Pair Production and the Light-front Vacuum

90
Pair Production and the Light-front Vacuum Ramin Ghorbani Ghomeshi

Transcript of Pair Production and the Light-front Vacuum

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Pair Production and the

Light-front Vacuum

Ramin Ghorbani Ghomeshi

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Department of Physics

Umea University

SE-901 87 Umea, Sweden

Thesis for the degree of Master of Science in Physics

c© Ramin Ghorbani Ghomeshi 2013

Cover background image: Original artwork by Josh Yoder. www.jungol.net

Cover design: The hypersurface Σ : x+ = 0 defining the front form (c.f. page 13)

Typeset in LATEX using PT1.cls 2010/12/02, v1.20

Electronic version available at http://umu.diva-portal.org/

This work is protected in accordance with the copyright law (URL 1960:729).

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Optimis parentibus

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Contents

Abstract page vii

Preface viii

Acknowledgment ix

1 Strong field theory 1

1.1 Nonlinear quantum vacuum processes 2

1.2 Pair creation 6

1.3 Summary 7

2 Introductory light-front field theory 9

2.1 Dirac’s forms of quantization 10

2.2 Light-front dynamics 12

2.2.1 Light-cone coordinates 14

2.2.2 Light-front vacuum properties 15

2.3 Summary 18

3 Free theories on the light-front 19

3.1 Free scalar field 19

3.2 Free fermion field 21

3.3 Summary 23

4 LF quantization of a fermion in a background field in (1+1) dimensions 24

4.1 Classical solution 26

4.2 Quantization 27

4.3 Zero-mode issue 29

4.4 Summary 33

5 Discrete Light-Cone Quantization 35

5.1 Quantization 36

5.2 Zero-mode issue 38

5.3 Summary 39

6 Tomaras–Tsamis–Woodard solution 40

6.1 Methodology 40

6.2 The model and its solution in Woodard’s notation 42

6.3 Quantization 44

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vi Contents

6.4 Pair production on the light-front 44

6.5 Summary 47

7 Alternative to Tomaras–Tsamis–Woodard solution 48

7.1 Quantum mechanical path integral 48

7.2 Path integral formulation for a scalar particle 49

7.2.1 Pair creation 49

7.3 Path integral for a scalar particle on the light-front 52

7.4 Summary 52

Appendix A Conventions and side calculations 53

A.1 Light-cone coordinates and gauge conventions 53

A.2 Side calculations 53

A.2.1 Derivation of the anti-commutation relation for the Dirac

spinors on the light-front 53

A.2.2 The generators of Poincare algebra for a free fermion field 57

Notes 59

References 60

Subject index 77

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Abstract

Dominated by Heisenberg’s uncertainty principle, vacuum is not quantum me-

chanically an empty void, i.e. virtual pairs of particles appear and disappear

persistently. This nonlinearity subsequently provokes a number of phenom-

ena which can only be practically observed by going to a high-intensity regime. Pair

production beyond the so-called Sauter-Schwinger limit, which is roughly the field

intensity threshold for pairs to show up copiously, is such a nonlinear vacuum phe-

nomenon. From the viewpoint of Dirac’s front form of Hamiltonian dynamics, how-

ever, vacuum turns out to be trivial. This triviality would suggest that Schwinger

pair production is not possible. Of course, this is only up to zero modes. While

the instant form of relativistic dynamics has already been at least theoretically

well-played out, the way is still open for investigating the front form.

The aim of this thesis is to explore the properties of such a contradictory aspect

of quantum vacuum in two different forms of relativistic dynamics and hence to

investigate the possibility of finding a way to resolve this ambiguity. This exercise

is largely based on the application of field quantization to light-front dynamics. In

this regard, some concepts within strong field theory and light-front quantization

which are fundamental to our survey have been introduced, the order of magnitude

of a few important quantum electrodynamical quantities have been fixed and the

basic information on a small number of nonlinear vacuum phenomena has been

identified.

Light-front quantization of simple bosonic and fermionic systems, in particular,

the light-front quantization of a fermion in a background electromagnetic field in

(1 + 1) dimensions is given. The light-front vacuum appears to be trivial also in

this particular case. Amongst all suggested methods to resolve the aforementioned

ambiguity, the discrete light-cone quantization (DLCQ) method is applied to the

Dirac equation in (1 + 1) dimensions. Furthermore, the Tomaras-Tsamis-Woodard

(TTW) solution, which expresses a method to resolve the zero-mode issue, is also

revisited. Finally, the path integral formulation of quantum mechanics is discussed

and, as an alternative to TTW solution, it is proposed that the worldline approach

in the light-front framework may shed light on different aspects of the TTW solu-

tion and give a clearer picture of the light-front vacuum and the pair production

phenomenon on the light-front.

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Preface

Since the invention of quantum electrodynamics (QED) as an effort to unify the

special theory of relativity and quantum mechanics in the late 1920s (Dirac,

1927), quantum vacuum has emerged as an extremely interesting medium

with remarkable properties to investigate. QED has been extremely successful in

explaining the physical phenomena involving the interaction between light and mat-

ter. Extremely accurate predictions of quantities like the Lamb shift of the energy

levels of hydrogen (Lamb and Retherford, 1947) and the anomalous magnetic mo-

ment of the electron (Foley and Kusch, 1948) appeared as the first testimonials

of the full agreement between quantum mechanics and special relativity through

QED and are included among the most well-verified predictions in physics (Bethe,

1947; Odom et al., 2006; Gabrielse et al., 2006, 2007). While several aspects of

this modern theory have experimentally been well-substantiated in the high-energy

low intensity regime so far, a few interesting ones in the low-energy high intensity

regime of QED, where the nonlinearity of the quantum vacuum shows up, are left

to be verified. Many different processes have already been proposed that their ver-

ification may confirm the theories about quantum vacuum structure and the high

intensity sector of QED. Upon approaching appropriate high fields, the Schwinger

pair production phenomenon is one of the most important ones which will be the

subject of careful experimental tests.

Research on this medium promises to find even a new physics beyond the Stan-

dard Model. Studying the pair production phenomenon on the front form of rela-

tivistic dynamics revealed a theoretical issue. The light-front vacuum appeared to

be trivial. This would imply that the Schwinger pairs are not allowed to pop out

of the vacuum, while they clearly must be able to be produced. Therefore, some-

thing has gone wrong. Since this thesis concerns the Schwinger pair production

phenomenon on the light-front, our survey starts from simple strong-field processes

and goes over the light-front field theory to look into such contradictory aspects

of quantum vacuum in different forms of relativistic dynamics and then probes the

possible ways that might enable us to resolve such an ambiguity. We use natural

units ~ = c = 1.

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Acknowledgment

By enrolling at Umea University, I unexpectedly embarked on a long-term

journey not only to Sweden but also to other European countries. During

this rather extended period of time, many people helped and supported me

without whom this project could not have been accomplished.

First and foremost, I would like to express my sincere gratitude to my super-

visor Anton Ilderton for introducing me to this interesting and exciting topic in

theoretical physics, his continuous support and tolerating my eccentric way of do-

ing physics. I would also like to warmly thank my examiner Mattias Marklund,

firstly for introducing me to Anton and secondly for his kind advices and critical

comments on the final version of my thesis draft.

Roger Halling was the one whose constant encouragement and support helped

me to firmly take the very first steps on my way to getting admission to Umea

University and to start my studies here without any stress and tension. I avail this

opportunity to express my admiration for the noble task that he has undertaken

as the Director of International Relations. I would also like to extend my sincere

regards to all the members of staff at the Department of Physics for their timely

support. In particular, I would like to thank Michael Bradley, Andrei Shelankov,

Jørgen Rammer and Gert Brodin who taught me different aspects of fundamental

physics and to express my gratefulness and reverence to my fellow Master’s student

and specially my office-mates Sahar Shirazi, Oskar Janson and Yong Leung who

were great sources of encouragement and made my time in office enjoyable and

memorable.

Making use of the opportunity provided for me initially by Umea University to

attend the international Master’s programme in physics, meanwhile, I could also

participate in the prestigious Erasmus Mundus AtoSiM Master’s Course (AtoSiM)

operated jointly by a consortium of three European universities which provides a

high qualification in the field of computer modeling. I feel personally obliged and

take the opportunity to thank Ralf Everaers and Samantha Barendson, the scientific

and administrative coordinators of AtoSiM programme, as the representatives of

all their colleagues in this course for all their helps and kindnesses, and specially

my AtoSiM thesis supervisor at Sapienza University of Rome, Andrea Giansanti,

to whom I am profoundly grateful.

I would like to deeply acknowledge the generosity of the editorial division of

the Cambridge University Press for giving me the right to modify and use their

pretty LATEX template, PT1.cls, to typeset my thesis. I would also like to express

my gratitude to Josh Yoder (www.jungol.net) who gave me the right to use his

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x Acknowledgment

original artwork as the background image on the cover page of my thesis report.

It is also to be noted that Figures 1.2 to 1.6 have been created using JaxoDraw

(Binosi and Theußl, 2004; Binosi et al., 2009).

I am extremely indebted to Faustine Spillebout and her family for all their kind-

ness, persistent support, hospitality and providing me with a comfortable and calm

place to work on my thesis during my stay in Mulhouse and Tours in France.

In my last trip back to Umea, I was welcomed by couples of friends, Mehdi

Khosravinia, Elnaz Hosseinkhah, Hamid Reza Barzegar and Aliyeh Moghaddam,

and spent my first few weeks in their places. I am thankful and fortunate to get

constant encouragement, support and help from all these nice friends.

I would also like to express my full appreciation to my roommate Mehdi Shahmo-

hammadi for his continuing support this year. I also sincerely express my feelings

of obligation to my fellow students at the Department of Physics: Tiva Sharifi,

Avazeh Hashemloo, Atieh Mirshahvalad, Amir Asadpoor, Narges Mortezaei, El-

ham Abdollahi, Zeynab Kolahi and Amir Khodabakhsh. I am also deeply grateful

to my friends, from those who have already left Umea or who are still here, for

keeping in touch, their helps and supports. I would like to list their names, how-

ever, the list is long and I just name a few ones as the representatives: Ali Beygi,

Amin Beygi, Ava Hossein Zadeh, Fatemeh Damghani, Bahareh Mirhadi and Yaser

Khani. I am also very thankful to Milad Tanha, Dariush Shabani and specially

Kasra Katibeh and his family for all Christmas fun we had together and Aliakbar

Farmahini Farahani and Mansour Royan for the facilities they left for us after their

departure. These friends formed my small family in Umea and their friendship will

be memorable forever.

I also gratefully thank Omid Amini for correspondence.

Last but not least, my special thanks go to my family who always valued educa-

tion above everything else, for all their love, unconditional supports and continual

efforts to make a calm and enjoyable space-time for me to work efficiently during

my whole life.

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1 Strong field theory

There have always been insoluble problems of great interest in the physics

of the current era, however, the number of constituents required to make a

problem “insoluble” has decreased with the increasing complexities of the

theories considered. The progress of physics in twentieth century has transmuted

the concerns about the insolubility of three-body problem in Newtonian mechanics

to the concerns about the problem of zero bodies (vacuum) in quantum field theory

(Mattuck, 1976). Being the lowest possible energy state of quantum field, ruled over

by the uncertainty principle and mass-energy equivalence (Figure 1.1), vacuum has

technically a quite different definition in quantum mechanics. In such a medium,

according to quantum theory, pairs of virtual particles of all types are allowed to be

created and annihilated spontaneously – vacuum fluctuation (Figure 1.2). Although

the existence of such fluctuations cannot generally be detected in a direct manner,

however, vacuum acquires a nonlinear nature due to these fluctuations that can

exhibit detectable effects which might be magnified by an external disturbance

(Klein, 1929; Sauter, 1931; Heisenberg and Euler, 1936). This disturbance can be

brought forth by applying an external electromagnetic field or imposing a boundary

condition.

Quantum

Field Theory

Special

Relativity

Vacuum

Fluctuation

Quantum

Mechanics

UncertaintyPrinciple

Mass-EnergyEquivalence

tFig. 1.1 A schematic which shows how Quantum Field Theory formed out of quantum

mechanics and special relativity merging.

1

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2 Strong field theory

As mentioned in the preface, QED theory has already been fairly tested in its

low-intensity, high energy regime. Study of the nonlinear effects of quantum vacuum

in the low-energy, high intensity regime, nevertheless, paves the way to probe QED

in its non-perturbative realm. Such investigations aimed not only at giving insight

into the validity of QED itself but also at looking for a probable new physics (Gies,

2008, 2009).

The nonlinear effects are then expected to be observed in the presence of an

strong electromagnetic field (except the Casimir effect, see Section 1.1). Although

such strong fields might be naturally found within some astrophysical systems,

nonetheless, in laboratory scales, lasers are of the few sources available for gener-

ating stronger fields than present in normal environments1. New laser techniques,

appearing after the birth of Chirped Pulse Amplification (CPA) technique (Strick-

land and Mourou, 1985), have been highly promising to supply strong enough fields

for nonlinear vacuum studies in the near future (Tajima and Mourou, 2002; Shen

and Yu, 2002; Bulanov et al., 2003; Melissinos, 2009; Marklund, 2010; Marklund

et al., 2011; Di Piazza et al., 2012a,b). As it can be seen, strong-field processes

should also have a counter effect on the internal behavior of astrophysical systems.

Thus, high-power lasers may even enable us to model the astrophysical plasma

conditions in the laboratory (Remington, 2005; Marklund and Shukla, 2006).

We roughly count a few of such quantum vacuum processes in the following. A

much more detailed discussion can, however, be found in (Milonni, 1994; Heinzl

and Ilderton, 2008; Marklund and Lundin, 2009; Lundin, 2010; Ilderton, 2012) and

the references therein.

1.1 Nonlinear quantum vacuum processes

Based on the above discussions, a typical vacuum diagram is shown in Figure 1.2

(Mandl and Shaw, 2010). However, the effects due to the nonlinearity of quantum

vacuum are stimulated in the presence of an external disturbance. A few of such

effects have been summarized in the following:

tFig. 1.2 A vacuum diagram.

1 Undulators and heavy ion collisions are other examples of the available sources.

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3 Nonlinear quantum vacuum processes

• The Casimir effect

As stated before, one way to disturb the vacuum is to impose boundary condi-

tions. This boundary condition can be in the form of two uncharged perfectly

conducting plates which are placed a few micrometers apart, parallel to each

other in vacuum. Virtual photons are the main virtual particles produced due

to the vacuum fluctuations. As the quanta of the electromagnetic field, the ap-

pearance and annihilation of these virtual photons imply the fluctuation of an

electromagnetic field in the quantum vacuum. From electrodynamics, we know

that only the normal modes of the electromagnetic field, which form a discrete

mode spectrum, can fit the distance between the plates, while any mode can

exist outside – forming a continuous mode spectrum (Figure 1.3). Thus, only

these normal modes do contribute to vacuum energy in between the plates

whereas the contribution to vacuum energy outside the plates comes from the

aforementioned continuous mode spectrum consisting of every mode. As the

plates are moved closer, number of such normal modes decreases which implies

that the energy density decreases in between the plates. Therefore, the energy

density will be lower than the outside and a finite attractive force between the

plates will appear due to the change in energy. Casimir showed in his paper

that this attractive force (per unit area) has the following form (Casimir, 1948;

Casimir and Polder, 1948),

F (d) = −0.0013 d−4 N.m−2 , (1.1)

where d is the distance between the plates measured in microns. This, for in-

stance, implies an attractive force of 0.0013 Newtons for two 1 × 1 m plates

which are separated by 1 µm (Milonni and Shih, 1992). This effect was then

generalized to the case of parallel plates of dielectrics (Lifshitz, 1956) and early

experiments supported the existence of such an attractive force qualitatively

(Deriagin and Abrikosova, 1957a,b; Sparnaay, 1958; van Blokland and Over-

beek, 1978). Later on, different aspects of the Casimir effect have been studied

in more detail and high precision experiments have been proposed and set up

to test it (Bordag et al., 2001) and even some applications due to this effect

have been developed (Serry et al., 1998; Buks and Roukes, 2001; Chan et al.,

2001; Palasantzas and De Hosson, 2005). Recently a group of scientists re-

ported the observation of the dynamical Casimir effect (Wilson et al., 2011),

which had been predicted some 40 years ago (Moore, 1970).

• Vacuum birefringence

A strong external field modifies the vacuum fluctuations such that the quan-

tum vacuum, as a medium, acquires different non-trivial refractive indices for

different polarization modes of a probe photon and, hence, the phase velocity

is different for photons of different polarizations. This is the so-called vacuum

birefringence phenomenon (Toll, 1952; Heyl and Hernquist, 1997; Heinzl and

Schroder, 2006; Heinzl and Ilderton, 2009; Ilderton, 2012).

