ · SW ::= rˆ · σ~ dµ(p) a (p, sη)a(p, sξ) − b (p, sη)b(p, sξ) (4.38) 2 ηξ η,ξ

where dµ(p) is the Lorentz invariant integration measure, which in this project is chosen as q 3 ¡ ¢ d p 2 2 dµ(p) = 2(2p0) . The difference lies in the expression 1 + β (rˆ · pˆ) − 1 from equation (4.37), in which it is equal to one when one is considering the helicity direction (rˆ = pˆ). It can be Sec. 4.2 Unitary Transformations on the Dirac Spinors 67

~ concluded that the Dirac spin operator is equivalent to the Wigner spin operator rˆ · SW up to a proportional factor, and these agree with each other when one is considering the helicity direction.

In conclusion, it is possible to modify the Dirac spin operator so that the Wigner spin operator is derived. Therefore, one can conclude that the Dirac spin operator is a better choice than its Wigner counterpart in evaluating the multi-particles spins since the Dirac spin operator does not restrict itself to helicity direction.

Chapter Summary

It is shown in this chapter that the field-theoretic realization of the spin angular momentum as shown in equation (3.25) is actually equivalent to the Pauli-Lubanski vector Wcµ under appropriate Lorentz transformation. The Dirac spinors, which is the momentum eigenfunctions

5 µ of the Dirac equation, are also helicity basis for the spin projection iγ (γµw ). In the non- relativistic limit, it is shown that the relativistic spin projection for the Dirac spinors reverts to the Pauli spin formalism [22]. It has been shown by way of choosing appropriate coordinate representation for the Poincar´egenerators that the Pauli-Lubanski vector Wcµ becomes well- defined for the bispinors. Note that the Pauli-Lubanski vector does not qualify as the relativistic spin operator since the algebra does not close (refer to equation (4.23)). However, a slight bk 1 −1 k cµ bk modification in which SP = m [L (p)] µ W allows one to define SP as the relativistic spin b b b operator that satisfies the Lie algebra for the spin angular momentum (SP × SP = iSP ). Last but not least, it has been shown that the Dirac spin operator is equivalent to the Wigner spin q ¡ ¢ operator up to a factor of 1 + β2 (rˆ · pˆ)2 − 1 in its integral. The Wigner spin operator is obtained exactly when one considers the helicity direction. Chapter 5

Conclusion

The Dirac spin operator has been derived using the Noether’s symmetry principle in the context of rotational invariance between different coordinate systems. As standard quantum mechanics dictate, the angular momentum is a conserved quantity in which the spin refers to the intrinsic angular momentum of the system. The field-theoretic realization of the angular momentum, given in terms of the covariant total angular momentum tensor Mµν is given below: Z ½ ¾ 1 h i M = d3x π (x) [Σ ] ϕ (x) − Θ0 x − Θ0 x = L + S . (5.1) µν 2 m µν mn n µ ν ν µ µν µν

As such, the three-dimensional orbital and spin operators are defined by a contraction of the covariant angular momentum tensor, and they are given by the following equations (to quote equations (1.65a, 1.65b, 1.65c ): h i k def 1 ijk 1 ijk M = ² Mij = ² Lij + Sij (5.2a) 2 Z ½ 2 ¾ i ³ ¡ ¢ ¡ ¢´ ³ ¡ ¢ ¡ ¢´ Lk = ²ijk d3x Ψ† x ∂ Ψ − x ∂ Ψ − x ∂ Ψ† − x ∂ Ψ† Ψ 4 i j j i i j j i Z ½ ¾ i ³ ¡ ¢´k ³ ¡ ¢´k = d3x Ψ† x × ∇Ψ − x × ∇Ψ† Ψ (5.2b) 2 Z ½ h i ¾ k i ijk 3 † S = ² d x Ψ γi, γj Ψ 16 − Z ½ ¾ 1 = d3x Ψ† Ξk Ψ . (5.2c) 4

In addition, the Lorentz transformation on the momentum of the particle has been shown to be restricted in the SU(2) group, since the spin operator is changes to: Z ½ ¾ ³ ´ X d3p U(Λ) rˆ · S~ U †(Λ) = Θaa a†(p , s )a(p , s ) + b(p , s )b†(p , s ) . (5.3) 2(Λp)0 mn Λ m Λ n Λ m Λ n m,n

