Download Pair Production and the Light-front Vacuum

Pair Production and the
Light-front Vacuum
Ramin Ghorbani Ghomeshi
Department of Physics
Umeå University
SE - 901 87 Umeå, 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).
Optimis parentibus
Contents
Abstract
Preface
Acknowledgment
v
page vii
viii
ix
1 Strong field theory
1.1 Nonlinear quantum vacuum processes
1.2 Pair creation
1.3 Summary
1
2
6
7
2 Introductory light-front field theory
2.1 Dirac’s forms of quantization
2.2 Light-front dynamics
2.2.1 Light-cone coordinates
2.2.2 Light-front vacuum properties
2.3 Summary
9
10
12
14
15
18
3 Free
3.1
3.2
3.3
19
19
21
23
theories on the light-front
Free scalar field
Free fermion field
Summary
4 LF quantization of a fermion in a background field in (1+1) dimensions
4.1 Classical solution
4.2 Quantization
4.3 Zero-mode issue
4.4 Summary
24
26
27
29
33
5 Discrete Light-Cone Quantization
5.1 Quantization
5.2 Zero-mode issue
5.3 Summary
35
36
38
39
6 Tomaras–Tsamis–Woodard solution
6.1 Methodology
6.2 The model and its solution in Woodard’s notation
6.3 Quantization
40
40
42
44
Contents
vi
6.4
6.5
Pair production on the light-front
Summary
44
47
7 Alternative to Tomaras–Tsamis–Woodard solution
7.1 Quantum mechanical path integral
7.2 Path integral formulation for a scalar particle
7.2.1 Pair creation
7.3 Path integral for a scalar particle on the light-front
7.4 Summary
48
48
49
49
52
52
Appendix A
Conventions and side calculations
A.1 Light-cone coordinates and gauge conventions
A.2 Side calculations
A.2.1 Derivation of the anti-commutation relation for the Dirac
spinors on the light-front
A.2.2 The generators of Poincaré algebra for a free fermion field
53
53
53
Notes
References
Subject index
59
60
77
53
57
Abstract
ominated by Heisenberg’s uncertainty principle, vacuum is not quantum mechanically an empty void, i.e. virtual pairs of particles appear and disappear
persistently. This nonlinearity subsequently provokes a number of phenomena 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 phenomenon. From the viewpoint of Dirac’s front form of Hamiltonian dynamics, however, 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 solution and give a clearer picture of the light-front vacuum and the pair production
phenomenon on the light-front.
D
vii
Preface
ince 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 matter. Extremely accurate predictions of quantities like the Lamb shift of the energy
levels of hydrogen (Lamb and Retherford, 1947) and the anomalous magnetic moment 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 verification 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 Standard Model. Studying the pair production phenomenon on the front form of relativistic 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, something 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.
S
viii
Acknowledgment
y enrolling at Umeå 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 supervisor Anton Ilderton for introducing me to this interesting and exciting topic in
theoretical physics, his continuous support and tolerating my eccentric way of doing 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 Umeå
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 Umeå 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
B
ix
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 kindness, 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 Umeå, 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 Shahmohammadi 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, Elham Abdollahi, Zeynab Kolahi and Amir Khodabakhsh. I am also deeply grateful
to my friends, from those who have already left Umeå or who are still here, for
keeping in touch, their helps and supports. I would like to list their names, however, 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 Umeå 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 education 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.
1
Strong field theory
here 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.
T
Special
Relativity
Quantum
Mechanics
Quantum
Field Theory
Uncertainty
Principle
Mass-Energy
Equivalence
Vacuum
Fluctuation
t
Fig. 1.1
1
A schematic which shows how Quantum Field Theory formed out of quantum
mechanics and special relativity merging.
Strong field theory
2
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 generating stronger fields than present in normal environments1 . New laser techniques,
appearing after the birth of Chirped Pulse Amplification (CPA) technique (Strickland 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:
t
Fig. 1.2
A vacuum diagram.
1
Undulators and heavy ion collisions are other examples of the available sources.
3
Nonlinear quantum vacuum processes
• The Casimir effect
As stated before, one way to disturb the vacuum is to impose boundary conditions. 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 appearance 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 instance, 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 Overbeek, 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 reported 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 quantum 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
Schröder, 2006; Heinzl and Ilderton, 2009; Ilderton, 2012).
Strong field theory
4
d
t
Fig. 1.3
The Casimir effect, schematically.
• Photon-photon scattering
A linear system generally satisfies two requirements: superposition and homogeneity, c.f. (Hoffman and Kunze, 1971) for example. The superposition principle 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; Lundström et al., 2006; Archibald et al., 2008).
t
Fig. 1.4
Photon-photon scattering diagram. Double lines represent the dressed propagators
due to particles in background field.
Nonlinear quantum vacuum processes
5
• 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 Kämpfer,
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 nonlinear 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).
γ
e−
e−
t
Fig. 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 mutually 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; Soljačić 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 behavior 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 (Mendonça et al., 1998; Mendonça,
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).
Strong field theory
6
2001; Mendonça 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 electromagnetic 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 critical 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
m2 c3
e~
≈ 1.3 × 1018 V/m ,
Ec =
(1.2)
where m here is the mass of electron. This process occurs with a probability proportional 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
4
The Compton wavelength is defined as λC = ~/mc for a particle of mass m (Compton, 1923), so
that for a field with a wavelength smaller than this value for a special particle, the field quanta
will have energies well above the rest-mass energy of that particle and particle-antiparticle pair
creation becomes abundant.
http://www.xfel.eu/
Summary
7
k
q
t
Fig. 1.6
k−q
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 (∼ 10−4 Ec ) 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; Schützhold 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; Kohlfürst 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; Schützhold and Unruh, 2011; Belgiorno
et al., 2011).
1.3 Summary
In a video by CERN8 , Peter Higgs well-summarizes the idea this chapter is based
upon: “When you look at a vacuum in a quantum theory of fields, it isn’t exactly
nothing”. Unification of quantum mechanics and the special relativity theory represents a new picture of vacuum in which “vacuum is no longer quite as empty as
5
6
7
8
http://www.hiperlaser.org/
http://www.extreme-light-infrastructure.eu/
http://www.xcels.iapras.ru/
Meet Peter Higgs: http://cds.cern.ch/record/1019670
8
Strong field theory
it is used to be”, but virtual particle pairs are allowed to be spontaneously created
and annihilated. This phenomenon is called vacuum fluctuation which gives the
vacuum a nonlinear characteristic. These fluctuations cannot be observed directly.
However, they give rise to a set of nonlinear effects in the presence of an external
disturbance which can confirm their existence indirectly. This external disturbance
can 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 of
which, for instance, the Casimir effect, vacuum birefringence and Schwinger pair
production, which is the break down of highly polarized vacuum into real pairs
due to the presence of a strong external electric field, can be named. Normally a
quite strong external electromagnetic field is required for such nonlinear vacuum
phenomena to happen detectably. Such a critical field strength, e.g., for pair production phenomenon has already been calculated by Schwinger and turned out to
be of the order of 1018 V/m. With the appearance of modern laser facilities, high
field intensities up to 10−4 Ec will hopefully be reached in the near future and it
may become possible to verify the Strong-Field QED effects directly.
2
Introductory light-front field theory
ith the appearance of modern theories of physics in twentieth century, a
rather new field in physics came gradually into existence that was trying 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 relativistic 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 particle 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. field 1 . 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 operator 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
theory 2 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).
W
1
2
9
For a nice review of the underlying principles of QFT, see (Wilczek, 1999; Tong, 2006).
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.
Introductory light-front field theory
10
Table 2.1 All possible choices of hypersurfaces Σ : τ = const. with transitive
action of the stability group GΣ · d denotes the dimension of GΣ , that is, the
number of kinematical Poincaré generators; x⊥ ≡ (x1 , x2 )
namea
instant
Σ
τ
0
x =0
t
light front
0
3
x +x =0
hyperboloid
x20
2
hyperboloid
x20
hyperboloid
x20 − x21 = a2 > 0, x0 > 0
a
d
6
3
t + x /c
2
0
− x = a > 0, x > 0
⊥ 2
2
0
− (x ) = a > 0, x > 0
2
2
7
2
2
2 1/2
(t − x /c − a /c )
2
⊥ 2
2
2
2 1/2
6
(t − (x ) /c − a /c )
4
(t2 − x21 /c2 − a2 /c2 )1/2
4
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 mathematically 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 “transform as irreducible representations of the Lorentz and Poincaré 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 dynamical system and invariance under certain transformations, one may naturally
refer to the concept of full Poincaré group in order to study the full relativistic
invariance of a system. Poincaré group packs dealing with all the above-mentioned
fundamental quantities in a set of algebraic equations. This group is generated by
the four-momentum P µ and the generalized angular momentum M µν . These generators 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 =
M µν =
Z
ZΣ
Σ
d3 x T 0µ ,
d3 x xµ T 0ν − xν T 0µ ,
(2.1a)
(2.1b)
where T µν is the energy-momentum tensor. Hence, the relations for Poincaré algebra, which is the Lie algebra1 of the Poincaré group, can be written in terms of P µ
11
Dirac’s forms of quantization
Box 2.1
Metric tensors corresponding to Dirac’s forms of Hamiltonian dynamics
The instant form
gµν
The front form


