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HHPPPXXX hhhh (((( ��� IFT Instituto de F́ısica Teórica

Universidade Estadual Paulista

TESE DE DOUTORAMENTO IFT–T.002/18

On equivalence of Scalar Quantum Electrodynamics via

Duffin-Kemmer-Petiau and Klein-Gordon-Fock formalism

using Causal Perturbation Theory approach

Jhosep Victorino Beltrán Ramirez

Orientador

Prof. Bruto Max Pimentel Escobar

March 2018





Dedicated to Victorino and Ana.





Acknowledgement

I thank CAPES for the scholarship support during the 4 years. I finish this stage hoping

that the policy in Science and Technology of Brazil will continue.

I thank Professor B. M. Pimentel for the confidence of being able to undertake

this project and for his supervision. With him we are three salmons in the American

Continent.

I also thank Professor G. Scharf for the electronic correspondence on some doubts

about Causal Perturbation Theory which has been developed.

I would like to thank my wife Milagros for the support and company away from her

family.

I thank my brother Johel for his support.

Finally, but not less important, I thank my parents Victorino and Ana. Without

them we would not had been able to survive the internal war in Peru





Abstract

In this Thesis we use Causal Perturbation Theory to study Scalar Quantum Elec-

trodynamics with Duffin-Kemmer-Petiau fields. We determine the differential cross

sections at the tree level, the vacuum polarization tensor, self energy function and

the normalizability of the theory. After that, we compare our results with those ones

obtained via Klein-Gordon-Fock fields determining that they are not completely equiv-

alent.

Keyword: Causal Perturbation Theory; Scalar Quantum Electrodynamics; DKP.

Research field: 1.05.01.01-0;1.05.02.01-7; 1.05.03.01-3





Resumo

Nesta tese utilizamos a Teoria de Perturbação Causal para estudar a Eletrodinâmica

Quântica Escalar com os campos de Duffin-Kemmer-Petiau. Determinamos as seções

de choque diferenciais no ńıvel da árvore, o tensor de polarização do vácuo, a função

de auto energia e a renormalizabilidade da teoria. Depois disso, comparamos nossos

resultados com os obtidos através dos campos de Klein-Gordon-Fock, determinando

que eles não são completamente equivalentes.

Palavras Chaves: Teoria de Perturbação Causal; Eletrodinâmica Quântica Escalar;

DKP.

Áreas do conhecimento: 1.05.01.01-0;1.05.02.01-7; 1.05.03.01-3





Contents

1 Introduction 1

2 Elementary Theory of Distributions 5

2.1 The necessity of distribution and its definition . . . . . . . . . . . . . . 5

2.2 Properties of distributions and the space of test functions T . . . . . . 7

2.3 Product of two distributions . . . . . . . . . . . . . . . . . . . . . . . . 10

2.4 Fourier transform and convolution of distributions . . . . . . . . . . . . 10

3 Causal Perturbation Theory 13

3.1 Axioms of Causal Perturbation Theory . . . . . . . . . . . . . . . . . . 14

3.1.1 Axiom I: Boundary Condition . . . . . . . . . . . . . . . . . . . 15

3.1.2 Axiom II: Base Term and Perturbative Gauge Invariance . . . . 16

3.1.3 Axiom III: Poincaré Invariance . . . . . . . . . . . . . . . . . . 16

3.1.4 Axiom IV: Causality . . . . . . . . . . . . . . . . . . . . . . . . 16

3.2 Iterative construction of S-Matrix . . . . . . . . . . . . . . . . . . . . . 17

3.2.1 Properties of the n-point distributions . . . . . . . . . . . . . . 17

3.2.2 From Tn´1 to Tn . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.2.3 Supports of the retarded Rn and advanced An distributions . . . 20

3.2.4 The causal distribution Dn . . . . . . . . . . . . . . . . . . . . . 21

3.3 Causal splitting Procedure . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.3.1 Numerical distribution dn . . . . . . . . . . . . . . . . . . . . . 22

vii



viii CONTENTS

3.3.2 Singular and Regular distributions . . . . . . . . . . . . . . . . 23

3.3.3 Uniqueness of the retarded part rpxq . . . . . . . . . . . . . . . 28

3.4 Causal-splitting procedure in momentum space . . . . . . . . . . . . . . 28

3.4.1 Regular distribution Case . . . . . . . . . . . . . . . . . . . . . 29

3.4.2 Singular distribution Case . . . . . . . . . . . . . . . . . . . . . 30

4 Quantized free Fields and Perturvative Gauge Invariance 35

4.1 Electromagnetic Field . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

4.2 Duffin-Kemmer-Petiau fields . . . . . . . . . . . . . . . . . . . . . . . . 38

4.2.1 Spxq function . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

4.3 Fermionic Scalar (Ghost) Fields . . . . . . . . . . . . . . . . . . . . . . 43

4.4 Perturbative Gauge Invariance . . . . . . . . . . . . . . . . . . . . . . . 43

5 Scattering processes of Scalar Quantum Electrodynamics at the tree-

level 47

5.1 Definition of term T1 for SDKP via Perturbation Gauge Invariance at

first order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

5.2 Scattering of DKP scalar particle by static external field . . . . . . . . 49

5.3 Causal distribution in the second order D2px, yq . . . . . . . . . . . . . 54

5.4 Moller scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

5.4.1 Causal splitting of D0 . . . . . . . . . . . . . . . . . . . . . . . 58

5.4.2 Computation of differential cross section . . . . . . . . . . . . . 59

5.5 Compton scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

5.5.1 Causal splitting of Spx´ yq . . . . . . . . . . . . . . . . . . . . 64

5.5.2 Fixation of constant C . . . . . . . . . . . . . . . . . . . . . . . 67

5.5.3 Computation of the differential cross section . . . . . . . . . . . 69

6 Radiative Corrections. 77

6.1 Vacuum polarization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

6.2 Self-Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83



CONTENTS ix

7 (Re)Normalizability of SDKP 99

7.1 Order of singularity of the intermediate distributions by an independent

contraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

7.1.1 Normalization of vacuum polarization tensor . . . . . . . . . . . 104

7.1.2 The non-renormalizability of Self Energy sector . . . . . . . . . 108

7.1.3 The non-renormalizability of Photon-Photon scattering . . . . . 109

7.2 The „ pψ̄ψq2 term . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

8 Conclusions and perspectives 111

A Computations for the General theory 115

A.1 Causality of intermediate distributions . . . . . . . . . . . . . . . . . . 115

A.2 Wick Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

A.3 Power counting function ρpxq . . . . . . . . . . . . . . . . . . . . . . . 117

A.4 Normalized solution for the retarded numerical distribution . . . . . . . 118

A.5 Central splitting solution . . . . . . . . . . . . . . . . . . . . . . . . . . 119

A.6 Symmetry of retarded formulas . . . . . . . . . . . . . . . . . . . . . . 121

A.6.1 Regular Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

A.6.2 Singular Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

B Calculation of differential cross sections using wave packets 125





Chapter 1

Introduction

The Duffin-Kemmer-Petiu (DKP) theory is based on the idea of obtaining a first order

relativistic equation to model photons. This idea was implemented during 1934 by L.

de Broglie, who considered that the photon was composed of two leptons and used a

product of Dirac γ-matrices to construct a similar equation but with square β-matrices

of order 16 [1, 2].

During the years 1936 to 1939 G. Petiau, R. J. Duffin and N. Kemmer [3–5] in-

dividually found that the 16 ˆ 16 β-matrices had three irreducible representations of

dimensions 1, 5 and 10. The representation of order 1 is trivial, the order 5 and 10

representation allow modeling scalar and spin-1 particles respectively1.

After World War II, many calculations were performed on scalar quantum electro-

dynamics using the DKP (SDKP) and Klein-Gordon-Fock (SQED) fields. The main

intention was to determine the differences between the two approaches, but up to 1-loop

corrections all differential cross sections were the same [7–10]. Therefore, the belief on

the total equivalence between both approaches was established in the scientific commu-

nity.

However, in May 1971 the doubt about the equivalence between the two scalar

particle theories was revived. In reference [11], E. Fischbach and collaborators found

different results for the broken-symmetry parameter in the kaon semi-leptonic decay

k Ñ π ` l ` ν. The difference comes from the presence of two mesons with different

masses and from the fact that the SUp3q broken-symmetry process is sensitive to the

field dimensions which takes the value 3{2 for DKP and 1 for KGF fields. Furthermore,

1A good historical development of the Duffin-Kemmer-Petiau equation is in [6].

1



2 1. Introduction

the result obtained via DKP was surprisingly closer to the experimental data than that

obtained via KGF formalism. Currently, the calculation is performed considering the

compositional nature of k and π confirming the values obtained using DKP fields [12].

In 2000, V. Ya. Fainberg and B. M. Pimentel did a systematic study of S-matrices

obtained from SDKP and SQED via minimal coupling procedure with an external or

quantized electromagnetic field. They constructed the functional generator of the Green

functions to quantize the DKP theory. After that, they used the LSZ reduction formula

to determine the matrix elements of the S-matrix [13].

The results of V. Ya. Fainberg and B. M. Pimentel were positive but not conclusive.

The equivalence between SDKP and SQE does not include the sector of diagrams

generated by the self-interaction term „ pφ˚φq2 and diagrams without the presence

of external photons. The authors suggested the inclusion of analogous self-interacting

term with DKP fields proportional to pψ̄Pψq2, where P is a projector that eliminates

the DKP vectorial sector [14].

In the new millennium a rebirth in the interest of DKP theory comes from its advan-

tages compared with the KGF fields. For example, the greater number of combinations

of the DKP fields to generate self-interacting terms [14] has been used to determine an-

alytical solutions of the DKP equation in presence of different kinds of potentials, see

for example [15]. In addition, the DKP theory has been used to study their interactions

in Riemann and Riemann-Cartan spaces [16–20], to study confinement in QCD [21],

as well as applied to covariant Hamiltonian dynamics [22] and to the study of spin-1

particles in the Abelian monopole field [23].

The inclusion of the missing sectors in the article [13] of V. Ya. Fainberg and B.

M. Pimentel is the main objective of this thesis. For this goal, we are going to use an

axiomatic formalism known as Causal Perturbation Theory (CPT). The decision to use

CPT has been taken because of the results obtained by M. Dütsch, F. Krahe and G.

Scharf about SQED [26]. In the framework of CPT they demonstrated the unitarity,

gauge invariance and normalizability without the second order self-interaction term

„ pφ˚φq2. The latter does not mean that the interaction term is missing in CPT,

on the contrary the approach recovers all that sector using a physical property called

perturbative gauge invariance. We must mention that the study of SDKP was initiated

by J. T. Lunardi et al. [24, 25], therefore this thesis could be seen as an extension of

those papers.

CPT is an approach that treats the quantized fields as operator value distributions



3

and constructs the S-matrix as a formal series using two fundamental physical princi-

ples: Causality and perturbative gauge invariance. Each step is mathematically well

defined within the framework of distributions theory. The main point of this formalism

is to avoid the ill-defined product of distributions at the same point such as those that

floods the formalism based on Feynman diagrams and that we believe are the ones that

generate UV divergences.

The origin of CPT started in 1973 when H. Epstein and V. Glaser wrote their

article entitled “The role of locality in perturbation theory” [27] where they devel-

oped an iterative construction of the S-matrix taking as advantage the causal support

of the propagators to determine their advanced and retarded part. Ten years later,

G. Scharf began to apply the approach to study Quantum Electrodynamics (QED)

obtaining a finite theory, in other words, free UV and infrared divergences! [28–37].

From the striking results in QED, G. Scharf and collaborators applied CPT to study

other quantum field theories as Yang-Mills [38–43], Higgs boson [44], Electroweak the-

ory [45,46], Super-symmetry [47–50] and Quantum Gravity [51–61]. On the other side

of the Atlantic ocean, B. M. Pimentel and Collaborators applied CPT in General Quan-

tum Electrodynamics (GQED) [62], Light front Dynamics [63, 64], SDKP [65], gauge

Thirring model [66,67], and QED3 [68, 69].

This thesis is organized in the following form. In the second chapter, we summarize

the concepts of Distribution Theory which we believe necessary to understand CPT. In

Chapter 3, CPT is introduced in generality to be applied to any quantum field theory.

In the fourth chapter, we develop the quantum properties of free DKP, electromagnetic

and fermionic scalar ghost fields to be applied in SDKP and to develop gauge invari-

ance at the quantum level. In the fifth chapter, we use perturbative gauge invariance

to determine the base term of S-Matrix, after that, we determine the differential cross

section of a DKP particle scattered by external electromagnetic field and for the Moller

and Compton scattering process. In the sixth chapter, we compute the vacuum polar-

ization tensor and the self energy function. In the seventh chapter, we will study the

renormalizability of the theory. Finally, in the eight chapter, we write our conclusions

and perspectives.





Chapter 2

Elementary Theory of Distributions

Mathematical discovery is subversive and always ready to overthrow

taboos, and it depends very little on established powers.

Laurent Schwartz

In 1950-51 Laurent Schwartz published Théorie des Distributions [70], a treatise in

two volumes where he constructs systematically the concept of Distribution1. Although

this mathematical tool defines correctly many “functions” used in physics allowing con-

sistent calculation, it has not yet been adopted by the community in all its potentiality.

Following the subversive spirit of L. Schwartz, in this thesis we will use the Bogoliubov-

Epstein-Glaser or CPT approach to solve Quantum Field Theory. CPT uses the theory

of distributions framework in the construction of S-matrix. For this reason, we dedicate

this chapter to present the necessary concepts about Schwartz’s theory.

2.1 The necessity of distribution and its definition

In 1927 P. A. M. Dirac introduce the delta symbol δpxq [72] with the following properties

δpxq “

$

&

%

0, x ‰ 0

8, x “ 0
,

ż

dxδpxq “ 1. (2.1)

1The Theory of Distributions is also known as Theory of Generalized Functions which was the name

that S. L. Sobolev proposed in his study of Cauchy’s problem in hyperbolic equations [71].

5



6 2. Elementary Theory of Distributions

Taking into consideration functional analysis we can prove that the two properties

in (2.1) are in contradiction. Knowing the latter, Dirac said “Strictly, of course, δpxq

is not a proper function of x, but can be regarded only as a limit of a certain sequence

of functions”, but again, in the context of function analysis, this limit does not exist.

The necessity of the Dirac delta function δpxq is a consequence to fix a physical

quantity in a point of space. For example consider the density ρpxq of a point particle

of mass 1. We can understand this quantity as the limit of a sequence of spheres

densities ρεpxq with less and less radius ε but same mass 1. This sequence of densities

have the values

ρεpxq “

$

&

%

1
4
3
πε3
, }x} ď ε

0, }x} ą ε
(2.2)

and we can note that in the limit εÑ 0 we have ρεpxq Ñ δpxq “ ρpxq, but the integral

in all space is null, which have no physical sense for a density [73, 74]. To solve this

problem, we define the weak limit.

Definition 2.1 Consider the continuous fpxq, with x P Rn. For a sequence of func-

tions ρεpxq, the function ρpxq is called the weak limit of that sequence if @fpxq

lim
εÑ0

ż

dxρεpxqfpxq “

ż

dxρpxqfpxq. (2.3)

It is straightforward to show that for the sequence (2.2) we have

lim
εÑ0

ż

dxρεpxqfpxq “

ż

dxδpxqfpxq “ fp0q. (2.4)

The functional result in (2.4) is in concordance with the second Dirac condition in

(2.1), because in that case δpxq must be understood multiplied by a constant function

fpxq “ 1. Consequently, the Dirac delta function δpxq must not be used as a function

in the sense of functional analysis because its mathematical behavior is to map the

function fpxq to its functional fp0q

δ : fpxq Ñ fp0q. (2.5)

The Dirac delta function is not the unique mathematical entity that acts as func-

tional on definite space of functions. For example we have the Heaviside step function

or the principal value operator. This kind of functionals are called distributions or

generalized functions.