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4 Strong field theory

d

tFig. 1.3 The Casimir effect, schematically.

• Photon-photon scattering

A linear system generally satisfies two requirements: superposition and homo-

geneity, c.f. (Hoffman and Kunze, 1971) for example. The superposition princi-

ple necessitates the waves (here, photons) propagating in such a linear system

to be indifferent to each other, as it has already been taken for granted in

classical physics. Since the quantum vacuum is a nonlinear medium, however,

this principle may not hold. As a result, there might be an interaction between

the propagating photons and virtual electron-positron pairs of the quantum

vacuum. Therefore, quantum vacuum fluctuations may appear as mediators

interacting with them exchanges energy and momentum between the photons.

In other words, photon-photn scattering may happen via vacuum fluctuations.

At very high laser beam intensities, there would also be a non-zero probability

for multiple photons to interact with vacuum fluctuations at the same time and

a smaller number photons with higher frequencies come out of the interaction

process (Fedotov and Narozhny, 2007). In other words, high-order harmonics

may be generated during this high-intensity nonlinear vacuum process. This

has been a hot topic of research in recent years (Brodin et al., 2001; Eriksson

et al., 2004; Brodin et al., 2006; Lundstrom et al., 2006; Archibald et al., 2008).

tFig. 1.4 Photon-photon scattering diagram. Double lines represent the dressed propagators

due to particles in background field.

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5 Nonlinear quantum vacuum processes

• Nonlinear Compton scattering

In a strong background field, an electron can simply emit a photon and digress

from its initial direction of motion (Nikishov and Ritus, 1964; Harvey et al.,

2009; Boca and Florescu, 2009a,b; Heinzl et al., 2010a; Seipt and Kampfer,

2011; Mackenroth and Di Piazza, 2011). This simple nonlinear process, at

higher orders, appears as part of more complicated processes like trident pair

production in which an either virtual or real photon that is created by a nonlin-

ear Compton scattering itself, creates a pair of electron-positron via stimulated

pair production2 (Ilderton, 2011, 2012), or cascades in which the nonlinear

Compton scattering and stimulated pair production occur consecutively for a

number of times (Fedotov et al., 2010b,a; Sokolov et al., 2010; Elkina et al.,

2011). Nonlinear Compton scattering has been experimentally verified (Bula

et al., 1996).

e−

e−

γ

tFig. 1.5 Nonlinear Compton scattering diagram.

• Self-lensing effects

This term clearly refers to those kind of effects that arise from the self-affecting

characteristic of a strong pulse of light in the quantum vacuum. As might be

expected, modified properties of an electromagnetically disturbed vacuum mu-

tually modifies the way the disturbing electromagnetic pulse itself propagates

in the vacuum. Under certain circumstances this effect may result in a few

subsequent effects, e.g. photon splitting (Adler, 1971) or the formation of light

bullets (Brodin et al., 2003). Discussion on more such effects can be found in

(Rozanov, 1998; Soljacic and Segev, 2000; Marklund et al., 2003; Shukla and

Eliasson, 2004; Marklund and Lundin, 2009).

• Photon acceleration

The quantum vacuum fluctuations in the presence of a strong background field

causes the vacuum to look like a rippling medium with respect to the density

distribution of the virtual electron-positron pairs at various instants. This be-

havior mimics the plasma oscillations. As a result, the group velocity of a test

photon propagating in such a medium will continually change and, hence, its

frequency will also shift consequently. This recurrent change of group velocity

naturally denotes a photon acceleration (Mendonca et al., 1998; Mendonca,

2 Pair creation due to a high energy photon (Heinzl et al., 2010b; Ilderton, 2012), which was

experimentally addressed in SLAC Experiment 144 for the first time (Bamber et al., 1999).

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6 Strong field theory

2001; Mendonca et al., 2006).

Many other nonlinear effects have already been introduced in quantum vacuum.

We have just roughly discussed a few of them above. A more complete list can be

found, e.g., in (Marklund and Lundin, 2009). One more effect, which constitutes

the keystone of our survey, has been left to be introduced: pair creation.

1.2 Pair creation

Amongst all other quantum vacuum processes, spontaneous pair production has

been one of the most popular one in the literature of different fields (Pioline and

Troost, 2005; Marklund et al., 2006; Kim and Page, 2008; Ruffini et al., 2010;

Garriga et al., 2012; Chernodub, 2012). As mentioned before, an external electro-

magnetic field will modify the distribution of virtual electron-positron pairs. This

modification can be thought of as vacuum polarization (Figure 1.6). The virtual

e−e+ pair can gain energy from this external electric field to become real particles

(Dunne, 2009). This happens for a virtual electron, for instance, if the energy gained

by this electron from the external field in traversing one Compton wavelength3

amounts to its rest-mass energy. Thus, if the electric field strength surpasses a crit-

ical value, the vacuum will break down spontaneously into electron-positron pairs

(Figure 1.6). This critical field strength is called Sauter-Schwinger limit (Sauter,

1931; Schwinger, 1951a) and is given by

Ec =m2c3

e~≈ 1.3 × 1018 V/m , (1.2)

where m here is the mass of electron. This process occurs with a probability propor-

tional to exp(−πEc/E) which implicitly shows this process is exponentially drops

off in the weak fields limit.

Lasers are the most powerful high-intensity electromagnetic field generators in

laboratory scales. It is possible to construct a region at the intersection of two or

more coherent laser beams wherein only a strong electric field exists, but not any

magnetic one. (Roberts et al., 2002). Although the critical electric field strength is

not accessible at the present time, the next generation high-power laser facilities,

such as the European X-ray Free Electron Laser (XFEL)4, the European High

3 The Compton wavelength is defined as λC = ~/mc for a particle of mass m (Compton, 1923), sothat for a field with a wavelength smaller than this value for a special particle, the field quantawill have energies well above the rest-mass energy of that particle and particle-antiparticle paircreation becomes abundant.

4 http://www.xfel.eu/

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7 Summary

q

k

k − q

tFig. 1.6 Left: vacuum polarization. Right: vacuum breaks down into real e−e+ pairs above the

Sauter-Schwinger limit.

Power laser Energy Research facility (HiPER)5, Extreme Light Infrastructure (ELI)

project6 and the project running at the Exawatt Center for Extreme Light Studies

(XCELS)7 will hopefully be able to approach the field intensities (∼ 10−4Ec) a few

orders below the field intensity threshold above which pair production rate becomes

significant and may enable us to directly investigate the ultra-high intensity sector of

the QED theory (Roberts et al., 2002; Schutzhold et al., 2008; Dunne et al., 2009;

Heinzl and Ilderton, 2009; Ilderton et al., 2011). The Schwinger pair production

theory is based on a constant electric field whereas laser systems normally generate

rapidly-alernating electromagnetic fields. The effect of such alternating, pulsed,

and in some cases inhomogeneous, electromagnetic fields on Schwinger mechanism

of pair production and Sauter-Schwinger limit has already been investigated to a

great extent and different setups to verify this process experimentally has already

been proposed (Alkofer et al., 2001; Narozhny et al., 2004; Di Piazza, 2004; Dunne

and Schubert, 2005; Kim and Page, 2006; Kleinert et al., 2008; Hebenstreit et al.,

2008; Allor et al., 2008; Hebenstreit et al., 2009; Chervyakov and Kleinert, 2009;

Hebenstreit et al., 2011b; Dumlu and Dunne, 2011b; Hebenstreit et al., 2011a;

Chervyakov and Kleinert, 2011; Kim et al., 2012; Kohlfurst et al., 2012; Gonoskov

et al., 2013). The profound effect of the pair production process under strong fields

in large-scale universe events has been predicted some 40 years ago (Hawking,

1974, 1975; Unruh, 1976) and, although still under dispute, has been claimed to be

observed recently (Belgiorno et al., 2010; Schutzhold and Unruh, 2011; Belgiorno

et al., 2011).

1.3 Summary

In a video by CERN8, Peter Higgs well-summarizes the idea this chapter is basedupon: “When you look at a vacuum in a quantum theory of fields, it isn’t exactlynothing”. Unification of quantum mechanics and the special relativity theory rep-resents a new picture of vacuum in which “vacuum is no longer quite as empty as

5 http://www.hiperlaser.org/6 http://www.extreme-light-infrastructure.eu/7 http://www.xcels.iapras.ru/8 Meet Peter Higgs: http://cds.cern.ch/record/1019670

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8 Strong field theory

it is used to be”, but virtual particle pairs are allowed to be spontaneously createdand annihilated. This phenomenon is called vacuum fluctuation which gives thevacuum a nonlinear characteristic. These fluctuations cannot be observed directly.However, they give rise to a set of nonlinear effects in the presence of an externaldisturbance which can confirm their existence indirectly. This external disturbancecan be of the form of an external electromagnetic field or a boundary condition.These kinds of disturbances induce a bunch of new physical effects to happen ofwhich, for instance, the Casimir effect, vacuum birefringence and Schwinger pairproduction, which is the break down of highly polarized vacuum into real pairsdue to the presence of a strong external electric field, can be named. Normally aquite strong external electromagnetic field is required for such nonlinear vacuumphenomena to happen detectably. Such a critical field strength, e.g., for pair pro-duction phenomenon has already been calculated by Schwinger and turned out tobe of the order of 1018 V/m. With the appearance of modern laser facilities, highfield intensities up to 10−4Ec will hopefully be reached in the near future and itmay become possible to verify the Strong-Field QED effects directly.

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2 Introductory light-front field theory

With the appearance of modern theories of physics in twentieth century, a

rather new field in physics came gradually into existence that was try-

ing to find new ways to describe the different physical characteristics

and behaviors of elementary particles. Subatomic scales and relativistic speeds of

elementary particles drew attention to the need for a consistent combination of the

two apparently distinct modern theories of the twentieth-century physics: relativity

theory and quantum mechanics. Attempts in unifying these two theories, however,

pushed the physicists off an effort at a quantum description of a single relativis-

tic particle into an inherently many-body theory. Amongst all the motivations to

come by such a relativistic many-body theory, the demands for locality, ubiquitous

particle identicality, non-conservativity of particle number (specially when a parti-

cle is localized within a distance of the order of its Compton wavelength) and the

necessity of anti-particles can be addressed. The nature of such particles finally

showed that they should actually be considered as subordinate identities derived

from a more comprehensive concept, i.e. field1. It was found out that the problem

was originated from the fact that space and time had been treated very differently

in quantum mechanics. The former is consistently represented by a Hermitian op-

erator while the latter, which is also an observable like space, enters into the theory

just as a label. This task, which seemed hard to accomplish in the beginning, could

finally be fulfilled by treating both space and time equally as labels rather than

operators (Srednicki, 2007). This approach led us to the concept of quantum field

theory2 in which at least one degree of freedom was assigned to each point x in

space. These degrees of freedom are basically functions of space and time.

Furthermore, studies of the free relativistic point particle showed that the choice

of time parameter within special relativity corresponds to a gauge fixing and is not

unique. The procedure of choosing a time parameter naturally leads to a (3 + 1)-

foliation of space-time into space (hypersurfaces of equal-time, τ = const.) and time

(with a direction orthogonal to these equal-time hypersurfaces). In a reasonable

choice of time, however, the equal-time hypersurface Σ should intersect any possible

world-line (existence criterion) once and only once (uniqueness criterion) in order

to be consistent with causality (Heinzl, 1998, 2001).

1 For a nice review of the underlying principles of QFT, see (Wilczek, 1999; Tong, 2006).2 The alternative approach in which time is promoted to an operator can motivate a theory based

on world-sheets that leads us to the much more complicated concept of string theory (Srednicki,2007; Zwiebach, 2004). It is to be noted, however, that this approach by no means represents

how string theory was actually developed.

9

Page 20: Pair Production and the Light-front Vacuum

10 Introductory light-front field theory

Table 2.1 All possible choices of hypersurfaces Σ : τ = const. with transitiveaction of the stability group GΣ · d denotes the dimension of GΣ , that is, the

number of kinematical Poincare generators; x⊥ ≡ (x1, x2)

namea Σ τ d

instant x0 = 0 t 6

light front x0 + x3 = 0 t+ x3/c 7

hyperboloid x20 − x2 = a2 > 0, x0 > 0 (t2 − x2/c2 − a2/c2)1/2 6

hyperboloid x20 − (x⊥)2 = a2 > 0, x0 > 0 (t2 − (x⊥)2/c2 − a2/c2)1/2 4

hyperboloid x20 − x2

1 = a2 > 0, x0 > 0 (t2 − x21/c

2 − a2/c2)1/2 4

a Note: table has been taken from (Heinzl, 2001).

2.1 Dirac’s forms of quantization

Symmetry, conservation law and invariance are of key concepts in modern physics,

which are fundamentally interconnected. The group structure of the set consisting

of all symmetry operations in a system suggests that the best way to mathemat-

ically treat the symmetries and invariants is to use the group theory (Arfken and

Weber, 2005; Carmichael, 1956). Indeed, all fields in a quantum field theory “trans-

form as irreducible representations of the Lorentz and Poincare groups and some

isospin group” (Kaku, 1993). Intuitively, we know that energy, 3 momenta, 3 angular

momenta and 3 boosts (i.e. Lorentz transformations) comporise ten fundamental

quantities that characterize a dynamical system (Harindranath, 1997). Since the

conservation of these quantities generally addresses underlying symmetries in a dy-

namical system and invariance under certain transformations, one may naturally

refer to the concept of full Poincare group in order to study the full relativistic

invariance of a system. Poincare group packs dealing with all the above-mentioned

fundamental quantities in a set of algebraic equations. This group is generated by

the four-momentum Pµ and the generalized angular momentum Mµν . These gener-

ators have the following form in our conventional framework (in the next section, we

will see that this conventional framework corresponds to a certain (3+1)-foliation

of space-time which is called the instant form of Hamiltonian dynamics) (Heinzl,

2001)

Pµ =

Σ

d3xT 0µ , (2.1a)

Mµν =

Σ

d3x(xµT 0ν − xνT 0µ

), (2.1b)

where T µν is the energy-momentum tensor. Hence, the relations for Poincare alge-

bra, which is the Lie algebra1 of the Poincare group, can be written in terms of Pµ

Page 21: Pair Production and the Light-front Vacuum

11 Dirac’s forms of quantization

Box 2.1 Metric tensors corresponding to Dirac’s forms of Hamiltonian dynamics

The instant form The front form The point form

gµν =

1 0 0 0

0 −1 0 0

0 0 −1 0

0 0 0 −1

gµν =

0 0 0 12

0 −1 0 0

0 0 −1 012

0 0 0

gµν =

1 0 0 12

0 −τ2 0 0

0 0 −τ2 sinh2 ω 0

0 0 0 −τ2 sinh2 ω sin2 θ

Note: the contents of this box have been taken from (Pauli, 2000).

and Mµν as follows (Ryder, 1985; Weinberg, 1995)

[Pµ, P ν ] = 0 , (2.2a)

[Pµ,Mρσ] = i (gµρP σ − gµσP ρ) , (2.2b)

[Mµν ,Mρσ] = i (−gµρMνσ + gµσMνρ − gνσMµρ + gνρMµσ) , (2.2c)

or equivalently as

Pµ, P ν = 0 , (2.3a)

Mµν , P ρ = gνρPµ − gµρP ν , (2.3b)

Mµν ,Mρσ = gµσMνρ − gµρMνσ − gνσMµρ + gνρMµσ . (2.3c)

Going back to the case of (3 + 1)-foliation of space-time, we realize that the dy-

namical evolution of a system, i.e. development in τ , is technically determined by

the structure of those Poincare group generators which correspondingly map the

initial data hypersurface Σ to another hypersurface Σ′ at a later time τ ′ (Flem-

ing, 1991). Such generators are fairly called dynamical. In contrast, those Poincare

group generators under which the hypersurface Σ is left invariant are called kine-

matical and they form a subgroup of the Poincare group called stability group2

GΣ of Σ (Heinzl, 2001). This group is closely associated with the topology of the

hypersurface.

Further studies by Dirac (Dirac, 1949, 1950) showed that only three different

foliations of space-time and, therefore, three forms of initial data hypersurfaces are

essentially possible. He called them the instant, front and point forms. Later on,

two more possible choices were added to this list (Leutwyler and Stern, 1978). The

list of all possible choices of hypersurfaces with transitive action3 of the stability

group, GΣ, has been summarized in Table 2.1 which has been clipped from (Heinzl,

2001). As it can be seen in Table 2.1, all forms obey the correspondence principle

in the non-relativistic limit of c→ ∞.