0
1 0
0
0
0
0 −1 0
0



=
 gµν = 
0 0 −1 0 
0
1
0 0
0 −1

0
0
−1 0
0 −1
0
0
2
The point form

1

1
0
0 −τ 2

=
0
0
−τ 2
0
0
2
0

 gµν
0
0

1
0
2

0
0



sinh2 ω
0
2
2
2
0
−τ sinh ω sin θ
Note: the contents of this box have been taken from (Pauli, 2000).
and M µν as follows (Ryder, 1985; Weinberg, 1995)
[P µ , P ν ] = 0 ,
µ
ρσ
µν
ρσ
[P , M
[M
,M
(2.2a)
µρ
σ
] = i (g P − g
µρ
] = i (−g M
νσ
µσ
ρ
P ),
+g
µσ
µρ
ν
M
(2.2b)
νρ
−g
νσ
M
µρ
νρ
+g M
µσ
),
(2.2c)
or equivalently as
{P µ , P ν } = 0 ,
{M
{M
µν
µν
ρ
νρ
ρσ
µσ
(2.3a)
µ
,P } = g P − g P ,
,M
}=g
M
νρ
µρ
−g M
(2.3b)
νσ
−g
νσ
M
µρ
νρ
+g M
µσ
.
(2.3c)
Going back to the case of (3 + 1)-foliation of space-time, we realize that the dynamical evolution of a system, i.e. development in τ , is technically determined by
the structure of those Poincaré group generators which correspondingly map the
initial data hypersurface Σ to another hypersurface Σ′ at a later time τ ′ (Fleming, 1991). Such generators are fairly called dynamical. In contrast, those Poincaré
group generators under which the hypersurface Σ is left invariant are called kinematical and they form a subgroup of the Poincaré group called stability group 2
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
Introductory light-front field theory
12
conventional quantum field theory had been formulated. The front form is the hypersurface, 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 : xµ xµ = const., lying inside future lightcone. 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 stability 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 Poincaré 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 actionoriented 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 picture 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).
x0
x0
x0
Σ : xµxµ = const.
Σ : x0 = 0
3
x3
x
x3
Σ : x+ = 0
x1, x2
x1, x2
t
Fig. 2.1
Left: the instant form. Middle: the front form. Right: the point form.
x1, x2
Introductory light-front field theory
14
2.2.1 Light-cone coordinates
As we already noted, (3+1)-foliation of space-time based on the front form proposes working in a new coordinate system which is called light-cone coordinates5 .
Converting to this coordinate is not a Lorentz transformation, but a general coordinate 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)
x0
x+
x−
x3
t
Fig. 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
1
1
(2.6)
a · b = gµν aµ bν = a+ b− + a− b+ − a1 b1 − a2 b2 .
2
2
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 reasonable 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).
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.
with the mass-shell constraint on the
Its combination
+ −
⊥ 2
2
light-cone, k k − k
= m , 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 nonrelativistic 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 theory. 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 interactions 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 nonzero probability for Schwinger pairs to be created (Sauter, 1931; Schwinger, 1951a).
In fact, this phenomenon had been studied conventionally in the instant form
of
µ
0
Hamiltonian dynamics. The ordinary kinetic momentum, i.e. p in p = p , p , can
have both negative and positive values in the instant form. Therefore, we may find
many excited states, like â†k â†−k , with zero valued kinetic momenta (k + (−k) = 0)
Table 2.2 Vacuum structure from the instant and front forms of relativistic dynamics point of view
:H: ∼
Quantum vacuum of the free theory in the instant form of relativistic dynamics; Fµν = 0
Quantum vacuum in the instant form in the presence of
an electromagnetic background
field; Fµν 6= 0
+ dˆ†p dˆp
:H: |0i = 0
|0i ≡ vacuum state of
the free theory
Although virtual pairs are allowed to be
created, no real pairs are come into existence 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 annihilation operators appear. This means, in
other words, that particle number is conserved ([H, N ] = 0).
The appearance of such terms indicate
that particle number is not conserved
([H, N ] 6= 0) and as a result, pairs of
:H: |0i = |f i
particles-antiparticles can be created.
|f i ≡ ψ0 |0i + ψ1 |pairi + Here, N is the number operator. Note
that p can acquire both positive and negψ2 |two pairsi + · · ·
ative values (c.f. Section 4.3).
:H: ∼ b̂†p dˆ†−p + etc.
:H: ∼
Light-front vacuum in the presence of an electromagnetic background field; Fµν 6= 0
b̂†p b̂p
b̂†k− b̂k−
+ dˆ†k− dˆk−
:H: |Ωi = 0
|Ωi ≡ 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− conservation can hold. Thus, only terms involving
the same number of creation and annihilation operators remain in the Hamiltonian of the system on the light-front.
Hence, there is no fluctuation.
Light-front dynamics
17
that can mix with the vacuum6 (Burkardt, 2002). Therefore, vacuum of the interacting 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 operator (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, Q̂, acts on the
vacuum state, |Ωi, it satisfies
Q̂ |Ωi = 0
(2.10)
if and only if there exists a local conservation law for its associated generalized fourcurrent 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
7
We distinguish between the canonical momentum k and kinetic momentum p following the
convention made in (Kluger et al., 1992). In the case of a free field, the canonical momentum
coincides with the kinetic momentum.
Of course, with the exception of zero-modes, namely the modes with k− = 0.
18
Introductory light-front field theory
2.3 Summary
By the appearance of quantum field theory, it was discovered that the choice of
time parameter within special relativity is not unique and any attempt to choose
a time parameter leads to a foliation of space-time into space and time. Dirac