68 Sec. 5.0 Unitary Transformations on the Dirac Spinors 69

Here the Lorentz transformation has been shown to transform as a unitary matrix from the SU(2) group, in which given as ·½ m h³ ´ i h i Θaa = c† cos Ω Ωb × (rˆ + ² pˆ) × Ωb − sin Ω Ωb × (rˆ + ² pˆ) mn p0 m p p r p p p r ¾ ¸ h ¡ ¢i b b + Ωp · rˆ + ²rpˆ Ωp ¦ σ~ cn m h i ≡ c† r · σ~ c (5.4) p0 m Λ n

Based on the theory of induced representation of the Poincar´egenerators, one can define c cα b bα another Casimir invariant WαW for the Poincar´egroup aside from the mass term PαP . It has been shown that the Pauli-Lubanski vector Wcα allows one to define a relativistic spin operator bi 1 −1 i cµ by choosing SP = m [L (p)] µ W as the relativistic spin operator. It has been shown in b b b b equation (4.22) that the operator SP satisfy the spin commutation relation (SP × SP = iSP ). This has been verified for Poincar´egenerators in which no fixed coordinate representation has been chosen. If one chooses the momentum representation, then the Pauli-Lubanski vector from equation (4.16c) can be used to define a relativistic spin operator for the Dirac spinors.

Equations (5.2a, 5.2b, 5.2c) are so universal that if one writes down the (2jn + 1) states of

the spin-jn particles, it is possible to furnish the complete orbital angular momentum and spin operators.

As a final conclusion, it is possible to re-write the Dirac spin operator into the following expression: s Z · ¸ · ¸ X h i d3p |p| 2³ ´ ~ 2 † † :: rˆ · S :: = rˆ · σ~ 1 + (rˆ · pˆ) − 1 a (p, si)a(p, sj) − b (p, si)b(p, sj) . ij 2(2p0) p0 i,j (5.5) The Wigner spin operator is well-known as the spin operator for Dirac field in the helicity direc- tion (rˆ = pˆ). However, it is shown in this thesis that the Dirac spin operator is a generalization of the Wigner spin operator when the measurement axis rˆ is arbitrary. In addition, it is easy to extend the spin operator to multi-particle states since the creation/annihilation operators for indistinguishable particles can be incorporated into the quantum fields. In passing, it is inter- esting to note that this has implications on how to measure the spin states of indistinguishable particles and also connections to the spin-statistics theorem. However, this is beyond the scope of this thesis. References

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[12] Wu-Ki Tung, Group Theory in Physics (World Scientific, Singapore (1985)).

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70 [16] S. Weinberg, The Quantum Theory of Fields. Vol. 1 (Cambridge University Press, United Kingdom (1995)).

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71 APPENDIX A

Unitary Representations of the Poincar´eGroup

Consider the following transformation (parameterized by the variable α) on the space-time coordinates xµ:

µ 0µ µ ν µ µ x −→ x = Λ ν x + a ≡ f (x; α).

Under an infinitesimal transformation, ¯ ∂f µ(x; α)¯ dxµ = ¯ dαk k ¯ ∂α all αk=0 so the wavefunction will transform accordingly (by Taylor’s expansion)

Ψ(x) −→ Ψ0(x) ≡ Ψ(x + dx)

∼ µ ∂ = Ψ(x) + dx µ [Ψ(x)] · ∂x ¯ ¸ ∂f µ(x; α)¯ ∂ = Ψ(x) + dαi ¯ Ψ(x) ∂αk ¯ ∂xµ · all αk¯=0 ¸ ∂f µ(x; α)¯ ∂ = Ψ(x) + idαk −i ¯ Ψ(x) (A-1) ∂αk ¯ ∂xµ | {z all αk=0 } Gk (generator) in which the generator for the parameter αk is given by ¯ ∂f µ(x; α)¯ ∂ G = −i ¯ . (A-2) k k ¯ µ ∂α all αk=0 ∂x

For a simple translation, ¯ ∂f µ(x; α)¯ x0µ ≡ f µ(x; aµ, µ = 0, 1, 2, 3) = xµ + aµ ∴ ¯ = δµ ν ¯ ν ∂a aµ=0 ∂ ∂ =⇒ Pˆ = −i δµ = −i = −i ∂ . ν ν ∂xµ ∂xν ν For a finite translation aµ, the wavefunction will transform as ³ ´ 0 µ ˆ Ψ(x) −→ Ψ (x) = exp −ia Pµ Ψ(x) (A-3)

0 ˆ Note that the generators P = −i ∂0 and Pk = −i ∂k are known as the Hamiltonian and momentum operators and they are Hermitian since their associated eigenvalues are defined to

72 be real. This implies that the representation for a finite translation acting on the wavefunction given as U(I, a) = exp(−ia · Pˆ) is actually unitary.