2.2. Properties of distributions and the space of test functions T 7

Definition 2.2 A distribution T is a continuous linear functional on space functions

T where the elements f P T are called test functions

T : T Ñ C. (2.6)

The definition 2.2 implies the fulfillment of the following conditions:

1. For each test function fpxq P T , the complex functional value associated is de-

noted by xT, fpxqy.

2. @ tλ1, λ2u P C, @ tf1pxq, f2pxqu P T , xT, λ1f1 ` λ2f2y “ λ1xT, f1y ` λ2xT, f2y

3. If a sequence fi P T converge to a function fpxq P T , then the sequence xT, fiy

converge to xT, fy

Another important concept, associated with the nature of distribution, is the sup-

port. We will define two kinds of support, one that belongs to test functions and another

that belongs to distributions.

Definition 2.3 The support of a test function fpxq is the compact set of points supppfq

where fpxq ‰ 0.

Definition 2.4 The support of a distribution T is the complement of reunion of open

set points supppT q where xT, fy “ 0 for all test functions f .

In general, the distributions T represents the physical law that we want to investigate

and to test. The test functions fpxq are the representation of the external agent which

fluctuate around the point x (supp(f)) where we want to test the physical law.

2.2 Properties of distributions and the space of test

functions T

The space of test functions T appear naturally to give a mathematically well defined

definition of a distribution. The linearity condition of the distribution mapping implies

that T must be a vector space, which means that a distribution T is an element of

the continuous dual space T 1. The continuity property for the mapping T : T Ñ C



8 2. Elementary Theory of Distributions

point out that T must have an inner product x, y to define a norm to use the Cauchy

condition2 [75].

The inner product, necessary to have a well defined theory, guides us to choose

L2 space3 as our first option to construct T . Actually, every function gpxq P L2 is a

distribution over the test function space T “ L2. But there is one problem, the Dirac

delta function do not belongs to L2 space [76]. Furthermore, non continuous functions

belongs to L2, which means that if we take T “ L2 then δpxq R T 1 .

Taking into account the inner product in L2, we can classify the distributions T P T 1

in regulars and singulars.

Definition 2.5 A distribution T is regular if xT, fy could be written as the inner prod-

uct of L2 space

xT, fy “

ż

T pxqfpxqdnx, (2.7)

in other case the distribution is singular.

The latter definition means that δpxq is a singular distribution and the notation

(2.4) is just symbolic. Nevertheless, all properties of distributions could be obtained

from the integral representation (2.7).

To include all the singular distributions, we could use the Schwartz space SpRnq as

space of test functions.

Definition 2.6 The test function space SpRnq is the set of infinite differentiable func-

tions fpxq P C8 that fulfill the following property

lim
}x}Ñ8

}x}k}Dlfpxq} “ 0 (2.8)

for all k P N and l P Nm
0 .4

2We say that a sequence of functionsfnpxq converges to fpxq if for every ε ą 0, exist N P N that

}fm ´ fn} ă ε for every m,n ą N.
3The square-integrable space L2 is that where the inner product is define as xgpxq, fpxqy “

ş

dnxg˚pxqfpxq ă 8.
4The differential operator Dl is defined as

Dlf “
Bl1`...`lm

Bxl1Bxl2 . . . Bxlm
fpxq, l “ pl1, . . . , lmq.



2.2. Properties of distributions and the space of test functions T 9

The property (2.8) tell us that the elements of S decrease faster than any power of

}x}´1. Furthermore, S has the important property that the Fourier transformation of its

elements also belongs to S [77]. This is important because in Quantum Mechanics any

state function must be independent of working in configuration or momentum space.

But the “nice” behavior of S is not enough to define the derivative of a distribution.

It is necessary to define a sub space of S called the close support space C80 pRmq.

Definition 2.7 The space C80 pRmq is the set of functions fpxq P C8 with a compact

support Ω Ă Rm.

It is straightforward to demonstrate that C80 pRmq Ă S and, because of that, S 1 Ă
C810 pRmq. From a physical point of view the distributions T P C810 pRmq express the fact

that it is not possible to define a physical quantity at a point, but in a region around

that point.

Now, considering two distributions T1, T2 P C
81
0 pRmq, they have the following prop-

erties:

• Addition

xT1 ` T2, fy “ xT1, fy ` xT2, fy.

• Multiplication by a complex α

xαT, fy “ xT, αfy “ αxT, fy.

• Translation by a vector xa P Rm

xαT px` xaq, fpxqy “ xαT pxq, fpx´ xaqy.

• Linear transformation of the independent variables x ÞÑ Λx where Rm

xT pΛxq, fpxqy “
1

|detrΛs|
xT pxq, fpΛ´1xqy

• Derivative

x
BnT

Bxni
, fpxqy “ xT, p´1qn

Bnfpxq

Bxni
y

It is possible to extend all these properties to distributions in S 1, considering fp˘8q “

0 for fpxq P S. The product of distributions is an important point and we will present

in the next section.



10 2. Elementary Theory of Distributions

2.3 Product of two distributions

To define the product of two distribution, we have to take one of them as a reference

to investigate the characteristics of the other distribution. Therefore, consider a distri-

bution T P C810 , after that, if we multiply T with a distribution gpxq we want that the

complex value xTg, fy exist. For the latter objective is necessary that

xTg, fy “ xT, fgy. (2.9)

To fulfill the condition (2.9), it is sufficient that the product fpxqgpxq belongs to C80 .

Consequently, if fpxq P C80 , we need the function gpxq to be infinitively differentiable to

guarantee a well define product. In this thesis we will work with this type of products.

Another kind of product, which is well defined, is the tensorial product. For a two

distributions T1 and T2 and a test function fpx, yq over S ˆ S we define the product

T1pxq ˆ T2pyq as

xT1pxq ˆ T2pyq, fpx, yqy ” xT1pxq, xT2pyq, fpx, yqyy “ xT2pyq, xT1pxq, fpx, yqyy (2.10)

2.4 Fourier transform and convolution of distribu-

tions

For an unidimensional function fptq, the direct f̂ptq and inverse f̌ptq Fourier transform

are defined as

f̂ppq “ p2πq´
1
2

ż

dteip.tfptq, (2.11)

f̌ppq “ p2πq´
1
2

ż

dte´ip.tfptq. (2.12)

For a distribution T , its Fourier transformation T̂ goes to the test function as follows

xT̂ , fy ” xT, f̂y. (2.13)

From (2.13), we can see that the following property is true

xT, fy “ xT̂ , f̌y. (2.14)

The convolution product f ˚ g in one dimension is defined as

tf ˚ guptq ”

ż

dxfpt´ xqgpxq “

ż

dxfpxqgpt´ xq. (2.15)



2.4. Fourier transform and convolution of distributions 11

Regarding the definition (2.15), it is straightforward to determine the relation with

the Fourier transform of a product

fg
Ź

ppq “ p2πq´
1
2 tf̂ ˚ ĝuppq, (2.16)

fg

Ź

ppq “ p2πq´
1
2 tf̌ ˚ ǧuppq, (2.17)

f ˚ g
Ź

ppq “ p2πq´
1
2 f̂ppqĝppq, (2.18)

f ˚ g

Ź

ppq “ p2πq´
1
2 f̌ppqǧppq. (2.19)

For all distributions, again we generalize the following property of regular distribu-

tions

xF ˚G, fy “

ż

dxtF ˚Gupxqfpxq “

ż

dxfpxq

ż

dyF pyqGpx´ yq

“

ż

dyF pyq

ż

dxGpx´ yqfpxq “

ż

dyF pyq

ż

dξGpξqfpξ ` yq

(2.20)

Then, with the help of (2.10), the convolution of two distributions T1 ˚ T2 is defined as

xT1 ˚ T2, fy ” xT1pxq ˆ T2pyq, fpx` yqy (2.21)

In m dimensions, all these properties and definitions are

f̂ppq “ p2πq´
m
2

ż

dteip.tfptq, (2.22)

f̌ppq “ p2πq´
m
2

ż

dte´ip.tfptq, (2.23)

xT̂ , fy ” xT, f̂y, (2.24)

xT, fy “ xT̂ , f̌y, (2.25)

tf ˚ guptq ”

ż

dmxfpt´ xqgpxq “

ż

dmxfpxqgpt´ xq, (2.26)

fg
Ź

ppq “ p2πq´
m
2 tf̂ ˚ ĝuppq, (2.27)

fg

Ź

ppq “ p2πq´
m
2 tf̌ ˚ ǧuppq, (2.28)

f ˚ g
Ź

ppq “ p2πq´
m
2 f̂ppqĝppq, (2.29)

f ˚ g

Ź

ppq “ p2πq´
m
2 f̌ppqǧppq, (2.30)

xT1 ˚ T2, fy ” xT1pxq ˆ T2pyq, fpx` yqy. (2.31)





Chapter 3

Causal Perturbation Theory

The latter is one of the most important papers in quantum field theory.

However, for a long time, only a few specialists noticed this important

approach to quantum field theory.

Eberhard Zeidler, writing about the seminal paper of H. Epstein and V.

Glaser [27] in his book [78]

CPT is an axiomatic approach for solving QFT where ill-defined mathematical quan-

tities or computations are avoided due to the use of the theory of distributions (or gen-

eralized function theory) to give the correct mathematical nature to quantum fields as

operator value distributions (OVD). [79,80]

It focuses in the causality property to construct the S-Matrix as a formal perturba-

tive power series in the coupling constant [27, 37, 81], leaving other physical properties

for the end of computation. Even more, the CPT methodology is free of ultraviolet

divergencies as a consequence of not containing ill-defined product of operator value

distributions (OPV) in the same Minkowski space-time point!

In this Chapter we develop the fundamental tools to construct the S-matrix following

the references [37] and [81].

13



14 3. Causal Perturbation Theory

3.1 Axioms of Causal Perturbation Theory

CPT works directly constructing the scattering operator 1 S. In this sense, it follows

the Heisenberg program for QFT [82] which consider the in and out Fock spaces Fin

and Fout respectively. The space Fin is the set of all multi-particle states |φyin before

the scattering process and Fout is the set of all multi-particle states |ψyout after. Then,

the operator S is defined as the bijective application

S : Fin Ñ Fout. (3.1)

Consequently, the transition from the in to out state is

S : |φyin Ñ |ψyout ” S|φyin, (3.2)

and the transition amplitude A from the state |φyin to |ψyout is computed as

Ap|φyin, |ψyoutq ” p|ψyout, S|φyinq “out xψ|S|φyin. (3.3)

But in contrast to the usual construction of S via temporal order product, H. Epstein

and V. Glaser used the formalism developed by Bogoliubov [83] where a test function

is introduced to give the correct mathematical nature to the quantum fields as OVD

which appear in the temporal product2.

Bogoliubov uses a function gpxq P r0, 1s to control the long range interaction which

causes the infrared divergences. The function gpxq is named switching on-off function.

If in a space-time region gpxq “ 0 the interaction is switched-off, if 0 ă gpxq ă 1 the

interaction is partially switched-on, and if gpxq “ 1 the interaction is fully switched-on.

We choose gpxq “ 0 for x0 “ ˘8, and gpxq ‰ 0 for a time interval x0 P rton, toff s

where the interaction scattering is stronger. Furthermore, gpxq must belong to C80 or

S to allow the derivatives of singular OVDs. Via this reasoning, we conclude that the

operator S must be a functional of g

S “ Srgs. (3.4)

1The S-Matrix and the scattering operator are different concepts but intimately related. The S-

Matrix is the collection of all possibles transition amplitudes in a scattering process, but in this work

we will use the two concepts as the same as usual.
2Remember from Chap. 2. that a distribution needs to be applicated in a test function space.



3.1. Axioms of Causal Perturbation Theory 15

Now, considering a perturbative construction of S, H. Epstein and V. Glaser [27]

propose the following ansatz as a formal series expansion

Srgs ” 1`

ż

d4x1T1px1qgpx1q `
1

2!

ż

d4x2d
4x1T2px1, x2qgpx1qgpx2q ` . . .

”

8
ÿ

0

1

n!

ż

d4x1 . . . d
4xnTnpx1, . . . , xnqgpx1q . . . gpxnq

” 1` T,

(3.5)

where } “ 1 “ c as usual and Tnpx1, . . . , xnq is called n-point distribution which are the

terms that we need to find. The factor n! is used to indicate the symmetry property of

Tn when a permutation of coordinates is done.

Using unitarity, Bogoliubov found that the Tn distributions were the temporal prod-

uct of interaction Lagrangian [83]. Instead of using the unitarity property of S-matrix,

H. Epstein and V. Glaser postulate four axioms to constrain Tn and then develop an

iterative construction with the guide of causality condition [27]. G. Scharf modernize

the approach and applicated it to QED [37].

3.1.1 Axiom I: Boundary Condition

This axiom constrains the spaces Fin and Fout. We postulate that in the temporal

limits t Ñ ˘8 the particle systems are asymptotically free, inclusive in the adiabatic

limit gpxq Ñ 1. Consequently, the two Fock spaces, in and out, are free multi-particle

spaces.

We can comment that in the adiabatic limit gpxq Ñ 1, (3.5) can be written as

Srgs “ 1`
8
ÿ

n“1

λnSn (3.6)

where λ is the coupling constant of the gauge theory and the convergence of the series

depends on its intensity.

The most important consequence from this axiom is that the theory is determined

via the free field operators, therefore the mass and charge in the free wave equations

are the physical quantities.



16 3. Causal Perturbation Theory

3.1.2 Axiom II: Base Term and Perturbative Gauge Invariance

We postulate that the construction of the S-matrix, as a formal series (3.5), will be

done inductively from the definition of the first term T1pxq which will be different for

every gauge theory.

The systematized methodology to construct the term T1pxq is the major contribu-

tion of G. Scharf and collaborators to CPT. In a series of articles [30, 36, 38–61], they

determine the conditions for the gauge invariant transformation for every term of the

series (3.5), and apply it to construct Yang-Mills and Electroweak theories. They call

this formalism Perturbative Gauge Invariance (PGI), and it complements CPT. PGI

will be developed in Chapter 5.

3.1.3 Axiom III: Poincaré Invariance

Similar to the usual framework, CPT must be invariant under translation and Lorentz

transformation. These transformations must be done on the test functions gpxq because

of the functional nature of S-matrix.

If an observator O uses the test function gpxq to study a physical phenomenon, then

an observator O1 translated to x ` a, must use a test function ga “ gpx ´ aq. Or if O1

is boosted or rotated to Λx, the test function must be gΛ “ gpΛ´1xq. In both cases the

S-matrix must be invariant.

If Upa,Λq is a representation of translation a or Lorentz transformation Λ in Fock

space F , then the transformation rule of S-matrix is

S 1 “ Upa,ΛqSU´1
pa,Λq, (3.7)

Therefore, the Poincaré invariance implies

S “ Upa,ΛqSU´1
pa,Λq, (3.8)

3.1.4 Axiom IV: Causality

This is the principal axiom to construct the S-matrix. CPT postulates that there exists

a parameter which order the evolution of events in space-time.

In this thesis we will use the temporal parameter x0 to order the S-matrix scattering

events. Then, because of the functional dependence on the switch on-off test functions

gpxq, we will time order S regarding gpxq.



3.2. Iterative construction of S-Matrix 17

Definition 3.1 By considering two test functions g1pxq and g2pxq with disjoint sup-

ports, then if @x1 P supprg1s and @x2 P supprg2s we can define the time-ordering rule

x0
1 ă x0

2 ñ supprg1s ă supprg2s. (3.9)

Now, if for a reference system, the S-matrix depends on two test functions g1 and g2,

the time-ordering rule supprg1s ă supprg2s implies the following causal decomposition

Srg1 ` g2s “ Srg2sSrg1s. (3.10)

3.2 Iterative construction of S-Matrix

Regarding the four axioms of CPT, we proceed to construct term by term the pertur-

bative series (3.5). Of course, the first step is to define the one point distribution T1pxq.