showed 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 which
the so-called conventional quantum field theory has been formulated. The front
form, which is a hyper-plane tangent to the light-cone, constitutes the keystone of
the current survey. Light-front dynamics presents a set of peculiar features which
do not have any analogous structure in the instant form. For instance, the boost
and Galilei invariance can be mentioned. However, the most remarkable feature of
the front form of Hamiltonian dynamics is the triviality of its vacuum which, apart
from showing a few fundamental disparities compared to the instant form (like
giving no signature of Schwinger pairs, which will be discussed in more detail in the
next chapters), seems to be extremely promising in dealing with the quantum field
theories that suffer from the complexities of the ground states in the instant form
of Hamiltonian dynamics. In other words, light-front vacuum is simple and, with
the exception of zero-modes (modes with k− = 0), no other excited state can mix
with it. Although the inclusion of zero-modes means that the light-front vacuum is
not actually pure trivial, however, it is in fact essential for getting many quantum
vacuum processes right. Thus, engaging a Hamiltonian approach in the front form
seems to reduce the complexities that appeared with this approach in the instant
form.
3
Free theories on the light-front
e 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).
W
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)
1
1
(∂µ φ)2 − m2 φ2 ,
(3.1)
2
2
turns into the following relation when it is expressed in light-front framework in
(1 + 1) dimensions:
L=
1 + −
1
∂ φ ∂ φ − m2 φ2 .
(3.2)
2
2
The equation of motion, following from the Euler-Lagrange equation, can be written
as
∂ + ∂ − + m2 φ = 0 .
(3.3)
L=
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
Free theories on the light-front
20
• 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
Z
i h
1
d3 k
† ik·x −ik·x
√
âk e
+ âk e
,
(3.4)
φ(x) =
0
(2π)3 2E
k
k =Ek
with the only non-vanishing commutation relation of [âk , â†k′ ] = (2π)3 δ (3) (k − k′ ).
This gives a canonical commutation relation
[φ(x) , φ(y)]
Z
Z
h
i
h
i
d3 k
1
d3 k ′
†
†
−ik·x+ik′ ·y
ik·x−ik′ ·y
′
√
=
â
,
â
e
+
â
,
â
e
′
k
k
k
k
(2π)3
(2π)3 2 Ek Ek′
Z
3
d k 1
e−ik·(x−y) − eik·(x−y)
=
(2π)3 2Ek
Z
d3 k 1 ik·(x−y)
0
e
sin
k
(x
−
y
)
.
(3.5)
= −i
0
0
0
(2π)3 Ek
k =Ek
As a result, the equal-x+ commutation relation turns out to have the following form
in (1+1) dimensions (Heinzl, 2001)
i
[φ(x) , φ(y)]x+ =y+ =τ = − sgn(x− − y − ) ,
4
(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).
Free fermion field
21
Rohrlich, 1971; Chang et al., 1973)
Z ∞
dk + φ(x) =
a(k) e−ik·x + a† (k) eik·x ,
+ (2π)
2k
0
(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 Poincaré algebra (2.3) for such a field in Fock representation in (1+1) dimensions
P
+
P−
K−
Z
dk + †
a (k) a(k) ,
4π
Z
dk + m2 †
a (k) a(k) ,
=
4πk + k +
Z
∂ †
dk +
a
(k)
k + a(k) ,
=
4πk + ∂k +
=
(3.10a)
(3.10b)
(3.10c)
where P + , P − and K − are corresponding to momentum operator, Hamiltonian operator 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 Poincaré 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
(3.11)
L = Ψ i∂/ − m Ψ ,
/ ≡ γ µ Aµ . This Lagrangian gives
where the Feynman slash notation is defined as A
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 + − i − +
γ ∂ + γ ∂ −m Ψ = 0,
(3.13)
2
2
Free theories on the light-front
22
Introducing projection operators as in Table A.1, ψ ± = Λ± Ψ, and after some
algebra (Harindranath, 2000), we get
i∂ + ψ − = γ 0 mψ + ,
(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 +
ψ .
i∂ +
(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 +
n
o
†
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 expansion of the free fermion field is given by (Kogut and Soper, 1970; Chang et al.,
1973)
Z
dk + X (3.17)
bs (k) ϕs (k) e−ik·x + d†s (k) κs (k) eik·x ,
Ψ(x) =
2k + (2π) s
where
n
o
bs (k) , b†s′ (k ′ ) = 2(2π) k + δ(k − k ′ ) ,
n
o
ds (k) , d†s′ (k ′ ) = 2(2π) k + δ(k − k ′ ) ,
(3.18a)
(3.18b)
and all other anti-commutation relations vanish.
The generators of Poincaré algebra in the front form are almost similar to those
of the instant form (2.1)
Z
P µ = 12
dx− d2 x⊥ T +µ ,
(3.19a)
Σ
Z
dx− d2 x⊥ xµ T +ν − xν T +µ ,
(3.19b)
M µν = 12
Σ
1
2
where the factor 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 = ∂ ν Ψ
∂L
∂L
+
∂ ν Ψ − g µν L .
∂
(∂
∂ ∂µ Ψ
µ Ψ)
(3.20)
Therefore, the generators of Poincaré algebra for a free fermion field in terms of the
23
Summary
dynamical fermion field ψ + in (1+1) dimensions are given by
Z
†
+
P =
dx− ψ + i∂ + ψ + ,
Z
2
† m
P− =
dx− ψ +
ψ+ ,
+
i∂
Z
2
− +†
+ +
+ +† m
−
−
+
1
x ψ i∂ ψ − x ψ
dx
K = 2
,
ψ
i∂ +
(3.21a)
(3.21b)
(3.21c)
where P + , P − and K − are again corresponding to momentum operator, Hamiltonian 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 this
chapter. It was shown that the light-front quantization procedure is almost nonstandard, because the systems appeared as first-order in derivatives. Due to this
fact, the constrained and dynamical fields were emerged in the cases of free fermion
field. The equal-x+ (anti-)commutation relations were also showing a somewhat
different characteristics compared to their analogous structures in the instant form.
The fermion field, which is our main focus, will be more scrutinized in the next
chapter.
LF quantization of a fermion in a
background field in (1+1) dimensions
4
ince 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
S=
2
Z
/ − m)Ψ.
dx+ dx− Ψ(iD
(4.1)
where, following the (3+1)-dimensional case, the Dirac spinor can be written as
" #
ψ1
Ψ=
.
(4.2)
ψ2
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