For the rotation acting on the space-time coordinates, note that the time coordinate is unaffected.

0k k k i x ≡ f (x; α) = [R(nˆ, θ )] i x = xk + (1 − cos θ)(nˆ × (nˆ × x))k + sin θ (nˆ × x)k ¯ ∂f k(x; α)¯ ∂ ∂ ∴ nˆ · Jˆ = − i ¯ = − i (nˆ × x)k ∂θ ¯ ∂xk ∂xk θ=0 · ¸ ∂ ∂ = − i ²ijk(nˆ) x = (nˆ)i i ²ijk x i j ∂xk j ∂xk ∂ =⇒ Jˆi = i ²ijk x (A-4) j ∂xk

j in which we have made use of the relation xj = −x . It can be seen that the generators for rotation are equivalent to the generators for the SO(3), which are Hermitian. Thus the representation for a finite rotation acting on the wavefunction is unitary and it is given by U(R) = exp(−i θ nˆ · Jˆ).

Consider a Lorentz boost along the x-direction         00 0 0 1  x   cosh ξ sinh ξ 0 0   x   x cosh ξ + x sinh ξ           01     1   0 1   x   sinh ξ cosh ξ 0 0   x   x sinh ξ + x cosh ξ    =     =            x02   0 0 1 0   x2   x2          x03 0 0 0 1 x3 x3 p in which cosh ξ = γ, sinh ξ = βγ where γ = 1/ 1 − β2 and β = v/c. In four-vector notation, we have

0µ µ ν µ x = Λ νx ≡ f (x; ξ) and the associated generator is given as ¯ ¯ ∂f µ(x; ξ)¯ ∂ ∂Λ µ ¯ ∂ Kˆ = −i ¯ = −i ν ¯ xν 1 ¯ µ ¯ µ ∂ξ ξ=0 ∂x ∂ξ ξ=0 ∂x " ¯ ¯ ¯ ¯ # ∂Λ 0 ¯ ∂ ∂Λ 0 ¯ ∂ ∂Λ 1 ¯ ∂ ∂Λ 1 ¯ ∂ = −i 0 ¯ x0 + 1 ¯ x1 + 0 ¯ x0 + 1 ¯ x1 ¯ 0 ¯ 0 ¯ 1 ¯ 1 ∂ξ ξ=0 ∂x ∂ξ ξ=0 ∂x ∂ξ ξ=0 ∂x ∂ξ ξ=0 ∂x · ¸ · ¸ ∂ ∂ ∂ ∂ = −i x1 + x0 = i x − x . (A-5) ∂x0 ∂x1 1 ∂x0 0 ∂x1

73 Similarly, · ¸ ∂ ∂ Kˆ = i x − x = i (x ∂ − x ∂ ). (A-6) i i ∂x0 0 ∂xi i 0 0 i ˆ Consider that the wavefunction of the system is given by Ψ(x, t), hence the action of Ki on the state is · ¸ ∂ ∂ Kˆ Ψ(x, t) = i x Ψ(x, t) − x Ψ(x, t) i i ∂x0 0 ∂xi h i† ˆ 6= KiΨ(x, t) .

ˆ This implies that the generator for the Lorentz boost, Ki is not Hermitian, and hence the i ˆ exponentiation of the generator, exp(−i ξ Ki) will not be unitary. The representation of the Lorentz boost acting on the wavefunction is not unitary and hence is not trace-preserving.

We can summarize the effects of the rotations and the Lorentz boost into one second-rank covariant tensor: ˆ Mλµ = i (xλ ∂µ − xµ ∂λ) (A-7)

ˆi 1 ijk ˆ ˆ ˆ in which J = 2 ² Mjk and Ki = Mi0. These generators obey the following commutation relations, which characterize the Lie algebra of the Poincar´egroup: h i ˆ ˆ ˆ Ji, Jj = i ²ijk Jk (A-8a) h i ˆ ˆ ˆ Ji, Kj = i ²ijk Kk (A-8b) h i ˆ ˆ ˆ Ki, Kj = −i ²ijk Jk (A-8c) h i ˆ ˆ ˆ Ji, Pj = i ²ijk Pk (A-8d) h i ˆ ˆ ˆ0 Ki, Pj = i δij P (A-8e) h i ˆ ˆ0 ˆ Ki, P = i Pi (A-8f) h i h i h i ˆ ˆ0 ˆ ˆ0 ˆ ˆ Ji, P = Pi, P = Pi, Pj = 0. (A-8g)