Geometrically, the instant form is exactly what we have been familiar with,

namely the celebrated equal usual time hypersurface, Σ : x0 = 0, on which the

Page 22: Pair Production and the Light-front Vacuum

12 Introductory light-front field theory

conventional quantum field theory had been formulated. The front form is the hy-

persurface, Σ : x+ ≡ x0 + x3 = 0, in space-time which is tangent to the light-cone.

It is similar to the wave front of a plane wave advancing in x3 direction with the

velocity of light. That is why it is called the ‘front’ form4. Finally, the point form is

a Lorentz-invariant hyper-hyperboloid, Σ : xµxµ = const., lying inside future light-

cone. Three inequivalent forms of Hamiltonian dynamics have been illustrated in

Figure 2.1 and their corresponding metrics have been given in Box 2.1 taken from

(Pauli, 2000).

According to the above definitions, it appears that there is an isomorphism4

between the stability group of any space-like hypersurface and the six-parameter

Euclidean group of spatial translations and rotations (Fleming, 1991). Therefore,

the instant form has six stability group members; d = 6 in Table 2.1. The front

form and the point form appeared to have seven-parameter and six-parameter sta-

bility groups. As a result, the dynamical evolution of the instant form, front form

and point form will be determined by the structure of only four, three and four

independent Poincare group generators, respectively.

It appears conclusively that the more the the number of stability group members,

the higher the degree of symmetry of the hypersurface is (Heinzl, 2001). Therefore,

there naturally would be interests in further practice with the front form that has

the biggest stability group.

2.2 Light-front dynamics

Dirac’s Hamiltonian approach in covariant theories (Dirac, 1949, 1950), which

seemed more convenient for dealing with the structure of bound states in atomic

and subatomic systems, was overshadowed for a long time by Feynman’s action-

oriented approach, which in turn was more suitable for deriving the cross sections.

However, this approach revealed new features of Hamiltonian dynamics that could

describe the dynamical evolution of a system much simpler. The front form, with

the largest stability group, was the most spectacular and interesting achievement

of this approach, which was rediscovered later on in the context of high energy

physics (Fubini and Furlan, 1965; Weinberg, 1966, 1967; Dashen and Gell-Mann,

1966; Lipkin and Meshkov, 1966) and was applied to the case of “constituent pic-

ture of the hadron” to avoid the complexity of the ground state (vacuum) in QCD

(Bjorken, 1969; Feynman, 1972; Kalloniatis, 1995). After that, it was applied to a

wider range of cases, either to cope with the complexities arose in the conventional

approach (Wilson, 1990; Perry et al., 1990; Brodsky and Pauli, 1991; Wilson et al.,

1994; Brodsky, 1998) or trying to get a better understanding of available theories

(Witten, 1983, 1984).

4 It is also alternatively referred to as the null-plane (Neville and Rohrlich, 1971; Coester, 1992).

Page 23: Pair Production and the Light-front Vacuum

x1,x2

x3

x0

Σ : x0= 0

x1,x2

x3

x0

Σ : x+= 0

x1,x2

x3

x0

Σ : xµxµ = const.

tFig. 2.1 Left: the instant form. Middle: the front form. Right: the point form.

Page 24: Pair Production and the Light-front Vacuum

14 Introductory light-front field theory

2.2.1 Light-cone coordinates

As we already noted, (3+1)-foliation of space-time based on the front form pro-

poses working in a new coordinate system which is called light-cone coordinates5.

Converting to this coordinate is not a Lorentz transformation, but a general coor-

dinate transformation. The light-cone coordinates can be defined, in dim. ≥ 2, as

the world-line of light traveling in ±x3 direction at x0 = 0 (see Figure 2.2):

x± ≡ x0 ± x3 . (2.4)

x3

x0

x− x+

tFig. 2.2 Light-cone coordinate axes x± compared with usual space-time axes.

Other coordinates do not change. Therefore, we simply show them as x⊥ ≡(x1, x2

).

While either x+ or x− can basically be considered as time and the other one as

space, we take x+ as light-cone time and x− as light-cone space. This coordinate

transformation is correspondingly applied to any vector (or tensor) as well. Thus,

a vector aµ is transformed to light-cone coordinates as

a± ≡ a0 ± a3 , (2.5)

while the other components remain unchanged. Based on this fact and considering

the front form metric gµν in Box 2.1, scalar product in such a coordinate system is

defined as

a · b = gµνaµbν =

1

2a+b− +

1

2a−b+ − a1b1 − a2b2 . (2.6)

Energy and momentum are similarly defined on the light-cone as

p± ≡ p0 ± p3 . (2.7)

Since in k · x product, k− is conjugated with the light-cone time x+ it seems rea-

sonable to take it as the energy on the light-cone and, naturally, k+ as momentum

on the light-cone. Note that for a particle moving in x3 direction with velocity v,

the light-cone velocity turns out to be dx−/dx+ = (1 − v)/(1 + v). Obviously, the

5 It is also frequently referred to in the literature by other names like infinite momentum frame

(Fubini and Furlan, 1965; Weinberg, 1966, 1967; Soper, 1971).

Page 25: Pair Production and the Light-front Vacuum

15 Light-front dynamics

light-cone velocity can range from 0 to ∞ for a particle which is traveling with the

speed of light in the x3 or −x3 directions, respectively. As an advantage of these

coordinates, they offer very simple transformation under boosts along x3 axis which

is quite useful in high energy physics.

Converting to light-cone framework causes a few peculiar features to appear. The

first interesting characteristic is that for an on-mass shell particle, we will have

k+ ≥ 0 . (2.8)

This simple condition culminates in some profound changes in our conventional

viewpoint towards QFT. Its combination with the mass-shell constraint on the

light-cone, k+k− −(k⊥)2

= m2, gives a dispersion relation of the form of

k− =

(k⊥)2

+m2

k+, (2.9)

for an on-mass shell particle, which has a number of interesting features of which

we may mention, e.g., the absence of any square root factor in such a relativistic

dispersion relation. A more complete list of such interesting features can be found

in (Harindranath, 2000). Another unusual feature of light-cone dynamics is the

separation of relative and center of mass motion of a relativistic many body system

much in the same way these two motions decouple from each other in a non-

relativistic many body system. A pedagogical summary on light-cone methods can

be found in (Collins, 1997).

2.2.2 Light-front vacuum properties

Different definitions have already been presented for vacuum in quantum field the-

ory. So far, we have seen one of such definitions in the beginning of Chapter 1 and

two more, which are among the most popular definitions for a vacuum state, have

been summarized in (Fleming, 1991). Any definition we take at the outset, by a

common-sense approach towards the vacuum as a physical medium, we expect it to

behave similarly in different forms of Hamiltonian dynamics. This is actually the

case in the absence of any interaction. However, the situation is different when in-

teractions come into play. As we have already seen in Chapter 1, in the presence of

interactions (specifically, a strong background electric field), there should be a non-

zero probability for Schwinger pairs to be created (Sauter, 1931; Schwinger, 1951a).

In fact, this phenomenon had been studied conventionally in the instant form of

Hamiltonian dynamics. The ordinary kinetic momentum, i.e. p in pµ =(p0,p

), can

have both negative and positive values in the instant form. Therefore, we may find

many excited states, like a†ka†−k, with zero valued kinetic momenta (k + (−k) = 0)

Page 26: Pair Production and the Light-front Vacuum

Table 2.2 Vacuum structure from the instant and front forms of relativistic dynamics point of view

Quantum vacuum of the free the-

ory in the instant form of rela-

tivistic dynamics; Fµν = 0

:H : ∼ b†pbp + d†pdp

:H : |0〉 = 0

|0〉 ≡ vacuum state of

the free theory

Although virtual pairs are allowed to be

created, no real pairs are come into exis-

tence in a free theory. This can be seen

from the normal-ordered Hamiltonian of

the system in which only terms involving

the same number of creation and annihi-

lation operators appear. This means, in

other words, that particle number is con-

served ([H,N ] = 0).

Quantum vacuum in the in-

stant form in the presence of

an electromagnetic background

field; Fµν 6= 0

:H : ∼ b†pd†−p

+ etc.

:H : |0〉 = |f〉

|f〉 ≡ ψ0 |0〉+ψ1 |pair〉+

ψ2 |two pairs〉+ · · ·

The appearance of such terms indicate

that particle number is not conserved

([H,N ] 6= 0) and as a result, pairs of

particles-antiparticles can be created.

Here, N is the number operator. Note

that p can acquire both positive and neg-

ative values (c.f. Section 4.3).

Light-front vacuum in the pres-

ence of an electromagnetic back-

ground field; Fµν 6= 0

:H : ∼ b†k−bk−

+ d†k−dk−

:H : |Ω〉 = 0

|Ω〉 ≡ vacuum of the

interacting theory

The appearance of terms involving delta

functions of the form of, e.g. δ(k− + k′−)

together with Equation (2.8) prevents

pairs to appear such that k− conserva-

tion can hold. Thus, only terms involving

the same number of creation and anni-

hilation operators remain in the Hamil-

tonian of the system on the light-front.

Hence, there is no fluctuation.

Page 27: Pair Production and the Light-front Vacuum

17 Light-front dynamics

that can mix with the vacuum6 (Burkardt, 2002). Therefore, vacuum of the interact-

ing theory is very complicated. When we study this problem in front form, however,

we find out that the light-front vacuum, which is a ground state of the free theory,

remains a ground state of the full theory as well. It means that Schwinger pairs

do not have any room to appear in the front form treatment. Actually, it comes

to know that, due to the non-negative spectrum of light-cone momentum opera-

tor (2.8), the emergence/disappearance of any quanta from/into light-front vacuum

would be accompanied by a violation of light-cone momentum conservation which

prevents such processes to occur (Fleming, 1991). In other words, except for pure

zero mode excitations, k− = 0, all the other excited states will have non-zero value

longitudinal momenta and therefore cannot mix with the vacuum. Thus, as the most

spectacular feature of the light-front dynamics, the vacuum turns out to be trivial,

i.e. stable7 (see Table 2.2; this case will be discussed in more detail in Section 4.3).

It is worth pointing out that zero-modes are high energy modes and have to be

properly treated in a way (Lenz et al., 1991). The discrete light-cone quantization

(DLCQ) method, which will be reviewed in Chapter 5, has been proposed to resolve

such a zero-mode issue. Having considered the general definitions available for the

vacuum state of a field theory, one naturally expects to encounter with surprising

subsequent features in light-front vacuum. For instance, it is shown that Coleman’s

theorem (Coleman, 1966) breaks down in null-plane quantization (Fleming, 1991).

This theorem simply states that when a generalized charge operator, Q, acts on the

vacuum state, |Ω〉, it satisfies

Q |Ω〉 = 0 (2.10)

if and only if there exists a local conservation law for its associated generalized four-

current density (or correspondingly a continuous local symmetry in the theory).

Nevertheless, it has been demonstrated that Equation (2.10) can hold in light-front

vacuum even if no local symmetry exists (Fleming, 1991). Briefly speaking, it can be

shown almost for all explicit cases that, with a few exceptions in some aspects, the

null-plane quantum field theory is equivalent to instant form quantum field theory

(Brodsky et al., 1998) and it is equally appropriate for the field theory quantization

(Srivastava, 1998).

As roughly stated before, it turned out that the front form is less cumbersome in

coping with the vacuum state of some quantum field theories (Brodsky et al., 1998).

Actually, the reason is that vacuum is simple in front form. In the next chapters

we will see how different fields will be quantized on the light-front.

6 We distinguish between the canonical momentum k and kinetic momentum p following theconvention made in (Kluger et al., 1992). In the case of a free field, the canonical momentumcoincides with the kinetic momentum.

7 Of course, with the exception of zero-modes, namely the modes with k− = 0.

Page 28: Pair Production and the Light-front Vacuum

18 Introductory light-front field theory

2.3 Summary

By the appearance of quantum field theory, it was discovered that the choice oftime parameter within special relativity is not unique and any attempt to choosea time parameter leads to a foliation of space-time into space and time. Diracshowed that only three distinct foliations of space-time, corresponding to instant,front and point forms, are essentially possible (c.f. Table 2.1 and Figure 2.1).Among these, the instant form is the one we are already familiar with, on whichthe so-called conventional quantum field theory has been formulated. The frontform, which is a hyper-plane tangent to the light-cone, constitutes the keystone ofthe current survey. Light-front dynamics presents a set of peculiar features whichdo not have any analogous structure in the instant form. For instance, the boostand Galilei invariance can be mentioned. However, the most remarkable feature ofthe front form of Hamiltonian dynamics is the triviality of its vacuum which, apartfrom showing a few fundamental disparities compared to the instant form (likegiving no signature of Schwinger pairs, which will be discussed in more detail in thenext chapters), seems to be extremely promising in dealing with the quantum fieldtheories that suffer from the complexities of the ground states in the instant formof Hamiltonian dynamics. In other words, light-front vacuum is simple and, withthe exception of zero-modes (modes with k− = 0), no other excited state can mixwith it. Although the inclusion of zero-modes means that the light-front vacuum isnot actually pure trivial, however, it is in fact essential for getting many quantumvacuum processes right. Thus, engaging a Hamiltonian approach in the front formseems to reduce the complexities that appeared with this approach in the instantform.

Page 29: Pair Production and the Light-front Vacuum

3 Free theories on the light-front

We have seen in Chapter 2 that, contrary to the vacuum structure in the

instant form, the light-front vacuum is trivial which makes using of the

Hamiltonian approach simpler. In this chapter, we are going to see how

different fields are quantized on the light-front. Different quantization methods

on the light-front, like Schwinger’s quantum action principle (Schwinger, 1951b,

1953a,b) or the method due to Faddeev and Jackiw (Faddeev and Jackiw, 1988;

Jackiw, 1993), have been comprehensively discussed in (Heinzl, 2001).

To review the basics of light-front quantization, we start with the quantization of

free fields. The evolution of every single degree of freedom in free theories does not

depend on the other degrees of freedom. For the details behind these calculations,

one can turn to the reviews by (Brodsky et al., 1998), (Harindranath, 1997) or

(Heinzl, 2001).

3.1 Free scalar field

As the simplest relativistic free theory, we consider the classical Klein-Gordon (KG)

equation for a real scalar field. The Lagrangian density for this field (Peskin and

Schroeder, 1995)

L =1

2(∂µφ)2 − 1

2m2φ2 , (3.1)

turns into the following relation when it is expressed in light-front framework in

(1 + 1) dimensions:

L =1

2∂+φ∂−φ− 1

2m2φ2 . (3.2)

The equation of motion, following from the Euler-Lagrange equation, can be written

as(∂+ ∂− +m2

)φ = 0 . (3.3)

There are a few characteristics involved in this equation of motion that are briefly

summarized below:

• It is first-order in the time derivative;

• The conjugate momentum for this system is constrained and not dynamical;

19

Page 30: Pair Production and the Light-front Vacuum

20 Free theories on the light-front

• Other quantization methods like the aforementioned method due to Faddeev

and Jackiw (Faddeev and Jackiw, 1988; Jackiw, 1993) should normally be used

to treat such a system with a constrained conjugate momentum rather than

the conventional canonical quantization formalism.

A simple trick here is, however, to make use of the already known equal usual

time commutation relation to construct the equal-x+ commutation relation for this

system (Harindranath, 2000). A similar calculation of this type, which has been

done for the case of a free fermion field, can be found in Appendix A. The mode

expansion for a free scalar field in the instant form in (3+1) dimensions is written

as below

φ(x) =

∫d3k

(2π)31√2Ek

[ak e

−ik·x + a†k eik·x] ∣∣∣∣

k0=Ek

, (3.4)

with the only non-vanishing commutation relation of [ak , a†k′ ] = (2π)3 δ(3)(k−k′).

This gives a canonical commutation relation

[φ(x) , φ(y)]

=

∫d3k

(2π)3

∫d3k′

(2π)31

2√EkEk′

([ak , a

†k′

]e−ik·x+ik′·y +

[a†k , ak′

]eik·x−ik′·y

)

=

∫d3k

(2π)31

2Ek

(e−ik·(x−y) − eik·(x−y)

)

= −i∫

d3k

(2π)31

Ek

eik·(x−y) sin(k0(x0 − y0)

) ∣∣∣∣k0=Ek

. (3.5)

As a result, the equal-x+ commutation relation turns out to have the following form

in (1+1) dimensions (Heinzl, 2001)

[φ(x) , φ(y)]x+=y+=τ = − i

4sgn(x− − y−) , (3.6)

where the antisymmetric Green function sgn(x−) is defined such that it satisfies

∂− sgn(x−) = 2δ(x−) . (3.7)

Equation (3.6) is obviously different from the analogous commutation relation in the

instant form (3.5) for equal usual time in which [φ(x) , φ(y)]x0=y0 = 0 to satisfy the

condition of microscopic causality. However, for the case of x+ = y+ in light-front

dynamics, the two fields are separated by a light-like distance. Thus, the associated

commutation relation has not necessarily to vanish.