#
"
#
"
0 −i
0 i
0
1
γ =
,γ =
.
(4.3)
i 0
i 0
2
We introduce γ ± = γ 0 ± γ 1 such that (γ ± ) = 0 and the projection operators
2
Λ± = 41 γ ∓ γ ± , which have the properties of (Λ± ) = Λ± and Λ± Λ∓ = 0. Also we
have γ 0 γ ± = 2Λ± . Hence,
"
#
"
#
0 0
0 −2i
+
−
γ =
,γ =
.
(4.4)
2i 0
0
0
The projection operators are, thus, given by
"
#
"
1 0
0
Λ+ =
, Λ− =
0 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 ) .
1
24
(4.6b)
A detailed discussion on gamma matrices in various dimensions can be found in (Ortı́n, 2004).
LF quantization of a fermion in a background field in (1+1) dimensions
25
Introducing ψ± = Λ± Ψ, the field components are given by
" #
" #
ψ1
0
.
ψ+ =
and ψ− =
0
ψ2
(4.7)
Now, we may re-write the action in terms of field components
S=
=
=
=
=
Z
1
2Z
1
2
Z
1
2
Z
1
2
Z
1
2
dx+ dx− Ψ† γ 0 iγ + ∂+ + iγ − D− − m Ψ
dx+ dx− Ψ† 2iΛ+ ∂+ + 2iΛ−D− − mγ 0 Ψ
"
!
!
!# " #
i
h
2i 0
0 0
0
im
ψ1
†
†
+
−
∂+ +
D− +
dx dx ψ1 ψ2
0 0
0 2i
−im 0
ψ2
"
!
!
!#
i
h
2i∂+ ψ1
0
im ψ2
+
+
dx+ dx− ψ1† ψ2†
0
2iD− ψ2
−im ψ1
(4.8)
dx+ dx− 2i ψ1† ∂+ ψ1 + 2i ψ2† D− ψ2 + im ψ1† ψ2 − im ψ2† ψ1 .
From here on, we write ψ1 ≡ ψ+ and ψ2 ≡ ψ− to keep the notation simpler.
Therefore, we come up with
1
S=
2
Z
†
†
†
†
∂+ ψ+ + 2i ψ−
D− ψ− + im ψ+
ψ− − im ψ−
ψ+ (. 4.9)
dx+ dx− 2i ψ+
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 1
ψ+ .
2 [D− ]
(4.11)
Inserting (4.11) into (4.10a), we get the equation of motion for the dynamical field
ψ+
i∂+ ψ+ =
m2
ψ+ .
4[iD− ]
(4.12)
In the following, we try to solve (4.12) both classically and quantum mechanically.
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 ) =
Z
0
∞
i
−
−
dk− h
b(k− )e−ik− x φk− (x+ ) + d† (k− )eik− x κk− (x+ ) .
(2π)
(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 derivatives, we get
1
m2
φ,
4 k− − eA−
1
m2
i∂+ κ =
κ.
4 −k− − eA−
i∂+ φ =
Now we may easily find the classical solutions to this set of equations
!
Z +
im2 x
1
+
φk− (x ) = exp −
dy ,
4 0 k− − eA− (y)
!
+
2 Z x
im
1
κk− (x+ ) = exp −
dy .
4 0 −k− − eA− (y)
(4.14a)
(4.14b)
(4.15a)
(4.15b)
As it can be seen here, the singularity problem exists even in the case of classical 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 increasing 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 lightcone 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.
Quantization
27
A(x+ )
A(x+ )
∆x+
t
Fig. 4.1
x+
x+
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) φ̇a (x) − L(x),
(4.18a)
∂L
and sum over all fields is implicitly addressed in index a. As
∂ φ̇a
L = 0 on-shell, using (4.12) we have
where Πa (x) =
†
H = iψ+
∂+ ψ+
m2
†
ψ+ .
= ψ+
4[iD− ]
In other words, we want the following relation to be satisfied:
n
o
†
ψ+ (x+ , x− ), ψ+
(x+ , y − ) = δ(x− − y − ).
(4.19)
(4.20)
In this stage, b(k− ) and d(k− ) become operators with the following commutation
relations between them
n
o
ˆ − ) = 0,
b̂(k− ), d(k
(4.21a)
n
o
′
′
b̂(k− ), b̂† (k−
) = 2πδ(k− − k−
),
(4.21b)
n
o
ˆ − ), dˆ† (k ′ ) = 2πδ(k− − k ′ ).
(4.21c)
d(k
−
−
To verify if our above assumptions (4.21) are in agreement with Equation (4.20),
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+ ,
n
o
†
ψ+ (x+ , x− ), ψ+
(x+ , y − )
Z ∞
Z ∞
o
′ n
dk−
−
′
−
dk−
′
=
b̂(k− ), b̂† (k−
) e−ik− x eik− y φk− (x+ )φ†k′ (x+ )
−
(2π) 0 (2π)
0
n
o
′
y−
ˆ ′ ) eik− x− e−ik−
+ dˆ† (k− ), d(k
κk− (x+ )κ†k′ (x+ )
−
−
=
Z
∞
0
Z
dk− −ik− (x− −y− )
e
φk− (x+ )φ†k− (x+ )
(2π)
{z
}
|
=1
∞
dk− ik− (x− −y− )
+
e
κk− (x+ )κ†k− (x+ )
(2π)
{z
}
|
0
=1
Z 0
Z ∞
dk− −ik− (x− −y− )
dk− −ik− (x− −y− )
e
+
e
=
(2π)
(2π)
∞
0
Z ∞
dk− −ik− (x− −y− )
=
e
(2π)
∞
= δ(x− − y − ),
where, in first equality, we have made use of our assumption in which the anticommutation 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− Z
m2
†
ψ+
=
dx− ψ+
4[iD− ]
Z
Z ∞
Z ∞
i
′ h
−
dk−
dk−
ˆ − )e−ik− x− κ† (x+ )
=
dx−
b̂† (k− )eik− x φ†k− (x+ ) + d(k
k−
(2π) 0 (2π)
0
2
′
′
m
m2
′
−ik−
x−
+
† ′
ik−
x−
+
ˆ
′
′
·
b̂(k
)e
φ
d
(k
)e
κ
(x
)
+
(x
)
k−
k−
−
−
′ − eA )
′ − eA )
4(k−
4(−k−
−
−
Z ∞
Z
Z ∞
′
′
dk−
m2
dk−
−i(k−
−k− )x− †
′
b̂ (k− )b̂(k−
)φ†k− (x+ )
=
dx−
′ − eA ) e
(2π)
(2π)
4(k
−
0
0
−
′
−
m2
†
+
+
−i(k
†
′
+
− −k− )x
ˆ
ˆ
′ (x )
′ (x ) +
d(k− )d (k− )κk− (x )κk−
φk−
′ − eA ) e
4(−k−
−
Z ∞
m2
m2
dk−
ˆ − )dˆ† (k− ) ,
b̂† (k− )b̂(k− ) +
d(k
=
(2π) 4(k− − eA− )
4(−k− − eA− )
0
(4.22)
Zero-mode issue
29
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: =
Z
0
∞
dk−
(2π)
m2
m2
†
†
ˆ
ˆ
b̂ (k− )b̂(k− ) +
d (k− )d(k− ) .
4(k− − eA− )
4(k− + eA− )
(4.23)
4.3 Zero-mode issue
As once introduced in Table 2.2, we define the state |0i such that b̂(k− ) |0i =
ˆ − ) |0i = 0. This state, which is called the vacuum of the free theory, satisfies
d(k
H0 |0i = 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 |Ωi, it should satisfy the following
relation:
H |Ωi = 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 |ii in Figure 4.2 (left panel) to be the vacuum state of the
free theory |0i in the instant form at t = 0, we expect the state to evolve with
time to a final state |f i in which Schwinger pairs are present when the background
field is switched on, i.e. |f i = ψ0 |0i + ψ1 |pairi + ψ2 |two pairsi + · · · , 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 |ii = |0i at x+ = 0 evolves with light-cone time to
the final state |f i (Figure 4.2). Taking a closer look at Equation (4.23), however,
we find out that :H: |0i = 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 i = |0i. 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.
30
LF quantization of a fermion in a background field in (1+1) dimensions
|f i
t
|f i
x−
Background Field
|ii
x+
|ii
x
t
Fig. 4.2
Time evolution of an initial state |ii to a final state |f i 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 light2
cone time, thus eA− > 0 (Figure 4.3). Therefore, 4(k−m