ˆi ˆ j ˆ The rotation J and boost K generators can be written in covariant notation Mµν, and the commutation relations are then re-written as h i ³ ´ ˆ ˆ ˆ ˆ ˆ ˆ Mαβ, Mµν = −i ηµαMνβ − ηναMµβ + ηνβMµα − ηµβMνα (A-9a) h i ³ ´ ˆ ˆ ˆ ˆ Mµν, Pλ = −i ηνλPµ − ηµλPν (A-9b) h i ˆ ˆ Pµ, Pλ = 0. (A-9c)

74 APPENDIX B

Evaluation of the Poisson Bracket Relations for Field Generators

Consider the Poisson bracket relation between the four momenta, Pµ: Z µ ¶ 3 δPµ δPν δPµ δPν {Pµ , Pν}PB = d x − (B-1a) δϕm δπm δπm δϕm 0 0 in which Pλ ≡ Θλ = πm ∂λϕm − δλ L. The functional derivative of the momentum density, Pλ

with respect to the fields, ϕm is given by · ¸ δPλ ∂Pλ ∂ ∂Pλ ≡ − i = −∂λπm(x) (B-1b) δϕm ∂ϕm ∂x ∂(∂iϕm)

while the functional derivative of the momentum density, Pλ with respect to the conjugate field,

πm is · ¸ δPλ ∂Pλ ∂ ∂Pλ ≡ − i = ∂λϕm(x) . (B-1c) δπm ∂πm ∂x ∂(∂iπm) The Poisson bracket relation for the four-momentum becomes Z n o 3 {Pµ , Pν}PB = − d x ∂µπm(x) ∂νϕm(x) − ∂µϕm(x) ∂νπm(x) . (B-1d)

For the temporal coordinates, which means either µ, ν = 0, then the Poisson bracket relation will vanish because of equation (1.28) which states that any conserved quantity over time will vanish with the Hamiltonian, which is also given by P0. Then for the spatial coordinates µ = i, ν = j, the Poisson bracket gives rise to Z n o 3 {Pi, Pj}PB = − d x ∂iπm(x) ∂iϕm(x) − ∂iϕm(x) ∂jπm(x) Z n o 3 = − d x ∂i(πm∂jϕm) − πm∂i∂jϕm − ∂j(πm∂iϕm) + πm∂j∂iϕm = 0 (B-1e)

because they can be transformed into surface term integrations using Gauss Theorem, and they will vanish due to the boundary conditions. The Poisson bracket for the four-momenta can be summarized as

{Pµ , Pν}PB = 0 . (B-1f)

The Poisson bracket between the covariant total angular momentum and the four momentum is expressed as Z µ ¶ 3 δMµν δPσ δMµν δPσ {Mµν , Pσ}PB = d x − (B-2a) δϕm δπm δπm δϕm

75 where 1 £ ¤ M = π (x) [Σ ] ϕ (x) − Θ0 x − Θ0 x . (B-2b) µν 2 m µν mn n µ ν ν µ The functional derivatives of the covariant total angular momentum density with respect to the relevant fields ϕm, πm:

δMµν 1 λ λ ∂L ¡ 0 0 ¢ = πr [Σµν]rm + ∂λπm(δ µxν − δ νxµ) + ηiµδ ν − ηiνδ µ (B-2c) δϕm 2 ∂(∂iϕm) ³ ´ δMµν 1 = [Σµν]mn ϕn + xµ(∂νϕm) − xν(∂µϕm) . (B-2d) δπm 2 in which the Euler-Lagrange equation (1.22) have been used extensively. Utilizing the equations

