Spin-3/2 Field Theories and The Gauge Invariance

A Thesis

by

Haryanto Mangaratua Siahaan

20206014

In Partial Fulfillmentof the Requirements for the Degree

Magister Sains

Physics Study ProgramFaculty of Mathematics and Natural Sciences

Institut Teknologi BandungFebruary 20081

1Font size is in 11 pt (59 pages), where the original one is in 12 pt (66 pages).

Spin-3/2 Field Theories and The Gauge Invariance

Approved by:

Triyanta, Ph.D., Adviser

Date Approved

To Putri,

David,

and Victor.

iii

”Djoudjou ma tu Ahu, asa Hu alusi ho, asa Hupabotohon tu ho hata angka na

bolon dohot na songkal, angka na so binotomi”.

(Berserulah kepada-Ku, maka Aku akan menjawab engkau dan akan memberi-

tahukan kepadamu hal-hal yang besar dan yang tidak terpahami, yakni hal-hal

yang tidak kau ketahui)

Panurirang Djeremia (Yeremia) 32 : 3

”We all have dreams. In order to make dreams come into reality, it takes an

awful lot of determination, dedication, self-discipline, and effort”.

James Cleveland Owens (1913-1980)

”...

Ai tung so boi pe au inang da, marmido marjam tangan

Tarsongon dongan-dongan hi da, marsedan marberlian

Asal ma sahat gelleng hi da, sai sahat tu tujuan

Anakhon hi... do hamoraon di au ...”

Nahum Situmorang

iv

ABSTRACT

Spin-3/2 particle theory is still an unclear subject since its first formulation by W.

Rarita and J. Schwinger in 1941. The problems are about extra degree of freedom for

describing spin-3/2 particles and the lack of relativistic property of wave solutions when

interaction with external (electromagnetic) field is introduced (known as Velo-Zwanziger

problem). In this thesis we consider Rarita-Schwinger (RS) formalism as a standard descrip-

tion of spin-3/2 particle which widely used for the delta (1232) baryon resonance. Also, we

discuss a new approach by Napsuciale-Kirchbach (NK) based on the squared Pauli-Lubanski

(PL) eigen equation to describe the spin-3/2 field. Then we do a test of gauge invariance of

the theories by performing gauge transformation of polarization vector in the corresponding

Compton scattering amplitude. We find that both theories are gauge invariant in their full

propagators and vertices. Since NK formalism is free from Velo-Zwanziger problem, it seems

that this theory could be a better description of spin-3/2 particle after some experimental

verifications.

Keyword : Spin-3/2 theory, Rarita-Schwinger formalism, Napsuciale-Kirchbach formalism,

Compton scattering amplitude, gauge invariance.

v

ABSTRAK

Teori untuk partikel berspin-3/2 merupakan sebuah subjek yang belum begitu jelas

semenjak formulasi awalnya oleh W. Rarita dan J. Schwinger pada tahun 1941. Permasalahan-

permasalahan yang terkait yaitu mengenai derajat kebebasan ekstra dalam penggambaran

partikel-partikel berspin-3/2 serta pelanggaran terhadap sifat relativistik dari solusi gelom-

bang ketika suku interaksi dengan medan (elektromagnetik) eksternal dimasukkan (dike-

nal sebagai masalah Velo-Zwanziger (VZ)). Dalam thesis ini kita menggunakan formalisme

Rarita-Schwinger (RS) sebagai sebuah penjelasan standar dari partikel berspin-3/2 yang

telah digunakan secara luas untuk resonansi baryon delta (1232). Juga, kita membahas se-

buah pendekatan baru oleh Napsuciale dan Kirchbach (NK) berdasarkan persamaan eigen

dari kuadrat operator Pauli-Lubanski (PL) untuk menggambarkan pertikel berspin-3/2.

Kemudian kita melakukan sebuah tes keinvarianan gauge dari teori-teori ini melalui trans-

formasi vektor polarisasi foton di dalam amplitudo hamburan Compton yang terkait. Kita

menemukan bahwa kedua teori ini invarian gauge dalam propagator dan verteks penuhnya.

Karena formalsime NK bebas dari masalah VZ, maka sepertinya teori ini akan dapat men-

jadi pejelasan yang lebih baik dari partikel berspin-3/2 setelah beberapa verifikasi secara

eksperimen.

Kata kunci : Teori spin-3/2, formalisme Rarita-Schwinger, formalisme Napsuciale-Kirchbach,

amplitudo hamburan Compton, keinvarianan gauge.

vi

ACKNOWLEDGEMENTS

I am very grateful to my advisor, Triyanta, Ph.D. , for his supervision and teaching me

a great deal of physics (especially from his lectures on Classical Mechanics, Quantum Me-

chanics, and Quantum Field Theory). For answering a lot of questions, while giving me a

freedom to explore my own ideas. It has been a pleasure and an honor to be his student.

I am also grateful to F. P. Zen, D.Sc. and Dr. rer. nat. Bobby E. Gunara for sharing me

their knowledge in theoretical physics.

I thank to Jusak S. Kosasih, Ph.D. and Alexander A. Iskandar, Ph.D. for their willing-

ness to referee this thesis, and also for their comments.

I am also grateful to Terry Pilling, Ph.D. (North Dakota State University) and Prof.

M. Napsuciale (Univ. de Guanajuato) for very interesting discussions about spin-3/2 field

theory.

I would like to thank my friends Algemen, Sigit, Rinto Sinurat, Cin Pau, Benz P.J.,

Sony Pro, Arma (for providing me some papers from PROLA), and others. Also to my

colleagues (Mr. Guntur, Mr. Suwandi, Mr. Sony, Mr. J. Silalahi, Mr. Gunarto, Ms. Tyas,

and others) and students (my apology for merely fast speaking of mine) at Batununggal

St. Aloysius Junior High School for creating a very good working environment. Thanks to

our librarian, Ms. Silvi, for her assistance and moral support. Also to Mr. Daryat for his

helpful administration assistances. I also owe all the people in the GII Hok Im Tong night

prayer group special thanks.

I am thankful to my family (especially my parents), whose constant support was invalu-

able.

Above all, I thank God for making all the things possible.

vii

TABLE OF CONTENTS

DEDICATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii

ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v

ABSTRAK . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi

ACKNOWLEDGEMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii

I INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

II RARITA-SCHWINGER FORMALISM . . . . . . . . . . . . . . . . . . . 3

2.1 Free Rarita-Schwinger Field . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.2 Lagrangian and Propagator . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.3 Interaction with External Field . . . . . . . . . . . . . . . . . . . . . . . . 8

2.4 Feynman Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.5 Velo-Zwanziger Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

III NAPSUCIALE-KIRCHBACH FORMALISM . . . . . . . . . . . . . . . 15

3.1 Incompatibility of Rarita-Schwinger Formalism with Theory of Relativity . 15

3.2 Poincare Symmetry and Pauli Lubanski Operator . . . . . . . . . . . . . . 19

3.3 Equation of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.4 Free Lagrangian, Noether Current, and Propagator . . . . . . . . . . . . . 26

3.5 Interacting spin-3/2 field . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.6 Propagation and The Gyromagnetic Factor . . . . . . . . . . . . . . . . . . 32

3.7 Feynman Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

IV TEST OF THE GAUGE INVARIANCE . . . . . . . . . . . . . . . . . . . 36

4.1 The Importance of Gauge Invariance . . . . . . . . . . . . . . . . . . . . . 36

4.2 Compton Scattering Amplitude and Gauge Invariance in The R-S Formalism 37

4.3 Compton Scattering Amplitude and Gauge Invariance in The N-K Formalism 40

V CONCLUSION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

APPENDIX A — COURANT-HILBERT CRITERION FOR HYPER-BOLICITY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

viii

CHAPTER I

INTRODUCTION

Higher spin field theory, especially spin-3/2 has become one of an active research branch

of theoretical physics. Although nobody has ever observed such a single particle with spin-

3/2 in laboratory, theorists are still working in this field. At least since spin-3/2 particle

theory has a role in explaining gravitino in supergravity theory, even though it still has no

agreement from experiments. A more practical use of spin-3/2 field theory in these past

decades is to describe delta (1232) resonances which are produced, for example, in pion

nucleon reaction. Research activities in the fields include determination of the coupling

constant of spin-3/2 particle when it interacts with other fields or particles.

The spin-3/2 particle theory was first proposed by W. Rarita and J. Schwinger [1] based

on Fierz and Pauli construction of field theory with arbitrary spins. In the development,

it was found that this theory suffers some difficulties, mostly related to its extra spin-

1/2 representation that contained when we construct vector-spinor representation (a direct

product between a Lorentz vector and a Dirac bispinor). The other problem is about the

superluminal (exceed the speed of light) propagation of its wave solution in interaction with

”strong” external (electromagnetic) field. Recently, a new approach for describing particles

with spin-3/2 has been developed mainly by M. Napsuciale and M. Kircbach [2,3,4,5]. Their

formalism is based on the two Cassimir operators of the Poicare group, squared momenta,

squared Pauli-Lubanski operators, and their eigen functions which the later represents the

fields with exact spins and masses. In this formalism we have to deal with a second order

partial differentiation equation (which makes the theory is more complicated). The claim

of Napsuciale and Kirchbach that their theory is free from VZ problem attracts us to

study further. This theory seems to be a little bit ”out of the mainstream” in higher spin

theory development, but so far RS formulation (which widely considered as the mainstream

following Fierz-Pauli construction for general spin) still fails to give a good description of

1

spin-3/2 particle.

The flow of this thesis can be described by the following. In the Section 2 we will

give a short review of the RS formalism that is not much discussed in Quantum Field

Theory literatures. Indeed from many years since it was firstly proposed, people have

generalized the RS formalism (e.g. by Moldauer and Case [6], Kaloshin and Lomov [7,8],

and by Pilling [9,10]). But for the sake of simplicity, when interaction with external (EM)

field is introduced, we only discussed the original the RS formalism (the Lagrangian and

propagator). Then we write the Feynman rules of the RS theory. In the Section 3, we

will introduce the NK formalism. We start with an objection to the RS theory from its

kinematical framework. Then we derive the corresponding equation of motion and the

propagator in the NK formalism. Also, we will define the Feynman propagator that is

needed when we discuss the interaction. In the Section 4, as the main component of this

thesis, we test the gauge invariance of the theories. In the Section 5, we will give a conclusion

and discussion where future development of the theories can be done.

2

CHAPTER II

RARITA-SCHWINGER FORMALISM

This section mostly follows from [1, 9,11,12,13,14].