+eA− ) will always be positive
2
+
and 4(k−m
−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 Chapter 1, however, Schwinger showed that there should be a nonzero pair production
probability in an electric background field (Schwinger, 1951a). Therefore, this question 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 b̂ and dˆ do?
Do they rise or lower energy?
• What does
1
mean?
(k− − eA− )
eA−
eE
x+
t
Fig. 4.3
x+
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.
31
Zero-mode issue
Box 4.1
Commutation relation of Hamiltonian and the operators in our problem
h
i
h
i
2
2
†
:H: , b̂(k− ) = − 4(k−m
b̂(k
)
:H:
,
b̂
(k
)
= 4(k−m
b̂† (k− )
−
−
−eA− )
−eA− )
h
i
m2
ˆ −)
ˆ −) = −
d(k
:H: , d(k
4(k− +eA− )
h
i
:H: , dˆ† (k− ) =
m2
4(k− +eA− )
dˆ† (k− )
To see in which circumstances an operator rise or lower energy, we consider the
basic technique mentioned in the following Remark.
h
i
Remark Suppose that H |Ei = E |Ei. If H, Ô = δE Ô, then we have
h
i
H Ô |Ei =
H, Ô + Ô H |Ei
= δE Ô |Ei + E Ô |Ei
= (E + δE) Ô |Ei .
(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 b̂ in (4.26).
Z
h
i
′
:H: , b̂(k− ) =
=
=
=
Z
Z
∞
0
∞
0
∞
0
i
h
m2
dk−
′
b̂† (k− )b̂(k− ), b̂(k−
)
(2π) 4(k− − eA− )
n
o
−m2
dk−
′
b̂† (k− ), b̂(k−
) b̂(k− )
(2π) 4(k− − eA− )
dk−
−m2
′
2π δ(k− − k−
) b̂(k− )
(2π) 4(k− − eA− )
−m2
′
′ − eA ) b̂(k− ),
4(k−
−
(4.26)
where, in the third equality, we have made use of the useful Rrelation [AB, C] =
A{B, C} − {A, C}B, 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 b̂ destroys electrons and if k− < eA− (x+ ), b̂ adds something
to system; but what? It seems that for answering this question we have to look
1
. To this end, we need to digress a bit from our
for a key concept in
(k− − eA− )
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
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)
dxν
e
dpµ
= Fµν
.
(4.27)
dτ
c
dτ
In (1 + 1) dimensions, therefore, the set of equations of motion for such a particle
can be written in light-cone coordinates as below
mẍ+ = eF+− (x+ ) ẋ− ,
+
+
mẍ− = eF−+ (x ) ẋ .
(4.28a)
(4.28b)
Considering the fact that F+− ≡ ∂+ A− (x+ ) = −F−+ and that x± = 21 x∓ , this set
of equations may be re-written as
1
mẍ− = e ẋ− ∂+ A− (x+ ),
2
1
mẍ+ = −e ẋ+ ∂+ A− (x+ ),
2
d
A− (x+ ).
= −e
dτ
After doing one step integration, we get
e + + ẋ+ (τ ) = ẋ+ (0) −
A x (τ ) ,
m
(4.29a)
(4.29b)
(4.30)
which has been expressed in a completely contravariant form.
On the other hand, we know that mẋµ (τ ) = pµ and equation for the mass shell
in natural units reads as p2 = m2 . If we use the notation ẋµ (τ ) = uµ (τ ), then we
may conclude uµ uµ = 1, or in (1+1) dimensions
u+ u− = 1.
(4.31)
Plugging (4.30) in (4.31), thus we get
u− =
u+ (0)
−
e
m
1
.
A+ x+ (τ )
(4.32)
e + + 1
, if u+ (0) =
A x (τ ) , then
(k− − eA− )
m
−
−
0
3
+
u → ∞. What does it mean? We know that u = u − u 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
In this relation, which is similar to
In other words,
dx3
dx0
=−
.
dτ
dτ
(4.33)
dz
= −c.
dt
(4.34)
dx3
= −1, or
dx0
Hence, u− → ∞ means that particle is accelerated to the speed of light c. This
Summary
33
finding will help us when we study the pair production on the light-front in Chapter 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).
u−
u−
x+
t
Fig. 4.4
x+
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 background 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 components of which one is a dynamical field while the other one is a constrained field
which is determined at any x+ by the dynamical one. Furthermore, it has been
found out that in light-front framework, the vacuum state of the free theory is the
vacuum state of the interacting theory as well. In other words, light-front vacuum
is 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 becomes a creator after a certain time, which is a new ingredient compared to the
free 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. The
commutation relations of the normal-ordered Hamiltonian and the apparently creation and annihilation operators of the system has been summarized in Box 4.1.
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 the
operators in Box 4.1 and situation in which the particle is accelerated to the speed
of light c has also been discussed.
5
Discrete Light-Cone Quantization
here 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 antiperiodic 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 discussed 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
T
1
2
3
35
p+ = 0 notation has also been alternatively used in the literatures.
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., (Wilson
et al., 1994)
Periodic Boundary Condition
36
Discrete Light-Cone Quantization
further intrinsic advantages of the DLCQ method, which can be referred to. Despite 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 dynamics, 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 subsequently results in a discrete k− spectrum. Hence, the positive and negative momentum 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 (Chabysheva 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 boundary condition on our fermion field. Thus, space-time acquires the topological structure of a cylinder in DLCQ method (Figure 5.1), which means that the longitudinal
π
n
momenta will be discrete, k−
= L
(n − 12 ) (for n ∈ N). Then we make use of Fourier
decomposition again and write a general solution to the equation of motion of nonconstraint field (4.12), this time in a discrete manner. By using the anti-periodic
Quantization
37
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 excluding the mode with k− = 0 (Burkardt, 2002). Therefore, the ψ+ (x+ , x− ) function
can be written concisely as below