(B-1b, B-1c), we determine the Poisson brackets in terms of the independent fields ϕm(x), πm(x): Z ( · 3 1 λ λ ∂L {Mµν , Pσ}PB = d x πr [Σµν]rm + ∂λπm(δ µxν − δ νxµ) + 2 ∂(∂iϕm) ¸ · ¸ ) ¡ ¢ 1 ³ ´ × η δ0 − η δ0 ∂ ϕ + [Σ ] ϕ + x (∂ ϕ ) − x (∂ ϕ ) ∂ π (B-2e) iµ ν iν µ σ m 2 µν mn n µ ν m ν µ m σ m for different combinations involving the space-time coordinates µ, ν, σ. The first scenario in- volves all spatial coordinates for µ = i, ν = j (i 6= j), σ = k: Z ( 1 ³ ´ ³ ´ {M , P } = d3x [Σ ] ∂ π ϕ + x (∂ ϕ ) − x (∂ ϕ ) (∂ π ) ij k PB 2 ij mn k m n i j m j i m k m ) ³ ´ − xi(∂jπm) − xj(∂iπm) (∂kϕm) Z n o 3 = d x δikπm∂jϕm − δjkπm∂iϕm = δik Pj − δjk Pi (B-2f) after much simplification involving the disappearance of the surface terms integration and invoking the definition of the momentum generator. The second scenario involves the spatial coordinates in the angular momentum generator with the Hamiltonian µ = i, ν = j (i 6= j), σ = 0: Z ( 1 ³ ´ ³ ´ {M , P } = d3x [Σ ] ∂ ϕ π + (∂ π )x − (∂ π )x ϕ˙ ij 0 PB 2 ij mn 0 m n i m j j m i m ) h³ ´i + xi(∂jϕm) − xj(∂iϕm) π˙ m Z ½ ¾ 1 h i = ∂ d3x [Σ ] (π ϕ ) − π (∂ ϕ )x − (∂ ϕ )x 0 2 ij mn m n m i m j j m i

= ∂0Mij = 0 (B-2g)

76 since the total angular momentum is a conserved quantity. In the third scenario, the Lorentz boost generator will be evaluated with the Hamiltonian in which µ = k, ν = 0, σ = 0: Z ½ ³ ´ 3 1 ∂L {Mk0 , P0}PB = d x [Σk0]mn ∂0 ϕmπn + (∂kπm)x0 ϕ˙ m − π˙ m xk ϕ˙ m + k ϕ˙ m 2 ∂(∂ ϕm) ³ ´ ¾ + xk(ϕ ˙ m) − x0(∂kϕm) π˙ m Z 3 = ∂0 d x πm∂nϕm = Pn . (B-2h)

In the last instance, the Lorentz boost generator and the spatial momentum will be evaluated in the Poisson bracket µ = k, ν = 0, σ = j: Z ½ 1 ³ ´ {M , P } = d3x [Σ ] ∂ ϕ π + (∂ π )x (∂ ϕ ) − π˙ x (∂ ϕ ) k0 j PB 2 k0 mn j m n k m 0 j m m k j m ¾ ∂L ³ ´ + ∂ ϕ + x (ϕ ˙ ) − x (∂ ϕ ) ∂ π ∂(∂kϕ ) j m k m 0 k m j m Z ½ m ¾ 3 ¡ ¢ 0 k = d x − δkj πmϕ˙m − L − ∂0Θj xk + Θj Z 3 ¡ i ¢ = −δkjP0 + d x ∂i Θj xk = −δkjP0 (B-2i)

0 to which the continuity equation for the four momentum current has been employed ∂0Θj + i µ ∂iΘj = ∂µΘj = 0. To summarize, the Poisson bracket between the covariant total angular momentum and four momentum generators can be given by

{Mµν , Pσ}PB = ηνσPµ − ηµσPν . (B-2j)

The derivations for the Poisson bracket relation between the covariant total angular mo- menta are too complicated to be elucidated in this report. It can be summarized as

ˆ ˆ ˆ ˆ {Mαβ , Mµν}PB = ηναMµβ − ηµαMνβ + ηµβMνα − ηνβMµα . (B-3)

The Poisson brackets possess the same structure as the Lie algebra, which is given explicitly by equations (1.47a, 1.47b, 1.47c). This gives rise to the commutation of the fields in which they can be treated as operators obeying the Poincar´ealgebra.

77 APPENDIX C

The Wigner transformation corresponding to an arbitrary Lorentz transformation Λ is given by

W (Λ, p) = exp(iΩP ¦ J) (C-1)

k 1 ijk 1 0 0 1 jk where ΩP ≡ − 2 ² [ p0+m (piωj − pjωi )] and Jk = 2 ²ijkM in the absence of rotation. This can also be written as: p × ξ nˆ × p Ω ≡ − = ξ (C-2) P (p0 + m) (p0 + m)

i i 0 Here the boost parameters are represented by ξ = ξnˆ = ωi . For an infinitesimal variation in