2.1 Free Rarita-Schwinger Field

The quantities describing particles of integral spins are tensors with appropriate rank and

half-integral spins are described by spinors of multiple order. Spin-3/2 particle represented

by Rarita-Schwinger field is a tensor product between a first rank tensor and a Dirac

bispinor. Dirac bispinor can be denoted by SU(2) ⊕ SU(2) representation in the usual

way

ψ :(

12 , 0)⊕(0, 1

2

). (1)

A spin-1 field or a Lorentz vector Aµ can be constructed as the direct product(

12 , 0)⊗(0, 1

2

)by the following

Aµ :(

12 , 0)⊗(0, 1

2

)=(

12 ,

12

). (2)

Then spin-3/2 vector-spinor ψµ, according to Rarita and Schwinger [1], can be written as

ψµ ∼ Aµ ⊗ ψ :(

12 ,

12

)⊗((

12 , 0)⊕(0, 1

2

))=(1, 1

2

)⊕(0, 1

2

)⊕(

12 , 1)⊕(

12 , 0)

(3)

As can be seen in (3), the proper representation for spin-3/2 is contained in the direct sum

of(

12 , 1)

and(1, 1

2

). The direct sum

(0, 1

2

)⊕(

12 , 0)

in (3) shows that our representation

has one irreducible representation of spin-1/2, plus one that is contained in(1, 1

2

)⊕(

12 , 1).

Totally, we have one representation of spin-3/2 proper and two representations of spin-1/2

(in general are distinct).

Rarita and Schwinger proposed an equation to describe free spin-3/2 particle as the

following

(iγµ∂µ −m)ψν(x) = 0, (4)

3

with two constraints :

γµψµ = 0 and ∂µψµ = 0. (5)

These constraints are introduced to cancel spin-1/2 contribution in the equation of motion

and can be proved to have the relativistic transformation properties as the Dirac spinor [15].

In the following, we would classify two different ways to introduce explicit representation

Dirac spinor in the direct sum (3).

Using pµψµ = 0 as our primary constraint, we would identify

pµψµ ∼(0, 1

2

), (6)

where the complementary components are then given by(gµν − pµpν

p2

)ψν ∼

(1, 1

2

). (7)

The expression (7) has a zero contraction with pµ, suitable with one of the primary con-

straint. Hence the representation in (7) still contains spin-1/2 sector, we must construct an

appropriate projector of spin whose contraction with pµ is zero, but the contraction with

γµ is not zero. We can find it as the following

gµν − pµpν

p2= P (3/2)µν + P

(1/2)µν11 , (8)

where

P (3/2)µν = gµν − pµpν

p2− (pµ − γµγ · p) (pν − γ · pγν)

3p2(9)

is the projector associated with spin-3/2 proper and

P(1/2)µν11 =

(pµ − γµγ · p) (pν − γ · pγν)3p2

(10)

is the projector onto the spin-1/2 component of(1, 1

2

)in the irreducible representation.

Also we can associate

P(1/2)µν22 =

pµpν

p2(11)

4

with(0, 1

2

)representation. To describe the transition of one irreducible representation of

spin-1/2 into another and allow us to mix the two representations, one can define [16] the

operators as the following

P(1/2)µν12 = −(pµ − γµγ · p) pν

√3p2

, (12)

and

P(1/2)µν21 = −p

µ (pν − γ · pγν)√3p2

. (13)

One can check that these transition operators satisfy P(1/2)µνAB P

(1/2)BA να = P

(1/2)µAA α for

A,B = {1, 2} and A 6= B. Also P (3/2), P (1/2)µν11 , and P (1/2)µν

22 are mutually orthogonal and

one can write an expansion of identity as the following

gµν = P (3/2)µν + P(1/2)µν11 + P

(1/2)µν22 . (14)

Expression (14) shows us that ψµ contains spin-1/2 particles representations,

ψµ = gµνψν =(P (3/2)µν + P

(1/2)µν11 + P

(1/2)µν22

)ψν .

Finally, this particular set of irreducible representations suffers an unphysical singularity

at p2 = 0 which affects

gµν − pµpν

p2.

Then, using γµψµ = 0 as the primary constraint, we identify

γµψµ ∼(0, 1

2

), (15)

since γµψµ also transforms in the same way as Dirac spinor. Then the complementary

component is given by (gµν − 1

4γµγν

)ψν ∼

(1, 1

2

), (16)

which has a zero contraction with γµ. Implementation of γµψµ = 0 as the primary constraint

for the(1, 1

2

)field contributions is referred as Rarita-Schwinger (RS) field [1]. In this

5

set representations, there is no objection from unphysical singularity as occurred in the

representation before. The RS field can be decomposed into

ψµ = (Rµν1 +Rµν

2 )ψν ≡ ψµ1 + ψµ

2 , (17)

where

Rµν1 = gµν − 1

4γµγν , (18)

and

Rµν2 =

14γµγν . (19)

Rµν2 ψν contains no spin-3/2 components and to identify spin-1/2 within Rµν

1 ψν we can write

Rµν1 = P (3/2)µν +Rµν

3 , (20)

where

Rµν3 =

4(pµ − 1

4γµγ · p

) (pν − 1

4γ · pγν)

3p2. (21)

The tensor defined in (21) projects onto the subspace associated with spin-1/2 field from RS

field where before we perform this contraction, RS field still contains spin-1/2 field sectors.

To make the connection with this present treatment, we can rewrite P (3/2)µν as defined in

(9) in another form by the following

P (3/2)µν = gµν − 14γµγν −

4(pµ − 1

4γµγ · p

) (pν − 1

4γ · pγν)

3p2. (22)

P (3/2), R2, and R3are mutually orthogonal and therefore R2 and R3 correspond to alterna-

tive irreducible representation of the two spin-1/2 fields within this set treatment,

R2 +R3 = P(1/2)11 + P

(1/2)22 . (23)

The representation of spin-1/2, shown in (15), produces an unsingular compliment part and

is considered as a better basic step to cunstruc the spin-3/2 theory.

6

2.2 Lagrangian and Propagator

A general form of the possible Lagrangian related to equation (4) is given by [6,17]

L = ψµΛµνψν , (24)

where

Λµν = (γ · p−m) gµν +A (γµpν + pµγν) +12(3A2 + 2A+ 1

)γµγ · pγν

+m(3A2 + 3A+ 1

)γµγν . (25)

By variation of ψµ from (24), we will get an equation of motion

Λµνψν = 0, (26)

and any one set of Lagrangians obtained from (24) guarantee the existence of subsidiary

conditions γ · ψ = 0 and p · ψ = 0 by contracting the equation of motion with γµ and pµ

(except for A = −12). This Lagrangian is invariant under ”point transformation ” as defined

by following

ψµ → ψ′µ = (gµν + aγµγν)ψν , (27)

A→ A′ =A− 2a1 + 4a

, (28)

where a and A are defined as arbitrary real constant parameters. As Rarita and Schwinger

had stated on their paper [1], a possible Lagrangian which permits a relatively simple

expression of the equation of motion in the presence of electromagnetic fields can be written

in the following form

Λµν = (γ · p−m) gµν − (γµpν + pµγν) + γµ (γ · p+m) γν , (29)

where we have used A = −1 for expression (25). The full equation of motion from the

tensor (29) can be read of as

(γ · p−m)ψµ − (γµpν + pµγν)ψν + γµ (γ · p+m) γνψν = 0, (30)

7

or

(Hµν −m (gµν − γµγν))ψν = 0, (31)

where

Hµν = γ · pgµν − (γµpν + pµγν) + γµγ · pγν ≡ Hµν(p). (32)

The propagator Πµα related to Lagrangian (25) must satisfy

ΠµαΛαν = gνµ (33)

before we impose any constraints. For RS equation, we can find the propagator has the

following form (well known RS propagator)

Π(RS)µν =

(γ · p+m) ∆(RS)µν

p2 −m2, (34)

with ∆(RS)µν = gµν − 1

3γµγν − 2

3m−2pµpν − 1

3m−1 (γµpν − γνpµ).

2.3 Interaction with External Field

Interaction with external (electromagnetic) fields can be introduced by changing standard

derivative with covariant derivative in our Lagrangian, i∂µ → i∂µ − eAµ ≡ πµ. For the

sake of simplification, we will use the Rarita-Schwinger Lagrangian to analyze spin-3/2

propagation in presence of external field, following derivation by Velo and Zwanziger [11].

Then the Lagrangian for interacting Rarita-Schwinger field can be read of as the following

L = ψµ {(γ · π −m) gµν − (γµπν + πµγν) + γµ (γ · π +m) γν}ψν , (35)

or can also be written in a more compact form as

L = ψµ (Γαπα −B)µν ψν , (36)

where

(Γαπα)µν = gµνγ · π − (γµπν + πµγν) + γµγ · πγν = γ5εµαβνγαπβ (37)

and

Bµν = m (gµν − γµγν) . (38)

8

One can check that the symmetry of the transformations (27) and (28) no longer holds.

Using the Euler-Lagrange equation, one can write the equation of motion as following

(Γαπα −B)µν ψν = 0. (39)

We can find that for µ = 0, the equation (39) has no equation for ψ0. This fact shows us

that for µ = 0, we get a constraint rather than an equation of motion.

(Γαπα −B)0β ψβ =

(πj −

(γkπ

k −m)γj

)ψj = 0 (40)

According to Dirac analysis on the constrained system, we denote this constraint also as the

primary constraint [18]. The primary constraint is a condition that imposed in the system

or arises from the structure of the Lagrangian itself. In the situation related to expression

(40), our constraint appears from the equation of motion itself (Lagrangian). This evidence

is analogue to the zero time component of conjugate momentum, pµ = −F 0µ → p0 = 0,

in Maxwell field covariant theory before we apply any gauge fixing prescription1. The

other primary constraint appears by imposing such a condition, for example is the Lorentz

Gauge. This constraint is introduced into the formalism to reduce the degree of freedom of

our vector field beside to get such a relativistic wave equation.

We can get the equation of motion for complete ψµ covariantly by multiplying equation

(39) by γµ and πµ,

2 (γαπαγν − πν)ψν + 3mγνψν = 0, (41)

and

−ieγ5γµFµνψν −m (πν − γµγµγν)ψν = 0. (42)

In equation (39) we have used the dual of electromagnetic field strength tensor2 which is

transformed from the original one as Fµν = 12εµναβF

αβ where εµναβ is the 4 indices Levi-

Civita tensor3. Substitution of (41) to (42) yields the secondary constraint as the following

γµψµ = − 2ie

3m2γ5γαFαβψ

β , (43)

1F µν is the electromagnetic field strength tensor2Fαβ = ∂αAβ − ∂βAα3ε0123 = +1

9

which shows us clearly the dependence of ψ0 to ψk. Evidently, equation (43) also breaks

the primary constraint that has been used before in the non-interacting case, γµψµ = 0.