X
X

π
1
1
π
1
bn e−i L (n− 2 )x +
ψ+ (x+ , x− ) = √ 
d†n ei L (n+ 2 )x  .
2L n≥1
n≥0
−
−
(5.1)
It is easy to check that
(5.2)
ψ+ x+ , x− = −L = −ψ+ x+ , x− = +L .
o
o n
n
Now, bn and dn become operators and we, similarly, assume b̂n , dˆn′ = b̂†n , dˆ†n′ =
0 and guess that
o
n
b̂n , b̂†n′ = δnn′ ,
n
o
dˆn , dˆ†n′ = δnn′ .
(5.3a)
(5.3b)
x+
x−
x− = L
x− = −L
t
Fig. 5.1
Topological structure of space-time in DLCQ method when the spatial light-cone
coordinate is compactified as |x− | ≤ L.
Discrete Light-Cone Quantization
38
Again to verify if our guess and assumption give the reasonable results, we compute the equal-time anti-commutation relation for the one-component fermion field
ψ+ here,
n
o
†
ψ+ (x+ , x− ), ψ+
(x+ , y − )
o
−
′
−
1 X X n
π
π
1
1
=
b̂n , b̂†n′ e−i L (n− 2 )x ei L (n − 2 )y
2L
n≥1 n′ ≥1
1 X X n ˆ† ˆ o i π (n+ 1 )x− −i π (n′ + 1 )y−
2
2
+
e L
dn , dn′ e L
2L
n≥0 n′ ≥0
1 X −i π (−n− 1 )(x− −y− )
1 X −i π (n− 1 )(x− −y− )
2
2
+
=
e L
e L
2L
2L
=
n≥1
∞
X
n≥0
1
π
1
ei L (n+ 2 )(x
2L n=−∞
π
= ei 2L (x
−
−y − )
= δ(x− − y − ),
−
−
−y )
∞
1 X in π (x− −y− )
e L
2L n=−∞
where, in first equality, we have made use of our assumption in which the anticommutation relation of different operators vanishes. The summation in fourth
π
(x− −
equality is the series representation of Dirac’s delta function in which −π < L
−
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− =
Z
L
−L
=
X
n≥1
†
dx− ψ+
m2
ψ+
4[iD− ]
m2
π
(n − 21 ) − eA−
4 L
!
b̂†n b̂n +
X
n≥0
−m2
π
(n + 12 ) + eA−
4 L
!
dˆn dˆ†n .
(5.4)
Summary
39
After normal ordering, the Hamiltonian is given by
:H: =
X
n≥1
m2
π
4 L (n − 21 ) − eA−
!
b̂†n b̂n +
X
n≥0
m2
π
4 L (n + 12 ) + eA−
!
dˆ†n dˆn .
(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: |Ωi = 0) and,
depending on the value of A− , the same divergences occur as in the normal (noncompact) 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 problem 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
n
n
that e.g. k−
is never zero, it still looks like something can go wrong since k−
− eA−
can still vanish.
6
Tomaras–Tsamis–Woodard solution
slightly 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 contradiction 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 equation in an arbitrary background field and studying the back-reaction phenomenon
analytically. This method seems to resolve the traditional ambiguity at k− = 0.
A
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
40
Despite this claim, it has not been taken quite arbitrary, but has been considered to be monotonically increasing such that k− − eA− = 0 at most only once when the pair production
phenomenon is discussed.
Methodology
41
Table 6.1 Woodard’s light-cone coordinate and gauge conventions
Coordinatesa
+
x ≡
√1
2
0
x +x
1
−
x ≡
0
√1
2
x −x
Metric components
1
Scalar product
a µ bµ = a + b− + a − b+
Vector components
a+ = a−
g+− = g−+ = 1
Gamma matricesb
#
"
"
0
1
0
0
i
γ =
γ =
0 −1
−σ i
γ 0 γ µ† γ 0 = γ µ
in (1+1)
−−−−−−−→
dimensions
σi
0
#
"
0
γ5 = γ =
1
5
ηµν = (+1, −1)
#
−i
0
{γµ , γν } = 2ηµν
"
1
3
σ =
0
Light-front gamma matrices
γ + ≡ √12 γ 0 + γ 1
γ − ≡ √12 γ 0 − γ 1
Dirac spinor
b
ψ± ≡ P± Ψ
Derivatives
#
0
−1
γ±
Light-front spinor projection operators
P± ≡ 21 I ± γ 0 γ 1 = 12 (I ± γ5 ) = 12 γ ∓ γ ±
a
(i = 1, 2, 3)
γ5 † = γ5
Pauli matrices
"
#
"
0 1
0
1
2
σ =
σ =
i
1 0
Ψ = ψ+ + ψ−
#
1
0
a− = a+
2
∂± ≡ ∂/∂x±
=0
{γ + , γ − } = 2
Auxiliary relations
TrP± = 1
Anti-lightcone gauge
A+ = 0
Note: this table is based on conventions made in (Tomaras et al., 2001).
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 homogeneous 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.
Tomaras–Tsamis–Woodard solution
42
6.2 The model and its solution in Woodard’s
notation
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+ = −
Z
x+
du E(u) ,
(6.2)
0
with the initial value of zero. Thus, analogous to (4.10), the system of equations of
motion can be written as follows
m −
γ ψ− ,
2
m
(i∂− − eA− )ψ− = γ + ψ+ .
2
i∂+ ψ+ =
(6.3a)
(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
Z +
im x
ψ+ (x+ , x− ) = ψ+ (0, x− ) −
du γ − ψ− (u, x− ) ,
(6.4a)
2 0
+
−
ψ− (x+ , x− ) = e−ieA− (x )(x +L) ψ− (x+ , −L)
Z −
+
−
im x
dv e−ieA− (x )(x −v) γ + ψ+ (x+ , v) .
−
2 −L
(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 solutions 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 ) =
Z
i
− mγ −
2
∞
dv
−L
Z
0
x+
Z
∞
−∞
du e
dk + i(k+ +i/L)(v−x− )
e
2π
−ieA− (u)(v+L)
(
E[eA− ](0, x+ ; k + ) ψ+ (0, v)
+
+
)
E[eA− ](u, x ; k ) ψ− (u, −L) ,
(6.5)
43
The model and its solution in Woodard’s notation
t
x+
x−
x
−L
t
Fig. 6.1
The domain of solution (Tomaras et al., 2001).
ψ− (x+ , x− ) = e−ieA− (x
+
)(x− +L)
ψ− (x+ , −L) +
Z
∞
dv
Z
∞
dk +
2π
−L
−∞
(
+
−
ei(k +i/L)(v−x )
γ + ψ+ (0, v) E[eA− ](0, x+ ; k + )
× +
k − eA− (x+ ) + i/L
)
Z x+
−ieA− (u)(v+L)
+
+
du e
E[eA− ](u, x ; k ) ψ− (u, −L) ,
−im
(6.6)
0
where the functional E[eA− ] has been defined as
"
i
E[eA− ](u, x+ ; k + ) ≡ exp − m2
2
Z
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 k+ −eA−1(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.
Tomaras–Tsamis–Woodard solution
44
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 operators. As a result, the field operators obey the following anti-commutation relations
(Tomaras et al., 2001)
n
o
1
†
(6.8a)
ψ+ (x+ , x− ), ψ+
(x+ , y − ) = √ P+ δ(x− − y − ) ,
2
n
o
1
†
(6.8b)
ψ− (x+ , x− ), ψ−
(y + , x− ) = √ P− δ(x+ − y + ) .
2
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, ψe+ , is defined as follows
Z
dk + −ix− (k+ +i/L) e
ψ+ (x+ , k + ) ,
(6.9)
e
ψ+ (x+ , x− ) ≡
2π
or expressed explicitly as
(
Z ∞
+
i
+
+
i(k
+i/L)v
ψe+ (x , k ) ≡
dv e
E[eA− ](0, x+ ; k + ) ψ+ (0, v) − mγ −
2
−L
)
Z x+
du e−ieA− (u)(v+L) E[eA− ](u, x+ ; k + ) ψ− (u, −L) .
×
(6.10)
0
However, this ψe+ (x+ , k + ) operator turns out to be an eigenoperator of the lightfront Hamiltonian away from the singular point at k + = eA− (x+ ) in the largeL 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 ψe+ (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
Pair production on the light-front
45
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
s
s
1 − β ′+
1 + β ′−
−
+
p
,
p =
p ,
(6.11)
p =
1+β
1−β
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 lightcone 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.
46
Tomaras–Tsamis–Woodard solution
e+ creation region
eA−
free theory region
k+
x+
X(k+ )
e− annihilation region
t
Fig. 6.2
Particle creation (light gray) and annihilation (dark gray) regions for a specific k +
mode.
e−
t
e+
x+
x−
x
t