ωµν, The transformation matrix according to Soo and Lin [11] is thus: µ ¶ µ ¶ µ ¶ |Ω | |Ω | Ω D(1/2)(W (Λ), p) = cos p I + i sin p p ¦ σ 1 2 2×2 2 |Ω | " # " p # µ ¶2 µ ¶ µ ¶3 µ ¶ 1 |Ωp| |Ωp| 1 |Ωp| Ωp ' 1 − I2×2 + i − ¦ σ 2! 2 2 3! 2 |Ωp| " µ ¶ # ·µ ¶ 1 ξ|nˆ × p | 2 ξ|nˆ × p | = 1 − I + i − 2! 2(p0 + m) 2×2 2(p0 + m) # µ ¶3 µ ¶ 1 ξ|nˆ × p | Ωp 0 ¦ σ . (C-3) 3! 2(p + m) |Ωp|

According to Doyeol [9], their transformation matrix is given as: µ ¶ µ ¶ Ω Ω D(1/2)(W (Λ, p)) = cos p I + i sin p (mˆ ¦ σ) (C-4) 2 2 2×2 2

p0 |p| Here cosh(η) = m and sinh η = m and mˆ = nˆ × pˆ. The corresponding infinitesimal Wigner representation matrix can be evaluated using the Taylor expansion with respect to ξ: ¯ ¯ ¯ ∂ ¯ ξ2 ∂2 ¯ ξ3 ∂3 ¯ D(1/2)(W (Λ, p)) ≡ D | + ξ [D ]¯ + [D ]¯ + [D ]¯ + ... 2 2 ξ=0 2 ¯ 2 2 ¯ 3 2 ¯ ∂ξ ξ=0 2! ∂ξ ξ=0 3! ∂ξ ξ=0

78 where

D2| = I2×2, ¯ξ=0 · µ ¶¸¯ · µ ¶¸¯ ¯ ¯ ¯ ∂ ¯ ∂ Ωp ¯ ∂ Ωp ¯ [D2]¯ = cos ¯ I2×2 + sin ¯ (imˆ ¦ σ) ∂ξ ξ=0 ∂ξ 2 ξ=0 ∂ξ 2 ξ=0 . . i sinh η = (σ ¦ mˆ ) 2(cosh η + 1) µ ¶ i i|nˆ × p | Ω ˆ p = 0 (n × p ¦ σ) ≡ 0 ¦ σ 2(p + m) 2(p + m) |Ωp|

The rest of the coefficients for the powers of ξ can be shown to be equivalent. Hence the expres- sions furnished by both Soo and Lin [11] and Doyeol [9] are similar, in which Soo’s approach has also shown the additional effect of rotation on the representation matrix D(1/2)(W (Λ, p)).

79 APPENDIX D

The Effect of Lorentz Transformations on the Dirac Spinors

The field operator furnished in chapter one (quoting from equation (1.73):

2 Z 3 ½ ¾ 1 X d p µ µ ˆ −ipµx † ipµx Ψ(x, t) = p a(p, si) u(p, si) e + b (p, si) v(p, si) e . (D-1) 3 2p (2π) i=1 0 The field operator is used to define a localized state by virtue of the following relation:

|x, ti = Ψˆ †(x, t)|0i (D-2) so that the quantum state can be projected onto its space-time coordinates. Therefore we are able to determine the behaviour of the Dirac wavefunction under a Poincar´etransformation by working out on how the space-time coordinates transform and also how would the mode coefficients and the Dirac spinors behave under the same transformation.

First of all, note that the unitary operator transform under the conditions given for the Poincar´egroup. A Poincar´etransformation comprises a Lorentz transformation (Λ) and a space-time translation (c): x −→ x0 = Λx + c (D-3) and the corresponding unitary transformation acting on the particle state is given by:

|p, sii −→ U(Λ, c)|p, sii ≡ U(I, c)U(0, Λ)|p, sii µ ¶ ³ ´ i = exp −icµPb exp − ωµνMc |p, s i µ 2 µν i

µ X ¡ ¢ −i(Λp)µc (jn) = e Dji W (Λ, p) |pΛ, sji j

µ X ¡ ¢ † −1 −i(Λp)µc (jn) † ⇒ U(Λ, c)a (p, si)U (Λ, c)|0i = e Dji W (Λ, p) a (pΛ, sj)|0i j

µ X ¡ ¢ † −1 −i(Λp)µc (jn) † ∴ U(Λ, c)a (p, si)U (Λ, c) = e Dji W (Λ, p) a (pΛ, sj) (D-4a) j

Note that (jn) denotes the quantum number of the spin describing the particle species. One can also extend the transformation relation for the anti-particle as well:

µ X ¡ ¢ † −1 −i(Λp)µc (jn) † U(Λ, c)b (p, si)U (Λ, c) = e Dji W (Λ, p) b (pΛ, sj) (D-4b) j