Furthermore, the (1+1)-dimensional version of the Fock space1 expansion for a

free scalar field in the light-front framework is written as (Leutwyler et al., 1970;

1 Defining the vacuum state and then creating other states by applying the creation operator on

it (Fock, 1932).

Page 31: Pair Production and the Light-front Vacuum

21 Free fermion field

Rohrlich, 1971; Chang et al., 1973)

φ(x) =

∫ ∞

0

dk+

2k+(2π)

[a(k) e−ik·x + a†(k) eik·x

], (3.8)

where[a(k) , a†(k′)

]= 2(2π) k+ δ(k− − k′−) , (3.9)

and all other commutation relations vanish.

To make our review a bit more inclusive, we just roughly mention the generators of

the Poincare algebra (2.3) for such a field in Fock representation in (1+1) dimensions

P+ =

∫dk+

4πa†(k) a(k) , (3.10a)

P− =

∫dk+

4πk+m2

k+a†(k) a(k) , (3.10b)

K− =

∫dk+

4πk+

(∂

∂k+a†(k)

)k+ a(k) , (3.10c)

where P+, P− and K− are corresponding to momentum operator, Hamiltonian op-

erator and the generator of boost at x+ = 0, respectively. The generators associated

with rotations will clearly vanish in a system with only one spatial dimension. For a

detailed discussion on different Poincare generators in (3+1)-dimensional light-front

dynamics and the commutation relations between them, refer to (Harindranath,

1997).

3.2 Free fermion field

Free fermion field is an example of a Lorentz invariant system with an equation

of motion which is first-order in derivatives. This system constitutes one of the

main parts of this study project and will be discussed in a more general case with

background field in the next chapter. Therefore, we are not going to spend that

much time on it here in this section. The Lagrangian density for a free fermion field

reads as

L = Ψ(i/∂ −m

)Ψ , (3.11)

where the Feynman slash notation is defined as /A ≡ γµAµ. This Lagrangian gives

an equation of motion of the following form for the system(i/∂ −m

)Ψ = 0 , (3.12)

which, considering the conventions made in Table A.1, can be expressed in (1+1)-

dimensional light-front framework as(i

2γ+∂− +

i

2γ−∂+ −m

)Ψ = 0 , (3.13)

Page 32: Pair Production and the Light-front Vacuum

22 Free theories on the light-front

Introducing projection operators as in Table A.1, ψ± = Λ±Ψ, and after some

algebra (Harindranath, 2000), we get

i∂+ψ− = γ0mψ+ , (3.14)

which shows that the ψ− is a constrained field which is determined by the dynamical

fermion field ψ+. Working out the equation of motion, we may derive the equation

of motion for ψ+ as well

i∂−ψ+ =m2

i∂+ψ+ . (3.15)

Once again, the simple trick mentioned in the previous section can be used to

give us the equal-x+ anti-commutation relation for the dynamical fermion field ψ+

ψ+(x), ψ+†

(y)

x+=y+=τ= Λ+δ(x− − y−) . (3.16)

The detailed calculations related to this part have been given in Appendix A.

Analogous to the case of free scalar field, the (1+1)-dimensional Fock space ex-

pansion of the free fermion field is given by (Kogut and Soper, 1970; Chang et al.,

1973)

Ψ(x) =

∫dk+

2k+(2π)

s

[bs(k)ϕs(k) e−ik·x + d†s(k)κs(k) eik·x

], (3.17)

wherebs(k) , b†s′(k

′)

= 2(2π) k+ δ(k − k′) , (3.18a)ds(k) , d†s′(k

′)

= 2(2π) k+ δ(k − k′) , (3.18b)

and all other anti-commutation relations vanish.

The generators of Poincare algebra in the front form are almost similar to those

of the instant form (2.1)

Pµ = 12

Σ

dx− d2x⊥ T+µ , (3.19a)

Mµν = 12

Σ

dx− d2x⊥(xµT+ν − xνT+µ

), (3.19b)

where the factor 12 is the Jacobian and has appeared as a result of transforming to

light-front framework (Heinzl, 2001). The energy-momentum tensor for a fermion

field is (see, for example, (Akhiezer and Berestetskii, 1965))

T µν = ∂νΨ∂L

∂(∂µΨ

) +∂L

∂ (∂µΨ)∂νΨ − gµνL . (3.20)

Therefore, the generators of Poincare algebra for a free fermion field in terms of the

Page 33: Pair Production and the Light-front Vacuum

23 Summary

dynamical fermion field ψ+ in (1+1) dimensions are given by

P+ =

∫dx− ψ+†

i∂+ ψ+ , (3.21a)

P− =

∫dx− ψ+† m2

i∂+ψ+ , (3.21b)

K− = 12

∫dx−

(x− ψ+†

i∂+ ψ+ − x+ ψ+† m2

i∂+ψ+

), (3.21c)

where P+, P− and K− are again corresponding to momentum operator, Hamil-

tonian operator and the generator of boost at x+ = 0, respectively; and we have

taken into consideration that L = 0 for on-shell fields.

3.3 Summary

The quantization of different fields on the light-cone was roughly reviewed in thischapter. It was shown that the light-front quantization procedure is almost non-standard, because the systems appeared as first-order in derivatives. Due to thisfact, the constrained and dynamical fields were emerged in the cases of free fermionfield. The equal-x+ (anti-)commutation relations were also showing a somewhatdifferent characteristics compared to their analogous structures in the instant form.The fermion field, which is our main focus, will be more scrutinized in the nextchapter.

Page 34: Pair Production and the Light-front Vacuum

4LF quantization of a fermion in a

background field in (1+1) dimensions

Since light-front quantization of a fermion in an electromagnetic background

field constitutes the core of our study, it is managed to be dealt with in a

separate chapter. The action of such a system in (1 + 1) dimensions reads as

follows

S =1

2

∫dx+dx− Ψ(i /D −m)Ψ. (4.1)

where, following the (3+1)-dimensional case, the Dirac spinor can be written as

Ψ =

[ψ1

ψ2

]. (4.2)

As a matter of convenience, the following representation of γ matrices, which is

a purely imaginary Majorana-Weyl representation of the two-dimensional Clifford

algebra1, is chosen

γ0 =

[0 −ii 0

], γ1 =

[0 i

i 0

]. (4.3)

We introduce γ± = γ0 ± γ1 such that (γ±)2

= 0 and the projection operators

Λ± = 14γ

∓γ±, which have the properties of (Λ±)2

= Λ± and Λ±Λ∓ = 0. Also we

have γ0γ± = 2Λ±. Hence,

γ+ =

[0 0

2i 0

], γ− =

[0 −2i

0 0

]. (4.4)

The projection operators are, thus, given by

Λ+ =

[1 0

0 0

], Λ− =

[0 0

0 1

]. (4.5)

The gauge covariant derivative components, in anti-lightcone gauge in which A+ =

0, are defined as below

D+ := ∂+ , (4.6a)

D− := ∂− + ieA−(x+) . (4.6b)

1 A detailed discussion on gamma matrices in various dimensions can be found in (Ortın, 2004).

24

Page 35: Pair Production and the Light-front Vacuum

25 LF quantization of a fermion in a background field in (1+1) dimensions

Introducing ψ± = Λ±Ψ, the field components are given by

ψ+ =

[ψ1

0

]and ψ− =

[0

ψ2

]. (4.7)

Now, we may re-write the action in terms of field components

S =1

2

∫dx+dx− Ψ†γ0

(iγ+∂+ + iγ−D− −m

=1

2

∫dx+dx− Ψ†

(2iΛ+∂+ + 2iΛ−D− −mγ0

=1

2

∫dx+dx−

[ψ†1 ψ†

2

] [(2i 0

0 0

)∂+ +

(0 0

0 2i

)D− +

(0 im

−im 0

)][ψ1

ψ2

]

=1

2

∫dx+dx−

[ψ†1 ψ†

2

] [(2i∂+ ψ1

0

)+

(0

2iD− ψ2

)+

(imψ2

−imψ1

)]

=1

2

∫dx+dx−

(2i ψ†

1∂+ψ1 + 2i ψ†2D−ψ2 + imψ†

1ψ2 − imψ†2ψ1

). (4.8)

From here on, we write ψ1 ≡ ψ+ and ψ2 ≡ ψ− to keep the notation simpler.

Therefore, we come up with

S =1

2

∫dx+dx−

(2i ψ†

+∂+ψ+ + 2i ψ†−D−ψ− + imψ†

+ψ− − imψ†−ψ+

).(4.9)

Considering the fact that the terms in the parentheses in Equation (4.9) form the

Lagrangian of the system, we may simply write the equations of motion following

from the Euler-Lagrange equation

2∂+ψ+ +mψ− = 0, (4.10a)

2D−ψ− −mψ+ = 0. (4.10b)

As it can be seen, ψ− is a constrained field which is determined at any x+ by ψ+:

ψ− =m

2

1

[D−]ψ+. (4.11)

Inserting (4.11) into (4.10a), we get the equation of motion for the dynamical field

ψ+

i∂+ψ+ =m2

4[iD−]ψ+. (4.12)

In the following, we try to solve (4.12) both classically and quantum mechanically.

Page 36: Pair Production and the Light-front Vacuum

26 LF quantization of a fermion in a background field in (1+1) dimensions

4.1 Classical solution

Using Fourier transform and considering the fact that k− ranges from zero to infinity

(c.f. Equations (2.8) and (2.9)), a general solution to the Equation (4.12) can be

written as

ψ+(x+, x−) =

∫ ∞

0

dk−(2π)

[b(k−)e−ik−x−

φk−(x+) + d†(k−)eik−x−

κk−(x+)

].

(4.13)

By plugging this general solution into the Equation (4.12) for the left hand side

(LHS) and right hand side (RHS) separately and taking the corresponding deriva-

tives, we get

i∂+φ =m2

4

1

k− − eA−

φ, (4.14a)

i∂+κ =m2

4

1

−k− − eA−

κ. (4.14b)

Now we may easily find the classical solutions to this set of equations

φk−(x+) = exp

(− im

2

4

∫ x+

0

1

k− − eA−(y)dy

), (4.15a)

κk−(x+) = exp

(− im

2

4

∫ x+

0

1

−k− − eA−(y)dy

). (4.15b)

As it can be seen here, the singularity problem exists even in the case of clas-

sical solutions. For instance, if we assume that A(0) = 0, then for φk−(x+) at

x+ = 0, the integration limits become zero, and the result would be φk−(x+) = 1.

If we, additionally, assume the background field A to be a monotonically increas-

ing function of x+ (see, e.g., Figure 4.1), we may still expect that for x+ & 0, i.e.

within a very short time interval ∆x+ bigger than zero, k− remains much bigger

than eA−, (k− ≫ eA−), and the solutions to be nonsingular. For bigger light-

cone times x+, however, we may expect that at some moment k− equals eA−, i.e.

for 0 < y < x+, k− − eA− = 0. In such a case, we encounter a singularity in our

solution, which should be regularized in a way. Similar discussion holds for κk(x+).

If we consider a background field of general form (see, e.g., Figure 4.1), the same

story holds except that there would be multiple zeros and the denominator changes

sign several times while in the case of a monotonically increasing background field

this happens only once.

Page 37: Pair Production and the Light-front Vacuum

27 Quantization

x+

A(x+)

∆x+ x+

A(x+)

tFig. 4.1 Left: a monotonically increasing background field. Right: a background field of general

form.

4.2 Quantization

Now, we follow the procedure of canonical quantization and regard ψ+(x+, x−)

and its momentum conjugate, Π(x+, x−), as Hermitian operators in the Heisenberg

picture, then impose the equal-time anti-commutation relation

ψ+(x+, x−),Π(x+, y−)

= iδ(x− − y−). (4.16)

Starting from the relation for the action (4.9) once again, we may write the canonical

momentum for ψ+ as below

Π(x+, x−) =∂L

∂ (∂+ψ+)

= iψ†+. (4.17)

We recall the general relation for the Hamiltonian density

H = Πa(x) φa(x) − L(x), (4.18a)

where Πa(x) =∂L∂φa

and sum over all fields is implicitly addressed in index a. As

L = 0 on-shell, using (4.12) we have

H = iψ†+∂+ψ+

= ψ†+

m2

4[iD−]ψ+ . (4.19)

In other words, we want the following relation to be satisfied:ψ+(x+, x−), ψ†

+(x+, y−)

= δ(x− − y−). (4.20)

In this stage, b(k−) and d(k−) become operators with the following commutation

relations between themb(k−), d(k−)

= 0, (4.21a)

b(k−), b†(k′−)

= 2πδ(k− − k′−), (4.21b)

d(k−), d†(k′−)

= 2πδ(k− − k′−). (4.21c)

To verify if our above assumptions (4.21) are in agreement with Equation (4.20),

Page 38: Pair Production and the Light-front Vacuum

28 LF quantization of a fermion in a background field in (1+1) dimensions

we compute the equal-time anti-commutation relation for the one-component fermion

field ψ+,ψ+(x+, x−), ψ†

+(x+, y−)

=

∫ ∞

0

dk−(2π)

∫ ∞

0

dk′−(2π)

[b(k−), b†(k′−)

e−ik−x−

eik′−y−

φk−(x+)φ†k′

−(x+)

+d†(k−), d(k′−)

eik−x−

e−ik′−y−

κk−(x+)κ†k′

−(x+)

]

=

∫ ∞

0

dk−(2π)

e−ik−(x−−y−) φk−(x+)φ†k−

(x+)︸ ︷︷ ︸

=1

+

∫ ∞

0

dk−(2π)

eik−(x−−y−) κk−(x+)κ†k−

(x+)︸ ︷︷ ︸

=1

=

∫ ∞

0

dk−(2π)

e−ik−(x−−y−) +

∫ 0

dk−(2π)

e−ik−(x−−y−)

=

∫ ∞

dk−(2π)

e−ik−(x−−y−)

= δ(x− − y−),

where, in first equality, we have made use of our assumption in which the anti-

commutation relation of different operators vanishes and in getting the second

equality, we have made use of (4.21b) and (4.21c), and performed the integra-

tion over k′−. In third equality, we have first changed k− to −k− in the second term

and then exchanged the limits of the integral to retrieve the summation between

two terms. As it is shown, everything seems to be correct.

Using Equations (4.19), we can write the Hamiltonian of our system as below:

H−

=

∫dx− ψ†

+

m2

4[iD−]ψ+

=

∫dx−

∫ ∞

0

dk−(2π)

∫ ∞

0

dk′−(2π)

[b†(k−)eik−x−

φ†k−(x+) + d(k−)e−ik−x−

κ†k−(x+)

]

·[

m2

4(k′− − eA−)b(k′−)e−ik′

−x−

φk′−

(x+) +m2

4(−k′− − eA−)d†(k′−)eik

′−x−

κk′−

(x+)

]

=

∫dx−

∫ ∞

0

dk−(2π)

∫ ∞

0

dk′−(2π)

[m2

4(k′− − eA−)e−i(k′

−−k−)x−

b†(k−)b(k′−)φ†k−(x+)

φk′−

(x+) +m2

4(−k′− − eA−)e−i(k−−k′

−)x−

d(k−)d†(k′−)κ†k−(x+)κk′

−(x+)

]

=

∫ ∞

0

dk−(2π)

[m2

4(k− − eA−)b†(k−)b(k−) +

m2

4(−k− − eA−)d(k−)d†(k−)

],

(4.22)

Page 39: Pair Production and the Light-front Vacuum

29 Zero-mode issue

where, in third equality above, two terms have been disappeared in advance due to

the fact that they involve structural components like eik−x−

eik′−x−

or e−ik−x−

e−ik′−x−

that after integration over x− will result in Dirac’s delta functions of the form of

2π δ(k− +k′−) or 2π δ(−k−−k′−) which will vanish due to the fact that both k− and

k′− are positive definite in their corresponding integration intervals. Performing the

integrations over x− and k′− and making use of (4.21), much in the same way as

we did for calculating the anti-commutation relation above, we finally end up with

the last result. After normal ordering, the Hamiltonian is given by

:H : =

∫ ∞

0

dk−(2π)

[m2

4(k− − eA−)b†(k−)b(k−) +

m2

4(k− + eA−)d†(k−)d(k−)

].