Also by inserting equation (43) to (42), we can gather another useful relation

πµψµ = −

(γαπ

α +32m

)23m−2ieγ5γµFµβψ

β. (44)

To get a complete equation that describe Rarita-Schwinger field, we can reinsert equation

(43) and (44) into (39) which yields

(γαπα −m)ψµ +(πµ +

12mγµ

)2ie3m2

γ5γαFαβψβ = 0, (45)

which can also be written in a hermitian form by again inserting (43) and (44) as the

following

(γαπα −m)ψµ +(πµ +

12mγµ

)2ie3m2

γ5γαFαβψβ +

2ie3m2

Fµνγνγ5

(πα +

12mγα

)ψα

+2ie3m2

Fµβγβγ5 (γαπα + 2m)

2ie3m2

γ5γαFαβψβ = 0. (46)

The conserved vector current for interacting RS fields with external electromagnetic fields

can be found by splitting the total Lagrangian into its free and interacting part as the

following :

L = ψµ {((γ · p+ eγ ·A)−m) gµν − (γµ (pν + eAν) + (pµ + eAµ) γν)

+γµ ((γ · p+ γ ·A) +m) γν}ψν

= Lfree + eψµ {γ ·Agµν − (γµAν +Aµγν) + γµγ ·Aγν}ψν︸ ︷︷ ︸i nt

= Lfree + ejαAα. (47)

The corresponding current for interacting RS field in its full form (constraints are not valid

anymore) can be read of as

jµ = ψα (γµgαβ − (γαgβµ + γβgαµ) + γαγµγβ)ψβ . (48)

10

2.4 Feynman Rules

Now we arrive at a section to depict the Feynman diagram and write its mathematical

descrption related to Compton scattering amplitude at the lowest order. The propagator

in the function of momentum p for free RS field is given by the expression (34). A repre-

sentative diagram for the RS propagator can be depicted as the following

: iΠ(RS)αβ =

i (γ · p+m)(gαβ − 1

3γαγβ −2pαpβ

3m2 − γαpβ−γβpα

3m

)p2 −m2

. (49)

Likewise, a corresponding diagram for the vertex related to the current (48) can be depicted

as :

:= ie (γµgαβ − (γαgβµ + γβgαµ) + γαγµγβ) ≡ ieYµαβ. (50)

As usual, we can add other forms of diagrams related to the initial or final state of RS fields

respectively as following :

: uβ (p)

and

: uβ (p)

11

2.5 Velo-Zwanziger Problem

The Velo-Zwanziger problem is an evidence where interacting RS fields only propagate

causally only in the ”weak-field case ”,(

23em

−2)2 ~B2 < 1, where ~B denotes the magnetic

field, e and m refer to the charge and mass of spin-3/2 particle respectively. In this sub-

section, we will re-derive this problem following the notion of original paper by Velo and

Zwanziger [11].

Velo and Zwanziger use the Courant-Hilbert prescription (Appendix A) to analyze the

interacting Rarita-Schwinger equation which should be unconditional relativistic in all of

condition. A relativistic wave equation should have a form of hyperbolic differential equa-

tion4. In general, a linier first order partial differential equation can be written as[Γµ

αβ

∂

∂xµ−Bαβ

]ψβ = 0. (51)

Equation for normals of the characteristic surfaces are determined by changing ∂µ → nµ

and the condition

D(n) =∣∣∣Γµ

αβnµ

∣∣∣ = 0. (52)

It is understood that both (51) and (52) for RS equation are 16× 16 matrix equation and

clearly from (52), the determinant D(n) is a polynomial in the component of nµ. Now we

would evaluate this determinant for equation (46). By Lorentz invariance properties, we

may take nµ = (n, 0, 0, 0) which left our evaluation of determinant only for the coefficient

of ∂/∂t in equation (46). After a slight rearrangement, we can write the determinant as

D (n) =∣∣∣∣(gα

µγ0 +

2ie3m2

Fµβγβγ5gα

0

)γ0

(gναγ

0 +2ie3m2

g0αγ

5γαF να

)n

∣∣∣∣ . (53)

We can compute this determinant explicitly by the following way where we can move out

the factor n for a while and multiply the final result with n16. First simplify our calculation

using the property of |AB| = |A| |B|. Using Dirac representation of gamma matrices, we

4Mathematically, every wave equation is categorized as a hyperbolic differential equation.

12

find∣∣γ0∣∣ = 1, and

∣∣∣gαµγ

0 + 2ie3m2 Fµβγ

βγ5gα0

∣∣∣ can be evaluated by the following way :

∣∣∣gαµγ

0 + 2ie3m2 Fµβγ

βγ5gα0

∣∣∣ =∣∣∣∣∣∣∣∣∣∣∣∣∣

γ0 + 2ie3m2 F0βγ

βγ5 0 0 0

2ie3m2 F1βγ

βγ5 γ0 0 0

2ie3m2 F2βγ

βγ5 0 γ0 0

2ie3m2 F3βγ

βγ5 0 0 γ0

∣∣∣∣∣∣∣∣∣∣∣∣∣. (54)

Using Cramer’s rule to find the determinant of n × n matrices, we can get the value of

expression (54) as(1 + 2e

3m2~B)4

. We have used the dual tensor of electromagnetic field

strength as

Fµν =

0 −B1 −B2 −B3

B1 0 E3 −E2

B2 −E3 0 E1

B3 E2 −E1 0

.

The 3-vector of magnetic field is denoted by ~B =(B1, B2, B3

), and electric field is ~E =(

E1, E2, E3).

The same way can be performed to get the value of∣∣∣gν

αγ0 + 2ie

3m2 g0αγ

5γαF να

∣∣∣ = (1− 2e3m2

~B)4

.

It is understood that determinant of (53) actually is a determinant of 16× 16 matrices. Fi-

nally, the total determinant can be written as the following∣∣∣∣(gαµγ

0 +2ie3m2

Fµβγβγ5gα

0

)γ0

(gναγ

0 +2ie3m2

g0αγ

5γαF να

)∣∣∣∣=(1 + 2e

3m2~B)4 (

1− 2e3m2

~B)4

=(1−

(2e

3m2

)2 ~B2)4. (55)

The final result of expression (53) is n16(1−

(2e

3m2

)2 ~B2)4

. The zero value of this deter-

minant describes the normals to the characteristic surfaces passing through each point.

Relativistic massive waves or particles should have the determinant value bigger than zero.

This condition evidently state that interacting massive RS field can only propagate relativis-

tically in the condition where(

2e3m2

)2 ~B2 < 1 (”weak-field”). In other words, the massive

interacting RS fields are ”conditional relativistic”. This problem also has some effects on the

quantization of interacting massive RS fields where no quantum theory can be constructed

for the ”strong-field” case (since the related equation ceases to be hyperbolic). The main

13

point in this subsection can be stated as follow. Transforming covariantly such a massive

wave equation does not guarantee the theory satisfies special relativity. We also have to

verify the propagation of the solution should not exceed the propagation of light.

14

CHAPTER III

NAPSUCIALE-KIRCHBACH FORMALISM

In this section we will discuss a new approach by M. Napsuciale and M. Kirchbach to

describe the dynamic of spin-3/2 particles. Their formalism is based on the Pauli-Lubanski

(PL) operator acting on an eigen function with definite spin and mass. As we know, PL

operator is one of the Casimir operators in the Poincare group. Poincare group reflects the

symmetry that occurs in the inhomogenous Lorentz transformation that should be satisfied

by a relativistic theory.

As the starting point, we will discuss an incompatibility of the Rarita-Schwinger for-

malism to describe relativistic spin-3/2 particles from its kinematical structure without

invoking any interaction (this part is based on ref. [19,20]). In the next subsection, we will

discuss the formalism of Napsuciale and Kirchbach in constructing a relativistic equation

for massive spin-3/2 particles. We follow their derivations from their series of papers in ref.

[2,3,4]. We also discuss the interaction of the spin-3/2 field based on this formalism with ex-

ternal electromagnetic fields. Following Velo and Zwanziger, they also use Courant-Hilbert

criterion to verify that their theory is unconditionally relativistic.

3.1 Incompatibility of Rarita-Schwinger Formalism with The-ory of Relativity

The complete RS representation space is a direct product of a spinor and a Lorentz vector.

In the rest frame, the spinor can be written as the following

ψ1 (0) =

1

0

1

0

ψ2 (0) =

0

1

0

1

ψ3 (0) =

1

0

−1

0

ψ4 (0) =

0

1

0

−1

, (56)

15

and the Lorentz vector in the(

12 ,

12

)representation can be read as

V1 (0) =

1

0

0

0

;V2 (0) =

1√2

0

1

1

0

;V3 (0) =

0

0

0

1

;V4 (0) =

1√2

0

1

−1

0

. (57)

The vectors in (57) can be understood as the eigen vectors of the squared spin operator

~J2 and Jz with the eigen values j (j + 1) and j respectively in(

12 ,

12

)representation. The

complete derivation is given in the ref. [20]. Denoting Dirac bispinor as

ψ (~p) =

φR (~p)

φL (~p)

=

K(0,

12

)(~p) 0

0 K(

12 ,0

)(~p)

φR (0)

φL (0)

, (58)

where K(0,

12

)(~p) and K

(12 ,0

)(~p) are the right and left handed boost, so for

(12 ,

12

)repre-

sentation we could write the related boost as K(

12 ,

12

)(~p) = K

(0,

12

)(~p) ⊗ K

(12 ,0

)(~p) which

explicitly could be written as

K(

12 ,

12

)(~p) =

12m (E +m)

[(E +m) I2×2 + ~σ · ~p]⊗ [(E +m) I2×2 − ~σ · ~p] . (59)

We have used ~σ as the three Pauli matrices, σ1 , σ2 , and σ3 in expression (59).