80 In addition, the following normalization conditions have been chosen as the anti-commutation relations for the creation/annihilation operators:

£ † 0 ¤ (3) 0 a(p, si), a (p , sj) + = 2p0 δ (p − p ) δij (D-5a)

£ † 0 ¤ (3) 0 b(p, si), b (p , sj) + = 2p0 δ (p − p ) δij (D-5b)

£ 0 ¤ £ 0 ¤ a(p, si), a(p , sj) + = b(p, si), b(p , sj) + = 0 (D-5c)

£ 0 ¤ £ † 0 ¤ a(p, si), b(p , sj) + = a(p, si), b (p , sj) + = 0 . (D-5d)

Secondly, we note that the quantum state (|ψi) can be written as a Fourier transform of its momentum eigenfunctions, that is

X Z 3 X Z 3 def d p k d p k † |ψi = ψ (p) |p, ski = ψ (p) a (p, sk)|0i (D-6) 2p0 2p0 k k

j where ψ (p) ≡ hp, sj|ψi is the associated momentum eigenfunction for the j-th spin state. The unitary operator representing the Poincar´etransformation operate on the quantum state in the following manner:

|ψi −→ |ψ0i = U(Λ, c)|ψi X Z 3 d p k † = U(Λ, c) ψ (p) a (p, sk)|0i 2p0 k Z 3 X µ X ¡ ¢ d p −i(Λp)µc (jn) k † = e Djk W (Λ, p) ψ (p) a (pΛ, sj)|0i 2p0 k j Z 3 0 X d p 0 µ X ¡ ¢ = e−i(p )µc D (jn) W (Λ, p) ψk(Λ−1p) |p0, s i 2p0 jk j k 0 j

Contracting both sides of the equation with hp, si|, we have:

Z 3 0 X d p 0 µ ¡ ¢ hp, s |ψ0i = e−i(p )µc D (jn) W (Λ, p) ψk(Λ−1p) hp, s |p0, s i i 2p0 jk i j j,k 0 Z 3 0 X d p 0 µ ¡ ¢ = e−i(p )µc D (jn) W (Λ, p) ψk(Λ−1p) (2p0)δ(3)(p − p0)δ 2p0 jk ik j,k 0

X µ ¡ ¢ 0i −ipµc (jn) j −1 ∴ ψ (p) = e Dji W (Λ, p) ψ (Λ p) . (D-7) j

At this juncture, the concept of a quantum wavefunction in quantum field theory is a localized function ascribed to the Dirac particles at unique space-time coordinates. From equation (D-6),

81 the wavefunction is given as: Z X d3p ψ(x) = hx, t|ψi = ψi(p) hx, t|p, s i 2p k i 0 Z 3 X d p µ ≡ ψi(p) e−ipµx 2p i 0 Z 3 d p µ ⇒ ψi(x) = ψi(p) e−ipµx . (D-8) 2p0 The wavefunction under the Poincar´etransformation is thus given as:

Z 3 d p µ ψ0i(x) ≡ ψ0i(p) e−ipµx 2p0 Z 3 d p X µ ¡ ¢ µ = e−ipµc D (jn) W (Λ, p) ψj(Λ−1p) e−ipµx 2p ji 0 j Changing the integration variable p to Λp0, and also making use of the identity Λp·x ≡ p·Λ−1x, we have X Z 3 0 (j )¡ ¢ d p 0 −1 µ ∴ ψ0i(x) = D n W (Λ, p) ψj(p0) e−ipµ(Λ (x+c)) ji 2p0 j 0 X ¡ ¢ ¡ ¢ (jn) j −1 = Dji W (Λ, p) ψ Λ (x + c) (D-9) j In the absence of the space-time translation vector cµ, the above equation can be converted into a general expression:

X h iα 0α ˆ β −1 ∴ ψ (x) = NΛ,p S(Λ) ψ (Λ x) β β X h iα 0α ˆ β ⇒ ψ (Λx) = NΛ,p S(Λ) ψ (x) (D-10) β β whereby Sˆ(Λ) is the representation of the Lorentz transformation in the external space. Notice that the summation index differed because the indices i, j indicate the spin degrees of freedom in the internal space, and α, β indicate the space-time components of the Dirac wavefunction. Last but not least, the normalization constant is present so as to ensure that all normalization conditions expounded in equations (D-5a, D-5b, D-5c, D-5d) are consistent with our formula- tion.