Fig. 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 probability of finding a positron with momentum p+ at time x+ , because, as it has been
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 ψe+ (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 + modes
which 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+ ) and
then become positron creators when k + < eA− (x+ ) due to the ever-increasing
characteristic of the function eA− (x+ ) with a positive value background electric
field. Correspondence between the time evolution of a state in light-cone gauge
and the infinite boost limit of the conventional time evolution of that state implies
that 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 instantaneous
event on the light-cone and that the newly created electron leaves the manifold
instantly. Therefore, only positrons remain on the manifold and can be seen. The
pair production probability can be computed in real time and it is turned out to be
of the form of (6.12) for 0 < p+ < eA− (x+ ).
Alternative to Tomaras–Tsamis–Woodard
solution
7
o get an insight into the different aspects of TTW solution, an alternative method can be chosen to investigate this problem. The worldline path
integral approach turns out to be an alternative way to look at pair production 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 results as TTW approach1 . Understanding this approach might help us to understand
what is going on with TTW solution and the vacuum.
T
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
Schrödinger 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 ) = hxf , tf | xi , ti i .
(7.1)
In 1948, Feynman formulated the path integral approach to non-relativistic quantum 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
#)
( Z
" Z xf
2
tf
dx
i
m
− V (x(τ ))
,
Dx(τ ) exp
dτ
hxf | e−iĤ(tf −ti )/~ |xi i =
~ ti
2 dτ
xi
(7.2)
where the symbol
1
48
R
Dx denotes the functional integral over all paths x(τ ) with end
Though the case is considered here for scalars, but that does not matter.
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 ) = xi
and 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)
Z ∞
Z x(T )=xf
RT
1 2
−m2 T
K[A](xf , tf ; xi , ti ) =
dT e
Dx(τ ) e− 0 dτ ( 4 ẋ +ieẋ·A(x(τ ))) ,
0
x(0)=xi
(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), semiclassical approach (Affleck et al., 1982), variational method (Alexandrou et al., 2000)
and numerical approach (Gies and Langfeld, 2001; Stamatescu and Schmidt, 2003;
Gies and Klingmüller, 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
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
∞
X
V
(−1)n
πnm2
2
Im Γ(E) = −
,
(7.4)
(eE)
exp −
16π 3
n2
eE
n=1
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 Im Γ ≈ 2 Im Γ .
(7.5)
The worldline path integral representation of the scalar QED one-loop effective
action turns out to be
Z
Z ∞
RT
1 2
dT −m2 T
e
Dx(τ ) e− 0 dτ ( 4 ẋ +ieẋ·A(x(τ ))) .
(7.6)
Γ[A] =
T
x(T )=x(0)
0
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 inhomogeneous 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 Klingmüller, 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
t
Fig. 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
the proper-time
performing
qR
1
2
integral and assuming a weak field approximation m 0 du ẋ ≫ 1 , one ends
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)
r
Z
√R
R
1
2π
1
du ẋ2 +ie 01 du A·ẋ
− m
0
,
(7.7)
Im Γ[A] =
Im
Dx e
m T0
where T0 is the stationary point of the proper-time integral
qR
1
du ẋ2
0
T0 =
.
m
(7.8)
Hence, this very last relation yields a new worldline action (Dunne and Schubert,
2005, 2006)
S=m
s
Z
1
du ẋ2
+ ie
0
Z
1
0
du A · ẋ ,
(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)
m qR
ẍµ
1
0
= ie Fµν ẋν .
(7.10)
du ẋ2
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)
E→0
Im Γ[A] ∼ 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 production 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 ;
52
Alternative to Tomaras–Tsamis–Woodard solution
• find the imaginary part of the effective action in the semiclassical approximation (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 the
light-front
In the previous sections we could see how the worldline approach to quantum mechanics 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 opportunity 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 Euclidean
space-time which can be considered as an alternative method to investigate the
pair-production phenomenon. The possibility of using this approach on the lightfront was proposed and briefly discussed. It seems that using such a method in the
case of light-front can shed light on different aspects of the TTW solution and the
light-front vacuum.
A
Conventions and side calculations
G
eneral 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 the
Dirac 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) =
XZ
s
Ψ† (x, t) =
XZ
s
d3 p
(2π)
3
2
d3 p
3
(2π) 2
r
r
m b(p, s) u(p, s) e−ip·x + d† (p, s) v(p, s) e+ip·x
Ep
(A.1a)
m †
b (p, s) u† (p, s) e+ip·x + d(p, s) v † (p, s) e−ip·x
Ep
Hence, for the anti-commutators of these fields we have
53
(A.1b)
Conventions and side calculations
54
Table A.1 Light-cone coordinate and gauge conventions
Coordinatesa
+
0
x ≡x +x
1
Scalar product
0
−
x ≡x −x
1
a µ bµ = a + b− + a − b+
Metric components
Vector components
g+− = g−+ = 1
a+ = a−
Gamma matricesb
#
"
"
0
0 −i
0
i
γ =
γ =
i
0
i
γ 0 γ µ† γ 0 = γ µ
in (1+1)
−−−−−−−→
dimensions
#
i
0
"
1
γ5 = γ =
0
#
0
−1
5
a− = a+
(i = 1, 2, 3)
γ5 † = γ5
ηµν = (+1, −1)
Pauli matrices
"
#
"
0 1
0
1
2
σ =
σ =
1 0
i
#
−i
0
{γµ , γν } = 2ηµν
σ3 =
"
1
0
#
0
−1
Derivatives
∂± ≡ ∂/∂x±
Light-front gamma matrices
γ+ ≡ γ0 + γ1
γ− ≡ γ0 − γ1
γ±
2
=0
Light-front spinor projection operators
Λ± =
1 ∓ ±
γ γ
4
2
(Λ± ) = Λ±
Dirac spinor
Ψ = ψ+ + ψ−
a
b
{γ + , γ − } = 2
Auxiliary relations
TrΛ± = 1
Anti-lightcone gauge
ψ± ≡ Λ± Ψ
A+ = 0
Note: this table is based on conventions made in all the chpaters, except Chapter 6.
A purely imaginary Majorana-Weyl representation of the two-dimensional Clifford algebra (Ortı́n, 2004).
n
o
Ψ̂(x, t), Ψ̂† (y, t′ )
Z
Z
r
r
′
d3 p′
m
m X n ′ ′
d3 p
× b̂(p , s ) uα (p′ , s′ ) e−ip ·x
=
3
3
′
E
E
(2π) 2
(2π) 2
p
p
s,s′
′
ˆ s) v † (p, s) e−ip·y
+ dˆ† (p′ , s′ ) vα (p′ , s′ ) e+ip ·x , b̂† (p, s) u†β (p, s) e+ip·y + d(p,
β
Z
i
d3 p m X h
†
†
−ip·(x−y)
+ip·(x−y)
=
u
(p,
s)
u
(p,
s)
e
+
v
(p,
s)
v
(p,
s)
e
α
α
β
β
(2π)3 Ep s
Z
d3 p m X h
=
uα (p, s) ūβ (p, s) γ 0 e−ip·(x−y)
(2π)3 Ep s
i
+vα (p, s) v̄β (p, s) γ 0 e+ip·(x−y)
o
Side calculations
55
Z
d3 p
(2π)3
Z
d3 p
=
(2π)3
Z
d3 p
=
(2π)3
Z
d3 p
=
(2π)3
Z
d3 p
=
(2π)3
=