From Wigner argument, we can get a wave function in(

12 ,

12

)representation with mo-

mentum ~p by boosting it from its rest frame. We focus only on the vectors in (57) which

are regarded as the spin-1 representations (actually only ξ = 1, 2, 3 which bring the exact

spin-1) and boost them by the following :

V1 (~p) = K(

12 ,

12

)(~p)V1 (0) =

12m (E +m)

(E +m− pz) (E +m+ pz)

− (px + ipy) (E +m+ pz)

(px + ipy) (E +m− pz)

− (px + ipy)2

,

V2 (~p) = K(

12 ,

12

)(~p)V2 (0) =

12m

√2 (E +m)

2 (px − ipy) pz(p2

z + 2 (m+ E) pz − p2x − p2

y + (E +m)2)

(p2

z − 2 (m+ E) pz − p2x − p2

y + (E +m)2)

2 (px + ipy) pz

,

16

V3 (~p) = K(

12 ,

12

)(~p)V3 (0) =

12m (E +m)

− (px − ipy)2

(px − ipy) (E +m+ pz)

− (px − ipy) (E +m− pz)

(E +m− pz) (E +m+ pz)

,

V4 (~p) = K(

12 ,

12

)(~p)V3 (0) =

12m

√2 (E +m)

−2 (px − ipy) (E +m)(p2

z + 2 (m+ E) pz + p2x + p2

y + (E +m)2)

−(p2

z − 2 (m+ E) pz + p2x + p2

y + (E +m)2)

2 (px + ipy) (E +m)

.

In the free RS framework, we apply ∂αAα = 0 to project out spin-1/2 contribution. Unfor-

tunately, our vectors above do not transform as a ”good” Lorentz vector, xµ = gµνxν . For

this reason, in their article Ahluwalia and Kircbach [19,20] introduce a Lorentz four vector

as the following

Aµξ (~p) ≡ SVξ (~p) , (60)

where S is a rotation matrix of the form

S =1√2

0 i −i 0

−i 0 0 i

1 0 0 1

0 i i 0

. (61)

The matrix defined in (58) comes from our necessity on the orthogonality and completeness

among the vectors. Recalling the standard Maxwell field theory, the polarization vectors

with wave momentum k, εµ (k, λ), satisfy orthonormal relation as the following

εµ (k, λ) εµ (k, ρ) = gλρ, (62)

for gλρ is flat metric tensor, diag (+1,−1,−1,−1). This polarization vectors also satisfy the

completeness relation

3∑λ=0

εµ (k, λ) εν (k, λ) = δµν (63)

17

in the rest frame. The same thing should also be valid for spin-1 field theory based on(

12 ,

12

)representation. By defining a new vector

Vξ (~p) = (Vξ (~p))† Λ, (64)

we could write the orthonormality relations as the following

Vξ (~p)Vξ (~p) =

−1forξ = 1, 2, 3

+1forξ = 4, (65)

and the completeness relation

V4 (~p)V4 (~p)−3∑

ξ=1

Vξ (~p)Vξ (~p) = I4×4. (66)

The Λ matrix is defined by

Λ =

−1 0 0 0

0 0 −1 0

0 −1 0 0

0 0 0 −1

, (67)

and we find an interesting result where SΛS−1 = diag (+1,−1,−1,−1).

After defining the Lorentz vectors Aµξ (~p), now we would check whether these vectors

satisfy the constraint that in momentum space it can be read as pµAµξ (~p) = 0. For the case

ξ = 1, as an example, we obtain

pµAµ1 (~p) = pµSV

µ1 (~p)

=1

2√

2m (E +m)[E,−px,−py,−pz]

0 i −i 0

−i 0 0 i

1 0 0 1

0 i i 0

(E +m− pz) (E +m+ pz)

− (px + ipy) (E +m+ pz)

(px + ipy) (E +m− pz)

− (px + ipy)2

=1

2√

2m (E +m)[E,−px,−py,−pz]

−2i (px + ipy) (E +m)

−i((E +m)2 − p2

z (px + ipy)2)

(E +m)2 − p2z − (px + ipy)

2

−2ipz (px + ipy)

18

=1

2√

2m (E +m)

{−2iE (px + ipy) (E +m) + ipx

((E +m)2 − p2

z (px + ipy)2)

−py

((E +m)2 − p2

z − (px + ipy)2)

+ 2ip2z (px + ipy)

}=

i (px + ipy)2√

2m (E +m)

{−2E (E +m) + (E +m)2 − p2

z + ipx (−i (px + ipy))

−py (i (px + ipy)) + 2p2z

}=

i (px + ipy)2√

2m (E +m)

{−E2 +m2 + p2

x + p2y + p2

z

}

=i (px + ipy)

2√

2m (E +m)

{m2 − pµp

µ}. (68)

Since by the on-mass shell condition, our result in (68) would be zero. This means for ξ = 1

the constraint pµAµξ (~p) = 0 is valid. The same calculation could be performed for the other

ξs. The results are

pµAµ2 (~p) =

−ipz

2√

2m (E +m)

{m2 − pµp

µ}, (69)

pµAµ3 (~p) =

−i (px − ipy)2√

2m (E +m)

{m2 − pµp

µ}, (70)

pµAµ4 (~p) =

i

2√

2m2 (E +m)pµp

µ. (71)

The strange result can be observed from expression (71) that does not have zero value

as expected. Thus for ξ = 4, the constraint fails to satisfy and this fact shows an incom-

patibility of supplementary condition in all of reference frames (that was imposed into the

theory by Rarita and Schwinger long time ago). This problem might be understood from

the evident that V4 brings the spin-0 value.

3.2 Poincare Symmetry and Pauli Lubanski Operator

The Poincare transformation is the homogenous Lorentz transformation (6 generators) plus

translation in 4 dimensions (4 generators). In general, the Poincare transformation in

space-time can be read as

g (a,Λ) = T (a) Λ, (72)

19

where T (a) is the translation and Λ is a homogenous Lorentz transformation. Using stan-

dard convention, we denote pµ as the four momenta and have a role as generators of trans-

lations. These generators commute each other,

[pµ, pν ] = 0. (73)

Homogenous Lorentz transformation in coordinate space can be written as x′µ = Λµνxν

where

Λµν = exp[− i

2ωµνLµν

], (74)

and Lµν = xµpν−xνpµ as the coordinate ”orbital” momentum. A field ψ (x) is transformed

by

ψ′ (x) = exp[− i

2ωµνMµν

]ψ (x) , (75)

where ωµν are arbitrary continuous parameters and Mµν is the total tensor of angular

momentum, Mµν = (xµpν − xνpµ) + Sµν . Here, Sµν is introduced as the intrinsic angular

momentum tensor. Mµν are totally anti-symmetric and satisfy the commutation relation

[Mµν ,Mρσ] = i (gµσMνρ − gµρMνσ + gνρMµσ − gνσMνρ) , (76)

and their commutation relations with the translation generators is given by

[pµ,Mρσ] = i (gµρpσ − gµσpρ) . (77)

To identify such a system or particle, we need information that does not change due

to transformations of the system or particle itself. In mathematical language, we need

operators that commute with all of generators of the group that represents the symmetry

of the system. Since they commute, the corresponding eigen values will not change for

any group transformations. These operators are known as Casimir operators. In Poincare

group, the Casimir operators should commute with generators of 4-translations and ho-

mogenous Lorentz transformation. There are two Casimir operators in Poincare group,

squared momentum operator p2 and squared Pauli-Lubanski (PL) operator, W 2, where the

PL operator is defined as

Wµ =12εµναβMναpβ, (78)

20

and transforms as Lorentz 4-vector in coordinate space. Evidently, one can construct such

an equation of motion just like what Napsuciale and Kirchbach had done for spin-3/2 that

would be discussed in this section.

The representation theory of Poincare Group had been worked out by Eugene P. Wigner

and falls into three classes (summarized in many of quantum field theory textbooks e.g.

[21]). Related to the NK formalism, we only need the first class of representation which can

be stated as follow : ” The eigen value of pµpµ ≡ m2 is a real positive number. The eigen

value of W 2 is −m2s (s+ 1) where s is the spin, which assumes to have discrete values

s = 0, 12 , 1,

32 , .... This representation is labeled by the mass m and the spin s. States within

the representation are distinguished by the third component of the spin (projection to the

axis of spin), sz = −s,−s+ 1, ..., s− 1, s, and the continuous eigen value of pi. Physically

a state corresponds to a particle of mass m, spin s, three momentum pi and spin projection

sz. Massive particle of spin s have 2s+ 1 degrees of freedom ”.

The squared PL operator as the second Casimir operator in the Poincare group can be

written as the following

W 2 =14εµαβγε

µπθρSαβpγSπθpρ

= −14δπθραβγS

αβpγSπθpρ

= −14

∣∣∣∣∣∣∣∣∣∣δπα δθ

α δρα

δπβ δθ

β δρβ

δπγ δθ

γ δργ

∣∣∣∣∣∣∣∣∣∣SαβpγSπθpρ

= −12SαβS

αβpγpγ +GαG

α, (79)

for Gα = Sαβpβ . In this section, we will use Latin alphabets to denote spinor indices, and

Greek alphabets for Lorentz indices.

The Homogenous Lorentz group generator in the vector space is given by

(Mµν

V

)αβ

= i(gµαg

νβ − gµ

βgνα

), (80)

and in spinor space

(Mµν

S

)ab

=12

(σµν)ab . (81)

21

The Lorentz indices in (80) can be easily understood from the tensor product in right hand

side of the equation. In the other side, the additional indices a, b, in (81) to express the

spin tensor of Dirac bispinor seem to be confusing. Since spinor indices are not related

to external coordinate as Lorentz indices, sometimes spinor indices are suppressed. For

instance, the wave function in standard Dirac equation is usually written in short as ψ (~r, t)

rather than ψa (~r, t) where the spinor index ”a” has been suppressed. So, the indices writing

method as in (81) should not make a confusion.

Then we can write the PL operator in the vector space as (Wµ)αβ = iεµαβνpν and in

spinor space (wµ)ab = 12 (γ5σµν)ab p

ν respectively. It yields the squared of PL operator in

vector and spinor space can be written as

(W2)αβ

= −2 (gαβgµν − gανgβµ) pµpν , (82)

and

(w2)ab

= −14

(σµν)ac (σνα)cb p

µpα. (83)

The PL operator in the vector-spinor space can be identified as the summation of the PL

operator in vector and spinor space. This is clearly due to the fact that the homogenous

Lorentz generators in the vector-spinor space are given by (Mµν)αβab =(Mµν

V

)αβδab +

gαβ

(Mµν

S

)ab

. This thought allow us to write the PL operator for vector-spinor space as

(Wµ)αβab = (Wµ)αβ δab + gαβ (wµ)ab , (84)

and its squared as

(W 2)αβab

=(W2)αβδab + (Wµ)αβ (wµ)ab + (wµ)ab (Wµ)αβ + gαβ

(w2)ab. (85)

Using the result in (82) and (83), and also the cross term in (85)

(Wµwµ + wµWµ)αβab = −12

(ελαβµγ

5 (σλν)ab + ελαβνγ5 (σλµ)ab

)pµpν , (86)

the squared PL operator in the vector-spinor space can be written as the following

(W 2)αβab

= Tαβabµνpµpν , (87)

22

where

Tαβabµν = −2 (gαβgµν − gανgβµ) δab −14gαβ (σρµσ

ρν)ab

−12

(ελαβµγ

5σλν + ελαβνγ5σλµ

)ab. (88)

3.3 Equation of Motion

The main prescription of NK formalism is constructing an equation of motion from an eigen

equation of the squared PL operator

W 2Ψ(m,s) = −p2s (s+ 1) Ψ(m,s), (89)

and the on-mass shell condition

p2Ψ(m,s) = m2Ψ(m,s). (90)

Clearly this equation is a matrix equation with appropriate dimension to the size of Ψ(m,s).