As mentioned in the axiom II, each gauge theory presents its own term T1pxq. We will

describe the construction of T1 for scalar QED in Chapter 5. In this section we describe

the second step which focuses in determining the term Tn from the knowledge of the

previous terms tTn´1, . . . , T1u.

3.2.1 Properties of the n-point distributions

Because the main elements to be computed are the n-point distributions Tn, it will

be useful to determine some of their properties that come from the properties of the

S-matrix:

1. The inverse S´1 will be determined in two forms, by inverting (3.5) as

S´1
“ p1` T q´1

“ 1`
8
ÿ

r“1

p´T qr, (3.11)

and as formal series

S´1
“ 1`

8
ÿ

n“1

1

n!

ż

d4x1 . . . d
4xn rTnpx1, . . . , xnqgpx1q . . . gpxnq, (3.12)

where rTnpx1, . . . , xnq is an n-point distribution that is symmetric under the per-

mutations of xi and 1 is the identity matrix. rTn is not the inverse of Tn, but it



18 3. Causal Perturbation Theory

can be determined as a function of the set tT1, . . . , Tnu using the fact that the

right-hand sides of equations (3.11) and (3.12) are equal

rTnpx1, . . . , xnq “
n
ÿ

r“1

p´1qr
ÿ

Pr

Tn1pX1q . . . TnrpXrq, (3.13)

where the sum is over all partitions Pr of the set X “ tx1, . . . , xnu in r disjoints

and not empty sub-sets Xi.

2. Making use of

1 “ SrgsS´1
rgs “ S´1

rgsSrgs, (3.14)

we obtain
ÿ

P 0
2

Tn1pXqT̃n´n1pZzXq “ 0, (3.15)

ÿ

P 0
2

Tn´n2pZzY qT̃n2pY q “ 0, (3.16)

ÿ

P 0
2

T̃n´n1pXqTn1pZzXq “ 0, (3.17)

ÿ

P 0
2

T̃n2pZzY qTn´n2pY q “ 0, (3.18)

where the sums run over all two partitions P 0
2 of the set Z “ tx1, . . . , xnu in two

disjoints sub-sets X and Y allowing the cases where X “ H or Y “ H.

3. From Poincaré invariance, we determine

Tnpx1, . . . , xnq “ Tnpx1 ` a, . . . , xn ` aq, (3.19)

T px1, . . . , xnq “ T pΛx1, . . . ,Λxnq. (3.20)

4. From causality, we can determine that the n-point distributions are well defined

time ordered product. If tx0
1, . . . , x

0
mu ą tx

0
m`1, . . . , x

0
nu, thus

Tnpx1, . . . , xm, xm`1, . . . , xnq “ Tmpx1, . . . , xmqTn´mpxm`1, . . . , xnq, (3.21)

and for rTn, we have

rTnpx1, . . . , xm, xm`1, . . . , xnq “ rTn´mpxm`1, . . . , xnqrTmpx1, . . . , xmq. (3.22)



3.2. Iterative construction of S-Matrix 19

3.2.2 From Tn´1 to Tn

In the computation of Tn, the main objective is to avoid the naive procedure to deter-

mine the advanced and retarded parts of a causal propagator via the multiplication to

a Heaviside step function Θptq. Because Θptq R C8, the naive product could not exist

as mentioned in Chapter 2.

First of all, from the knowledge of tTn´1, . . . , T1, T̃n´1, . . . , T̃1u, we define the inter-

mediate distributions A1n and R1n as

A1npx1, . . . , xnq ”
ÿ

P2

rTn1pXqTn´n1pY, xnq, (3.23)

R1npx1, . . . , xnq ”
ÿ

P2

Tn´n1pY, xnqrTn1pXq, (3.24)

where the sum runs over all partitions P2 of the set tx1, . . . , xn´1u in two non-empty

and disjoints sub-sets X and Y . This product is well define because it is done with

distributions defined in different space-points.

The next step is to extend the sums (3.23) and (3.24) allowing for the empty sub-set

X “ H

Anpx1, . . . , xnq ”
ÿ

P 0
2

rTn1pXqTn´n1pY, xnq, (3.25)

Rnpx1, . . . , xnq ”
ÿ

P 0
2

Tn´n1pY, xnqrTn1pXq, (3.26)

where T0 “ 1 “ T̃0 and P 0
2 represents the inclusion of empty sets. We will show that the

distributions An and Rn are the retarded and advanced distributions which we want to

determine. Furthermore, it is straightforward to rewrite the sums (3.25) and (3.26) as

Anpx1, . . . , xnq “ A1npx1, . . . , xnq ` Tnpx1, . . . , xnq, (3.27)

Rnpx1, . . . , xnq “ R1npx1, . . . , xnq ` Tnpx1, . . . , xnq. (3.28)

In equations (3.27) and (3.28) just R1n and A1n are known. If we determine An or

Rn through the use of another methodology, then we can determine the Tn by

Tnpx1, . . . , xnq “

$

&

%

Anpx1, . . . , xnq ´ A
1
npx1, . . . , xnq,

Rnpx1, . . . , xnq ´R
1
npx1, . . . , xnq.

(3.29)

The latter is possible in the framework of distribution theory.



20 3. Causal Perturbation Theory

3.2.3 Supports of the retarded Rn and advanced An distribu-

tions

The most important property of the distributions Rn and An is their support. To

identify this property, we are going to invoke the following theorem3

Theorem 3.1 Consider three sets of space points Y , P and Q such as Y “

P YQ, P ‰ H, P XQ “ H, |Y | “ n´1, and the point x such that x{PY , then:

• If tQ, xu ą P , |Q| “ n1, therefore

R1npY, xq “ ´Tn1`1pQ, xqTn´pn1`1qpP q (3.30)

• If tQ, xu ă P , |Q| “ n1, therefore

A1npY, xq “ ´Tn´pn1`1qpP qTn1`1pQ, xq (3.31)

Now, we can study the support of Rn. If Y “ tx1, . . . , xn´1u, then we can write

(3.28) as

RnpY, xnq “ R1npY, xnq ` TnpY, xnq. (3.32)

Now, we have three cases for time ordering the whole set tY, xnu
4:

• Case one: Y ą xn.

• Case two: xn ą Y .

• Case three: Q ą xn ą P , where Y “ P YQ.

In the second and third cases, we can use the theorem 3.1 to rewrite (3.32) in the

following form

RnpY, xnq “ ´Tn1`1pQ, xnqTn´pn1`1qpP q ` TnpP YQ, xnq, (3.33)

where if we use the causal decomposition for the n-point distribution, we get

RnpY, xnq “ ´Tn1`1pQ, xnqTn´pn1`1qpP q ` TnpP YQ, xnq

“ ´T pQ, xnqT pP q ` T pQ, xnqT pP q

“ 0

(3.34)

3The proof can be seen in Appendix A.1.
4For simplicity, we will write all causality conditions obviating the zero super-index.



3.2. Iterative construction of S-Matrix 21

In conclusion, from (3.34) the unique case where Rn ‰ 0 is for the time order

xn ă tx1, . . . , xn´1u, (3.35)

and for An, we can get the time ordering condition xn ą tx1, . . . , xn´1u in the non-null

case.

Considering the Lorentz invariance of the n-point distributions Tn, it is not difficult

to extend it to Rn and An. This is important because the causal condition (3.35) must

be the same for all reference system, and of course, the set tx1, . . . , xnu must be in the

light-cone with origin in xn.

To formalize the last deduction, we will define the 4n-dimension light-cone centered

in y as

Γ˘n pyq ” tpx1, . . . , xnq{xi P V̄
˘
pyqu, (3.36)

where V̄ ˘pyq are the closed forward and backward light-cone centered in y

V̄ `pyq ”
 

x{px´ yq2 ě 0, x0
ě y0

(

, (3.37)

V̄ ´pyq ”
 

x{px´ yq2 ě 0, x0
ď y0

(

, (3.38)

respectively.

Regarding (3.36), we conclude for the supports of Rn and An distributions the

following two properties

supprRnpx1, . . . , xnqs Ď Γ`n´1pxnq, (3.39)

supprAnpx1, . . . , xnqs Ď Γ´n´1pxnq, (3.40)

respectively.

3.2.4 The causal distribution Dn

The results (3.39) and (3.40), tell us that Rn and An are the retarded and advanced

parts of a subtraction

Dnpx1, . . . , xnq ” Rnpx1, . . . , xnq ´ Anpx1, . . . , xnq, (3.41)

where Dn is called causal distribution because its support will be the union of the

supports of Rn and An

supprDnpx1, . . . , xnqs Ď tΓ
`
n´1pxnq Y Γ´n´1pxnqu. (3.42)



22 3. Causal Perturbation Theory

The causal distribution is fully computable from (3.27) and (3.28) as

Dnpx1, . . . , xnq “ R1npx1, . . . , xnq ´ A
1
npx1, . . . , xnq (3.43)

The result (3.43), is the starting point for the computation of Tn. Because supprRnsX

supprAns “ txnu, it is possible to determine Rn or An splitting Dn with the specific

supports in the framework of distribution theory. The splitting process is called causal

splitting and will be developed in the next section.

3.3 Causal splitting Procedure

In the usual framework, the splitting of a causal distribution in its advanced or retarded

part is done by the naive multiplication by the Heaviside step function [84]. However,

this product is not always well defined because in quantum field theory there exist

causal singular distributions. As demonstrated by G. Scharf, in QED [37] this naive

procedure was the origin of ultraviolet divergences.

Then, we need to determine how to split correctly a causal distribution. First of all,

we must remember that, in the usual framework, the UV divergence is related with two

sources: the short distance behavior of the causal propagators and the bare physical

parameters as mass and charge of particles [85–92].

Because in CPT it is postulated that the mass and charge are the physical quantities,

this implies that we need to study the behavior of Dn in a vicinity of xn. The latter is

possible just in the numerical parts of the causal distribution Dn.

3.3.1 Numerical distribution dn

From the properties of the n-point distributions Tn and T̃n, it is not difficult to note

that the intermediate distributions R1 and A1 could be written as products of normal

order operators. Therefore, we have

D2px1, . . . , xnq “
ÿ

k

:
ź

j

Opxjq : dknpx1, . . . , xnq, (3.44)

where Opxjq represents all operator value distributions (OVD) and dknpx1, . . . , xnq is

the numerical part of each term in the sum (3.44) obtained via contractions of Wick



3.3. Causal splitting Procedure 23

Figure 3.1: Graph with four external legs and connected n points represented by dn.

theorem5. In general, we can represent each term of the summation (3.44) graphically,

the uncontracted operator value distribution fields represent external legs and the nu-

merical distributions dknpx1, . . . , xnq represent the connection of these legs as in Fig.

(3.1).

The numerical part dkn is what we will causal-split. Using the Poincaré invariance,

we can translate dn by xn obtaining

dknpx1, . . . , xnq “ dknpx1 ´ xn, . . . , xn´1 ´ xn, 0q ” dpx̃q, (3.45)

where we define dpx̃q as the general notation to denote each numerical distribution to

be split and x̃ “ px̃1, . . . , x̃n´1q where x̃i “ xi ´ xn.

From (3.45) we can note that the short distance behavior means the mathematical

behavior of dpx̃q in the limit x̃j Ñ 0. Furthermore, we can see that the UV divergence

problem is the ill-defined product with the Heaviside step functions Θpx0
j ´ x0

nq where

j “ 1, . . . , n´ 1 due to its ill defined limit lim
x0jÑx

0
n

Θpx0
j ´ x

0
nq.

3.3.2 Singular and Regular distributions

Following section (2.3), to causal split dpx̃q, we will construct the function χptq P C8

over R1

χptq ”

$

’

’

&

’

’

%

0 when t ď 0,

r0, 1s when 0 ă t ă 1,

1 when t ě 1.

(3.46)

5The Wick theorem is developed in Appendix (A.2).



24 3. Causal Perturbation Theory

There is no mathematical ill definitions in the product between χptq and any distri-

bution T P C810 . Furthermore, it is not difficult to note that Θptq will be constructed

as the limit

Θptq “ lim
αÑ0`

χp
t

α
q, (3.47)

this last property was the reason to construct χptq, because with its help we will multiply

χp
t

α
q by a causal distribution, then take the limit ॠ0` to obtain its retarded part.

We will generalize the definition (3.46) to m “ 4n ´ 4 dimensions with the help of

a retarded vector v P Γ`n´1p0q and define the function χαpx̃q as

χαpx̃q ” χp
v.x̃

α
q, (3.48)

where v “ pv1, . . . , vn´1q, and the product v.x̃ is defined as

v.x̃ ”
n´1
ÿ

i“1

gµνv
µ
i x̃

ν
i . (3.49)

Regarding (3.49), we can see that the space-like hyperplane

v.x̃ “ 0, (3.50)

split the causal support as show in Fig. (3.2).

From (3.48) and (3.49), we can compute that for all x̃i P V̄
´ we have lim

αÑ0`
χαpx̃q “ 0,

and for all x̃i P V̄
` we get lim

αÑ0`
χαpx̃q “ 1. This is the desired behavior to obtain the

retarded part rnpx̃q of dnpx̃q via the multiplication rnpx̃q “ χαpx̃qdnpx̃q. The problem

is to determine in which cases the following weak limit exists

xrnpx̃q, fpx̃qy “ lim
αÑ0`

xχαpx̃qdpx̃q, fpx̃qy, (3.51)

for all test functions fpx̃q P C80 .

As a Cauchy sequence labeled by α, we need to demonstrate that for all real value

ε ą 0, there exists a real value δ ă ε such that for all α and β, with values on the

interval 0 ă tα, βu ă δ, the following inequality is fulfilled

}xχβpx̃qdpx̃q, fpx̃qy ´ xχαpx̃qdpx̃q, fpx̃qy} ă ε. (3.52)

Taking β as β “ α{a, where a P R is fixed, and defining the function ψpx{aq as

ψ

ˆ

x̃

α

˙

“ χ

ˆ

v.x̃
α
a

˙

´ χ

ˆ

v.x̃

α

˙

, (3.53)



3.3. Causal splitting Procedure 25

Figure 3.2: Split of the causal support by the hyperplane v.x “ 0.

we can rewrite (3.52) in the following form

}xψp
x̃

α
qdpx̃q, fpx̃qy} ă ε. (3.54)

Because ψp
x̃

α
q P C8, it could be interchanged with fpx̃q

}xfpx̃qdpx̃q, ψp
x̃

α
qy} ă ε. (3.55)

In order to eliminate the α dependence of the new test function ψ, we can re-scale

the variable as x̃Ñ αx̃

}xfpαx̃qαmdpαx̃q, ψpx̃qy} ă ε. (3.56)

In the limit ॠ0`, we could think that the left hand side of (3.56) is null, but this

is not true for distributions dpαx̃q which increase faster than αm in the neighborhood

of α “ 0. For this reason, we introduce the function ρpαq to characterize the increase

behavior of dpαx̃q and define the quasi-asymptotic distribution d0pαx̃q.

Definition 3.2 A distribution dpxq P C 180 pR
mq has a quasi-asymptotic d0pxq

over x “ 0, if for a positive function ρpαq (α ą 0) the limit

lim
αÑ0`

xρpαqαmdpαxq, ψpxqy “ xd0pxq, ψpxqy ‰ 0, (3.57)

exists



26 3. Causal Perturbation Theory

With the help of (3.57), we can multiply and divide the left hand side of (3.56) by

ρpαq

xfpαx̃qαmdpαx̃q, ψpx̃qy “

“
1

ρpαq
xfpαx̃qρpαqαmdpαx̃q, ψpx̃qy

“
1

ρpαq

”

fp0qxρpαqαmdpαx̃q, ψpx̃qy `Dp1qfp0qxxρpαqαm`1dpαx̃q, ψpx̃qy ` . . .
ı

,

(3.58)

where, in the last equality, we did the Taylor series expansion for fpαxq around x “ 0.