(4.23)

4.3 Zero-mode issue

As once introduced in Table 2.2, we define the state |0〉 such that b(k−) |0〉 =

d(k−) |0〉 = 0. This state, which is called the vacuum of the free theory, satisfies

H0 |0〉 = 0; where H0 refers to the Hamiltonian of the free theory and is given

by the Equation (4.23) in the absence of any interaction (A− → 0). If we denote

the true vacuum of the interacting theory by |Ω〉, it should satisfy the following

relation:

H |Ω〉 = 0, (4.24)

where,H is the Hamiltonian of the interacting theory. The vacuum of the interacting

theory is normally different from the vacuum of the free theory. If, for example, we

take the initial state |i〉 in Figure 4.2 (left panel) to be the vacuum state of the

free theory |0〉 in the instant form at t = 0, we expect the state to evolve with

time to a final state |f〉 in which Schwinger pairs are present when the background

field is switched on, i.e. |f〉 = ψ0 |0〉 + ψ1 |pair〉 + ψ2 |two pairs〉 + · · · , where the

ψn denote the probability amplitudes to find n pairs in the final state. In fact,

in the interacting theory, there would be nonzero contributions from states with

various numbers of pairs. Since the physical vacuum should not depend on the

chosen framework, we expect to observe the same result in light-cone coordinates

as well when the initial state |i〉 = |0〉 at x+ = 0 evolves with light-cone time to

the final state |f〉 (Figure 4.2). Taking a closer look at Equation (4.23), however,

we find out that :H : |0〉 = 0 even if A 6= 0, which means that, in the front form,

the vacuum state of the free theory is the vacuum of the interacting theory as well

and |f〉 = |0〉. In other words, the light-front vacuum is trivial , i.e. stable (Heinzl,

2001). Briefly speaking, triviality of the light-front vacuum implies that no pairs

would be pop out of the vacuum.

Page 40: Pair Production and the Light-front Vacuum

30 LF quantization of a fermion in a background field in (1+1) dimensions

x

t

Background Field

|i〉

|f〉x− x+

|i〉

|f〉

tFig. 4.2 Time evolution of an initial state |i〉 to a final state |f〉 in a background field in the

instant form (left) and the front form (right).

For further discussion, we consider the background field to be an initially zero

valued electric field which turns on at light-cone time x+ = 0 and stays constant(E(x+) = const.

)(Figure 4.3). Hence, for the electromagnetic field tensor element

we have: F+− ∼ ∂+A− = E. If we choose A−(x+ = 0) = 0(which is the case,

since E(x+ = 0) = 0), it implies that eA− (≡ eEx+) always grows with light-

cone time, thus eA− > 0 (Figure 4.3). Therefore, m2

4(k−+eA−) will always be positive

and m2

4(k−−eA−) may, depending on x+, be positive or negative. Hence, Hamiltonian

changes sign, which might be a sign of pair production. Nevertheless, vacuum has

been shown to be stable, i.e. no pairs will be created. As it was mentioned in Chap-

ter 1, however, Schwinger showed that there should be a nonzero pair production

probability in an electric background field (Schwinger, 1951a). Therefore, this ques-

tion arises: what could we have done wrong? To take a closer look at this problem,

we will try to answer the following questions:

• What do b and d do?

Do they rise or lower energy?

• What does1

(k− − eA−)mean?

x+

eE

x+

eA−

tFig. 4.3 Left: electric background field E(x+) which is initially zero, then turns on at x+ = 0

and stays constant. Right: the resulting diagram for the electromagnetic vector

potential A− as a function of the light-cone time.

Page 41: Pair Production and the Light-front Vacuum

31 Zero-mode issue

Box 4.1 Commutation relation of Hamiltonian and the operators in our problem[:H: , b(k−)

]= − m2

4(k−−eA−)b(k−)

[:H: , b†(k−)

]= m2

4(k−−eA−)b†(k−)

[:H: , d(k−)

]= − m2

4(k−+eA−)d(k−)

[:H: , d†(k−)

]= m2

4(k−+eA−)d†(k−)

To see in which circumstances an operator rise or lower energy, we consider the

basic technique mentioned in the following Remark.

Remark Suppose that H |E〉 = E |E〉. If[H, O

]= δE O, then we have

H O |E〉 =([H, O

]+ OH

)|E〉

= δE O |E〉 + E O |E〉= (E + δE) O |E〉 . (4.25)

Hence, the commutation relation of the Hamiltonian of a system and an operator,

offers a possibility to answer the first question. The sign of δE determines if the

operator in question rise or lower energy. This relation has been calculated for

operator b in (4.26).

[:H : , b(k′−)

]=

∫ ∞

0

dk−(2π)

m2

4(k− − eA−)

[b†(k−)b(k−), b(k′−)

]

=

∫ ∞

0

dk−(2π)

−m2

4(k− − eA−)

b†(k−), b(k′−)

b(k−)

=

∫ ∞

0

dk−(2π)

−m2

4(k− − eA−)2π δ(k− − k′−) b(k−)

=−m2

4(k′− − eA−)b(k′−), (4.26)

where, in the third equality, we have made use of the useful relation [AB,C] =

AB,C − A,CB, and in the last equality, we have applied∫f(x)δ(x− a)dx =

f(a). All such results have been summarized in Box 4.1. As it can be seen here, e.g.,

if k− > eA−(x+), then b destroys electrons and if k− < eA−(x+), b adds something

to system; but what? It seems that for answering this question we have to look

for a key concept in1

(k− − eA−). To this end, we need to digress a bit from our

pathway and take a look at the relativistic particle motion in the presence of an

external electromagnetic field.

From classical field theory, we know that the equation of motion of a relativistic

Page 42: Pair Production and the Light-front Vacuum

32 LF quantization of a fermion in a background field in (1+1) dimensions

particle moving in an electromagnetic background field is given by the differential

form of the Lorentz force law (Lorentz, 1909)

dpµdτ

=e

cFµν

dxν

dτ. (4.27)

In (1 + 1) dimensions, therefore, the set of equations of motion for such a particle

can be written in light-cone coordinates as below

mx+ = eF+−(x+) x−, (4.28a)

mx− = eF−+(x+) x+. (4.28b)

Considering the fact that F+− ≡ ∂+A−(x+) = −F−+ and that x± = 12x

∓, this set

of equations may be re-written as

1

2mx− = e x− ∂+A−(x+), (4.29a)

1

2mx+ = −e x+ ∂+A−(x+),

= −e d

dτA−(x+). (4.29b)

After doing one step integration, we get

x+(τ) = x+(0) − e

mA+(x+(τ)

), (4.30)

which has been expressed in a completely contravariant form.

On the other hand, we know that mxµ(τ) = pµ and equation for the mass shell

in natural units reads as p2 = m2. If we use the notation xµ(τ) = uµ(τ), then we

may conclude uµuµ = 1, or in (1+1) dimensions

u+u− = 1. (4.31)

Plugging (4.30) in (4.31), thus we get

u− =1

u+(0) − em A+

(x+(τ)

) . (4.32)

In this relation, which is similar to1

(k− − eA−), if u+(0) =

e

mA+(x+(τ)

), then

u− → ∞. What does it mean? We know that u− = u0−u3 and u+ = u0 +u3. Now,

if u− → ∞, correspondingly u+ → 0, then from the latter we have u0 = −u3. This

relation can be re-written as follows

dx0

dτ= −dx3

dτ. (4.33)

In other words,dx3

dx0= −1, or

dz

dt= −c. (4.34)

Hence, u− → ∞ means that particle is accelerated to the speed of light c. This

Page 43: Pair Production and the Light-front Vacuum

33 Summary

finding will help us when we study the pair production on the light-front in Chap-

ter 6. At the end, Figure 4.4 roughly shows how u− changes with light-cone time,

on either side of the singularities, for the background fields shown in Figure 4.1,

respectively; plotted for a positive value of u+(0).

x+

u−

x+

u−

tFig. 4.4 The behavior of u− for background fields A− shown in Figure 4.1, correspondingly.

4.4 Summary

Light-front version of Dirac equation for a fermion in an electromagnetic back-ground field has been considered in an anti-lightcone gauge (A+ = 0) in (1+1)dimensions. It has been shown that the fermion field consisted of two field compo-nents of which one is a dynamical field while the other one is a constrained fieldwhich is determined at any x+ by the dynamical one. Furthermore, it has beenfound out that in light-front framework, the vacuum state of the free theory is thevacuum state of the interacting theory as well. In other words, light-front vacuumis trivial, i.e. stable, which means no Schwinger pairs will be created. Moreover,the role of the ladder operators, i.e. adding or subtracting energy from the system,varies with time. Thus, what one would usually consider to be an annihilator be-comes a creator after a certain time, which is a new ingredient compared to thefree theory. Depending on x+, the normal-ordered Hamiltonian of the system (4.23)may subsequently change sign which may then be a sign of pair creation itself. Thecommutation relations of the normal-ordered Hamiltonian and the apparently cre-ation and annihilation operators of the system has been summarized in Box 4.1.

Page 44: Pair Production and the Light-front Vacuum

34 LF quantization of a fermion in a background field in (1+1) dimensions

The probable link between the appearance of singularities in the coefficients of theoperators in Box 4.1 and situation in which the particle is accelerated to the speedof light c has also been discussed.

Page 45: Pair Production and the Light-front Vacuum

5 Discrete Light-Cone Quantization

There has been a few methods proposed to resolve the light-front ambiguity

at k− = 01. One of them, which is based on imposing a periodic or an anti-

periodic boundary condition on x− and discretizing a quantum field theory

in momentum space, is called discrete light-cone quantization (DLCQ). This method

was developed by (Maskawa and Yamawaki, 1976; Pauli and Brodsky, 1985a,b) as

a proposition to obtain non-perturbative solutions to field theories and was later

on applied to other fields like string/M-theory (Banks et al., 1997; Susskind, 1997;

Hyun et al., 1998; Antonuccio et al., 1998a,b; Hyun and Kiem, 1998; Aharony and

Berkooz, 1999), supergravity (Lifschytz, 1998; Hyun, 1998) and supersymmetry

(Antonuccio et al., 1998c; Lunin and Pinsky, 1999; Antonuccio et al., 1999). In this

method, which has been extremely successful in a large number of field theories

in (1+1) dimensions (Eller et al., 1987; Harindranath and Vary, 1987, 1988a,b,c;

Eller and Pauli, 1989; McCartor, 1988, 1991, 1994; Burkardt, 1989, 1993; Brodsky,

1988; Hornbostel et al., 1990; Hornbostel, 1988), the spatial light-cone coordinate,

x−, is compactified over a circle of length 2πR (here R = 2L, see Figure 5.1) that

gives a discrete spectrum for k−. In an alternative approach, “the field theory is

sometimes approximated by truncating the Fock space with the assumption that

the essential physics is described by a few excitations and that adding more Fock

space excitations only refines the initial approximation”2. (Brodsky et al., 1998)

The extension of DLCQ method to various quantum field theories has been dis-

cussed in (Brodsky et al., 1998). Being formulated in a Hamiltonian approach,

keeping the feasibility of doing the calculations in momentum representation and

equal light-cone time quantization are the main features of the DLCQ method and

it seeks its own aim in solving the light-front Hamiltonian eigenvalue problem.

Yukawa theory in (1+1) dimensions was the first quantum field theory that this

method was applied to and its great success in giving the mass spectrum and wave

function of this theory stimulated the motivation in applying this method to more

general QED(1+1) theories (Pauli and Brodsky, 1985b). However, the real trial for

DLCQ is to solve for bound states of four-dimensional theories. With the PBC3

feature of DLCQ, there is no need to make sure if all the fields tend to vanish

sufficiently fast at boundaries. (Brodsky and Pauli, 1991) has counted a number of

1 p+ = 0 notation has also been alternatively used in the literatures.2 Tamm-Dancoff method (TDA) (Tamm, 1945; Dancoff, 1950) is another method in which the

same strategy is followed. However, it will not be discussed in this survey. See, e.g., (Wilsonet al., 1994)

3 Periodic Boundary Condition

35

Page 46: Pair Production and the Light-front Vacuum

36 Discrete Light-Cone Quantization

further intrinsic advantages of the DLCQ method, which can be referred to. De-

spite the great success of the DLCQ method, it encounters with different problems

when it is applied to abelian and non-abelian quantum field theories. A number

of such problems have been counted in (Brodsky et al., 1998). Basic features of

DLCQ approach to light front have been discussed in (Maskawa and Yamawaki,

1976; Casher, 1976; Thorn, 1978; Pauli and Brodsky, 1985a; Bigatti and Susskind,

1998) and a nice overview of recent developments can be found in (Hiller, 2000).

Referring to Section 2.1, we recall that the front form of Hamiltonian dynamics

provides us with a simple description of the vacuum (Weinberg, 1966, 1967). By

applying a periodic boundary condition to the front form of Hamiltonian dynam-

ics, DLCQ utilizes an a priori convenient regularization of the infrared degrees of

freedom as well. As we have seen in Chapter 2, k− ≥ 0 for an on-mass shell particle

(Harindranath, 1997). The most important problem with light-cone quantization

is, however, due to the singularities that appear in the limit of k− → 0. One may

naturally feel these singularities may, in general, be regularized by putting a cut-off

either on k− or on x− which is conjugate to k− (Franke et al., 2006). The DLCQ

method actually chooses the strategy of treating x−; however, not by putting any

cut-off on it, but by imposing an anti-periodic boundary condition, which subse-

quently results in a discrete k− spectrum. Hence, the positive and negative momen-

tum modes will be decoupled (Antonuccio et al., 1999) and the zero-mode (n = 0)

will be separated from the other modes (n 6= 0) such that it can be treated easier

(Yamawaki, 1998). To have an understanding of vacuum effects in DLCQ approach,

however, it is necessary to include the zero momentum modes, which are usually

neglected, in this method. This case has, for instance, been discussed in (Chaby-

sheva and Hiller, 2009). Since both of these regularizations (namely, putting cut-off

on k− or using the DLCQ method) can break the Lorentz symmetry, the usual

perturbative renormalization may not be applied here (Franke et al., 2006).

5.1 Quantization

As it is known that DLCQ is one of the most powerful tools available to deal with

bound state problem (Perry et al., 1990; Brodsky et al., 1993) and that any (1+1)

quantum field theory can virtually be solved using this method (Pauli and Brodsky,

1985a; Brodsky and Pauli, 1991; Brodsky, 1998), we will naturally try to apply it

to our case.

Following (Franke et al., 2006), we compactify the spatial light-cone coordinate

|x−| ≤ L, according to the DLCQ prescription, and impose an anti-periodic bound-

ary condition on our fermion field. Thus, space-time acquires the topological struc-

ture of a cylinder in DLCQ method (Figure 5.1), which means that the longitudinal

momenta will be discrete, kn− = πL(n− 1

2 ) (for n ∈ N). Then we make use of Fourier

decomposition again and write a general solution to the equation of motion of non-

constraint field (4.12), this time in a discrete manner. By using the anti-periodic

Page 47: Pair Production and the Light-front Vacuum

37 Quantization

boundary condition we don’t need to be worried about the mode with k− = 0

anymore. Even, in many cases, the numerical problems converge faster with an

anti-periodic boundary condition compared to the periodic boundary condition ex-

cluding the mode with k− = 0 (Burkardt, 2002). Therefore, the ψ+(x+, x−) function

can be written concisely as below

ψ+(x+, x−) =1√2L

n≥1

bn e−i π

L(n− 1

2)x−

+∑

n≥0

d†n ei πL(n+ 1

2)x−

. (5.1)

It is easy to check that

ψ+

(x+, x− = −L

)= −ψ+

(x+, x− = +L

). (5.2)

Now, bn and dn become operators and we, similarly, assumebn, dn′

=b†n, d

†n′

=

0 and guess that

bn, b

†n′

= δnn′ , (5.3a)

dn, d

†n′

= δnn′ . (5.3b)

x− x+

x− = −L

x− = L

tFig. 5.1 Topological structure of space-time in DLCQ method when the spatial light-cone

coordinate is compactified as |x−| ≤ L.

Page 48: Pair Production and the Light-front Vacuum

38 Discrete Light-Cone Quantization

Again to verify if our guess and assumption give the reasonable results, we com-

pute the equal-time anti-commutation relation for the one-component fermion field

ψ+ here,ψ+(x+, x−), ψ†

+(x+, y−)

=1

2L

n≥1

n′≥1

bn, b

†n′

e−i π

L(n− 1

2)x−

eiπL(n′− 1

2)y−

+1

2L

n≥0

n′≥0

d†n, dn′

ei

πL(n+ 1

2)x−

e−i πL(n′+ 1

2)y−

=1

2L

n≥1

e−i πL(n− 1

2)(x−−y−) +

1

2L

n≥0

e−i πL(−n− 1

2)(x−−y−)

=1

2L

∞∑

n=−∞

eiπL(n+ 1

2)(x−−y−)

= eiπ2L

(x−−y−) 1

2L

∞∑

n=−∞

einπL(x−−y−)

= δ(x− − y−),

where, in first equality, we have made use of our assumption in which the anti-

commutation relation of different operators vanishes. The summation in fourth

equality is the series representation of Dirac’s delta function in which −π < πL(x−−

y−) < π. Hence, everything seems correct so far.