The NK formalism also applies the same representation as in RS, where the wave function

for spin-3/2 field is a direct product between Lorentz vector and Dirac bispinor that yields 16

degrees of freedom (the size of Ψ(m,s) ). Because this number exceed our necessity to describe

particle with spin-3/2 which needs only 8 degree of freedoms, Rarita and Schwinger imposed

the supplementary condition to reduce this degree of freedom. In the NK formalism, one

does not need to impose the same constraints from the beginning as RS did. Instead, we

can take another way to get such a good theory of spin-3/2 particles which will be discussed

below.

First, we decompose our Ψ(m,s) into its Poincare invariant1 as the following

Ψ(m,(sL,sR)) → Ψ(m,|sL−sR|) ⊕Ψ(m,|sL−sR|+1) ⊕ ...⊕Ψ(m,(sL+sR)), (91)

and related to our purpose to describe spin-3/2 particle, expression (91) can be written as

ψµ = Aµ ⊗ ψ →[Ψ

(m,

32

)](2)

⊕[Ψ

(m,

12

)](4)

. (92)

1Poincare invariant denotes an object that is invariant under Poincare transformation. One can labelthis object by its constant mass and spin.

23

The first term of direct product in expression (92) corresponds to representation of particle

with definite mass and spin-3/2 which each of them has four degrees of freedom (related to

the description of massive particle and antiparticle with spin-3/2). The extra two distinct

Dirac particles are described by the rest term in expression (92).

Then, to get an equation, we could define the projection operator (projector) that

project the related wave function with definite mass and spin back to itself and have zero

result if it works to other different wave functions. This projector has an analogy with some

projectors that had discussed in RS formalism before. These projectors can be constructed

from equation (89). After some simple algebraic manipulation, we could write

P (m,s)(p)Ψ(m,s) = Ψ(m,s), (93)

where

P (m,s)(p) = − 12s

(W 2

m2+ s (s− 1)

p2

m21n×n

). (94)

1n×n stands for an identity matrix with appropriate size n. From (102), we are allowed to

write that P (m,s−1)(p)Ψ(m,s−1) = Ψ(m,s−1), so using the fact that P (m,s) and P (m,s−1) are

orthogonal, we find the following condition

P (m,s)(p)Ψ(m,s−1) = 0. (95)

P (m,s−1) can be defined as

P (m,s−1) =12s

(W 2

m2+ s (s+ 1)

p2

m21n×n

), (96)

and finally we can write these following properties

[P (m,s)(p)

]2= P (m,s)(p);

[P (m,s−1)(p)

]2= P (m,s−1)(p) ,

P (m,s)(p)P (m,s−1)(p) = 0;P (m,s)(p) + P (m,s−1)(p) = 1n×n.

For s = 32 , the related projector has the form

P

(m,

32

)(p) = −1

3

(W 2

m2+

3p2

4m2116×16

). (97)

24

The size 16× 16 in identity matrix in (97) denotes that we deal with matrix equation with

this size. This is reasonable since our W 2 indeed in 16× 16 matrix form. The equation of

motion following (94) can be written as[−1

3

(W 2 +

34p2116×16

)−m2116×16

]ψ = 0, (98)

where the ψ as defined in (92) and the indices has been suppressed. The covariant form

can be read as [−1

3Tαβµνp

µpν −(

14p2 +m2

)δαβ

]ψβ = 0, (99)

where the spinor indices has been suppressed. A better form of an equation for free spin-3/2

particle can be written as

[−Kαβ +m2gαβ

]ψβ = 0, (100)

with Kαβ ≡ Γαβµνpµpν and

Γαβµν =23

(gαβgµν − gανgβµ) +16

(ελαβµγ

5σλν + ελαβνγ5σλµ

)

+112σλµσ

λν gαβ4 −

14gµνgαβ . (101)

The tensor Kαβ has the following properties

pαKαβ = 0 ; γαKαβ = 0 ; Kαβpβ = 0 ; Kαβγ

β = 0. (102)

Expression (100) can be written in detail as[(−p2 +m2

)gαβ +

23pβpα +

13

(pαγβ + pβγα) γ · p− 13γαγ · pγβγ · p

]ψβ = 0. (103)

Using properties as stated in (102), it is easy to verify these relations

pµψµ = 0, (104)

γµψµ = 0, (105)

[p2 −m2

]ψµ = 0. (106)

25

Expression (106) as the on-mass shell condition shows us that our wave function satisfies

relativistic mass energy relation. The other two as stated in (104) and (105) reminding

us the supplementary conditions imposed in Rarita-Schwinger formalism. We regard both

γµψµ = 0 and pµψµ = 0 as the conditions or constraints in Napsuciale-Kirchbach formalism.

Different from the existence of both constraints in RS formalism as the imposed conditions

from its beginning, the constraints in NK formalism come from the equation of motion. We

could say that in NK formalism, these constraints appear naturally.

3.4 Free Lagrangian, Noether Current, and Propagator

The corresponding Lagrangian related to equation of motion (100) can be read in a hermitian

form as

Lfree = −12

[(∂µψα

)Γαβµν∂

νψβ +(∂νψβ

)Γαβµν∂

µψα]

+m2ψαψα, (107)

where

Γαβµν = γ0 (Γαβµν)† γ0. (108)

As we can check that Γαβµν = Γαβνµ, (107) can be read as

Lfree = −(∂µψα

)Γαβµν∂

νψβ +m2ψαψα. (109)

The related Noether current can be obtained as

jµ =(i∂νψα

)Γαβνµψ

β − ψαΓαβµνi∂νψβ, (110)

where it satisfies the continuity equation, ∂µjµ = 0.

The propagator Παβ for free spin-3/2 particle in NK formalism, can be found by com-

puting the inverse of(−Kαβ +m2δαβ

). Doing some calculations yields

Παβ =1

(p2 −m2 + iε)

[−gαβ +

23pαpβ +

13m2

(pαγβ + pβγα) γ · p

− 13m2

γαγ · pγβγ · p]. (111)

26

For short, we can write the propagator as Παβ = ∆αβ

(p2 −m2 + iε

)−1. The tensor ∆αβ

can be decomposed into new projectors

∆αβ = −P(

32

)αβ +

p2 −m2

m2P

(12

)αβ , (112)

where

P

(32

)αβ = −1

3

(W 2

αβ

p2+

34gαβ

); P

(12

)αβ =

W 2αβ

3p2+

54gαβ . (113)

P

(32

)αβ to P

(m,

32

)αβ as P

(m,

32

)αβ (p)ψβ = p2

m2 P

(32

)αβ ψβ = ψα. This means that in the mass shell

condition, the projectors satisfy P

(m,

32

)αβ (p)ψβ = P

(32

)αβ ψβ.

Since Παβ is the inverse of(−Kαβ +m2δαβ

), that is

Πµα(−Kαν +m2δαν

)= gµ

ν ,

we can easily write

−Πµα(p)Kαν(p) = gµν −Πµ

ν (p)m2. (114)

Note that both Πµα and Kαν are function of momenta p, restricting that this relation only

valids for tensors with the same momentum components.

3.5 Interacting spin-3/2 field

To introduce the interaction with external electromagnetic field, we replace the standard

partial derivative ∂µ in (114) with the covariant derivative Dµ = ∂µ + ieAµ with +e as the

charge of the particle. Then our Lagrangian has the following form

L = −(Dµ†ψα

)ΓαβµνD

νψβ +m2ψαψα, (115)

which can be decomposed into the free part and the interacting part of the Lagrangian,

L = Lfree + Lint. The interacting part can be written as follow

Lint = ie[(∂νψα

)Γαβνµψ

β − ψαΓαβµν∂νψβ

]Aµ − e2ψαΓαβµνψ

βAµAν

= ejµAµ − e2ψαΓαβµνψ

βAµAν (116)

27

Expanding the wave function ψβ(x) into uβ(p) exp [−ip · x] allowing us to write the transi-

tion current in momentum space as

jµ(p′, p) = −uα(p′)(Γαβνµp

′ν + Γαβµνpν)uβ(p). (117)

From equation (100) for free case, we can also write the equation of motion of interacting

field by changing standard momentum pµ = −i∂µ to πµ = −iDµ = pµ + eAµ,

[−Γαβµνπ

µπν +m2gαβ

]ψβ = 0. (118)

When we follow the way of ref. [11] in verifying the hyperbolicity of equation (124), we find

that this equation still suffers the same problem. Until this point, we have not found any

good resolution for relativistic massive spin-3/2 interacting theory.

The key idea to solve this problem is to find the right gyromagnetic factor (g-factor)

related to our spin-3/2 particle in magnetic field. First, we should check the g-factor value

in our theory. The g-factor is a constant coupled with a tensor product between spin tensor

and EM field strength tensor in the interacting equation of motion. For example, in Dirac

theory we can recognize the g-factor is 2 as the following way.

(iγ · ∂ − eγ ·A+m) (iγ · ∂ − eγ ·A−m)ψ =((iγ · ∂ − eγ ·A)2 −m2

)ψ

=

(i∂ − eA)2 − e

2σµνFµν︸ ︷︷ ︸

spin dependent

−m2

ψ = 0.

The spin dependent term above is a term where the g-factor lives. We can rewrite this term

as

−g e2σµν

2Fµν = −g e

2MµνFµν ,

where we set g = 2. We denote that the Lorentz generator for Dirac bispinor isMµν = 12σ

µν .

We could find the g-factor for the interacting spin-3/2 field theory, similarly.