In (3.58), after the first term, we have factors proportional to αm`i, with i “ 1, 2, . . .,

which decrease more rapidly than ρpαqdpαxq. Then, in the limit αÑ 0`, we have

xfpαx̃qαmdpαx̃q, ψpx̃qy «
fp0q

ρpαq
xd0px̃q, ψpx̃qy. (3.59)

As shown in Appendix (A.3), we could use the result (A.13) to replace ρpαq “

αωLpαq in (3.59)

xfpαx̃qαmdpαx̃q, ψpx̃qy «
fp0q

αωLpαq
xd0px̃q, ψpx̃qy, (3.60)

where ω P R`, and Lpαq is a slow varying or quasi-constant function of α in the

neighborhood of α “ 0.

From (3.60), we can conclude that the condition (3.56) is fulfilled, for all test func-

tions fpxq, just in the case where ω ă 0. For ω ě 0, the condition is fulfilled for a finite

subgroup of C80 . To show the latter we go back to the Taylor expansion of the test

function fpxq in (3.58)

xfpαx̃qαmdpαx̃q, ψpx̃qy “

“
1

αωLpαq

”

ω
ÿ

|l|“0

1

l!
rDlf sp0qxxlρpαqαm`ldpαx̃q, ψpx̃qy

`

8
ÿ

|l|“ω`1

1

l!
rDlf sp0qxxlρpαqαm`ldpαx̃q, ψpx̃qy

ı

“

ω
ÿ

|l|“0

1

αω´lLpαq

1

l!
rDlf sp0qxxlρpαqαmdpαx̃q, ψpx̃qy

`

8
ÿ

|l|“ω`1

αl´ω

Lpαq

1

l!
rDlf sp0qxxlρpαqαmdpαx̃q, ψpx̃qy,

(3.61)



3.3. Causal splitting Procedure 27

where xl “ xl11 . . . x
ln
n with l “ l1 ` . . .` ln, and the sum runs over all possible l.

From (3.61), we can note that in the case ω ě 0, only the test functions where

rDlf sp0q “ 0 with l “ 1, . . . , ω allow the condition (3.56).

Therefore, in order to causal-split dpx̃q valid for all test functions f , we could define

the projection

W : f ÑWf, (3.62)

Wf “ fpxq ´ wpxq
ω
ÿ

|l|“0

1

l!
rDlf sp0qxl, (3.63)

where wpxq P S has the following properties wp0q “ 1 and rDνwsp0q “ 0 for all

ν “ 1, . . . , ω. Over the new test functions Wf the existence condition (3.56) is valid.

In conclusion, in order to determine the retarded part rn via (3.51), we need to

compute the quantity ω first. For its importance and nature, ω is knowing as order

of singularity because it could be used to classify the distributions as regular or

singular in the cases where ω ă 0 or ω ě 0, respectively. As demonstrated by G.

Scharf et al., ω is the formal form of the superficial degree of divergence used in the

standard formalism based on Feynman diagrams.

In summary, to obtain the retarded part rpx̃q of a causal distribution dpx̃q, we need

to follow these steps:

1. Determine the power counting function ρpxq to obtain the quasi-asymptotic dis-

tribution d0pxq defined in (3.57).

2. Determine the order of singularity ω via

lim
αÑ0`

ρpaαq

ρpαq
“ aω. (3.64)

3. If ω ă 0, the retarded part rpx̃q is obtained from

xrpx̃q, fpx̃qy “ lim
αÑ0`

xχp
v.x̃

α
qdpx̃q, fpx̃qy “ xΘpv.x̃qdpx̃q, fpx̃qy. (3.65)

4. If ω ě 0, the retarded part rpx̃q is obtained from

xrpx̃q, fpx̃qy “ lim
αÑ0`

xχp
v.x̃

α
qdpx̃q,Wfpx̃qy “ xΘpv.x̃qdpx̃q,Wfpx̃qy. (3.66)



28 3. Causal Perturbation Theory

3.3.3 Uniqueness of the retarded part rpxq

In the regular case, the solution for rpxq is unique. In this section we want to show

that in the singular case it is not.

First of all, we want to emphasize the characteristics of projected test functions

Wfpxq. From (3.63), it is not difficult to rewrite Wfpxq as

Wf “ xω`1gpxq. (3.67)

Then, the following property is fulfilled

Dl
pWfq

ˇ

ˇ

ˇ

x“0
“ 0, for all l “ 0, . . . , ω. (3.68)

Now, we can define the retarded part r̃pxq

r̃pxq ” rpxq `
ω
ÿ

l“0

ClD
lδpxq, (3.69)

where Cl are constants. By construction, we can show that r̃pxq generates the same

result as r

xr̃, fpxqy “ xΘpxqdpxq `
ω
ÿ

l“0

ClD
lδpxq,Wfpxqy

“ xΘpxqdpxq,Wfpxqy `
ω
ÿ

l“0

ClxD
lδpxq,Wfpxqy

“ xΘpxqdpxq,Wfpxqy “ xrpxq, fpxqy

(3.70)

The result (3.70) demonstrated that in the singular case ω ě 0, the most general

solution for the retarded part of dpxq is (3.69).

3.4 Causal-splitting procedure in momentum space

As in the standard framework, we are going to present the computation of the retarded

part rpxq in momentum space using the properties described in section 2.4.



3.4. Causal-splitting procedure in momentum space 29

3.4.1 Regular distribution Case

In a regular distribution case, using (3.65) we have

xr̂ppq, f̌ppqy “ xΘpv.x̃qdpx̃q
Ź

ppq, f̌ppqy

“ p2πq´
m
2 xΘ̂ppq ˚ d̂ppq, f̌ppqy

“ xp2πq´
m
2

ż

dmkΘ̂pp´ kqd̂pkq, f̌ppqy,

(3.71)

then

r̂ppq “ p2πq´
m
2

ż

dmkΘ̂pp´ kqd̂pkq. (3.72)

In order to determine Θ̂pqq, we choose a vector v “ p1,0, 0, . . .q where 0 tell us that

the first three spacial coordinates of v are null. Then, we have

Θ̂pqq “ p2πq
m
2
´1δpq1, q2, . . . , qn´1q

i

q0
1 ` i0

`
. (3.73)

Replacing (3.73) into (3.72), we obtain

r̂ppq “ p2πq´1

ż

dk0
1

id̂pk0
1,p1, . . . , pn´1q

p0
1 ´ k

0
1 ` i0

`
. (3.74)

Regarding that pi P tΓ
`
1 Y Γ´1 u and making the substitution k0

1 “ t1p
0
1 in (3.74), we

obtain

r̂ppq “ p2πq´1Sgnpp0
1q

ż

dt1
id̂pt1p

0
1,p1, . . . , pn´1q

1´ t1 ` iSgnpp0
1q0

`
, (3.75)

where Sgn represents the sign function.

Apparently, in (3.75), we lost the covariance but because d̂ppq is Lorentz invariant

we can make the computation in a reference system where p1 “ 0, then making the

boost ptp0
1,0q Ñ ptp0

1, tp1q we will obtain

r̂ppq “ p2πq´1Sgnpp0
1q

ż

dt1
id̂pt1p1, . . . , pn´1q

1´ t1 ` iSgnpp0
1q0

`
. (3.76)

The result (3.76) shows that in the computation of r̂ppq we could choose the variables

tp2, p3, . . . , pn´1u arbitrarily. Of course, if we take v “ p0, . . . , v0
j “ 1,vj “ 0, . . . , 0q, we

finally obtain a momentum dependence on pj P tΓ
`
1 Y Γ´1 u.



30 3. Causal Perturbation Theory

To obtain r̂ppq independent of a specific variable pj, we must multiply (3.71) by

n´ 1 step functions
n´1
ś

j“0

Θpx0
jq giving us the following formula

r̂ppq “

ˆ

i

2π

˙n´1

Sgnp
n´1
ź

j“0

p0
jqˆ

ˆ

ż n´1
ź

j“0

dtj

«

n´1
ź

j“0

1

1´ tj ` iSgnpp0
jq0

`

ff

d̂pt.pq,

(3.77)

where t.p “ t1.p1 ` . . .` tn´1.pn´1 and p P tΓ`n´1 Y Γ´n´1u.

For a two-point retarded part, the formula (3.77) will be

r̂ppq “
i

2π
Sgnpp0

q

8
ż

´8

dt
d̂ptpq

1´ t` iSgnpp0q0`
, p P Γ`1 Y Γ´1 . (3.78)

3.4.2 Singular distribution Case

Similarly to (3.71), in the singular case we have

xr̂ppq, f̌ppqy “ xΘpv.x̃qdpx̃q
Ź

ppq,Wf

Ź

ppqy

“ p2πq´
m
2 xΘ̂ppq ˚ d̂ppq,Wf

Ź

ppqy.
(3.79)

From (3.63), we can compute the term Wf

Ź

ppq

Wf

Ź

ppq “
”

fpxq ´ wpxq
ω
ř

|l|“0

1
l!
rDlf sp0qxl

Ź

ı

ppq

“ f̌ppq ´
ω
ÿ

|l|“0

1

l!
rDlf sp0q

”

wpxqxl

Ź

ı

ppq

“ f̌ppq ´
ω
ÿ

|l|“0

1

l!
rDlf sp0qrilDl

pw̌ppqs.

(3.80)

The term rDlf sp0q could be written as

rDlf sp0q “ xδpxq,Dlfpxqy “ p´1qlxDlδpxq, fpxqy

“ p´1qlxp2πq´
m
2 p´ip1ql, f̌pp1qy “ p2πq´

m
2 xpip1ql, f̌pp1qy,

(3.81)

and replacing into (3.80), we obtain



3.4. Causal-splitting procedure in momentum space 31

Wf

Ź

ppq “ f̌ppq ´
ω
ÿ

|l|“0

1

l!
p2πq´

m
2 xpip1ql, f̌pp1qyrilDl

pw̌ppqs

“ f̌ppq ´ p2πq´
m
2

ω
ÿ

|l|“0

p´1ql

l!
xp1l, f̌pp1qyrDl

pw̌ppqs.

(3.82)

Now, replacing (3.82) into (3.79), we get for the term xr̂ppq, f̌ppqy the following result

xr̂ppq, f̌ppqy

“ p2πq´
m
2 xΘ̂pkq, xd̂pp´ kq, f̌ppqyy

´ p2πq´m
ω
ÿ

|l|“0

p´1ql

l!
xΘ̂pkq, xd̂pp´ kq,Dl

pw̌ppqyyxp
1l, f̌pp1qy

“ p2πq´
m
2 xΘ̂pkq, xd̂pp´ kq, f̌ppqyy

´ p2πq´m
ω
ÿ

|l|“0

p´1ql

l!
xΘ̂pkq, xd̂pp1 ´ kq,Dl

p1w̌pp
1
qyyxpl, f̌ppqy,

(3.83)

where in the last line we interchange p and p1.

In (3.83), the distribution result with the step function could be written as an

integral. Also we could factorize the test function f̌ppq and obtain

xr̂ppq, f̌ppqy “ p2πq´
m
2

ż

dkΘ̂pkqxd̂pp´ kq, f̌ppqy

´ p2πq´m
ω
ÿ

|l|“0

p´1ql

l!

ż

dkΘ̂pkqxd̂pp1 ´ kq,Dl
p1w̌pp

1
qyxpl, f̌ppqy

“ xp2πq´
m
2

ż

dkΘ̂pkqd̂pp´ kq, f̌ppqy

´ xp2πq´m
ż

dkΘ̂pkq
ω
ÿ

|l|“0

p´1ql

l!
xd̂pp1 ´ kq,Dl

p1w̌pp
1
qypl, f̌ppqy.

(3.84)



32 3. Causal Perturbation Theory

Again, comparing the two sides of equation (3.84), we get the formula

r̂ppq “ p2πq´
m
2

ż

dkΘ̂pkqd̂pp´ kq

´ p2πq´m
ż

dkΘ̂pkq
ω
ÿ

|l|“0

plp´1ql

l!
xd̂pp1 ´ kq,Dl

p1w̌pp
1
qy

“ p2πq´
m
2

ż

dkΘ̂pkqd̂pp´ kq

´ p2πq´m
ż

dkΘ̂pkq
ω
ÿ

|l|“0

pl

l!
xDl

p1 d̂pp
1
´ kq, w̌pp1qy

“ p2πq´
m
2

ż

dkΘ̂pkqd̂pp´ kq

´ p2πq´m
ż

dkΘ̂pkq
ω
ÿ

|l|“0

pl

l!

ż

dp1rDl
p1 d̂pp

1
´ kqsw̌pp1q,

(3.85)

where in the second equality we used the definition of distribution’s derivative, and in

the second term of third equality we wrote the distribution result as an integral over p1.

The formula (3.85) depend on function w̌ppq. We could eliminate the latter depen-

dence regarding the non-uniqueness of rpxq. In the momentum space, the most general

solution r̂ppq will be obtained from the Fourier transformation of (3.69)

ˆ̃rppq “ r̂ppq `
ω
ÿ

l“0

Ĉlp
l. (3.86)

The formula (3.86) tells us that we can add to r̂ppq any polynomial, of degree equal

or less than ω, to obtain an equivalent solution. The latter property allows us to define

the normalized solution r̂qppq in the following form

r̂qppq “ r̂ppq ´
ω
ÿ

b“0

pp´ qqb

b!
rDbr̂spqq Ø rDbr̂qspqq “ 0 for all b ď ω, (3.87)

where q P Rm is a fixed point.

In Appendix A.4, we show the computation to get the following explicit form for

r̂qppq

r̂qppq “ p2πq
´m

2

ż

dkΘ̂pkq
”

d̂pp´ kq ´
ω
ÿ

b“0

pp´ qqb

b!
Db
qd̂pq ´ kq

ı

. (3.88)

Because q could be any point of Rm, we define the central splitting solution



3.4. Causal-splitting procedure in momentum space 33

r̂0ppq when we choose q “ 0

r̂0ppq “ p2πq
´m

2

ż

dkΘ̂pkq
”

d̂pp´ kq ´
ω
ÿ

b“0

pb

b!
Db
qd̂pq ´ kq

ˇ

ˇ

ˇ

q“0

ı

. (3.89)

As show in Appendix A.5, we could find a nice formula for r̂0ppq by taking into ac-

count the indetermination of vector v in the construction of the Heaviside step function.

For a two-point retarded distribution, the solution takes the following form

r̂0ppq “
i

2π
Sgnpp0

q

ż

dt
d̂ptpq

pt´ i0`qω`1p1´ t` iSgnpp0q0`q
. (3.90)

Summarizing, in the momentum space, the procedure to obtain the retarded part

of a causal distribution is:

1. Compute the Fourier transform of numerical causal distribution d̂ppq.

2. Determine the power counting function ρpαq via (3.57), which has a momentum

space version as follows

Definition 3.3 A distribution d̂ppq P C 180 pR
mq has a quasi-asymptotic d̂0ppq

over p “ 8, if for a positive function ρpαq (α ą 0) there exists the limit

lim
αÑ0
xρpαqd̂p

p

α
q, ψ̌ppqy “ xd̂0ppq, ψ̌ppqy ‰ 0. (3.91)

3. Obtain the order of singularity ω via

lim
αÑ0`

ρpaαq

ρpαq
“ aω. (3.92)

4. If the numerical causal distribution d̂ppq is regular ω ă 0, the retarded part,

normalized in the origin and at second order, is given by the following formula

r̂0ppq “
i

2π
Sgnpp0

q

8
ż

´8

dt
d̂ptpq

1´ t` iSgnpp0q0`
, p P Γ`1 Y Γ´1 . (3.93)

5. For the singular case ω ě 0, the most general solution for the retarded part in

second order is

ˆ̃r0ppq “
i

2π
Sgnpp0

q

ż

dt
d̂ptpq

pt´ i0`qω`1p1´ t` iSgnpp0q0`q
`

ω
ÿ

l“0

Ĉlp
l, (3.94)

where the constants Ĉl are not defined by the causal splitting procedure.