5.2 Zero-mode issue

As we have seen in Section 4.3, the light-front vacuum turned out to be trivial. In

this section, we are going to revisit this problem from a discrete fermion field point

of view. To this end, we start by calculating the Hamiltonian of our system in this

viewpoint. Using the Equations (4.19) and (5.1), we can write the Hamiltonian of

the system as below:

H− =

∫ L

−L

dx− ψ†+

m2

4[iD−]ψ+

=∑

n≥1

(m2

4(πL(n− 1

2 ) − eA−

))b†nbn +

n≥0

(−m2

4(πL(n+ 1

2 ) + eA−

))dnd

†n .

(5.4)

Page 49: Pair Production and the Light-front Vacuum

39 Summary

After normal ordering, the Hamiltonian is given by

:H : =∑

n≥1

(m2

4(πL (n− 1

2 ) − eA−

))b†nbn +

n≥0

(m2

4(πL(n+ 1

2 ) + eA−

))d†ndn.

(5.5)

If we compare (5.5) with (4.23), we find out that they are similar with respect to the

fact that equal numbers of both creation and annihilation operators appear in each

term of these normal-ordered Hamiltonians. Therefore, the vacuum is trivial even if

we apply the DLCQ method to our case in ligh-cone coordinates (:H : |Ω〉 = 0) and,

depending on the value of A−, the same divergences occur as in the normal (non-

compact) theory. Thus, we have the same potential conflict between pair production

results and triviality of the vacuum as in the continuum theory.

5.3 Summary

The discrete light-cone quantization (DLCQ) method has been applied to our prob-

lem as a promising method to resolve the light-front ambiguity at p+ = 0. In this

method the zero-modes are either excluded by the PBC from the outset or are

explicit and directly treatable. As it can be seen in (5.5), however, the number

of creation and annihilation operators in each term of the Hamiltonian are equal,

which means the vacuum remains trivial when it operates on the vacuum state of

the full theory. Thus, going to DLCQ method does not seem, in this particular case,

to give us any insight into the zero-mode issue. Essentially due to the fact that

even if you exclude zero modes at the outset by imposing boundary conditions, so

that e.g. kn− is never zero, it still looks like something can go wrong since kn−−eA−

can still vanish.

Page 50: Pair Production and the Light-front Vacuum

6 Tomaras–Tsamis–Woodard solution

Aslightly different viewpoint towards the phenomenon of electron-positron

pair production in an external electric field is epitomized in defining the

back reaction phenomenon. This phenomenon, as a natural consequence of

formulating quantum field theory on a non-trivial gauge field or metric background

(Tomaras et al., 2000), states that “when a prepared state is initially released in

the presence of a homogeneous electric field, electron-positron pairs emerge from

the vacuum to form a current which diminishes the electric field” (Tomaras et al.,

2001). As we have seen in Chapter 4, light-front quantization of a fermion field in

the presence of an electromagnetic background field indicates that the light-front

vacuum is trivial. This finding implies as a corollary that, contrary to Schwinger’s

prediction (Schwinger, 1951a), particle pairs don’t show up. This apparent contra-

diction with such a well-established phenomenon (Brezin and Itzykson, 1970; Troup

and Perlman, 1972; Casher et al., 1979; Affleck et al., 1982; Bialynicki-Birula et al.,

1991; Kluger et al., 1992; Best and Eisenberg, 1993; Gavrilov and Gitman, 1996;

Kluger et al., 1998; Greiner et al., 1985; Fradkin et al., 1991; Damour and Ruffini,

1975) stimulates the motivation for a more precise investigation of light-cone QED

in a homogeneous electric background by Woodard et al. In this chapter, we take a

look at the method proposed by (Tomaras et al., 2001) in solving the Dirac equa-

tion in an arbitrary background field and studying the back-reaction phenomenon

analytically. This method seems to resolve the traditional ambiguity at k− = 0.

6.1 Methodology

Woodard et al. have considered Dirac electron of arbitrary mass in a homogeneous

electric background field in Heisenberg picture. Light-cone time dependence of this

electric background field has also been claimed to be arbitrary in the original paper1

and the solution is expected to be applicable in any space-time dimension. The

initial value operators are unorthodoxly defined here on both light-cone time and

space at x+ = 0 and x− = −L, respectively. The essential nature of providing the

initial value data on both surfaces of constant x+ and x− had even been noted

in previous works (Neville and Rohrlich, 1971; Rohrlich, 1971; McCartor, 1988).

1 Despite this claim, it has not been taken quite arbitrary, but has been considered to be mono-tonically increasing such that k− − eA− = 0 at most only once when the pair production

phenomenon is discussed.

40

Page 51: Pair Production and the Light-front Vacuum

41 Methodology

Table 6.1 Woodard’s light-cone coordinate and gauge conventions

Coordinatesa Scalar product

x+ ≡ 1√2

(x0 + x1

)x− ≡ 1√

2

(x0 − x1

)aµbµ = a+b− + a−b+

Metric components Vector components

g+− = g−+ = 1 a+ = a− a− = a+

Gamma matricesb

γ0 =

[1 0

0 −1

]γi =

[0 σi

−σi 0

]γ5 = γ5 =

[0 1

1 0

](i = 1, 2, 3)

γ0㵆γ0 = γµ γ5† = γ5

in (1+1)−−−−−−−→dimensions

ηµν = (+1,−1) γµ, γν = 2ηµν

Pauli matrices Derivatives

σ1 =

[0 1

1 0

]σ2 =

[0 −i

i 0

]σ3 =

[1 0

0 −1

]∂± ≡ ∂/∂x±

Light-front gamma matrices

γ+ ≡ 1√2

(γ0 + γ1

)γ− ≡ 1√

2

(γ0 − γ1

) (γ±)2 = 0 γ+, γ− = 2

Light-front spinor projection operators Auxiliary relations

P± ≡ 12

(I ± γ0γ1

)= 1

2(I ± γ5) =

12γ∓γ± TrP± = 1

Dirac spinor Anti-lightcone gauge

Ψ = ψ+ + ψ− ψ± ≡ P±Ψ A+ = 0

a Note: this table is based on conventions made in (Tomaras et al., 2001).b Gamma matrices are chosen here to be the ones used by (Bjorken and Drell, 1965).

Although this idea was initially applied to the case of back reaction phenomenon in

an earlier study (Tomaras et al., 2000), however, it turned out to yield a solution

which was only valid in the distributional limit of L → ∞ and, subsequently,

restricted to be applicable in evaluating the expectation values of non-singular

operators. Soon after that, however, Woodard and his colleagues published a paper

in which this restriction had been removed (Tomaras et al., 2001). We follow this

last method in our survey here, once more in (1 + 1) dimensions.

To stay consistent with the notation used by Woodard et al. in this chapter, we

introduce the light-cone coordinate and gauge conventions as in Table 6.1. Having

known that the mode functions of Dirac theory in any x+-dependent homoge-

neous electric background field are simple (Artru and Czyzewski, 1998; Srinivasan

and Padmanabhan, 1999b,a; Tomaras et al., 2000), we set to solve the model in

Woodard’s manner.

Page 52: Pair Production and the Light-front Vacuum

42 Tomaras–Tsamis–Woodard solution

6.2 The model and its solution in Woodard’snotation

Starting from (4.1) and considering the conventions in Table 6.1, the Dirac equation

in anti-lightcone gauge with an arbitrary A−(x+) background field can once more

be written as

(γ+i∂+ + γ−iD− −m

)Ψ = 0 , (6.1)

where D− = ∂− + ieA−, in which

A−

(x+)

= −∫ x+

0

duE(u) , (6.2)

with the initial value of zero. Thus, analogous to (4.10), the system of equations of

motion can be written as follows

i∂+ψ+ =m

2γ−ψ− , (6.3a)

(i∂− − eA−)ψ− =m

2γ+ψ+ . (6.3b)

Instead of using the elimination method and writing Fourier transform readily,

Equations (6.3a) and (6.3b) have been chosen to simply be integrated. To this end,

(6.3a) is directly integrated and both sides of (6.3b) are multiplied by eieA−(x+)x−

and then simply integrated to give

ψ+(x+, x−) = ψ+(0, x−) − im

2

∫ x+

0

du γ−ψ−(u, x−) , (6.4a)

ψ−(x+, x−) = e−ieA−(x+)(x−+L) ψ−(x+,−L)

− im2

∫ x−

−L

dv e−ieA−(x+)(x−−v) γ+ψ+(x+, v) . (6.4b)

Therefore, we are left with a set of coupled equations for ψ±(x+, x−) which, as it

was intended, are expressed in terms of initial value data ψ+(x+ = 0, x− > −L)

and ψ−(x+ > 0, x− = −L). Hence, the domain of solution for ψ±(x+, x−) is limited

to the shaded area of space-time shown in Figure 6.1. Treating the Equations (6.4a)

and (6.4b) iteratively, after some algebra, culminates in the following general so-

lutions for the Dirac equation on the light-cone in the presence of an arbitrary

background electric field as characterized in (6.2), (Tomaras et al., 2001)

ψ+(x+, x−) =

∫ ∞

−L

dv

∫ ∞

−∞

dk+

2πei(k

++i/L)(v−x−)

E [eA−](0, x+; k+)ψ+(0, v)

− i

2mγ−

∫ x+

0

du e−ieA−(u)(v+L)E [eA−](u, x+; k+)ψ−(u,−L)

, (6.5)

Page 53: Pair Production and the Light-front Vacuum

43 The model and its solution in Woodard’s notation

x

t

x− x+

−L

tFig. 6.1 The domain of solution (Tomaras et al., 2001).

ψ−(x+, x−) = e−ieA−(x+)(x−+L) ψ−(x+,−L) +

∫ ∞

−L

dv

∫ ∞

−∞

dk+

× ei(k++i/L)(v−x−)

k+ − eA−(x+) + i/L

γ+ ψ+(0, v) E [eA−](0, x+; k+)

−im∫ x+

0

du e−ieA−(u)(v+L) E [eA−](u, x+; k+)ψ−(u,−L)

, (6.6)

where the functional E [eA−] has been defined as

E [eA−](u, x+; k+) ≡ exp

[− i

2m2

∫ x+

u

du′

k+ − eA−(u′) + i/L

]. (6.7)

Note that the first term in (6.6) has been added adequately to satisfy the equation

of motion (6.3b) at x− = −L. Furthermore, k+ in the above relations and k−,

which is mostly used in the previous chapters, are conceptually equivalent. As it

can be seen easily, the solutions for ψ± fulfill the Dirac equation. Nevertheless, it is

worth pointing out that providing initial value data on both x+ and x− means we

are not really working on the light-front anymore. However, it is to be noted that

while the appearance of i/L in the denominator of 1k+−eA−(u)+i/L certainly does

indeed regulate the singularity, it is not immediately clear why you have to take

i/L and not i/2L or just iǫ for some small ǫ.

Page 54: Pair Production and the Light-front Vacuum

44 Tomaras–Tsamis–Woodard solution

6.3 Quantization

Having known the way through which the general solutions for ψ±(x+, x−) depend

upon the initial value data, the algebra of any operator at any space-time point

can, in turn, be determined by the canonical quantization of the initial value oper-

ators. As a result, the field operators obey the following anti-commutation relations

(Tomaras et al., 2001)ψ+(x+, x−), ψ†

+(x+, y−)

=1√2P+δ(x

− − y−) , (6.8a)

ψ−(x+, x−), ψ†

−(y+, x−)

=1√2P−δ(x

+ − y+) . (6.8b)

To have a general idea of how the projection operators P± appear in the above

relations, refer to a similar calculation that has been done in Appendix A. Validity of

these relations as well as recovery of some known results such as Schwinger’s vacuum

persistence amplitude in the L → ∞ limit entirely depend on taking operators on

x− = −L into account (Tomaras et al., 2001).

In the next steps, a few physical quantities such as the probability for pair creation

can be computed. For computational purposes, it is convenient to work in Fourier

space. The Fourier transform, ψ+, is defined as follows

ψ+(x+, x−) ≡∫

dk+

2πe−ix−(k++i/L) ψ+(x+, k+) , (6.9)

or expressed explicitly as

ψ+(x+, k+) ≡∫ ∞

−L

dv ei(k++i/L)v

E [eA−](0, x+; k+)ψ+(0, v) − i

2mγ−

×∫ x+

0

du e−ieA−(u)(v+L)E [eA−](u, x+; k+)ψ−(u,−L)

. (6.10)

However, this ψ+(x+, k+) operator turns out to be an eigenoperator of the light-

front Hamiltonian away from the singular point at k+ = eA−(x+) in the large-

L limit. Further calculations shows that, for k+ < eA−(x+), this eigenoperator

creates positrons of momentum k+ with amplitude 2−1/4 and for k+ > eA−(x+),

it destroys electrons of momentum k+ with the same amplitude (Tomaras et al.,

2001; Woodard, 2001).

6.4 Pair production on the light-front

Particle creation and annihilation phenomena by virtue of ψ+(x+, k+) operator in

the large-L limit was roughly discussed in the previous section. As it can be seen

in Equation (6.2) and was also shown for a specific case in Chapter 4, however, the

Page 55: Pair Production and the Light-front Vacuum

45 Pair production on the light-front

function eA−(x+) increases monotonically from zero at x+ = 0 provided that the

electric field remains positive4 (Figure 4.3). Hence, for x+ < 0 the theory is reduced

to its free field version and of no interest for the case of pair production phenomenon.

Since, according to (2.8), for on-mass shell particles in the light-front framework we

have k+ ≥ 0, the particle production phenomenon can only be physically discussed

for the modes with positive constant canonical momenta, k+’s. Thus, considering

the positive sign of eA−(x+), though k+ goes mathematically over the (−∞,+∞)

interval in (6.5), only the modes with positive k+ have the chance to come physically

into play. On the other hand, it has been shown that light-cone time evolution can

be regarded as the infinite boost limit of conventional time evolution (Kogut and

Soper, 1970). If we assume two inertial frames; one at rest (S) and the other one

(S′) boosted to the velocity β in the −x direction, Lorentz transformation between

the two systems entails

p+ =

√1 − β

1 + βp′+ , p− =

√1 + β

1 − βp′− , (6.11)

for momentum transformation (Woodard, 2001, 2002), where p± are the light-cone

momenta measured by an observer in the frame at rest (S) and p′± are the light-

cone momenta measured by an observer in the boosted frame (S′). Hence, if the

observer in the boosted frame (S′) observes a particle creation phenomenon due

to the background field in his inertial frame and measures the light-cone momenta

p′± for this event, these values will be finite and non-zero in his frame (Woodard,

2001, 2002). The observer in the frame at rest (S), however, will according to (6.11)

measure p± for this event in his corresponding inertial frame. Obviously, when β

approaches unity, p+ goes to zero. Therefore, the physical (kinetic) momentum

p+ = k+ − eA−(x+) on the light-cone, which is analogous to the above-mentioned

boosted frame, is measured to be zero. Hence, p+ = 0 or in other words, k+ =

A−(x+) denotes the instant of pair creation on the light-front. All in all, according

to (Tomaras et al., 2001) “modes with positive k+ start out as electron annihilation

operators and then become positron creators after the critical time x+ = X(k+) at

which eA−(x+) = k+. This is the phenomenon of pair creation”. The situation for

a specific k+ mode has been depicted in Figure 6.2.

Following the discussion made at the end of Chapter 4, therefore, electrons are

accelerated, in the boosted frame, to the speed of light in the negative x direction,

which is opposite to the direction of the electric field and parallel to the x− direction

in light-front frame. Thus, “they never evolve past a certain value of x+” (Woodard,

2001). In contrast, positrons appear to move at the speed of light in the positive x

direction, which means parallel to the x+ axis in light-front frame. The evolution

of a virtual e+e− pair has been depicted in Figure 6.3. Thus, we deduce that pair

creation phenomenon, on the light-front, is an instantaneous and singular event and

that we see only e+, because the newly created e− instantly leaves the manifold.

4 This is slightly different from the case we studied in Chapter 4. There, we took eE to remain

positive.

Page 56: Pair Production and the Light-front Vacuum

46 Tomaras–Tsamis–Woodard solution

x+

eA−

X(k+)

k+

e− annihilation region

e+ creation region

free theory region

tFig. 6.2 Particle creation (light gray) and annihilation (dark gray) regions for a specific k+

mode.

x

t

x+x−

e+e−

tFig. 6.3 The evolution of a e+e− pair (Tomaras et al., 2001).