The wavefunction arising from the collapse of the quantum state onto a localized state:

ψα(x) = h0|Ψˆ α(x, t)|ψi

= h0|U(Λ)Ψˆ α(x, t)U −1(Λ)U(Λ)|ψi = h0|U(Λ)Ψˆ α(x, t)U −1(Λ)|ψ0i . (D-11)

82 From equation (D-10), we have

X h iα α ˆ −1 0β ψ (x) = NΛ,p S(Λ ) ψ (Λx) β β X h iα ˆ −1 ˆ β 0 = NΛ,p S(Λ ) h0|Ψ (Λx)|ψ i β β X h iα ˆ −1 ˆ β ˆ α −1 ⇒ NΛ,p S(Λ ) Ψ (Λx) = U(Λ)Ψ (x, t)U (Λ) (D-12) β β Consider the LHS of equation (D-12): 2 Z ½ X h iα X h iα X 3 ˆ −1 ˆ β ˆ −1 1 d p β −ip·(Λx) S(Λ ) Ψ (Λx) = S(Λ ) p a(p, si) u (p, si) e β β 3 2p0 β β (2π) i=1 ¾ † β ip·(Λx) + b (p, si) v (p, si) e .

Now making use of the identity p · (Λx) ≡ (Λ−1p) · x, and also changing the integration variable p to Λp0, we have Z ½ X h iα X 3 0 h iα ˆ −1 ˆ β 1 d p ˆ −1 0 β 0 −ip0·x S(Λ ) Ψ (Λx) = p 0 S(Λ ) a(Λp , si) u (Λp , si) e β 3 2p β β i,β (2π) 0 ¾ † 0 β 0 ip0·x + b (Λp , si) v (Λp , si) e . (D-13a)

Consider the RHS of equation (D-12): ½ X2 Z 3 ˆ α −1 1 d p −1 α −ip·x U(Λ)Ψ (x, t)U (Λ) = p U(Λ)a(p, si)U (Λ) u (p, si) e (2π)3 2p0 i=1 ¾ † −1 α ip·x + U(Λ)b (p, si)U (Λ) v (p, si) e Z ½ X 3 ¡ ¢ 1 d p ∗(jn) α −ip·x = p D W (Λ, p) a(pΛ, sj) u (p, si) e 3 2p ji (2π) i,j 0 ¾ ¡ ¢ (jn) † α ip·x + Dji W (Λ, p) b (pΛ, sj) v (p, si) e . (D-13b)

By comparing equations (D-13a, D-13b), we arrive at the following relations: X ¡ ¢ X h iα ∗(jn) α ˆ −1 β ∴ Dji W (Λ, p) u (p, si) = NΛ,p S(Λ ) u (Λp, sj) (D-13c) β i β X ¡ ¢ X h iα (jn) α ˆ −1 β ∴ Dji W (Λ, p) v (p, si) = NΛ,p S(Λ ) v (Λp, sj) . (D-13d) β i β

To determine the constant NΛ,p , let us consider for example the complex transpose of equation (D-15a): X ¡ ¢ X h iβ (jn) ∗ ∗ ∗ ˆ† −1 Dji W (Λ, p) uα(p, si) = NΛ,p uβ(Λp, sj) S (Λ ) . α i β

83 Consider a pure Lorentz boost, Λ, along the nˆ-axis for the sake of simplification. In addition, the normalization condition derived in the equation (2.10) for the Wigner matrix is also being utilized. After some calculations, we have s p0 N = . (D-14) Λ,p (Λp)0

As a final conclusion, the transformations involving the Dirac spinors can be summarized as: s X ¡ ¢ p0 X h iα ∗(jn) α ˆ −1 β Dji W (Λ, p) u (p, si) = S(Λ ) u (Λp, sj) (D-15a) (Λp)0 β i β s X ¡ ¢ p0 X h iα (jn) α ˆ −1 β Dji W (Λ, p) v (p, si) = S(Λ ) v (Λp, sj) . (D-15b) (Λp)0 β i β

Note that the equations (D-15a, D-15b) are unambiguous because the representation for the Lorentz transformation (Sˆ(Λ)) is distinguished by the uniqueness of the arbitrary Lorentz trans- formation (Λ). As such, there is no need to decompose the arbitrary Lorentz transformation into well-defined Lorentz boosts and rotations that have been commonly used in various textbooks ³ ´ (jn) [12]. Therefore the matrix elements used to describe the Wigner rotation Dji (W (Λ, p)) are uniquely determined by the Lorentz transformation as well.

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