34 3. Causal Perturbation Theory

Of course, the uniqueness of S-matrix implies that one of the solution families (3.94)

is the real physical one. The physical solution will be obtained by fixing the constants Ĉl

regarding physical properties of the theory such as gauge invariance, charge invariance,

particle masses, etc.



Chapter 4

Quantized free Fields and

Perturvative Gauge Invariance

Elementary particles are complicated real objects; free fields are simpler

mathematical ones. Nevertheless, free fields are the basis of quantum field

theory because the really interesting quantities like interacting fields and

scattering matrix (S-matrix) can be expanded in terms of free fields.

Günter Scharf

Free fields are solutions to the relativistic covariant homogeneous field equations

with a quantization rule. They are not physical objects because they do not model all

the properties of particles, but they are all we know how to solve. Fortunately, in the

case of electromagnetic interaction, the value of coupling constant is small enough to

allow for the expansion of the S-matrix in terms of free fields.

In this thesis we will study scalar QED as a Duffin-Kemmer-Petiau gauge theory

(SDKP) via CPT. Therefore, as mentioned in Chapter 3, we need to determine the

quantized electromagnetic and DKP free fields. In this Chapter we develop the latter.

Also, we show the properties of a fermionic scalar (Ghost) field to introduce the phys-

ical principle of Perturbative Gauge Invariance (PGI) in order to complement CPT.

Specifically, PGI is used to define the first term T1 of S-matrix expansion (3.5).

35



36 4. Quantized free Fields and Perturvative Gauge Invariance

4.1 Electromagnetic Field

The quantized electromagnetic field is modeled by the 4-potential Aµpxq which obeys

the relativistic wave equation

lAµ “ 0, l “ gµνBνBµ, gµν “ diagp`,´,´,´q, (4.1)

which is related with the Lorenz gauge condition BµA
µ
class “ 0 for a classical electro-

magnetic 4-potential. We will see that the latter is related to the physical Fock space

for transversal photons.

Taking into account (4.1) just as four massless Klein-Gordon-Fock equations, we

define the solutions as

A0
pxq “ p2πq´3{2

ż

d3k
?

2ω

`

c0
pkqe´ikx ´ c0

pkq:eikx
˘

, (4.2)

Aipxq “ p2πq´3{2

ż

d3k
?

2ω

`

cipkqe´ikx ` cipkq:eikx
˘

, (4.3)

where the operators cµpkq: and cµpkq are the creation and annihilation operators, re-

spectively, which follows the commutation relations

rcµpkq, cνpk1q:s “

#

δpk´ k1q for µ “ ν

0 for µ ‰ ν
. (4.4)

The minus sign in (4.2) has been chosen to lead to a mathematical consistent result

for the commutation of two electromagnetic 4-potentials1 components

“

Aαpxq, Aβpyq
‰

“ gαβiD0px´ yq, (4.5)

where D0px´ yq is the massless (m “ 0) Lorentz invariant Jordan-Pauli distribution

Dmpxq ”
i

p2πq3

ż

d4pδpp2
´m2

qsgnpp0
qe´ipx. (4.6)

1 If we do not use the minus sign we will obtain

“

Aαpxq, Aβpyq
‰

“ δαβ iD0px´ yq,

which is not correct because we have a second rank Lorentz tensor in the left hand side of the equation

and a scalar in the right. We could use the “indefinite metric” prescription to remedy the incoherence,

but in that case we would have negative states in the Hilbert space.



4.1. Electromagnetic Field 37

In order to determine how the Lorenz condition works at the operator level, note

that the covariant derivative of Aν gives the following result

BνA
ν
“ p2πq´3

ż

d3k

c

ω

2
r´i

ˆ

c0
pkq `

kj
ω
cjpkq

˙

e´ikx`

` i

ˆ

´c0
pkq: `

kj
ω
cjpkq:

˙

eikxs

“ p2πq´3

ż

d3k

c

ω

2
r´i

´

c0
pkq ` cj

||
pkq

¯

e´ikx`

` i
´

´c0
pkq: ` cj

||
pkq:

¯

eikxs,

(4.7)

where cj
||
pkq “

kj
ω
cjpkq is the annihilation operator for longitudinal photons.

Then, states |Φy P Fphys, which have neither longitudinal nor scalar polarized modes

photons, fulfill the following condition

xΦ|BνA
ν
|Φy “ 0, (4.8)

The constraint (4.8) is the quantum equivalent of the classical Lorentz condition.

We define the negative and positive frequency solution for Aµ as

Aµp`q “ p2πq´3{2

ż

d3k
?

2ω
cµpkq:eikx ˆ

$

&

%

1, for µ “ 1, 2, 3

´1, for µ “ 0,
(4.9)

Aµp´q “ p2πq´3{2

ż

d3k
?

2ω
cµpkqe´ikx. (4.10)

From (4.9), (4.10) and (4.4), we compute the following commutation relations

AµpxqAνpyq “ rAµp´qpxq, Aνp`qpyqs “ gµνiD
p`q

0 px´ yq, (4.11)

rAνp`qpxq, Aµp´qpyq, s “ gµνiD
p´q

0 px´ yq, (4.12)

where AµpxqAνpyq is the Wick contraction of two electromagnetic 4-field potentials (see

Appendix A.2), and where D
p`q

0 px ´ yq and D
p´q

0 px ´ yq are the positive and negative

part of Jordan-Pauli distribution

Dp`qm pxq ”
i

p2πq3

ż

d4pδpp2
´m2

qΘpp0
qe´ipx “

i

p2πq3

ż

d3p

2p0
e´ipx, (4.13)

Dp´qm pxq ”
´i

p2πq3

ż

d4pδpp2
´m2

qΘpp0
qeipx “

´i

p2πq3

ż

d3p

2p0
eipx, (4.14)

Dmpxq “ Dp`qm pxq `Dp´qm pxq. (4.15)



38 4. Quantized free Fields and Perturvative Gauge Invariance

4.2 Duffin-Kemmer-Petiau fields

DKP fields fulfill the Dirac like equation [3–5]

piβµBµ ´mqψpxq “ 0, (4.16)

where βµ represent four matrices which obey the following algebra

βµβνβρ ` βρβνβµ “ βµgνρ ` βρgµν . (4.17)

The algebra (4.17) has three irreducible representation of order 1, 5 and 10. The

representation of order 1 is trivial, the next order 5 represent scalar particles and the

order 10 represents spin-1 particles. For more details of historical development of the

DKP equation we refer to references [6, 14].

The equation (4.16) can be obtained from the Lagrangian density

LDKP “
i

2
ψpxqβµ

ÐÑ
Bµψpxq ´mψpxqψpxq, (4.18)

where the conjugate DKP field ψ̄pxq is obtained by

ψ̄pxq “ ψ:pxqη0, η0
“ 2pβ0

q
2
´ 1, (4.19)

and it obeys the equation

ψpxqpiβµ
ÐÝ
Bµ `mq “ 0. (4.20)

A particular solution for the βµ-matrices in its irreducible representation of order 5

is

β0
“

¨

˚

˚

˚

˚

˚

˚

˝

0 0 0 0 1

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

1 0 0 0 0

˛

‹

‹

‹

‹

‹

‹

‚

, β1
“

¨

˚

˚

˚

˚

˚

˚

˝

0 0 0 0 0

0 0 0 0 1

0 0 0 0 0

0 0 0 0 0

0 ´1 0 0 0

˛

‹

‹

‹

‹

‹

‹

‚

,

β2
“

¨

˚

˚

˚

˚

˚

˚

˝

0 0 0 0 0

0 0 0 0 0

0 0 0 0 1

0 0 0 0 0

0 0 ´1 0 0

˛

‹

‹

‹

‹

‹

‹

‚

, β3
“

¨

˚

˚

˚

˚

˚

˚

˝

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

0 0 0 0 1

0 0 0 ´1 0

˛

‹

‹

‹

‹

‹

‹

‚

.

(4.21)

.



4.2. Duffin-Kemmer-Petiau fields 39

A solution for the DKP field ψpxq in the scalar representation is given by

ψpxq “

ż

d3p

p2πq
3
2

appqu´ppqe´ipx `

ż

d3p

p2πq
3
2

b:ppqu`ppqeipx, (4.22)

ψpxq “

ż

d3p

p2πq
3
2

a:ppqu´ppqeipx `

ż

d3p

p2πq
3
2

bppqu`ppqe´ipx, (4.23)

where appq is the annihilation operator of a scalar particle and bppq is the annihilation

operator of an antiparticle. They obey the commutation relations

$

&

%

rappq, a:pp1qs “ δpp´ p1q,

rbppq, b:pp1qs “ δpp´ p1q,
(4.24)

and null for other commutations.

The factors u´ppq and u`ppq are five elements column vector normalized to get a

positive energy system as follows

u˘β0u˘ “ ¯1. (4.25)

From solution (4.22) we can define the positive and negative frequency solutions

ψp`q and ψp´q

ψp`qpxq ”

ż

d3p

p2πq
3
2

b:ppqu`ppqeipx, (4.26)

ψp´qpxq ”

ż

d3p

p2πq
3
2

appqu´ppqe´ipx, (4.27)

and by conjugation

ψ
p`q
pxq ”

ż

d3p

p2πq
3
2

a:ppqu´ppqeipx, (4.28)

ψ
p´q
pxq ”

ż

d3p

p2πq
3
2

bppqu`ppqe´ipx. (4.29)

For a global Up1q transformation δψpxq “ ieαψpxq, the conserved Noether current

jµ is

jµpxq “ e : ψpxqβµψpxq :, (4.30)

where e is the unit charge of a scalar particle and the double dots : . . . : mean a normal

ordering product, as usual, to normalize the vacuum expectation value of the current

as x0|jµpxq|0y “ 0.



40 4. Quantized free Fields and Perturvative Gauge Invariance

4.2.1 Spxq function

Now, we want to compute the commutation rψapxq, ψbpyqs. Using

ψpxq “ ψp`qpxq ` ψp´qpxq, (4.31)

we have

rψapxq, ψbpyqs “ rψ
p`q
a pxq, ψ

p´q

b pyqs ` rψp´qa pxq, ψ
p`q

b pyqs. (4.32)

The second commutation in the right hand side of (4.32) is

rψp´qa pxq, ψ
p`q

b pyqs “

ż

d3p

p2πq3
u´bppqu

´
a ppqe

´ippx´yq. (4.33)

In order to simplify the expression (4.33), we will determine the product u´bppqu
´
a ppq.

Replacing ψ
p`q
pxq from (4.28) into (4.20), we can obtain the following identity

ψ
p`q
pxqpiβµ

ÐÝ
Bµ `mq “ 0

ż

d3p

p2πq
3
2

a:ppqu´ppqeipxpiβµ
ÐÝ
Bµ `mq “ 0,

´

ż

d3p

p2πq
3
2

a:ppqu´ppqpβµpµ ´mqe
ipx
“ 0,

u´bppqpβ
µpµ ´mqbc “ 0.

(4.34)

Multiplying (4.34) by u´a ppq, we get

u´a ppqu
´
bppqpβ

µpµ ´mqbc “ 0. (4.35)

On the other hand, with the use of (4.17), we could obtain the identities

βµpµpβ
νpνβ

θpθ ´m
2
q “ 0,

βµpµpβ
νpν `mqpβ

θpθ ´mq “ 0.
(4.36)

Comparing (4.36) and (4.35), we finally have for u´a ppqu
´
bppq

u´a ppqu
´
bppq “ Crβµpµpβ

νpν `mqsab (4.37)

where C is a constant that we could compute using the normalization condition (4.25)



4.2. Duffin-Kemmer-Petiau fields 41

as follows
Trru´β0u´s “ 1

Trrβ0u´u´s “ 1

CTrrβ0βµpµpβ
νpν `mqs “ 1

CmpµTrrβ
0βµs “ 1

Cmpµ2gµ0
“ 1

C “
1

2mp0

(4.38)

where the following properties are used

Trrβµ1βµ2 . . . βµ2n´1s “ 0, (4.39)

Trrβµ1βµ2 . . . βµ2ns “ gµ1µ2gµ3µ4 . . . gµ2n´1µ2n ` gµ2µ3gµ4µ5 . . . gµ2nµ1 . (4.40)

Replacing (4.38) into (4.37), we have

u´a ppqu
´
bppq “

1

2mp0
rβµpµpβ

νpν `mqsab, (4.41)

and replacing the latter result into (4.33), we obtain

rψp´qa pxq, ψ
p`q

b pyqs “

ż

d3p

p2πq3
1

2mp0
r{pp{p`mqsabe

´ippx´yq, (4.42)

where we use the notation {p “ βµpµ. Following the same path, the first commutation

on the right hand side of (4.32) takes the following form

rψp`qa pxq, ψ
p´q

b pyqs “ ´

ż

d3p

p2πq3
1

2mp0
r{pp{p´mqsabe

ippx´yq. (4.43)

We can rewrite (4.42) as

rψp´qa pxq, ψ
p`q

b pyqs “
1

i
r

1

m
ri{Bpi{B `mqsabsri

ż

d3p

p2πq3
e´ippx´yq

2E
s, (4.44)

where we can identify the positive frequency part of Jordan-Pauli causal distribution

(4.13). Using the latter result (4.44), we define the positive frequency function Sp`q in

the following form

S
p`q

ab pxq ”
1

m
ri{Bpi{B `mqsabD

p`q
m pxq. (4.45)

Replacing (4.45) into (4.44) we obtain

rψp´qa pxq, ψ
p`q

b pyqs “
1

i
S
p`q

ab px´ yq. (4.46)



42 4. Quantized free Fields and Perturvative Gauge Invariance

Working in the same way with the commutation rψ
p`q
a pxq, ψ

p´q

b pyqs, it is possible to

write it as follow

rψp`qa pxq, ψ
p´q

b pyqs “

“ ´

ż

d3p

p2πq3
1

2mp0
r{pp{p´mqsabe

ippx´yq

“
1

i
r

1

m
ri{Bpi{B `mqsabsr´i

ż

d3p

p2πq3
1

2p0
eippx´yqs,

(4.47)

where we identify the negative frequency part of Jordan-Pauli causal distribution (4.14).

Then, we define the negative frequency function Sp´qpxq as

S
p´q

ab pxq ”
1

m
ri{Bpi{B `mqsabD

p´q
m pxq. (4.48)

Replacing (4.48) into (4.47), we obtain

rψp`qa pxq, ψ
p´q

b pyqs “
1

i
S
p´q

ab px´ yq. (4.49)

Finally, we define the function Spxq as

Spxq ” Sp`qpxq ` Sp´qpxq “
1

m
ri{Bpi{B `mqsDmpxq, (4.50)

and the commutation (4.32) takes the following form

rψapxq, ψbpyqs “
1

i
Sabpx´ yq. (4.51)

For future use, we will compute the Fourier transform of the Jordan-Pauli and Spxq

distributions. From (4.6), (4.13) and (4.14), the following formulas are clear

D̂mppq “
i

2π
δpp2

´m2
qSgnpp0

q, (4.52)

D̂p`qm ppq “
i

2π
δpp2

´m2
qΘpp0

q, (4.53)

D̂p´qm ppq “ ´
i

2π
δpp2

´m2
qΘp´p0

q. (4.54)

From (4.45), (4.48) and (4.50) it is straightforward to determine

Ŝ˘ppq “
˘i

p2πq
Θp˘p0

qδpp2
´m2

q
1

m
r{pp{p`mqs, (4.55)

Ŝppq “
i

p2πq
Sgnpp0

qδpp2
´m2

q
1

m
r{pp{p`mqs. (4.56)



4.3. Fermionic Scalar (Ghost) Fields 43

4.3 Fermionic Scalar (Ghost) Fields

In this section we will follow reference [93]. We define here two scalar fields upxq and

ũpxq

upxq ” p2πq´3{2

ż

d3p
?

2ω

`

d2ppqe
´ipx

` d1ppq
:eipx

˘

, (4.57)

ũpxq ” p2πq´3{2

ż

d3p
?