To conclude this chapter, we point out that computing the probability for pair

creation, Prob (e+), at time x+ > X(k+), considering the fact that A−(x+) has

been assumed to increase monotonically, gives a zero value for all p+ < 0; and for

0 < p+ < eA−(x+) a positron can be found with probability

Prob (e+) = e−2πλ(p+) , (6.12)

where the function λ(p+) has been introduced as below (Tomaras et al., 2001)

λ(p+) ≡ m2

2|e|E (X(p+)). (6.13)

Note that the probability for pair creation has been represented by the probabil-

ity of finding a positron with momentum p+ at time x+, because, as it has been

Page 57: Pair Production and the Light-front Vacuum

47 Summary

mentioned before, only the created positron of a pair production process remains

on the manifold and can be seen.

6.5 Summary

The Dirac equation (6.1) has been solved in Heisenberg picture for a massive

fermion in anti-lightcone gauge with a monotonically increasing background field

A−(x+), based on the light-cone coordinate and gauge conventions of Table 6.1.

The initial value data have unconventionally been defined on both light-cone time

and space at x+ = 0 and x− = −L respectively. The system of equations of motion

(6.3) has iteratively been solved to reveal how the general solutions (6.5) and (6.6)

for two field components ψ±(x+, x−) depend upon the initial value data.

Quantization of the field components shows, however, that the Fourier transform

of ψ+ plays a crucial role in pair production scenario and that ψ+(x+, k+) operator

creates positrons with amplitude 2−1/4 for k+ < eA−(x+) and destroys electrons

with the same amplitude for k+ > eA−(x+).It can also be inferred that the pair production phenomenon may only occur in

the case of those k+ modes which are accessible to eA−(x+), i.e. those k+ modeswhich may have the chance to equal eA− at some moment of time x+. As a result,the positive k+ modes act as electron annihilators as long as k+ > eA−(x+) andthen become positron creators when k+ < eA−(x+) due to the ever-increasingcharacteristic of the function eA−(x+) with a positive value background electricfield. Correspondence between the time evolution of a state in light-cone gaugeand the infinite boost limit of the conventional time evolution of that state impliesthat the instant of pair creation on the light-cone is defined by k+ = eA−(x+).Thus, the pair creation phenomenon appears to be a discrete and instantaneousevent on the light-cone and that the newly created electron leaves the manifoldinstantly. Therefore, only positrons remain on the manifold and can be seen. Thepair production probability can be computed in real time and it is turned out to beof the form of (6.12) for 0 < p+ < eA−(x+).

Page 58: Pair Production and the Light-front Vacuum

7Alternative to Tomaras–Tsamis–Woodard

solution

To get an insight into the different aspects of TTW solution, an alterna-

tive method can be chosen to investigate this problem. The worldline path

integral approach turns out to be an alternative way to look at pair produc-

tion phenomenon (Affleck et al., 1982; Kim and Page, 2002; Dunne and Schubert,

2005, 2006; Dunne and Wang, 2006; Dietrich and Dunne, 2007; Dumlu and Dunne,

2011a). Therefore, it might also be a good way to look at pair production on the

light-front. This current approach is expected to yield the same pair production re-

sults as TTW approach1. Understanding this approach might help us to understand

what is going on with TTW solution and the vacuum.

7.1 Quantum mechanical path integral

In quantum mechanics, the propagator of a quantum system in moving from an

initial point (xi, ti) to a final point (xf , tf ) in space-time can be defined in the

Schrodinger picture as the transition probability amplitude of that system between

the wave functions evaluated at those points and is obtained by the inner product

of the corresponding wave functions at these two points:

K(xf , tf ;xi, ti) = 〈xf , tf | xi, ti〉 . (7.1)

In 1948, Feynman formulated the path integral approach to non-relativistic quan-

tum mechanics (Feynman, 1948) based on an earlier work by Dirac (Dirac, 1945).

More specifically, he reformulated the transition probability amplitude in quantum

mechanics as sum over all paths which may connect two certain space-time points

(Feynman, 1948, 1950, 1951). These paths include all those which may go back in

time as well. He showed (Feynman, 1950, 1951), in a semiclassical approach, that the

above-mentioned propagator can be represented as a path integral in configuration

space as follows

〈xf | e−iH(tf−ti)/~ |xi〉 =

∫ xf

xi

Dx(τ) exp

i

~

∫ tf

ti

[m

2

(dx

)2

− V (x(τ))

],

(7.2)

where the symbol∫Dx denotes the functional integral over all paths x(τ) with end

1 Though the case is considered here for scalars, but that does not matter.

48

Page 59: Pair Production and the Light-front Vacuum

49 Path integral formulation for a scalar particle

points x(ti) = xi and x(tf ) = xf and the term in curly braces is the well-known

classical action of the system.

The Equation (7.2) is actually known as functional integral representation of the

path integral over all trajectories which hold the boundary condition x(ti) = xiand x(tf ) = xf . The contribution from each trajectory is, however, weighted by

the classical action of the system. Such path integrals are usually very hard to

evaluate due to the action-involving phases. This problem may be overcome by

doing oscillating integrals. To get an idea of how such path integrals are evaluated,

refer e.g. to (Feynman and Hibbs, 1965; Itzykson and Zuber, 1980; Chaichian and

Demichev, 2001; Kleinert, 2004).

7.2 Path integral formulation for a scalar particle

Feynman additionally found out that the S-matrix in quantum electrodynamics

can also be represented in terms of relativistic particle path integral (Feynman,

1950, 1951). Starting from the Klein-Gordon equation, it can be shown that the

propagator for a relativistic scalar particle between xi an xf in a background field

can be written in Euclidean space-time as (Schubert, 2012)

K[A](xf , tf ;xi, ti) =

∫ ∞

0

dT e−m2T

∫ x(T )=xf

x(0)=xi

Dx(τ) e−∫T

0dτ ( 1

4x2+iex·A(x(τ))) ,

(7.3)

where T is the Schwinger proper time parameter (Schubert, 1996) and m is the

mass of the particle.

Different methods have already been proposed to calculate such path integrals

which we will roughly count them out here just to have a comprehensive list. These

methods include string-inspired approach (Polyakov, 1987; Bern and Kosower, 1992;

Strassler, 1992; Reuter et al., 1997; Fliegner et al., 1998; Schubert, 2001), semiclas-

sical approach (Affleck et al., 1982), variational method (Alexandrou et al., 2000)

and numerical approach (Gies and Langfeld, 2001; Stamatescu and Schmidt, 2003;

Gies and Klingmuller, 2005). We are not going to do justice to all these approaches.

A fair scrutiny of different methods can be found in (Schubert, 1996, 2001; Dunne

and Schubert, 2005, 2006; Dunne and Wang, 2006; Dunne et al., 2006; Schubert,

2007; Dietrich and Dunne, 2007; Dumlu and Dunne, 2011a) and the references

therein. Amongst all these, we principally follow the semiclassical approach, which

is based on the worldline instanton concept, in our discussions in the following.

7.2.1 Pair creation

As it was frequently discussed in the previous chapters, a strong enough external

electric field can result in the spontaneous pair creation out of vacuum (Sauter,

1931; Heisenberg and Euler, 1936). In this respect, QED pair production in an

Page 60: Pair Production and the Light-front Vacuum

50 Alternative to Tomaras–Tsamis–Woodard solution

external field can be studied through the imaginary part of the effective action.

Schwinger showed that this imaginary part for the scalars in a constant electric

field at one-loop level is given by

Im Γ(E) = − V

16π3(eE)2

∞∑

n=1

(−1)n

n2exp

[−πnm

2

eE

], (7.4)

where V is the space-time volume (Schwinger, 1951a). Pair production rate is then

given by the first term of the above series (Nikishov, 1970) and higher order terms

can approximately be neglected for E < Ec. It is known that the pair production

probability is related to the imaginary part of the one-loop effective action as below

(Schwinger, 1951a)

Prob = 1 − e−2 ImΓ ≈ 2 Im Γ . (7.5)

The worldline path integral representation of the scalar QED one-loop effective

action turns out to be

Γ[A] =

∫ ∞

0

dT

Te−m2T

x(T )=x(0)

Dx(τ) e−∫T

0dτ ( 1

4x2+iex·A(x(τ))) . (7.6)

The path integral here is over all closed space-time loops with a length T ; hence,

these loops indeed represent the virtual particles. This approach clearly shows the

nonlocal characteristic of the pair production phenomenon, specially for inhomo-

geneous background fields, in the sense that although the dominant contribution

to the effective action path integral comes from a predominant path, but there

are also significant contributions coming from the paths squirming around it and

passing through the neighboring points (Gies and Klingmuller, 2005). Specifically,

these squiggling paths experience different field strengths at neighboring points in

an inhomogeneous background field (see Figure 7.1). We will see in the following

that, in the semiclassical approach, this path integral is calculated along this closed

predominant trajectory.

squirming paths

predominant path

Fµν

tFig. 7.1 A schematic of the predominant path and a few squirming paths around it in an

inhomogeneous background field Fµν .

After rescaling the proper time in Equation (7.6) and performing the proper-time

integral and assuming a weak field approximation

(m√∫ 1

0 du x2 ≫ 1

), one ends

Page 61: Pair Production and the Light-front Vacuum

51 Path integral formulation for a scalar particle

up with the following relation for the imaginary part of the effective action (Dunne

and Schubert, 2005, 2006)

Im Γ[A] =1

m

√2π

T0Im

∫Dx e−

(m√∫

1

0du x2+ie

∫1

0duA·x

)

, (7.7)

where T0 is the stationary point of the proper-time integral

T0 =

√∫ 1

0du x2

m. (7.8)

Hence, this very last relation yields a new worldline action (Dunne and Schubert,

2005, 2006)

S = m

√∫ 1

0

du x2 + ie

∫ 1

0

duA · x , (7.9)

which is stationary if the path is a worldline instanton or, in other words, a classical

solution to the following worldline loop equation of motion (Dunne and Schubert,

2005, 2006)

mxµ√∫ 1

0 du x2= ie Fµν xν . (7.10)

Note that the worldline instanton here is in fact the aforementioned predominant

path. Furthermore, the imaginary part of the effective action has the following

behavior in the semiclassical approximation (Dunne and Schubert, 2005, 2006)

Im Γ[A]E→0∼ e−S0 , (7.11)

where S0 is the worldline action (7.9) evaluated on the worldline instanton given by

(7.10). The application of this method to a wide variety of background fields can

be found in (Dunne and Schubert, 2005).

All in all, for any background field, the algorithm to calculate the pair produc-

tion probability for scalar particles using a worldline instanton approach can be

summarized as follows:

• find the classical solutions for the worldline loop equation of motion (7.10) for

desired gauge and background field;

• evaluate the worldline action (7.9) on the worldline instanton obtained in the

previous step. Let us call it S0;

Page 62: Pair Production and the Light-front Vacuum

52 Alternative to Tomaras–Tsamis–Woodard solution

• find the imaginary part of the effective action in the semiclassical approxima-

tion (7.11) using S0;

• insert this value in (7.5) to obtain the pair production rate.

Finally, it is worth pointing out that the worldline instanton approach for spinor

QED is also done in a very similar manner. We should just account for Fermi-Dirac

statistics of the spinor loop as well as a spin factor which covers all spin effects

(Schubert, 2012; Corradini, 2012).

7.3 Path integral for a scalar particle on thelight-front

In the previous sections we could see how the worldline approach to quantum me-

chanics would offer a tool to investigate the pair production phenomenon in the

Euclidean space-time. Therefore, we may expect that this approach can potentially

be applied to the light-front QED as well. In particular, it may provide an opportu-

nity to understand the pair production issue on the light-front better. The general

ideas here would be more or less the same as the previous section and we might start

with the concept of one-loop effective action (7.6). It is to be noted that Aµ here

would be our usual gauge potential, i.e. the anti-lightcone gauge and the boundary

conditions on the functional integral would be xµ(T ) = xµ(0). This path integral

can then be converted into the light-front framework considering the conventions

made in the Table A.1. The calculation steps are supposed to be more or less similar

to what we summarized above. However, we are not going to fairly consider it here.

At this stage, we just mention that this approach is expected to yield similar results

as TTW solution and may provide a clearer picture of the light-front vacuum and

its high-intensity processes.

7.4 Summary

We roughly reviewed the basics of the worldline approach mainly for the Euclideanspace-time which can be considered as an alternative method to investigate thepair-production phenomenon. The possibility of using this approach on the light-front was proposed and briefly discussed. It seems that using such a method in thecase of light-front can shed light on different aspects of the TTW solution and thelight-front vacuum.

Page 63: Pair Production and the Light-front Vacuum

A Conventions and side calculations

General conventions that have been used throughout the most chapters of

the current thesis have been summarized in this appendix. To make life

easier, solutions to a few physical quantities have also been assembled here.

A.1 Light-cone coordinates and gauge conventions

Except Chapters 6 and 7 in which the conventions of Table 6.1 have been held for

the purpose of consistent notation and easy reference, the conventions of Table A.1

have been used in the rest of this thesis.

A.2 Side calculations

In this section, we have tried to give a bit more details on a few relations which

were roughly stated or used in our study.

A.2.1 Derivation of the anti-commutation relation for theDirac spinors on the light-front

As the first step, anti-commutation relation for the free spin one-half field in equal

usual time will be calculated in the following. The Dirac spinors are defined in

Heisenberg picture as below (Bjorken and Drell, 1965)

Ψ(x, t) =∑

s

∫d3p

(2π)32

√m

Ep

[b(p, s)u(p, s) e−ip·x + d†(p, s) v(p, s) e+ip·x

]

(A.1a)

Ψ†(x, t) =∑

s

∫d3p

(2π)32

√m

Ep

[b†(p, s)u†(p, s) e+ip·x + d(p, s) v†(p, s) e−ip·x

]

(A.1b)

Hence, for the anti-commutators of these fields we have

53

Page 64: Pair Production and the Light-front Vacuum

54 Conventions and side calculations

Table A.1 Light-cone coordinate and gauge conventions

Coordinatesa Scalar product

x+ ≡ x0 + x1 x− ≡ x0 − x1 aµbµ = a+b− + a−b+

Metric components Vector components

g+− = g−+ = 1 a+ = a− a− = a+

Gamma matricesb

γ0 =

[0 −i

i 0

]γi =

[0 i

i 0

]γ5 = γ5 =

[1 0

0 −1

](i = 1, 2, 3)

γ0㵆γ0 = γµ γ5† = γ5

in (1+1)−−−−−−−→dimensions

ηµν = (+1,−1) γµ, γν = 2ηµν

Pauli matrices Derivatives

σ1 =

[0 1

1 0

]σ2 =

[0 −i

i 0

]σ3 =

[1 0

0 −1

]∂± ≡ ∂/∂x±

Light-front gamma matrices

γ+ ≡ γ0 + γ1 γ− ≡ γ0 − γ1(γ±)2 = 0 γ+, γ− = 2

Light-front spinor projection operators Auxiliary relations

Λ± = 14γ∓γ± (Λ±)

2 = Λ± TrΛ± = 1

Dirac spinor Anti-lightcone gauge

Ψ = ψ+ + ψ− ψ± ≡ Λ±Ψ A+ = 0

a Note: this table is based on conventions made in all the chpaters, except Chapter 6.b A purely imaginary Majorana-Weyl representation of the two-dimensional Clifford alge-

bra (Ortın, 2004).