Since the Lorentz generator (which will identify a term that contains g-factor) is a

totally anti-symmetric tensor, the g-factor would live in the anti-symmetric part of Γαβµν

in the equation of motion. We could split the tensor Γαβµν in the equation (118) into its

28

symmetric part, ΓSαβµν , and the anti-symmetric one, ΓA

αβµν , as the following way

ΓSαβµν =

12

(Γαβµν + Γαβνµ) , (119)

and

ΓAαβµν =

12

(Γαβµν − Γαβνµ) . (120)

Direct calculation with Γαβµν which is defined in (101) yields

ΓSαβµν = gαβgµν −

23

(gµαgνβ + gµβgνα) +16

[(gµαγν + gναγµ) γβ + γα (gµβγν + gνβγµ)]

−13γαγβgµν , (121)

and

ΓAαβµν =

13

[gµαgνβ − gµβgνα −

i

2gαβσµν

]= − i

3(Mµν)αβ . (122)

The tensor (Mµν)αβ in (122) denotes Lorentz generator in the vector-spinor space. This

tensor is a sum of the Lorentz generator in the vector and spinor space that can be written

based on (80) and (81) as

(Mµν)αβ =(Mµν

V

)αβ

+ gαβ

(Mµν

S

). (123)

For the sake of simplicity, spinorial indices have been suppressed. To get an exact value of

g-factor, we can only pay attention to the anti-symmetric part of our interacting equation

of motion.

ΓAαβµνπ

µπν = − i3

(Mµν)αβ πµπν

= − i3

(Mµν)αβ (−i∂µ + eAµ) (−i∂ν + eAν)

= − i3

(Mµν)αβ

(−∂µ∂ν − ie∂µAν − ieAµ∂ν + e2AµAν

)

=

i

3(Mµν)αβ ∂

µ∂ν − 23e

2(Mµν)αβ F

µν︸ ︷︷ ︸g−factor=2/3

− ie2

3(Mµν)αβ A

µAν

.

”The right” g-factor value for higher spin particles is still an unclear subject. Belinfante

conjectured that spin-s particle will bring the g-factor 1/s which is suitable for the result

29

above . But Weinberg stated a theorem that particles with spin higher than one should have

the g-factor = 2 in order to propagate causally. So, even though the equation as written in

(118) is not free from Velo-Zwanziger problem, we could still do such a ”gauging” to find a

better g-factor of the equation of motion that is free from Velo-Zwanziger problem. For this

purpose, we only need to focus again on the anti-symmetric part of our tensor Γαβµν . As

the starting point, we can write the most general Lorentz covariant anti-symmetric tensor

ΓAαβµν = −i

[aσµν

2gαβ + ib (gαµgβν − gανgβµ)

]+ ic (gαµσβν − gανσβµ)

+id (σαµgβν − σανgβµ) + ifεαβµνγ5, (124)

where a, b, c, d and f are arbitrary real parameters. Assuming f = 0 and c = d we have

ΓAαβµνπ

µπν = −i[aσµνπ

µπν

2gαβ − ebFαβ

]− 2iecFαβ + c (παγ · π − γ · ππα) γβ

+cγα (γ · ππβ − πβγ · π) , (125)

The total equation of motion,((

ΓAαβµν + ΓS

αβµν

)πµπν −m2gαβ

)ψβ = 0, can be written as[(

π2 −m2)gαβ − i

(aσµνπ

µπν

2gαβ − e

(b− 2c+

23

)Fαβ

)+

13

(γαγ · π − 4πα)πβ

+13(παγ · π − γαπ

2)γβ + ie

(16− c

)γµFµαγβ + ie

(16− c

)γαγ

µFβµ

]ψβ = 0. (126)

For interacting spin-3/2 particle theory, we wish to have a term in equation of motion which

contains g-factor as well as the electromagnetic field strength tensor

Lmag = −eg3

2

2ψα (Mµν)αβ ψ

βFµν

= −eg3

2

2ψα(i (gµαgνβ − gµβgνα) +

σµν

2gαβ

)Fµνψβ

= ig32ψα

(σµνπ

µπν

2gαβ − eFαβ

)ψβ. (127)

To get a term in (126) that has the form of (127), we can set

i(a1

2σµνπµπνgαβ − e

(b− 2c+ 2

3

)Fαβ

)30

in (126) where c = 16 and leave a and b as the g-factor for the spinor and vector parts of ψβ

respectively. To get a completely equal form to (127), one needs the equality between the

g-factor between spin-1/2 and spin-1 particle described in (126). Then we also have relation

between constants in (126) as g32

= a = b + 13 . Finally we have an interacting equation of

motion as [(π2 −m2

)gαβ − ig3

2

(σµνπ

µπν

2gαβ − eFαβ

)+

13

(γαγ · π − 4πα)πβ

+13(παγ · π − γαπ

2)γβ

]ψβ = 0. (128)

Here, the same thing occurs as when we deal with the RS formalism. Equation (128) is not

a genuine equation of motion since we would not get an equation for ψ0. In order to get

an equation for every component of ψβ, we can do a contraction of equation (128) with γβ

and πβ. Straight calculations yield

γ · ψ =ie

6m2

(3g3

2+ 2)(

Fµβγµ + iγ5γαFβα

)ψβ , (129)

and

π · ψ =ie

m2

[(1− g3

2

)(Fβµπ

µ + πµFβµ) + g32παFαβ + i

(g32

4+

16

)γ5[γαFβα, γ · π

]

+ie

(g32

4− 1

6

){γαFβα, γ · π}

]ψβ + ie

(g32

4− 1

6

)γν (Fνµπ

µ + πµFνµ) γ · ψ. (130)

It would be helpful if we rewrite (128) in the following form

(π2 −m2

)ψα − ig3

2

(σµνπ

µπν

2gαβ − eFαβ

)ψβ +

13

(γαγ · π − 4πα)π · ψ

+13(παγ · π − γαπ

2)γ · ψ = 0, (131)

where γ · ψ and π · ψ are as introduced in (129) and (130) respectively.

31

3.6 Propagation and The Gyromagnetic Factor

Now, we have to test the result in (136) whether it propagates causally or not. The first

step that should be done is denoting the characteristic determinant of the equation (136),

D (n) = |Kαβ |, where

Kαβ = n2gαβ +13

(γαγ · n− 4nα)Nβ +13(nαγ · n− γαn

2)Mβ, (132)

Nβ =ie

6m2

(3g3

2+ 2)(

Fµβγµ + iγ5γµFµβ

), (133)

and

Mβ =ie

m2

[(53−

3g32

2

)Fβµn

µ + i

(g32

4+

16

)γ5[γαFβα, γ · n

]

+

(g32

2− 1

3

)γνFνµn

µNβ

]. (134)

Expressions (132), (133), and (134) appear from equation (131) where Nβ related to γ · ψ

and Mβ to π · ψ. According to Courant-Hilbert prescription (Appendix A), we just change

the momentum pµ to the normal vector nµ and pay attention to the matrix coefficients

related to differential operators in the equation. The determinant of this matrix, denoted

by |Kαβ |, will show the hyperbolicity of the equation under some circumstances.

The determinant value of matrix described in (132) as obtained by creators of this theory

in ref. [4] can be written as

D(n) = n24

n2 − k2

(5g32− 2

4

)2

(n · F )2 + k2

(3g32

+ 2

4

)2 (n · F

)2

2

+k2

4

(3g32

+ 2

4

)2(5g32− 2

4

)2 (F · F

)2(n)2

×

n2 + k2

(3g32

+ 2

4

)2 [(n · F

)2− (n · F )2

]2

+k4

4

(3g32

+ 2

4

)2 (F · F

)2(n)2

. (135)

32

In expression (135), k = 23em

−2, (n · F )ν = nµFµν ,

(n · F

)ν= nµF

µν , F · F = FµνFµν ,

and (n · F )2 = (n · F )ν (n · F )ν . The Courant-Hilbert criterion in Velo-Zwanziger paper [11]

was simplified by choosing only the time component of the normal vector, so we only deal

with time-differential coefficients of the related equation.

Differently, in the series of their paper Napsuciale and Kirchbach use other method

which in principal also the same Courant-Hilbert criterion for hyperbolic partial differen-

tial equation. Napsuciale and Kirchbach determined whether the hyperbolic differential

equation is hyperbolic or not by the relation between the time component, n0, and spatial

components, ~n, of the normal vector from the determinant value of differential operator

coefficients matrix. If n0 is real for every real ~n, then the equation would be hyperbolic.

But if imaginary n0 exist for every real ~n, then the equation would not be hyperbolic.

Napsuciale and Kirchbach showed that Velo and Zwanziger problem can also be seen by

this following way. The determinant of interacting RS equation can be written in general

(without choosing the zero spatial components of normal vector) as

D(n) = n8

[n2 + k2

(F · n

)2]4

. (136)

The solution of for characteristic equation D(n) = 0 for n 6= 0 is n2 = −k2(F · n

)2which

potentially yields imaginary n0 for every real ~n, where nµ = (n0, ~n). The determinant in

(135) can be simplified by

nµFµνnµFµν − nµFµνnµF

µν = −12n2FµνF

µν ,

so we can write

D(n) =(n2)16((1− 2k2F · F

)2 +(2k2F · F

)2)2

. (137)

D(n) = 0 for expression (137) guarantees real n0 for every real ~n. So, it has proved

that gauged interacting equation of Napsuciale and Kirchbach is free from Velo-Zwanziger

problem. The final form of Γαβµν which yields the propagation of spin-3/2 field solution is

causal as can be read as

Γαβµν = ΓSαβµν + ΓA

αβµν , (138)

for ΓSαβµν = 5

3 (gαµgβν − gανgβµ)− iσµνgαβ + i6 (gαµσβν − gανσβµ − σαµgβν + σανgβµ).

33

3.7 Feynman Rules

Now we arrive at a section to depict the Feynman diagram and write its mathematical

description related to Compton scattering amplitude in the lowest order. The propagator

in the function of momentum p for free NK field can be depicted based on expression (111)

by following :

: iΠ(NK)αβ =

i[−gαβ + 2

3m2 pαpβ + 13m2 (pαγβ + pβγα) γ · p− 1

3m2γαγ · pγβγ · p]

(p2 −m2 + iε). (139)

The ”one-photon” vertex2 based on expression (117) has the form

: −ie(Γαβνµp

′ν + Γαβµνpν)≡ ieZαβµ(p′, p). (140)

As we can observe from the interacting part of Lagrangian, there is a term which contains

photon field in second order, AµAν . By this evidence, we should include an extra diagram

to describe the ”two-photon” vertex as the following

2We will use the terminology ”one-photon” and ”two-photon” vertex in this thesis for NK formalismrelated to the terms in the current that contains one and two photon field respectively.

34

: −ie2Γαβµν . (141)

As usual, we can also add other forms of diagram related to the initial or final state of RS

field as following

: uβ (p) ,

and

: uβ (p) .