2ω

`

´d1ppqe
´ipx

` d2ppq
:eipx

˘

, (4.58)

where the operators di and d:j are the annihilation and creation operators which satisfy

the following anticommutation relations

tdjppq, d
:

kpqqu “ δjkδpp´ qq. (4.59)

The positive and negative part of upxq and ũpxq are

up`qpxq “ p2πq´3{2

ż

d3p
?

2ω
d1ppq

:eipx, (4.60)

up´qpxq “ p2πq´3{2

ż

d3p
?

2ω
d2ppqe

´ipx, (4.61)

ũp`qpxq “ p2πq´3{2

ż

d3p
?

2ω
d2ppq

:eipx, (4.62)

ũp´qpxq “ ´p2πq´3{2

ż

d3p
?

2ω
d1ppqe

´ipx. (4.63)

From (4.59), the non-null anticommutaors are

tup´qpxq, ũp`qpyqu “ p2πq´3

ż

d3p

2E
e´ippx´yq “ ´iDp`qm px´ yq, (4.64)

tup`qpxq, ũp´qpyqu “ ´p2πq´3

ż

d3p

2E
eippx´yq “ ´iDp´qm px´ yq. (4.65)

The need for the introduction of the fields upxq and ũpxq is to construct a quantum

gauge theory in the next section.

4.4 Perturbative Gauge Invariance

As mentioned in Chapter 3, in order to begin the construction of the S-matrix, we need

to define the first nontrivial distribution term T1pxq in (3.5). In the usual approach



44 4. Quantized free Fields and Perturvative Gauge Invariance

T1pxq “ i : Lint :, where Lint is the interaction Lagrangian. In causal perturbation

theory this is not true.

As an example, we mention the case of SQED constructed with a complex scalar

field ϕpxq which obeys the Klein-Gordon-Fock equation pl`m2qϕpxq “ 0. In order to

obtain a gauge invariant theory, we consider ϕpxq as a classical field that will be coupled

to a classical electromagnetic field Aµpxq. Using the minimal coupling prescription, we

substitute the partial derivative in the free Lagrangian for ϕpxq with the covariant

derivative Dµ “ Bµ ` ieAµ, obtaining

Lint “ ´ieA
µ
pϕ˚
ÐÑ
Bµϕq ` e

2ϕ˚ϕAµAµ, (4.66)

where e represents the electric charge of the scalar particle.

The problem of using (4.66) to construct T1 is the second order term e2ϕ˚ϕAµAµ

which by construction must belong to T2 because in CPT the unit charge e represents

the physical charge and not a simple parameter. What is unquestionable is that T1

must be defined from the gauge invariance property but at the quantum level.

In general, a gauge transformation Aµpxq Ñ A1µpxq, implies that Aµpxq and A1µpxq

obey the same equation of motion. The latter is equivalent to obtain a transformation

where A1µpxq obey the same commutation relation as Aµpxq. This is possible with the

following transformation

A1µpxq “ e´iλQAµpxqeiλQ, (4.67)

where Q is called gauge charge .

By expanding the exponential operators, we obtain

A1µpxq “ Aµpxq ´ iλrQ,Aµpxqs `Opλ2
q. (4.68)

On the other hand, consider the following classical gauge transformation but at the

operator level

A1µpxq “ Aµpxq ` λBµupxq `Opλ2
q, (4.69)

where u is a free quantum field which obeys the massless Klein-Gordon-Fock equation

lupxq “ 0. (4.70)

For an infinitesimal parameter λ, by comparing (4.68) and (4.69), we can obtain an

equation which defines Q uniquely

rQ,Aµpxqs “ iBµupxq. (4.71)



4.4. Perturbative Gauge Invariance 45

The solution for Q from (4.71) is

Q “

ż

d3xrBνA
ν
B0u´ pB0BνA

ν
qus “

ż

d3xBνA
νÐÑ
B 0u, (4.72)

where the integral is evaluated over a hyperplane x0 “ constant.

If upxq is a fermionic scalar ghost field, we can obtain a nilpotent Q in the form

Q2
“

1

2
tQ,Qu “ 0, (4.73)

which can be used to construct a physical Fock space as we will show bellow.

Using (4.2), (4.3) and (4.57) we can obtain Q as

Q “

ż

d3kωpkqrpa‖pkq
:
´ a0pkq

:
qd2pkq ` d1pkq

:
pa‖pkq ` a0pkqqs

“

ż

d3kωpkqrc2pkq
:d2pkq ` d1pkq

:c1pkqs,

(4.74)

where

c1 “ a‖pkq ` a0pkq, c2 “ a‖pkq ´ a0pkq, (4.75)

are new operators which satisfy the usual commutation rule

rcipkq, c
:

jpk
1
qs “ δijδpk´ k1q. (4.76)

An important result for Q, stemming from (4.71), is the following identity

tQ:, Qu “ 2

ż

d3kω2
pkqrc:1c1 ` c

:

2c2 ` d
:

1d1 ` d
:

2d2s, (4.77)

where we can identify the number operators of non-physical particles. Consequently, we

could use the anticommutation tQ:, Qu to define the physical Fock space Fphys. Every

physical Fock state |Φy P Fphys must fulfill the following condition

tQ:, Qu|Φy “ 0. (4.78)

Now, returning to the quantum gauge invariance principle, we can see that the gauge

charge Q represents an infinitesimal gauge transformation generator. This allows us to

define the gauge derivative dQ for a product F of Bose fields and even number of ghost

fields and for a product G of Bose fields and odd number of ghost fields as follow

dQF ” rQ,F s, dQG ” tQ,Gu. (4.79)



46 4. Quantized free Fields and Perturvative Gauge Invariance

In order to obtain a gauge invariant theory, we demand that all n-point distributions

Tn must fulfill the following property

dQTnpx1, . . . , xnq “ i
n
ÿ

l“1

B

Bxµl
T µn{lpx1, . . . , xnq, (4.80)

where T µn{lpx1, . . . , xnq is the following time ordering product constructed by causal

perturbation theory

T µn{lpx1, . . . , xnq “ T tT1px1q . . . T
µ
1{1pxlq . . . T1pxnqu, (4.81)

and T µ1{1 is called the Q-vertex. The property (4.80) is called perturbative Gauge

invariance .



Chapter 5

Scattering processes of Scalar

Quantum Electrodynamics at the

tree-level

Hence most physicists are very satisfied with the situation. They say:

“Quantum electrodynamics is a good theory, and we do not have to worry

about it any more.” I must say that I am very dissatisfied with the

situation, because this so-called “good theory” does involve neglecting

infinities which appear in its equations, neglecting them in an arbitrary

way. This is just not sensible mathematics. Sensible mathematics involves

neglecting a quantity when it turns out to be small—not neglecting it just

because it is infinitely great and you do not want it!

P. A. M. Dirac

Here we begin to determine the equivalence between the two approaches to study

scalar QED. The first one via Klein-Gordon-Fock fields (SQED) and the second one via

Duffin-Kemmer-Petiau fields (SDKP). For this goal, we will follow the same spirit that

Fainberg and Pimentel used in [13] comparing the elements of the S-matrix.

As we mentioned in Chapter 1, we will use CPT to consider all sectors of SQED and

SDKP. As demonstrated by Scharf and collaborators in [26], it is not necessary to add

by hand the sectors generated by the vertices proportional to φ˚pxqφpxqAµpxqAµpxq and

pφ˚pxqφpxqq2. In CPT, these terms appears naturally from perturbative gauge invariance

and the causal splitting process. This is the power of CPT approach.

47



48 5. Scattering processes of Scalar Quantum Electrodynamics at the tree-level

In this chapter we compute the following process at the tree level only:

1. Scattering of scalar particle by external electromagnetic field.

2. Moller scattering.

3. Compton scattering.

Bhabha scattering can be studied by crossing properties from Moller process.

5.1 Definition of term T1 for SDKP via Perturba-

tion Gauge Invariance at first order

For a massless gauge field Aµpxq we have Q in the form (4.72) and the following gauge

transformations

dQA
µ
pxq “ iBµupxq, (5.1)

dQupxq “ 0, (5.2)

dQũpxq “ ´iBµA
µ
pxq. (5.3)

First of all, in order to determine T1pxq we can use (4.80) for n “ 1

dQT1px1q “ iBµT
µ
1{1px1q. (5.4)

Secondly, due to the adiabatic limit gpxq Ñ 1, the term T1pxq must contain all kinds

of interactions between gauge and matter fields. As shown by Scharf and collaborators

[38–41], for massless gauge fields Aµpxq only in the case with a collection Aµapxq, where

a “ 1, . . . , N , there are self interactions between gauge and ghost fields.

For SDKP we have only one massless gauge field, therefore T1 contain just the

interaction between electromagnetic and matter current jµ in the form

T
pSDKPq
1 px1q “ ijµpx1qAµpx1q. (5.5)

As usual, jµpxq contains DKP fields ψpxq and ψ̄pxq. Now, with the help of (5.1),

(5.2) and ((5.3)), the gauge derivative of (5.5) will take the following form

dQT
pSDKPq
1 px1q “ dQtij

µ
px1qAµpx1qu “ ijµpx1qdQAµpx1q “ ´j

µ
px1qBµupx1q. (5.6)



5.2. Scattering of DKP scalar particle by static external field 49

From (5.5) and (5.6), we can conclude that in order to obtain (5.4), the matter

current jµpxq must have null divergence. The latter is fulfilled if jµ is the DKP Noether

current (4.30) as follows

Bµj
µ
pxq “ 0 ùñ dQT

pSDKPq
1 px1q “ iBµT

µpSDKPq
1{1 “ iBµrij

µ
px1qupx1qs. (5.7)

Finally, replacing (4.30) in the expressions for the Q-vertex T
µpSDKPq
1{1 , we obtain the

following form for T1

T
µpSDKPq
1{1 “ ijµpx1qupx1q “ ie : ψpx1qβ

µψpx1q : upx1q (5.8)

T
pSDKPq
1 pxq “ ie : ψpx1qβ

µψpx1q : Aµpx1q. (5.9)

5.2 Scattering of DKP scalar particle by static ex-

ternal field

As a first application, we are going to determine the differential cross section dσ{dΩ in

the scattering of scalar by an external electromagnetic field Aextµ .

If the system includes an external field Aextµ , we need to make the substitution

Aµ Ñ Aµ ` A
ext
µ , then the perturbative expansion of S Matrix includes a term

S “ . . .`

ż

d4xie : ψpxqβµψpxq : Aextµ pxq ` . . . , (5.10)

this term is important in the case of a scattered scalar particle by this external field.

Because the initial and final states do not include creation (or annihilation) of photons,

these states are

|iny “ |Ψiy “

ż

d3p1Φipp1qa
:
pp1q|0y, (5.11)

|outy “ |Ψfy “

ż

d3p2Φf pp2qa
:
pp2q|0y, (5.12)

where a:pp1,2q are the creation operators for scalar particles with momenta p1,2 and Φi,f

are wave packets sharply picked at pi and pf which are the initial and final momentum

of the bunch of particles before and after the scattering, respectively.

Now, computing the scattering amplitude Sif “ xΨf |S|Ψiy, it is not difficult to see



50 5. Scattering processes of Scalar Quantum Electrodynamics at the tree-level

that the unique non-null result is

Sif “ ie

ż

d4xxΨf | : ψpxqβµψpxq : |ΨiyA
ext
µ pxq,

“ ie

ż

d4x

ż

d3p2

ż

d3p1Φ˚f pp2qΦipp1qˆ

ˆ x0|app2q : ψpxqβµψpxq : a:pp1q|0yA
ext
µ pxq.

(5.13)

In order to compute the term x0|app2q : ψpxqβµψpxq : a:pp1q|0y, we can use the

Wick theorem (see Appendix A.2). Therefore, in the Wick expansion only the term

with the two simultaneous contractions app2qψ̄pxq and ψpxqa:pp1q is not null. These

contractions are

app2qψpxq “ app2qψ
p`q
pxq “ apq2q

ż

d3p

p2πq
3
2

a:ppqu´ppqeipx

“

ż

d3p

p2πq
3
2

δpp2 ´ pqu´ppqeipx “
1

p2πq
3
2

u´pp2qe
ip2x,

(5.14)

ψpxqa:pp1q “ ψp´qpxqa:pp1q “

ż

d3p

p2πq
3
2

appqu´ppqe´ipxa:pp1q

“

ż

d3p

p2πq
3
2

δpp1 ´ pqu´ppqe´ipx “
1

p2πq
3
2

u´pp1qe
´ip1x.

(5.15)

With the help of (5.14) and (5.15), we obtain

x0|app2q : ψpxqβµψpxq : a:pp1q|0y “ x0|app2qψ̄pxqβ
µψpxqa:pp1q|0y

“
1

p2πq
3
2

u´pp2qe
ip2xβµ

1

p2πq
3
2

u´pp1qe
´ip1x

“
1

p2πq3
u´pp2qβ

µu´pp1qe
´ipp1´p2qx.

(5.16)

Replacing (5.16) into (5.13), the scattering amplitude takes the following form

Sif “ ie

ż

d4x

ż

d3p2

ż

d3p1Φ˚f pp2qΦipp1q
1

p2πq3
u´pp2qβ

µu´pp1qe
´ipp1´p2qxAextµ pxq

“ ie

ż

d3p2

ż

d3p1Φ˚f pp2qΦipp1q
1

p2πq3
u´pp2qβ

µu´pp1q

ż

d4xe´ipp1´p2qxAextµ pxq.

(5.17)

Considering a static electromagnetic field, we can replace Aextµ pxq “ Aextµ pxq and



5.2. Scattering of DKP scalar particle by static external field 51

evaluate the integral in x0 to obtain

Sif “
ie

p2πq2

ż

d3p2

ż

d3p1Φ˚f pp2qΦipp1qu´ppf qβ
µu´pp1qδpE1 ´ E2q

ż

d3xe´ipp2´p1qxAextµ pxq

“
ie

p2πq2

ż

d3p2

ż

d3p1Φ˚f pp2qΦipp1qu´pp2qβ
µu´pp1qδpE1 ´ E2qp2πq

3
2 µpp2 ´ p1q

“

ż

d3p2

ż

d3p1Φ˚f pp2qΦipp1qMif pp1,p2qδpE1 ´ E2q,

(5.18)

where

Âpp2 ´ p1q “ p2πq
´ 3

2

ż

d3xe´ipp2´p1qxAextµ pxq, (5.19)

Mif pp1,p2q “
ie

p2πq
1
2

u´pp2qβ
µu´pp1qÂpp2 ´ p1q. (5.20)

With the computation of Sif , we will determine the probability transition Pif defined

as

Pif “ |Sif |
2. (5.21)

Replacing (5.18) into (5.21), we have

Pif “

ż

d3p1d
3p2Φf pp2qS̃

˚
if pp1,p2qΦ

˚
i pp1q

ż

d3p11d
3p12Φ˚f pp

1
2qS̃if pp

1
1,p

1
2qΦipp

1
1q, (5.22)

where

S̃if “Mif pp1,p2qδpE1 ´ E2q. (5.23)

Summing over all possible final states

ÿ

f

Pif “

ż

d3p1d
3p2S̃

˚
if pp1,p2qΦ

˚
i pp1q

ż

d3p11d
3p12S̃if pp

1
1,p

1
2qΦipp

1
1q
ÿ

f

Φf pp2qΦ
˚
f pp

1
2q

“

ż

d3p1d
3p2S̃

˚
if pp1,p2qΦ

˚
i pp1q

ż

d3p11d
3p12S̃if pp

1
1,p

1
2qΦipp

1
1qδpp2 ´ p12q

“

ż

d3p1d
3p2S̃

˚
if pp1,p2qΦ

˚
i pp1q

ż

d3p11S̃if pp
1
1,p2qΦipp

1
1q

“

ż

d3p1d
3p2M

˚
if pp1,p2qδpE1 ´ E2qΦ

˚
i pp1q

ż

d3p11Mif pp
1
1,p2qδpE

1
1 ´ E2qΦipp

1
1q

(5.24)

where in the last line we used (5.23).