Ψ(x, t), Ψ†(y, t′)

=

∫d3p′

(2π)32

∫d3p

(2π)32

√m

Ep′

√m

Ep

s,s′

×b(p′, s′)uα(p′, s′) e−ip′·x

+ d†(p′, s′) vα(p′, s′) e+ip′·x , b†(p, s)u†β(p, s) e+ip·y + d(p, s) v†β(p, s) e−ip·y

=

∫d3p

(2π)3m

Ep

s

[uα(p, s)u†β(p, s) e−ip·(x−y) + vα(p, s) v†β(p, s) e+ip·(x−y)

]

=

∫d3p

(2π)3m

Ep

s

[uα(p, s) uβ(p, s) γ0 e−ip·(x−y)

+vα(p, s) vβ(p, s) γ0 e+ip·(x−y)]

Page 65: Pair Production and the Light-front Vacuum

55 Side calculations

=

∫d3p

(2π)31

2Ep

[(/p+m)αβ γ

0 e−ip·(x−y) + (/p−m)αβ γ0 e+ip·(x−y)

]

=

∫d3p

(2π)31

2Ep

[(/p+m)αβ γ

0 e−ip·(x−y) − (−/p+m)αβ γ0 e+ip·(x−y)

]

=

∫d3p

(2π)31

2Ep

[(γµpµ +m)αβ γ

0 e−ip·(x−y) − (−γµpµ +m)αβ γ0 e+ip·(x−y)

]

=

∫d3p

(2π)31

2Ep

[(iγµ∂µ +m)αβ γ

0 e−ip·(x−y) − (−iγµ∂µ +m)αβ γ0 e+ip·(x−y)

]

=

∫d3p

(2π)31

2Ep

i(γ0∂0︸︷︷︸

=0

+γi∂i) +m

αβ

γ0 e−ip·(x−y)

−i(γ0∂0︸︷︷︸

=0

+γi∂i) +m

αβ

γ0 e+ip·(x−y)

=

∫d3p

(2π)31

2Ep

[(iγi∂i +m

)αβ

γ0 e−ip·(x−y) −(−iγi∂i +m

)αβ

γ0 e+ip·(x−y)]

=

∫d3p

(2π)31

2Ep

[(γipi +m

)αβ

γ0 e−ip·(x−y) −(γipi +m

)αβ

γ0 e+ip·(x−y)]

= (γipi +m)αβ γ0

∫d3p

(2π)31

2Ep

[e−ip·(x−y) − e+ip·(x−y)

]

= (iγi∂i +m)αβ γ0

∫d3p

(2π)31

2Ep

[e−ip·(x−y) − e+ip·(x−y)

]

= (i/∂x +m)αβ γ0

∫d3p

(2π)31

2Ep

[e−ip·(x−y) − e+ip·(x−y)

]

= (i/∂x +m) γ0 i∆(x− y) . (A.2)

where ∆(x − y) denotes the Pauli-Jordan function defined as (Bjorken and Drell,

1965)

∆(x− y) = −i∫

d3p

(2π)31

2Ep

[e−ip·(x−y) − e+ip·(x−y)

], (A.3)

and we have made use of the equalities u† = uγ0 ,∑suα(p, s) uβ(p, s) =

(/p+m

2m

)

αβ

and∑svα(p, s) vβ(p, s) =

(/p−m

2m

)

αβand took into consideration that only terms

involving b, b† and d†, d contribute.

Making use of the current result, now we may find the equal x+ anti-commutation

relation of ψ+ and ψ+†. Before starting the calculations, however, it is worth point-

ing out a few useful characteristics of the Lorentz invariant Pauli-Jordan function

which will be employed in the following. A more detailed discussion on this function

can be found in (Bjorken and Drell, 1965). Such properties include the following

boundary conditions that hold at vanishing time difference (Greiner and Reinhardt,

Page 66: Pair Production and the Light-front Vacuum

56 Conventions and side calculations

1996)

∆(0,x) = 0 , (A.4a)

∂0 ∆(x0,x)∣∣x0=0

= −δ(3)(x) , (A.4b)

∂i ∆(x0,x)∣∣x0=0

= 0 . (A.4c)

Now we may proceed by multiplying both sides of the above equation

Ψ(x, t),Ψ†(y, t′)

= (i/∂x +m) γ0 i∆(x− y) , (A.5)

by the projection operator Λ+ from left and right

Λ+

Ψ(x, t),Ψ†(y, t′)

Λ+

︸ ︷︷ ︸LHS

= Λ+(i/∂x +m) γ0 i∆(x− y)Λ+

︸ ︷︷ ︸RHS

. (A.6)

The left hand side and right hand side of this equality are calculated in the following

LHS : Λ+(

Ψ(x, t)Ψ†(y, t′) + Ψ†(y, t′)Ψ(x, t))

Λ+

= Λ+Ψ(x, t)Ψ†(y, t′)Λ+ + Λ+Ψ†(y, t′)Ψ(x, t)Λ+

=(Λ+Ψ(x, t)

) (Ψ†(y, t′)Λ+†

)+ Ψ†(y, t′)Λ+Λ+Ψ(x, t)

=(Λ+Ψ(x, t)

) (Λ+Ψ(y, t′)

)†+(

Ψ†(y, t′)Λ+†)

︸ ︷︷ ︸(Λ+Ψ(y,t′))†

(Λ+Ψ(x, t)

)

= ψ+(x, x+)ψ+†(y, y+) + ψ+†

(y, y+)ψ+(x, x+)

=ψ+(x, x+), ψ+†

(y, y+), (A.7)

RHS : Λ+(i/∂x +m)αβ γ0 i∆(x− y)Λ+

= Λ+(iγ0∂0 + iγi∂i +m)αβ γ0 i∆(x− y)Λ+

= Λ+(−γ0∂0 − γi∂i + im)αβ γ0 ∆(x − y)Λ+

∣∣∣∣x0=y0

= Λ+

−γ0 ∂0 ∆(x− y)︸ ︷︷ ︸

−δ3(x−y)

−γi ∂i ∆(x− y)︸ ︷︷ ︸=0

+im∆(x− y)︸ ︷︷ ︸=0

αβ

γ0 Λ+

∣∣∣∣x0=y0

= Λ+ (γ0)2︸ ︷︷ ︸=1

δ3(x− y) Λ+

= (Λ+)2 δ3(x− y)

= Λ+ δ3(x − y) . (A.8)

Thus, we are left with the following anti-commutation relationψ+(x), ψ+†

(y)

x+=y+= Λ+ δ3(x − y) , (A.9)

Page 67: Pair Production and the Light-front Vacuum

57 Side calculations

or, in other words, we have

ψ+(x), ψ+†

(y)

x+=y+= Λ+ δ(x− − y−) δ2(x⊥ − y⊥) . (A.10)

A.2.2 The generators of Poincare algebra for a free fermionfield

Using the Equations (3.15), (3.19) and (3.20), the generators of Poincare algebra

for a free fermion field in (1+1) dimensions can primarily be written as follows

P+ = 12

Σ

dx− T++ , (A.11a)

P− = 12

Σ

dx− T+− , (A.11b)

K− = 12 M

+−

= 14

Σ

dx−(x− T++ − x+ T+−

). (A.11c)

Rewriting the Lagrangian density of the free fermion field as below

L =i

2Ψγ+∂−Ψ +

i

2Ψγ−∂+Ψ −mΨΨ , (A.12)

we may start to calculate T++ and T+−. Hence, we have

T+− = ∂−Ψ∂L

∂(12∂

−Ψ)

︸ ︷︷ ︸=0

+∂L

∂(12∂

−Ψ) ∂−Ψ − g+−L︸ ︷︷ ︸

=0

= iΨγ+∂−Ψ

= iΨ† γ0γ+︸ ︷︷ ︸2Λ+

∂−Ψ

= 2iΨ†Λ+∂−Ψ

= 2iΨ†(Λ+)2∂−Ψ

= 2i(Λ+Ψ

)†∂−(Λ+Ψ

)

= 2iψ+†∂−ψ+

= 2ψ+† m2

i∂+ψ+ . (A.13)

Page 68: Pair Production and the Light-front Vacuum

58 Conventions and side calculations

Similarly, for T++ we have

T++ = ∂+Ψ∂L

∂(12∂

−Ψ)

︸ ︷︷ ︸=0

+∂L

∂(12∂

−Ψ) ∂+Ψ − g++L︸ ︷︷ ︸

=0

= iΨγ+∂+Ψ

= 2iψ+†∂−ψ+ . (A.14)

In the end, substituting T++ and T+− in Equations (A.11), we come up with the

final results seen before as Equations (3.21).

Page 69: Pair Production and the Light-front Vacuum

Notes

Chapter 2

1 A set G which has a group structure and is also a n-dimensional ‘differentiablemanifold’ is said to be a Lie group of real dimension n (Isham, 1989).

2 If we fix some point x0 in set X, the stability group Gx0of G at the point x0 is

technically defined as (Isham, 1989)

Gx0:= g ∈ G | gx0 = x0.

3 If ∀x, y ∈ X, ∃g ∈ G : gx = y, the action of group G on set X is called transitive

(Isham, 1989).4 If ∀a, b ∈ G1 there exist a bijection i : G1 → G2 for which

i(ab) = i(a).i(b) ,

the map i is called an isomorphism of G1 with G2, where · denotes the combinationlaw in group G2 (Isham, 1989).

59

Page 70: Pair Production and the Light-front Vacuum

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Page 87: Pair Production and the Light-front Vacuum

Subject index

γ matrices, 24light-front, 53Majorana-Weyl representation, 24, 53

N, 36

action, 24classical, 49effective, 50one-loop effective, 50, 52worldline, 51

algebraLie, 10Poincare, 10, 21, 22, 57

anomalous magnetic moment, viii

anti-commutation relation, 38, 44, 53antisymmetric Green function, 20approach

numerical, 49semiclassical, 49string-inspired, 49worldline, 52

approximationsemiclassical, 51weak field, 50

bijection, 59boost, 10, 15, 45

generator, 21, 23boost and Galilei invariance, 18bound states, 12boundary condition, 1, 3, 35, 49, 52

anti-periodic, 35periodic, 35, 36

canonicalmomentum, 45

cascades, 5Casimir effect, 3, 8

dynamical∼, 3causality, 9Clifford algebra, 24, 53coherent laser beams, 6Coleman’s theorem, 17commutation relation

equal time, 20equal-x+, 20

Compton wavelength, 6, 9conducting plates, 3

constituent picture of the hadron, 12

constraintmass-shell, 15

correspondence principle, 11covariant

theory, 12cross section, 12cut-off, 36

Diracspinor, 53

Dirac equation, 33, 40, 42, 47Dirac’s

delta function, 29, 38forms, 10

DLCQ, vii, 17, 35

eigenoperator, 44electron-positron pair, 40

virtual, 5, 6electron-positron pairs

virtual, 4energy levels of hydrogen, viiienergy-momentum tensor, 10, 22equation

∼ of motion, 25Dirac, 33, 40, 42, 47Euler-Lagrange, 25Klein-Gordon, 49

equation of motion, 22Euler-Lagrange equation, 19excitation, 35excited states, 15existence, 9expectation value, 41external disturbance, 1, 8

Faddeev and Jackiw method, 19, 20Fermi-Dirac statistics, 52Feynman slash notation, 21field

background, vii, 5, 15, 24, 32, 40, 49–51componen, 25constrained, 23, 25, 33critical, 6dynamical, 23, 25, 33electric, 6, 30electromagnetic, 6, 7external, 1, 6, 8, 40, 49

fermion, 28, 33, 36, 40

77

Page 88: Pair Production and the Light-front Vacuum

78 Subject index

dynamical, 22, 23free fermion, 20, 21, 57free scalar, 20

gauge, 40homogeneous, 40

inhomogeneous, 50intensity, 7

non-constraint, 36quantization, viireal scalar, 19

strong, 2tensor, 30

field theoryclassical, 31

quantum, 1, 9, 15, 35, 40conventional, 12null-plane, 17

Fockrepresentation, 21

space expansion, 20, 22foliation

of space-time, 9, 11, 14, 18form

front, vii, viii, 11, 12, 14, 18, 22, 29, 36

instant, 10–12, 15, 18, 19, 22, 29point, 11, 12, 18

Fourierdecomposition, 36space, 44

transform, 26, 44functional integral, 48, 52

gamma matrices, 41, 54gauge

anti-lightcone, 24, 33, 42, 52conventions, 53

covariant derivative, 24fixing, 9

generatordynamical, 11kinematical, 11

ground state, 12group

actiontransitive, 11, 59

combination law, 59Euclidean, 12generator, 10

isomorphic, 59isospin, 10

Lie, 59Lorentz, 10

Poincare, 10generator, 11, 12, 21

stability, 11, 12, 59

theory, 10

Hamiltonian

dynamics, vii, 12, 15, 18, 36normal-ordered, 29, 33, 39

Heisenberg

picture, 27, 40, 47, 53

uncertainty principle, vii

high energy physics, 12, 15

high-order harmonics generation, 4

homogeneity, 4

hyper-hyperboloid, 12

hypersurface, 11equal-time, 9

space-like, 12

infinite momentum frame, 14

interacting theory, 17

isomorphism, 12, 59

Klein-Gordon equation, 19

laboratory astrophysics, 2

Lagrangian, 21, 25

density, 19, 21, 57

Lamb shift, viii

laser techniques, 2

Chirped Pulse Amplification, 2light and matter interaction, viii

light bullets formation, 5

light-cone

coordinates, 14, 29, 32, 36, 53

dynamics, 15

energy, 14

framework, 15

momentum, 14, 17quantization, 36

space, 14, 40

time, 14, 30, 33, 40

velocity, 14

light-fron

framework, 52

light-front, vii, 48

γ matrices, 53

ambiguity, viii, 35dynamics, vii, 12, 20

framework, vii, 19

Hamiltonian, 35

projection operator, 53

QED, 52

quantization, vii, 23, 24, 40

vacuum, vii, viii, 15, 17, 19, 33, 40, 52

light-like distance, 20

locality, 9Lorentz

force, 32

invariant, 12, 21

transformation, 10, 14, 45

manifold, 45, 47

differentiable, 59

many body system

Page 89: Pair Production and the Light-front Vacuum

79 Subject index

non-relativistic, 15relativistic, 15

many-body theory, 9mass-energy equivalence, 1microscopic causality, 20mode function, 41momentum

angular, 10canonical, 17, 27conjugate, 19, 27four-∼, 10kinetic, 15, 17, 45light-cone, 17operator, 21physical, 45

non-perturbative, 35solution, 35

non-relativisticlimit, 11many body system, 15quantum mechanics, 48

normal modes, 3

on-shell, 27one-loop effective, 50operator

annihilation, 45creation, 20, 45Hamiltonian, 21, 23Hermitian, 27initial value, 40momentum, 21non-singular, 41projection, 22, 24

oscillating integral, 49

pair creation, 6, 49pair production, 40, 48, 50, 52

probability, 50, 51rate, 50spontaneous, 6stimulated, 5trident, 5

particleaccelerated, 32anti∼, 9identical, 9on-mass shell, 15, 36point, 9relativistic, 32virtual, 1, 50

path integral, 48, 50path integral approach, 48Pauli-Jordan function, 55phenomenon

back reaction, 40nonlinear vacuum ∼, vii

pair production, vii, 44, 50

photon acceleration, 5photon splitting, 5plane wave, 12plasma oscillation, 5predominant path, 50, 51predominant trajectory, 50propagator, 48, 49proper time, 50

integral, 50, 51

QED, viiinon-perturbative, 2strong-field∼, 8

QED(1+1), 35quantization

canonical, 20, 27free field, 19light-front, 23, 24, 40non-standard, 23null-plane, 17

quantumelectrodynamics, 49system, 48vacuum, viii

quantum field theoryabelian, 36non-abelian, 36

quantum mechanics, 7, 9, 48non-relativistic, 48

regularizationinfrared, 36

relativisticdispersion relation, 15dynamics, viii, 16invariance, 10many body system, 15many-body theory, 9particle, 49particle point, 9scalar particle, 49speed, 9

renormalizationperturbative, 36

representationfunctional integral, 49irreducible, 10momentum, 35path integral, 48, 50

rest-mass energy, 6

Sauter-Schwinger limit, vii, 6, 7scattering

nonlinear Compton, 5photon-photon, 4

Schrodinger picture, 48Schwinger

mechanism, 7

pair production, vii, viii, 7, 8

Page 90: Pair Production and the Light-front Vacuum

80 Subject index

pairs, viii, 15, 17, 18, 29, 33proper time parameter, 49

Schwinger’squantum action principle, 19vacuum persistence amplitude, 44

self-lensing effects, 5semiclassical approach, 48singularity, 26, 34, 36space

configuration, 48Euclidean, 59

n-dimensional, 59Fock, 35momentum, 35

space-time, 36, 48Euclidean, 49, 52loop, 50volume, 50

special relativity, viii, 7, 9spectrum

discrete k−, 36mass, 35

spin factor, 52spinor loop, 52squirming path, 50string theory, 9string/M-theory, 35strong-field

processes, 2theory, vii

structuretopological, 36

supergravity, 35superposition, 4supersymmetry, 35symmetry, 10

Tamm-Dancoff method, 35Tomaras-Tsamis-Woodard (TTW) solution,

vii, 40, 48, 52alternative, 48

topological structure of space-time, 37topology, 11trajectory, 49transition probability amplitude, 48

uncertainly principle, 1uniqueness, 9

vacuum, 36, 48, 49birefringence, 3fluctuation, 1, 3–5, 8free theory, 29interacting theory, 29light-front, vii, 15, 17, 19, 33, 38, 40, 52nonlinear, 2, 8nonlinear nature, 1physical, 29

polarization, 6

quantum, vii, 2, 4, 5structure, viii

stable, 17, 29state, 17trivial, 17, 29, 39

vacuum birefringence, 8variational method, 49velocity

group∼, 5virtual

photon, 3

wave function, 35worldline

action, 51approach, 52instanton, 49, 51loop equation of motion, 51path integral approach, 48

worldsheet, 9

Yukawa theory, 35

zero body problem, 1zero-mode, vii, 36, 39

excitations, 17issue, vii, 17, 29, 38, 39