35

CHAPTER IV

TEST OF THE GAUGE INVARIANCE

4.1 The Importance of Gauge Invariance

Electrodynamics is an interaction in nature which determines the overall structure of atoms

and molecules and the way in which they radiate and absorb light. In other words, every

particle that build an atom must be affected by electromagnetic fields. When an interaction

with external electromagnetic fields is introduced to a matter field (usually by covariant

derivative prescription), the total equation clearly will contain a term of photon fields (which

are interpreted as electromagnetic fields).

One of the properties related to photon field equation is its gauge invariance. The

equation would not change after transforming field variables (which have the gauge freedom)

as a function of time and space. Definitely, the amount of these transformation is infinite

and we can say that the system exhibit an infinite degree of freedom. In practical we can

observe that free Lagrangian of photon field is remain invariant under performing gauge

transformation δAµ = ∂µθ(x) where θ(x) is an arbitrary function of space and time.

Explicit consequence of gauge invariance in an interacting quantum field theory can be

observed from the effect of transforming photon state in the S-matrix element. For any

physical process involving an outgoing photon with momentum k and polarization vector

εµ(k) in the final state, the Feynman amplitude can be written in the form M = Mµε∗µ(k).

The S-matrix element can be written as

⟨f ′, γ(k)

∣∣− i

∫d4xjµ(x)Aµ(x) + ... |i〉 , (142)

where γ(k) denotes the photon (in the final state) and f ′ as a particle that would appear.

Applying an analogy to one-particle state, the form of (142) can read as

−iε∗µ(k)∫d4x

eik·x

N

⟨f ′∣∣ jµ(x) + ... |i〉 , (143)

36

where N is a normalization factor (not significance in this discussion). If one replaces ε∗µ(k)

above with kµ, the expression (143) will be read as

−ikµ

∫d4x

eik·x

N

⟨f ′∣∣ jµ(x) + ... |i〉

= −∫d4x

(∂µeik·x

)N

⟨f ′∣∣ jµ(x) + ... |i〉

= −∫d4x∂µ

(eik·x

N

⟨f ′∣∣ jµ(x) + ... |i〉

)︸ ︷︷ ︸

surface term

+∫d4x

(eik·x

)N

⟨f ′∣∣ ∂µjµ(x) + ... |i〉 .

By the conservation of current, ∂µjµ(x) = 0, we can observe that

−ikµ

∫d4xN−1eik·x

⟨f ′∣∣ jµ(x) + ... |i〉 = 0.

In other words, after replacing the polarization vector with a corresponding photon mo-

mentum vector in the S-matrix, the amplitude goes to zero, kµMµ = 0. Further we can

also check that after gauge transformation of the polarization vector, εµ(k) → εµ(k) + kµζ

(where ζ is arbitrary complex parameter), the corresponding amplitude should be remain

invariant since the coefficient of ζ goes to zero.

4.2 Compton Scattering Amplitude and Gauge Invariance inThe R-S Formalism

The two lowest order diagrams for Compton scattering process in Rarita-Schwinger formal-

ism can be depicted as the following

37

Figure 1

and

Figure 2

The two pictures above can be understood mathematically from the fact that explicit S-

matrix related to the corresponding current (in Rarita-Schwinger formalism) has the fol-

lowing form

S =∫d4xd4yei(kf ·x−ki·y) 〈pf |T : jµ (x)Aµ :: jν (y)Aν : |pi〉,

where T denotes time ordered product, and ”::” as the normal ordering symbol that acting

on operators within it. Pictorially, the two diagrams above can be considered as ”event”

that might be happened (probabilistic) on the lowest order for the elementary process that

is depicted as follow1

1p and k denote the momenta of particle and photon respectively, α for particle’s spin, and ε for photon’spolarization.

38

Figure 3

The relation between momentums from diagrams above can be easily understood as Q =

p+ q = p′ − q′ and Q′ = p− q′ = p′ − q. The amplitude for fig. 1 reads

iM1 = uα (ieYναβ) iΠβδ(Q) (ieYµγδ)uδε∗ν(q′)εµ(q)

or

M1 = −e2uα (Yναβ) Πβγ(Q) (Yµγδ)uδε∗ν(q′)εµ(q). (144)

For fig. 2, the corresponding mathematical form is

M2 = −e2uα (Yµαβ) Πβγ(Q′) (Yνγδ)uδε∗ν(q′)εµ(q). (145)

The total amplitude can be read as

M = −e2uα[(Yναβ) Πβγ(Q) (Yµγδ) + (Yµαβ) Πβγ(Q′) (Yνγδ)

]uδε∗ν(q′)εµ(q). (146)

In the expressions above, Πβδ(Q) denotes the propagator Π(RS)βδ as a function of mo-

mentum Q = p+ q. Gauge transformations of the polarization vectors

ε∗ν(q′) → ε∗ν(q′) + q′νζ ′∗, (147)

εµ(q) → εµ(q) + qµζ, (148)

yields the difference between the total amplitude after and before transformation as

δM = −e2uα[(Yναβ) Πβγ(Q) (Yµγδ) + (Yµαβ) Πβγ(Q′) (Yνγδ)

]

×(ε∗νqµζ + εµq′νζ ′∗ + q′νζ ′∗qµζ

). (149)

39

We can decompose expression (149) into components correspond to the parameter ζ, ζ ′∗,

and ζζ ′∗ by the following way

δM = C1ζ + C2ζ′∗ + C3ζζ

′∗. (150)

Each coefficient C in (150) should be zero and can be checked as follows

C1 = −e2uα[(Yναβε

∗ν) Πβγ(Q) (Yµγδqµ) + (Yµαβq

µ) Πβγ(Q′) (Yνγδε∗ν)]uδ.

We can use the relation Yµγδqµ = Hγδ(Q) − Hγδ(p), where tensor Hγδ as defined in (32)

with corresponding momentum and q = Q − p. Also it is clear that Hµν(p)uν(p) =

m (gµν − γµγν)uν(p) and Hµν(p)Πνα(p) = gαµ +m (gµν − γµγν) Πνα(p). Then the first term

in C1 can be computed as follows

−e2uα(p′) (Yναβε∗ν) Πβγ(Q) (Yµγδq

µ)uδ(p)

= −e2uα(p′) (Yναβε∗ν) Πβγ(Q) (Hγδ (Q)−Hγδ (p))uδ(p)

= −e2uα(p′) (Yναβε∗ν)(gβδ u

δ(p) + Πβγ(Q)m (gγδ − γγγδ)uδ(p)

−Πβγ(Q)m (gγδ − γγγδ)uδ(p))

= −e2uα(p′) (Yναδε∗ν)uδ(p). (151)

The rest term of C1 is found to be +e2uα(p′) (Yναδε∗ν)uδ(p), so its total is vanished.

The same method can be performed to find C1 and C2 and the results also vanish. So we

have found that for the case of Compton scattering, the spin-3/2 field theory from Rarita-

Schwinger formalism show the gauge invariance property in its full propagator (without

imposing any subsidiary conditions).

4.3 Compton Scattering Amplitude and Gauge Invariance inThe N-K Formalism

Beside the two lowest order diagrams as presented in the subsection before, we must include

the other two extra diagrams which represent interaction with photon in the second order.

40

The corresponding diagram can be depicted as follows

Figure 4

and by changing ”the photon wiggly lines” as

Figure 5

The two diagrams above can be considered as the result of normal ordering process that is

occurred in the scattering matrix. : Aµ (x)Aν (y) : will produce the terms in the amplitude

that can be written as εµ (q′) εν (q) ei(q′·x−q·y) + εµ (q) εν (q′) e−i(q·x−q′·y) which show us the

nature of extra diagrams above. Alternatively, one can associate a factor two to the ”two-

photon” vertex as stated in (141) for the corresponding Feynman rules2. This alternative is

an analogy to Feynman rules evaluation for scalar electrodynamics theory (for example ref.

[24]). Clearly, these two methods will produce the same results. Based on Feynman rules

that have been defined in (139), (140), and (141), the mathematical form of total amplitude

can be read as

M = −e2uα(p′)[Zαβν(p′, Q)Πβγ(Q)Zγδµ(Q, p) + Zαβµ(p′, Q′)Πβγ(Q′)Zγδν(Q′, p)

+ (Γαδνµ + Γαδµν)]uδ(p)ε∗ν(q′)εµ(q). (152)

Before we show the proof of the gauge invariance property from the amplitude (152) in

its full propagator for spin-3/2 particle, here some relations that would be useful in our

2In this case, we don’t have to depict two ”distinct” diagrams for interaction which included ”two-photon”vertex.

41

computation later. From expression (114), we have

−Πµα(p)Kαν(p) = gµν −Πµ

ν (p)m2. (153)

Then we can also write that

Zαβν(p′, Q)q′ν = Zαβν(p′, Q)(p′ν −Qν

)= −Γαβµνp

′µ (p′ν −Qν)− ΓαβνµQ

µ(p′ν −Qν

)= −Γαβµνp

′µp′ν + ΓαβνµQµQν

= −Kαβ(p′) +Kαβ(Q). (154)

From the corresponding equation of motion, it is clear that

uα(p′)Kαβ(p′) = m2uβ . (155)

The total amplitude as described in (159) can be decomposed into components for each

corresponding diagrams as follows

M =(M(1)

ν + M(2)ν + M(3)

ν + M(4)ν

)εν(q), (156)

where

M(1)µ = −e2uα(p′)

[Zαβν(p′, Q)Πβγ(Q)Zγδµ(Q, p)

]uδ(p)ε∗ν(q′), (157)

M(2)µ = −e2uα(p′)

[Zαβµ(p′, Q′)Πβγ(Q′)Zγδν(Q′, p)

]uδ(p)ε∗ν(q′), (158)

M(3)µ = −e2uα(p′) [Γαδνµ]uδ(p)ε∗ν(q′), (159)

M(4)µ = −e2uα(p′) [Γαδµν ]uδ(p)ε∗ν(q′). (160)

Contraction between M(1)µ with photon momentum qµ can be performed by the following

way

M(1)µ qµ = −e2uα(p′)

[Zαβν(p′, Q)Πβγ(Q)Zγδµ(Q, p)qµ

]uδ(p)ε∗ν(q′)

42

= −e2uα(p′)[Zαβν(p′, Q)Πβγ(Q) (−Kγδ(Q) +Kγδ(p))

]uδ(p)ε∗ν(q′)

= −e2uα(p′)[Zαβν(p′, Q)

(−Πβ

δ (Q)m2↙ + gβδ + Πβγ(Q)m2g↙γδ

)]uδ(p)ε∗ν(q′)

= −e2uα(p′)[Zαβδ(p′, Q)

]uδ(p)ε∗ν(q′), (161)

where to get the last result we have used the relations in (153), (154), and (155). The same

way can be applied for M(2)µ as the following

M(2)µ qµ = −e2uα(p′)