Using the fact that the function Φipp1q is sharply peaked around pi and considering

that its width is too small compared with the scale of varying of Mif , we can rewrite



52 5. Scattering processes of Scalar Quantum Electrodynamics at the tree-level

(5.24) as follows

ÿ

f

Pif “

ż

d3p2|Mif ppi,p2q|
2

ż

d3p1d
3p11δpE1 ´ E2qΦ

˚
i pp1qδpE

1
1 ´ E2qΦipp

1
1q. (5.25)

Computing the p2 integral in spherical coordinates, we obtain

ÿ

f

Pif “

ż

dΩ2|Mif ppi,p2q|
2

ż

|p2|
2d|p2|

ż

d3p1d
3p11δpE1 ´ E2qΦ

˚
i pp1qδpE

1
1 ´ E2qΦipp

1
1q

“

ż

dΩ2|Mif ppi,p2q

ż

|p2|E2dE2

ż

d3p1d
3p11δpE1 ´ E2qΦ

˚
i pp1qδpE

1
1 ´ E2qΦipp

1
1q

“

ż

dΩ2|Mif ppi,p2q|
2

ż

d3p1d
3p11Φ˚i pp1qΦipp

1
1q

ż

|p2|E2dE2δpE1 ´ E2qδpE
1
1 ´ E2q

“

ż

dΩ2|Mif ppi,p2q|
2

ż

d3p1d
3p11Φ˚i pp1qΦipp

1
1q|pi|EiδpE

1
1 ´ E1q.

(5.26)

Replacing the integral form of the delta function δpE 11´E1q “ p2πq
´1

ş

dte´ipE
1
1´E1qt

into (5.26), we can rewrite it as

ÿ

f

Pif “

ż

dΩ2|Mif ppi,p2q|
2
|pi|Ei

ż

d3p1d
3p11Φ˚i pp1qΦipp

1
1qδpE

1
1 ´ E1q

“

ż

dΩ2|Mif ppi,p2q|
2
|pi|Ei

ż

d3p1d
3p11Φ˚i pp1qΦipp

1
1qp2πq

´1

ż

dte´ipE
1
1´E1qt

“

ż

dΩ2|Mif ppi,p2q|
2
|pi|Eip2πq

2

ż

dtp2πq´
3
2

ż

d3p1Φ˚i pp1qe
iE1tp2πq´

3
2

ż

d3p11Φipp
1
1qe

´iE11t

“

ż

dΩ2|Mif ppi,p2q|
2
|pi|Eip2πq

2

ż

dt

”

p2πq´
3
2

ż

d3p1Φ˚i pp1qe
ipE1t´p1xq

ı

x“0

”

p2πq´
3
2

ż

d3p11Φipp
1
1qe

´ipE11t´p
1
1xq

ı

x“0

“

ż

dΩ2|Mif ppi,p2q|
2
|pi|Ei|p2πq

2

ż

dt|Φpt,x “ 0q|2,

(5.27)

where Φpt,xq is the following free wave packet in x-space

Φpt,xq “ p2πq´
3
2

ż

d3qΦipqqe
´ipEqt´qxq. (5.28)

Considering that the velocity of the scattered particles is v, the wave packet have

the form

Φpt,xq “ Φ0px` vtq. (5.29)



5.2. Scattering of DKP scalar particle by static external field 53

Now, averaging (5.27) in a cylinder of radius R parallel to v using the wave packet

(5.29), we have

ÿ

f

Pif pRq “
1

πR2

ż

xKďR

d2xK

ż

dt|Φ0px` vtq|2
ż

dΩ2|Mif ppi,p2q|
2
|pi|Eip2πq

2. (5.30)

The cross section is defined as

σ “ lim
RÑ8

πR2
ÿ

f

Pif pRq. (5.31)

Then, replacing (5.30) into (5.31), we get

σ “

ż

xKďR

d2xK

ż

dt|Φ0pxK ` vtq|2
ż

dΩ2|Mif ppi,p2q|
2
|pi|Eip2πq

2

“
1

|v|

ż

d3x|Φ0|
2

ż

dΩ2|Mif ppi,p2q|
2
|pi|Eip2πq

2

“
1

|v|

ż

dΩ2|Mif ppi,p2q|
2
|pi|Eip2πq

2

“
|pi|Eip2πq

2

p
|pi|
Ei
q

ż

dΩ2|Mif ppi,p2q|
2

“ E2
i p2πq

2

ż

dΩ2|Mif ppi,p2q|
2,

(5.32)

which tell us that the differential cross section will take the following form

dσ

dΩ
“ p2πq2E2

i |Mppf ,piq|
2. (5.33)

Replacing (5.20) into (5.33), we get

dσ

dΩ
“

“ p2πq2E2
i r

ie

p2πq
1
2

u´ppf qβ
µu´ppiqµppf ´ piqsr´

ie

p2πq
1
2

u´ppiqβ
νu´ppf qÂνppi ´ pf qs

“ p2πqE2
i e

2
ru´mppf qu

´
appf qsβ

µ
adru

´
d ppiqu

´
lppiqsβ

ν
lmÂνppf ´ piqµppi ´ pf q

“
p2πqE2

i e
2

4m2pi0pf 0
Trr{pf p{pf `mqβ

µ
{pip{pi `mqβ

ν
sÂνppf ´ piqµppi ´ pf q

“
p2πqE2

i e
2

4pi0pf 0
rpµi p

ν
i ` p

ν
fp

µ
f ` p

µ
fp

ν
i ` p

ν
fp

µ
i sÂνppf ´ piqµppi ´ pf q,

(5.34)



54 5. Scattering processes of Scalar Quantum Electrodynamics at the tree-level

where (4.37) was used in the second line, and the trace identities (4.39) and (4.40) were

used in next ones.

For the potential Aµ, we will use the Coulomb potential. Therefore, only the 0-

component is not null and takes the following form

A0
pxq “

Ze

|x|
, Â0

ppq “

c

2

π

Ze

|p|2
, (5.35)

replacing this potential into (5.34), we have

dσ

dΩ
“
p2πqE2

i e
2

4pi0pf 0
rp0
i p

0
i ` p

0
fp

0
f ` p

0
fp

0
i ` p

0
fp

0
i sÂ0ppf ´ piqÂ0ppi ´ pf q

“
p2πqE2

i e
2

4EiEf
rEi ` Ef s

2
r
2

π

Z2e2

|pi ´ pf |4
s

“
Z2Eie

4

Ef

rEi ` Ef s
2

|pi ´ pf |4

“ Z2e4 4E2

|pi ´ pf |4

“ Z2e4 4E2

16|p|4 sin4pϑ{2q

“ Z2e4 E2

4|p|4 sin4pϑ{2q
.

(5.36)

The latter result is equivalent to that obtained in [10] using the usual approach.

5.3 Causal distribution in the second order D2px, yq

After setting T1pxq for SDKP, the next step is to compute the causal distribution

D2px, yq. Following (3.23) and (3.24), the intermediate distributions in second order

A12 and R12 take the following forms

A12px, yq “ T̃1pxqT1pyq “ ´T1pxqT1pyq, (5.37)

R12px, yq “ T1pyqT̃1pxq “ ´T1pyqT1pxq, (5.38)

where (3.13) was used.

Replacing (5.9) into (5.37) and (5.38), and using Wick theorem to obtain normal

ordered terms, we have



5.3. Causal distribution in the second order D2px, yq 55

A12px, yq “ ´T1pxqT1pyq

“ e2 : ψapxqβ
µ
abψbpxq :: ψcpyqβ

ν
cdψdpyq : AµpxqAνpyq

“ e2
r: ψapxqβ

µ
abψbpxqψcpyqβ

ν
cdψdpyq : ` : ψapxqβ

µ
abψbpxqψcpyqβ

ν
cdψdpyq :

` : ψapxqβ
µ
abψbpxqψcpyqβ

ν
cdψdpyq : ` : ψapxqβ

µ
abψbpxqψcpyqβ

ν
cdψdpyq :sˆ

ˆ r: AµpxqAνpyq : `AµpxqAνpyqs,

(5.39)

where the field contractions are

AµpxqAνpyq “ rAµp´qpxq, Aνp`qpyqs “ gµνiD
p`q

0 px´ yq, (5.40)

ψapxqψ̄bpyq “ rψ
p´q
pxq, ψ̄p`qpyqs “

1

i
S
p`q

ab px´ yq, (5.41)

ψ̄cpxqψdpyq “ rψ̄
p´q
pxq, ψp`qpyqs “ ´

1

i
S
p´q

dc py ´ xq. (5.42)

Replacing (5.40), (5.41) and (5.42) into (5.39), we obtain

A12px, yq “ `e
2βµabβ

ν
cd : ψapxqψbpxqψcpyqψdpyq : igµνD

p`q

0 px´ yq

´ e2βµabβ
ν
cd : ψbpxqψcpyq :

1

i
S
p´q

da py ´ xq : AµpxqAνpyq :

` e2βµabβ
ν
cd : ψapxqψdpyq :

1

i
S
p`q

bc px´ yq : AµpxqAνpyq :

´ e2βµabβ
ν
cd

1

i
S
p´q

da py ´ xq
1

i
S
p`q

bc px´ yq : AµpxqAνpyq :

´ e2βµabβ
ν
cd : ψbpxqψcpyq :

1

i
S
p´q

da py ´ xqigµνD
p`q

0 px´ yq

` e2βµabβ
ν
cd : ψapxqψdpyq :

1

i
S
p`q

bc px´ yqigµνD
p`q

0 px´ yq

´ e2βµabβ
ν
cd

1

i
S
p´q

da py ´ xq
1

i
S
p`q

bc px´ yqigµνD
p`q

0 px´ yq

` e2βµabβ
ν
cd : ψapxqψbpxqψcpyqψdpyq :: AµpxqAνpyq :,

(5.43)

which can be rewritten as

A12px, yq “ A
1p1q
2 px, yq ` A

1p2q
2 px, yq ` A

1p3q
2 px, yq ` A

1p4q
2 px, yq ` A

1p5q
2 px, yq

` e2βµabβ
ν
cd : ψapxqψbpxqψcpyqψdpyq :: AµpxqAνpyq :,

(5.44)

where

A
1p1q
2 px, yq “ `e2βµabβ

ν
cd : ψapxqψbpxqψcpyqψdpyq : igµνD

p`q

0 px´ yq, (5.45)



56 5. Scattering processes of Scalar Quantum Electrodynamics at the tree-level

A
1p2q
2 px, yq “ ´e2βµabβ

ν
cd : ψbpxqψcpyq :

1

i
S
p´q

da py ´ xq : AµpxqAνpyq :

` e2βµabβ
ν
cd : ψapxqψdpyq :

1

i
S
p`q

bc px´ yq : AµpxqAνpyq :,
(5.46)

A
1p3q
2 px, yq “ ´e2βµabβ

ν
cd

1

i
S
p´q

da py ´ xq
1

i
S
p`q

bc px´ yq : AµpxqAνpyq :, (5.47)

A
1p4q
2 px, yq “ ´e2βµabβ

ν
cd : ψbpxqψcpyq :

1

i
S
p´q

da py ´ xqigµνD
p`q

0 px´ yq

` e2βµabβ
ν
cd : ψapxqψdpyq :

1

i
S
p`q

bc px´ yqigµνD
p`q

0 px´ yq,
(5.48)

A
1p5q
2 px, yq “ ´e2βµabβ

ν
cd

1

i
S
p´q

da py ´ xq
1

i
S
p`q

bc px´ yqigµνD
p`q

0 px´ yq. (5.49)

Similarly for R12px, yq, we have

R12px, yq “ R
1p1q
2 py, xq `R

1p2q
2 py, xq `R

1p3q
2 py, xq `R

1p4q
2 py, xq `R

1p5q
2 py, xq

` e2βµabβ
ν
cd : ψapyqψbpyqψcpxqψdpxq :: AµpyqAνpxq :

(5.50)

where

R
1p1q
2 px, yq “ `e2βµabβ

ν
cd : ψapyqψbpyqψcpxqψdpxq : igµνD

p`q

0 py ´ xq, (5.51)

R
1p2q
2 px, yq “ ´e2βµabβ

ν
cd : ψbpyqψcpxq :

1

i
S
p´q

da px´ yq : AµpyqAνpxq :

` e2βµabβ
ν
cd : ψapyqψdpxq :

1

i
S
p`q

bc py ´ xq : AµpyqAνpxq :,
(5.52)

R
1p3q
2 py, xq “ ´e2βµabβ

ν
cd

1

i
S
p´q

da px´ yq
1

i
S
p`q

bc py ´ xq : AµpyqAνpxq :, (5.53)

R
1p4q
2 py, xq “ ´e2βµabβ

ν
cd : ψbpyqψcpxq :

1

i
S
p´q

da px´ yqigµνD
p`q

0 py ´ xq

` e2βµabβ
ν
cd : ψapyqψdpxq :

1

i
S
p`q

bc py ´ xqigµνD
p`q

0 py ´ xq,
(5.54)

R
1p5q
2 py, xq “ ´e2βµabβ

ν
cd

1

i
S
p´q

da px´ yq
1

i
S
p`q

bc py ´ xqigµνD
p`q

0 py ´ xq. (5.55)

The causal distribution D2 is obtained by the subtraction (3.43) as follows

D2px, yq “ R12px, yq ´ A
1
2px, yq “ D

p1q
2 `D

p2q
2 `D

p3q
2 `D

p4q
2 `D

p5q
2 (5.56)

where

D
p1q
2 “ ie2gµν : ψapyqβ

µ
abψbpyqψcpxqβ

ν
cdψdpxq :

´

D
p`q

0 py ´ xq ´D
p`q

0 px´ yq
¯

, (5.57)



5.4. Moller scattering 57

+
D

(1)

2 D
(2)

+

D
(3)

2 D
(4)

2 D
(5)

2

Figure 5.1: General graph for processes in the causal distribution D2px, yq.

D
p2q
2 “ie2 : ψcpxqβ

ν
cd

´

S
p`q

da px´ yq ` S
p´q

da px´ yq
¯

βµabψbpyq :: AµpyqAνpxq :

´ ie2 : ψapyqβ
µ
ab

´

S
p`q

bc py ´ xq ` S
p´q

bc py ´ xq
¯

βνcdψdpxq :: AµpyqAνpxq :,
(5.58)

D
p3q
2 “ e2TrrβµSp`qpy ´ xqβνSp´qpx´ yq ´ βνSp`qpx´ yqβµSp´qpy ´ xqsˆ

ˆ : AµpyqAνpxq :,
(5.59)

D
p4q
2 “ ´e2gµν : ψpxqβνrSp´qpx´ yqD

p`q

0 py ´ xq ` Sp`qpx´ yqD
p`q

0 px´ yqsβµψpyq :

` e2gµν : ψpyqβµrSp`qpy ´ xqD
p`q

0 py ´ xq ` Sp´qpy ´ xqD
p`q

0 px´ yqsβνψpxq :,

(5.60)

D
p5q
2 “ `e2βµSp`qpy ´ xqβνSp´qpx´ yqigµνD

p`q

0 py ´ xq

´ e2βµSp`qpx´ yqβνSp´qpy ´ xqigµνD
p`q

0 px´ yq.
(5.61)

Each term D
piq
2 represents different processes in the S-matrix and their diagrams

are represented in Fig. 5.1.

5.4 Moller scattering

Now, we will determine the differential cross section of Moller process which consist in

the elastic scattering of two scalar particles bppiq ` bpqiq Ñ bppf q ` bpqf q where pi,f and

qi,f are the initial and final momentum of particles after and before the interaction as

usual. Therefore, the in and out states take on the following form

|inMollery “ |Ψiy b |Φiy “

ż

d3p1d
3q1Ψipp1qΦipq1qa

:
pp1qa

:
pq1q|0y, (5.62)

|outMollery “ |Ψfy b |Φfy “

ż

d3p2d
3q2Ψf pp2qΦf pq2qa

:
pp2qa

:
pq2q|0y, (5.63)



58 5. Scattering processes of Scalar Quantum Electrodynamics at the tree-level

where tΨipp1q,Ψf pp2q,Φipq1q,Φf pq2qu are the wave packet functions sharply peaked in

tpi, pf , qi, qfu, respectively.