[Zαβµ(p′, Q′)qµΠβγ(Q′)Zγδν(Q′, p)

]uδ(p)ε∗ν(q′)

= −e2uα(p′)[(−Kαβ(p′) +Kαβ(Q′)

)Πβγ(Q′)Zγδν(Q′, p)

]uδ(p)ε∗ν(q′)

= −e2uα(p′)[(−m2gαβΠβγ(Q′)↙ − gγ

α +m2Πγα(Q′)↙

)Zγδν(Q′, p)

]uδ(p)ε∗ν(q′)

= +e2uα(p′)[Zαδν(Q′, p)

]uδ(p)ε∗ν(q′). (162)

Also for remaining amplitude terms, the contraction can be performed as

(M(3)

µ + M(4)µ

)qµ = −e2uα(p′) [Γαδνµ + Γαδµν ] qµuδ(p)ε∗ν(q′)

= −e2uα(p′)[Γαδνµ (Qµ − pµ) + Γαδµν

(p′µ −Q′µ)]uδ(p)ε∗ν(q′)

= −e2uα(p′)[ΓαδνµQ

µ + Γαδµνp′µ − Γαδνµp

µ − ΓαδµνQ′µ]uδ(p)ε∗ν(q′)

= −e2uα(p′)[−(ΓαδµνQ

′µ + Γαδνµpµ)

+(Γαδµνp

′µ + ΓαδνµQµ)]uδ(p)ε∗ν(q′)

= −e2uα(p′)[Zαδν(Q′, p)− Zαδν(p′, Q)

]uδ(p)ε∗ν(q′). (163)

To get the final expression in (163) we have used the definition (140),

−(Γαβνµp

′ν + Γαβµνpν)≡ Zαβµ(p′, p).

The sum of (161), (162), and (163) finally is found to be vanishing. Even though we only

calculate the coefficient of ζ, the same results would appear for the coefficient of ζ ′∗ and ζζ ′∗.

So the total difference of amplitude as the result of polarization vector gauge transformation

is vanished. This result is exactly as we expect to prove the gauge invariance of Napsuciale-

Kirchbach formalism.

43

CHAPTER V

CONCLUSION

Higher spin field (particularly spin-3/2) theory seems still an unclear object in theoretical

physics. This may be duo to no physical demands that need such a theory to exist. Until

today, our best understanding of a single particle with highest spin is the photon (masless

quantum particle with spin-1). In recent decade, physicists tried to approach baryon reso-

nances with spin-3/2 by using spin-3/2 particle theory (at least for the propagator and the

wave function). But still, we are confused whether this approach is the best way or not.

Even such a beautiful theory like supergravity which contains spin-3/2 description, fail to

find an agreement with experiments.

Even though the above skepticisms appears, finding such a consistent description for

particle with spin value 3/2 is an interesting object to be studied. A framework for spin-

3/2 particle that widely accepted by physicist is Rarita-Schwinger formalism. We have

discussed this formalism in Chapter 2. We have also discussed the Velo and Zwanziger

problem which arises when external fields are introduced into the theory.

In Chapter 3 we have introduced a new approach by Napsuciale and Kirchbach. We

discussed its framework to get the corresponding Feynman rules. In Chapter 4 we found

that both RS and NK formalism satisfy gauge invariance property only in its full propagator

(the spin-1/2 contributions are not removed out). The physical meaning of this phenomenon

(spin-1/2 contribution for interacting spin-3/2 particle) is still unclear and needs a further

investigation.

If the NK formalism for spin-3/2 particle is found to be better than the RS formalism,

it would be interesting to generalized NK theory into arbitrary spin. So far, finding such

a appropriate projectors correspond to every spin component of Poincare invariants in the

theory would not be a simple manner.

44

APPENDIX A

COURANT-HILBERT CRITERION FOR

HYPERBOLICITY

In this appendix, we review shortly on the sections that are related for determining the

hyperbolicity of an equation of motion in the form of partial differential equation (for

discussions below, it is understood that the word differential equation refers to partial

differential equation). Further verifications could refer directly to ref. [25].

A.1 Linear Differential Equation

We restrict our discussion up to second order differential equation where in general can be

written as

L [u] + d = 0, (164)

L [u] ≡n∑

i,k=0

aikuik +n∑

i=0

aiui + au. (165)

The coefficients aik, ai, a are given functions merely of the n + 1 independent variables

x0, x1, ..., xn (where later we would often distinguish x0 as t), uik ≡ ∂2u∂xi∂xk

, uk ≡ ∂u∂xk

, and

u is the solution. The characteristic condition that is written as

Q (φi, φk) =n∑

i,k=0

aikφiφk ≡ 0 (166)

is an equation for a surface C that contains initial data extension (characteristic serface).

These initial data consist of the values of u and the values of one exterior, say uφ =∑ni=0 uiφi.

Although the characteristic condition (166) has the form of the first order partial differ-

ential equation for φ, the function φ (x0, x1, ..., xn) need not satisfy this differential equation

identically. By definition, it must satisfy (166) only on C, that is, for φ = 0. However, if

45

C is given in the form x0 = ψ (x1, x2, ..., xn), then (166) represents a partial differential

equation for a function ψ of only n variables :

n∑i,k=1

aikψiψk − 2n∑

i=1

ai0ψi + a00 = 0. (167)

If in particular a00 = −1, ai0 = 0 for 1 ≤ i ≤ n, and aik is independent of x0 = t, then

(165) becomes

utt −n∑

i,k=1

aikuik + ... = 0, (168)

and the characteristic partial differential equation for ψ is

n∑i,k=1

aikψiψk = 1. (169)

Characteristic surfaces play a role as ”wave fronts”, that is, surfaces across which solutions

of (164) might suffer discontinuities (e.g. discontinuities of the second derivative). Related

to the special theory of relativity, the corresponding characteristic surface could be a (light)

cone and it separates two distinct worlds (inside cone related to ”our” world and outside

belongs to tachyonic particle).

A few simple invariance property of characteristic are important. Courant and Hilbert

assert: ”Characteristics are invariant with respect to arbitrary transformations of the in-

dependent variables”. We could see this assertion by following. Under a transformation

ξµ = ξµ (x0, x1, ..., xn) for µ = 0, 1, ..., n, u (x) may become ω (ξ). The result of this trans-

formation for equation (165) can be written as

L [u] ≡ L′ [u] + cu

=n∑

µ,ν=0

αµνωµν +n∑

µ=0

αµωµ + cω

≡ Λ [ω] ≡ Λ′ [ω] + cω.

This invariance is obvious from the conceptual meaning of the characteristic condition,

φ (x0, x1, ..., xn) = ψ (ξ0, ξ1, ..., ξn) = 0. We can verify that

αik =∑n

j,l=0ajl (∂ξi/∂xj) (∂ξk/∂xl),

46

so we have an identity that shows the invariance property of characteristic,

n∑i,k=0

aikφiφk =n∑

i,k=0

αikψiψk. (170)

A.2 Characteristic for Higher Order Operator and Hyper-bolicity

First we will use a new notation where

Dk = Dk =∂

∂xk(171)

and k = 0, ..., n. Let p = (p0, ..., pn) be any vector with n + 1 non-negative integer

components. Let ξ = (ξ0, ..., ξn) be any vector with n + 1 components. Then we define

ξp = ξp00 ...ξ

pnn , where ξ could be numbers or operators, e.g. ξ = D. The order of differential

operator D is denoted by |p| = p0 + ...+ pn.

With these notations, a differential equation of order m can be written in the following

form :

L [u] =∑|p|≤m

apDpu = f. (172)

a and f in the expression above may be constant or functions of the independent variables

x, or depend on x and partial derivative of u up to the order m − 1. Most equation

in mathematical physics appear as systems of k equations involving k unknown functions

u1, ..., uk, which can be collected and written in a matrix equation

L [u] =∑|p|≤m

ApDpu = f, (173)

where coefficients A are k×k square matrices, u and f is a column vector with k components.

Such as system of equations with order m can be written in this form. Special attention

will be paid for a system equations of first order

L [u] =n∑

i=0

AiDiu+Bu = f (174)

and of second order

L [u] =n∑

i,j=0

AijDiDju+n∑

i=0

AiDiu+Bu = f, (175)

47

with k × k matrices B, Ai, Aij (Aij = Aji). The corresponding characteristic equation for

first and second order respectively is given by

Q (ξ0, ..., ξn) =∣∣∣∑Aiξi

∣∣∣ = 0, (176)

and

Q (ξ0, ..., ξn) =∣∣∣∑Aijξiξj

∣∣∣ = 0. (177)

Before defining the hyperbolicity of an equation (or system equations), it useful to

restated that characteristic form (Q) and characteristic matrices (A), in practice, remain

invariant under coordinate transformation. In this statement, coordinate is regarded as the

independent variables related to Courant Hilbert assertion above. Related to the necessity

to solve such an equation, one could equate the solavability of Chaucy’s problem with the

concept of hyperbolicity.

Definition : At a point O the operator L [u] is called hyperbolic if there exist vectors ζ

through O such that every two dimensional plane π through ζ intersects the normal cone

(a cone that is constructed from characteristic condition) Q (ξ) = 0 in mk real and distinct

line.

The definition above can be stipulated as following : if θ is an arbitrary vector(not

parallel to ζ) then the line of points ξ = λζ + θ, λ being a parameter, must intersect the

normal cone in mk distinct real points; i.e., the equation for λ

Q (λζ + θ) = 0

must have mk real distinct roots. Space elements at O orthogonal to the vectors ζ are called

space-like, and ζ is called a space-like normal.

Until this point, we have given a short introduction of Courant-Hilbert criterion of

hyperbolicity. Based on materials above, we can understand that after transforming nµ

to a simple vector (n, 0, 0, 0), Velo and Zwanziger found for interacting Rarita-Schwinger

equation that n has real distinct roots only in the ”weak” field case. The same method had

performed by Napsuciale and Kirchbach for their interacting spin-3/2 equation (as discussed

in Section 4) without transforming the vector nµ.

48

As a simple example, we can consider the two dimensional wave equation as uxx +uyy−

utt = 0. The corresponding characteristic equation can be written as

φ2x + φ2

y − φ2t = 0,

and the equation of the normal cone in ξ, η, τ -space is

ξ2 + η2 − τ2 = 0,

that can be depicted as follows

The corresponding normal surface in the ξ, η-space is clearly a circle with unit radii.

The normal cone equation above guarantees that τ will has real roots for real ξ and η. This

evidence shows the hyperbolicity of the wave equation.

49

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51