Taking into account that the causal splitting procedure does not transform the quan-

tized fields, in the computation of scattering amplitude xoutMoeller|S|inMoellery just the

term coming from D
p1q
2 will be no-null. Consequently, we will determine the contribution

to T2px, yq coming from D
p1q
2 .

Using the property D
p`q

0 pxq “ ´D
p´q

0 p´xq, we can rewrite D
p1q
2 px, yq in the following

form

D
p1q
2 px, yq “ e2igµν : ψpyqβµψpyqψpxqβνψpxq : pD

p`q

0 py ´ xq ´D
p`q

0 px´ yqq

“ e2igµν : ψpyqβµψpyqψpxqβνψpxq : p´D
p´q

0 px´ yq ´D
p`q

0 px´ yqq

“ ´e2igµν : ψpyqβµψpyqψpxqβνψpxq : D0px´ yq,

(5.64)

where we can see that the numerical part of D
p1q
2 is D0px´ yq.

5.4.1 Causal splitting of D0

We will begin the causal splitting in momentum space. From (4.52), the Fourier trans-

formation of D0px´ yq has the following form

pD0ppq “
i

2π
δpp2

qSgnpp0
q (5.65)

The power counting function ρpαq for pD0ppq is determined using (3.91). With this

goal, we first have to note that the form of pD0p
p
α
q is

pD0p
p

α
q “

i

2π
δpp2α´2

qSgnpp0α´1
q

“
iα2

2π
δpp2

qSgnpp0α´1
q.

(5.66)

After that, it is not difficult to conclude that for ρpαq “ α´2 we obtain the following

no null limit

lim
αÑ0

ρpαq
A

pD0p
p

α
q, qfppq

E

“

A

pD0ppq, qfppq
E

‰ 0. (5.67)

From (3.92), the singular order of D̂0 is

ωrD̂0s “ ´2, (5.68)



5.4. Moller scattering 59

which means that pD0ppq is a regular distribution and its splitting in retarded and

advanced part is

D0px´ yq “ θpx0
´ y0

qD0px´ yq ´ θpy
0
´ x0

qD0px´ yq

“ Dret
0 px´ yq ´D

adv
0 px´ yq,

(5.69)

where Dret
0 px´ yq “ θpx0 ´ y0qD0px´ yq and Dadv

0 px´ yq “ θpy0 ´ x0qD0px´ yq.

From the splitting (5.69) and (5.64), the second order retarded distribution R
p1q
2 px, yq

is

R
p1q
2 px, yq “ ´e

2igµνβ
µ
abβ

ν
cd : ψapyqψbpyqψcpxqψdpxq : Dret

0 . (5.70)

Finally, using (3.29), (5.70) and (5.45) we are able to determine the contribution

T
p1q
2 px, yq for S-matrix coming from D

p1q
2 in the following form

T
p1q
2 px, yq “ R

p1q
2 ´R

1p1q
2

“ ´e2igµνβ
µ
abβ

ν
cd : ψapyqψbpyqψcpxqψdpxq : Dret

0 py ´ xq

´ re2βµabβ
ν
cd : ψapyqψbpyqψcpxqψdpxq : igµνD

p`q

0 py ´ xqs

“ ´e2igµνβ
µ
abβ

ν
cd : ψapyqψbpyqψcpxqψdpxq : pDret

0 px´ yq `D
p`q

0 py ´ xqq

“ ´e2igµνβ
µ
abβ

ν
cd : ψapyqψbpyqψcpxqψdpxq : pDret

0 px´ yq ´D
p´q

0 px´ yqq

“ ´e2igµνβ
µ
abβ

ν
cd : ψapyqψbpyqψcpxqψdpxq : DF

0 px´ yq,

(5.71)

where DF pxq ” Dret
0 pxq ´D

p´q

0 pxq is the well-known Feynman propagator for massless

scalar field.

5.4.2 Computation of differential cross section

The S-matrix term Sp1qpgq, which contributes in the computation of the differential

cross section for Moller scattering, takes the following form

Sp1qpgq “
1

2!

ż

d4yd4xT
p1q
2 px, yqgpxqgpyq. (5.72)

Recalling the equations (5.62) and (5.63), and taking the adiabatic limit for the

computations, we have the scattering amplitude as

S
pMoq
fi “ xoutMo|S|inMoy

“

ż

d3p2d
3q2

ż

d3p1d
3q1Ψ˚

f pp2qΦ
˚
f pq2qS̃

pMoq
if Ψipp1qΦipq1q

(5.73)



60 5. Scattering processes of Scalar Quantum Electrodynamics at the tree-level

where

S̃
pMoq
if “ x0|app2qapq2qS

p1qa:pp1qa
:
pq1q|0y

“ x0|app2qapq2q
1

2!

ż

d4yd4xTM2 px, yqa
:
pp1qa

:
pq1q|0y

“ ´
1

2!

ż

d4yd4xe2igµνβ
µ
abβ

ν
cdD

F
0 px´ yqˆ

ˆ x0|app2qapq2q : ψapyqψbpyqψcpxqψdpxq : a:pp1qa
:
pq1q|0y

“ ´
1

2!

ż

d4yd4xe2igµνβ
µ
abβ

ν
cdD

F
0 px´ yqˆ

ˆ x0|app2qapq2q : ψ
p`q

a pyqψ
p´q

b pyqψ
p`q

c pxqψ
p´q

d pxq : a:pp1qa
:
pq1q|0y

“ ´
1

2!

ż

d4yd4xe2igµνβ
µ
abβ

ν
cdD

F
0 px´ yqˆ

ˆ x0|app2qapq2qψ
p`q

a pyqψ
p`q

c pxqψ
p´q

b pyqψ
p´q

d pxqa:pp1qa
:
pq1q|0y.

(5.74)

Using Wick theorem and the contractions (5.14) and (5.15), we could reduce the

expression (5.74) as follows

S̃
pMoq
fi “ ´

1

2!

ż

d4yd4xe2igµνβ
µ
abβ

ν
cdD

F
0 px´ yqˆ

ˆ x0|
”

` app2qapq2qψ
p`q

a pyqψ
p`q

c pxqψ
p´q

b pyqψ
p´q

d pxqa:pp1qa
:
pq1q`

` app2qapq2qψ
p`q

a pyqψ
p`q

c pxqψ
p´q

b pyqψ
p´q

d pxqa:pp1qa
:
pq1q`

` app2qapq2qψ
p`q

a pyqψ
p`q

c pxqψ
p´q

b pyqψ
p´q

d pxqa:pp1qa
:
pq1q

` app2qapq2qψ
p`q

a pyqψ
p`q

c pxqψ
p´q

b pyqψ
p´q

d pxqa:pp1qa
:
pq1q

ı

ˆ |0y

“ ´
1

2!

ż

d4yd4xe2igµνβ
µ
abβ

ν
cdD

F
0 px´ yq

”

1

p2πq
3
2

u´app2qe
ip2y

1

p2πq
3
2

u´cpq2qe
iq2x

1

p2πq
3
2

u´b pp1qe
´ip1y

1

p2πq
3
2

u´d pq1qe
´iq1x

`
1

p2πq
3
2

u´app2qe
ip2y

1

p2πq
3
2

u´cpq2qe
iq2x

1

p2πq
3
2

u´d pp1qe
´ip1x

1

p2πq
3
2

u´b pq1qe
´iq1y

`
1

p2πq
3
2

u´cpp2qe
ip2x

1

p2πq
3
2

u´apq2qe
iq2y

1

p2πq
3
2

u´b pp1qe
´ip1y

1

p2πq
3
2

u´d pq1qe
´iq1x

`
1

p2πq
3
2

u´cpp2qe
ip2x

1

p2πq
3
2

u´apq2qe
iq2y

1

p2πq
3
2

u´d pp1qe
´ip1x

1

p2πq
3
2

u´b pq1qe
´iq1y

ı

.

(5.75)



5.4. Moller scattering 61

The four integrals in (5.75) can be evaluated by the following formula

ż

d4xd4yDF
0 px´ yqe

iAx`iBy

“

ż

d4reipA`Bqr
ż

d4uDF
0 puqe

ipA´B
2
qu

“ p2πq4δpA`Bq

ż

d4ur´p2πq´4

ż

d4k
e´iku

k2 ` i0
seip

A´B
2
qu

“ p2πq4δpA`Bqr´p2πq´4

ż

d4k
1

k2 ` i0

ż

d4ueip´k`
A´B

2
qu
s

“ p2πq4δpA`Bqr´

ż

d4k
1

k2 ` i0
δp´k `

A´B

2
qs

“ p2πq4δpA`Bqr´
1

A2 ` i0
s,

(5.76)

with the substitutions

x “ r `
u

2
, y “ r ´

u

2
, r “

x` y

2
, u “ x´ y. (5.77)

Therefore, (5.75) can be rewritten as

S̃
pMoq
fi “ δpq2 ´ q1 ` p2 ´ p1qM, (5.78)

where

M “
e2igµν
p2πq2

”

u´app2qβ
µ
abu

´
b pp1qu´cpq2qβ

ν
cdu

´
d pq1q

1

pq2 ´ q1q
2 ` i0

`

` u´app2qβ
µ
abu

´
b pq1qu´cpq2qβ

ν
cdu

´
d pp1q

1

pq2 ´ p1q
2 ` i0

ı

.

(5.79)

In Appendix B, we perform the computation of the differential cross section for a

general value of M. Here we will use its form in the center-of-mass reference

dσc.m
dΩ

“ p2πq2
E2

4
|M|2, (5.80)

where σc.m is the differential cross section in the center of mass reference.

The factor |M|2 for the Moller scattering is computed using (5.79) as follows



62 5. Scattering processes of Scalar Quantum Electrodynamics at the tree-level

|M|2 “

“
e4

p2πq4

”

u´ppf qβµu
´
ppiqu´pqf qβ

µu´pqiq
1

pqf ´ qiq2
`

` u´ppf qβωu
´
pqiqu´pqf qβ

ωu´ppiq
1

pqf ´ piq2

ı

ˆ

ˆ

”

u´ppiqβνu
´
ppf qu´pqiqβ

νu´pqf q
1

pqf ´ qiq2
`

` u´pqiqβαu
´
ppf qu´ppiqβ

αu´pqf q
1

pqf ´ piq2

ı

“
e4

p2πq4

” 1

16m4p0
fp

0
i q

0
fq

0
i

gµαgνω
pqf ´ qiq4

ˆ

ˆ Tr
 

{pf p{pf `mqβ
α
{pip{pi `mqβ

ω
(

Tr
 

{qf p{qf `mqβ
µ
{qip{qi `mqβ

ν
(

`

`
1

16m4p0
fp

0
i q

0
fq

0
i

gµαgνω
pqf ´ qiq2pqf ´ piq2

ˆ

ˆ Tr
 

{pf p{pf `mqβ
α
{pip{pi `mqβ

ν
{qf p{qf `mqβ

µ
{qip{qi `mqβ

ω
(

`

`
1

16m4p0
fq

0
i q

0
fp

0
i

gµαgνω
pqf ´ piq2pqf ´ qiq2

ˆ

ˆ Tr
 

{pf p{pf `mqβ
α
{qip{qi `mqβ

ν
{qf p{qf `mqβ

µ
{pip{pi `mqβ

ω
(

`

`
1

16m4p0
fq

0
i q

0
fp

0
i

gµαgνω
pqf ´ piq4

ˆ

ˆ Tr
 

{pf p{pf `mqβ
α
{qip{qi `mqβ

ω
(

Tr
 

{qf p{qf `mqβ
µ
{pip{pi `mqβ

ν
(

ı

.

(5.81)

The traces in (5.81) could be re-expressed with the help of properties (4.39) and

(4.40). We obtain from (5.81)

|M|2 “ e4

p2πq4
1

16E4
r
pi.qi ` pi.qf ` pf .qi ` pf .qf

pqf ´ qiq2
`
qi.pi ` qi.qf ` pf .pi ` pf .qf

pqf ´ piq2
s
2.

(5.82)

We can reduce this result further by taking into account the center-of-mass reference

frame. The result is

|M|2 “ e4

p2πq4
1

4E4

ˇ

ˇ

ˇ

ˇ

ppiqiq ` pqfqiq

pqf ´ piq2
`
pqipiq ` ppfqiq

ppf ´ piq2

ˇ

ˇ

ˇ

ˇ

2

. (5.83)

Using the Mandelstam variables

s “ ppi ` qiq
2
“ ppf ` qf q

2,
s

2
´m2

“ piqi “ pfqf , (5.84)



5.5. Compton scattering 63

t “ ppi ´ pf q
2
“ pqi ´ qf q

2, m2
´
t

2
“ pipf “ qiqf , (5.85)

u “ ppi ´ qf q
2
“ pqi ´ pf q

2, m2
´
u

2
“ piqf “ qipf , (5.86)

we can rewrite (5.83) as

|M|2 “ e4

p2πq4
1

16E4

ˇ

ˇ

ˇ

ˇ

s´ t

u
`
s´ u

t

ˇ

ˇ

ˇ

ˇ

2

. (5.87)

Replacing (5.87) into (5.80), we have

dσc.m
dΩ

“
α2

4s

ˇ

ˇ

ˇ

ˇ

s´ t

u
`
s´ u

t

ˇ

ˇ

ˇ

ˇ

2

, (5.88)

where α is the fine structure constant.

The result (5.88) is identical to that obtained by C. Itzykson and J. B. Zuber in [94]

using the usual approach, and by J. Beltran in [95] using CPT with Klein-Gordon-Fock

fields.

5.5 Compton scattering

The Compton scattering is the following process

b` γ Ñ b` γ, (5.89)

where b represents an scalar particle and γ a photon.

Using the creation and annihilation operator formalism, the in and out states will

take the following form

|inCompy “ |Ψiy b |Φiy

“

ż

d3p1d
3k1Ψipp1qΦipk1qa

:
pp1qεiνpk1qc

:
νpk1q|0y,

(5.90)

|outCompy “ |Ψfy b |Φfy

“

ż

d3p2d
3k2Ψf pp2qΦf pk2qa

:
pp2qεfµpk2qc

:
µpk2q|0y,

(5.91)

where Ψi,f pp1q and Φi,f pk2q are wave packet sharply peaked in pi,f and ki,f . Besides

εiν and εfµ are the initial and final vector polarization for photons.



64 5. Scattering processes of Scalar Quantum Electrodynamics at the tree-level

The transition matrix element Sfi “ xoutcomp|S|incompy is then

SCompfi “

ż

d3p2d
3k2

ż

d3p1d
3k1Ψ˚

f pp2qΦ
˚
f pk2qS̃

Comp
fi Ψipp1qΦipk1q, (5.92)

where

S̃Compfi “ x0|app2qεfµpk2qcµpk2qSa
:
pp1qεiνpk1qc

:
νpk1q|0y. (5.93)

Because of the creation and annihilation operators in (5.93), only the contribution

for S coming from D
p2q
2 will produce transition matrix element S̃Compfi non-null. There-

fore, we will focus on determining the contribution of T2 from D
p2q
2 which we rewrite as

follows
D
p2q
2 “e2i : ψpxqβνSpx´ yqβµψpyq :: AµpyqAνpxq :

´ e2i : ψpyqβµSpy ´ xqβνψpxq :: AµpyqAνpxq : .
(5.94)

To begin the causal splitting procedure, we can see in (5.94) that the numerical of

D
p2q
2 is Spx´ yq.

5.5.1 Causal splitting of Spx´ yq

In momentum space, the function Ŝppq is given by the following formula

pSabppq “
1

m
r{pp{p`mqsab pDmppq

“
i

2πm
r{pp{p`mqsabδpp

2
´m2

qSgnpp0
q.

(5.95)

In order to determine the order of singularity, we will compute the fo