=
=
1
2Ep
1
2Ep
1
2Ep
1
2Ep
i
h
(/
p + m)αβ γ 0 e−ip·(x−y) + (/p − m)αβ γ 0 e+ip·(x−y)
h
i
(/
p + m)αβ γ 0 e−ip·(x−y) − (−/p + m)αβ γ 0 e+ip·(x−y)
i
h
(γ µ pµ + m)αβ γ 0 e−ip·(x−y) − (−γ µ pµ + m)αβ γ 0 e+ip·(x−y)
i
h
(iγ µ ∂µ + m)αβ γ 0 e−ip·(x−y) − (−iγ µ ∂µ + m)αβ γ 0 e+ip·(x−y)


1  0
i(γ ∂0 +γ i ∂i ) + m
| {z }
2Ep
=0

− −i(γ 0 ∂0 +γ i ∂i ) + m
| {z }
Z
Z
=0
3
d p 1
(2π)3 2Ep
3
d p 1
(2π)3 2Ep
h
h
iγ i ∂i + m
γ i pi + m
Z
αβ
αβ
αβ
γ 0 e−ip·(x−y)
αβ

γ 0 e+ip·(x−y) 
γ 0 e−ip·(x−y) − −iγ i ∂i + m
γ 0 e−ip·(x−y) − γ i pi + m
αβ
i
d3 p 1 h −ip·(x−y)
e
− e+ip·(x−y)
3
(2π) 2Ep
Z
i
d3 p 1 h −ip·(x−y)
+ip·(x−y)
e
−
e
= (iγ i ∂i + m)αβ γ 0
(2π)3 2Ep
Z
i
d3 p 1 h −ip·(x−y)
+ip·(x−y)
= (i∂/x + m)αβ γ 0
e
−
e
(2π)3 2Ep
0
= (i∂/x + m) γ i∆(x − y) .
= (γ i pi + m)αβ γ 0
αβ
γ 0 e+ip·(x−y)
γ 0 e+ip·(x−y)
i
i
(A.2)
where ∆(x − y) denotes the Pauli-Jordan function defined as (Bjorken and Drell,
1965)
Z
i
d3 p 1 h −ip·(x−y)
+ip·(x−y)
,
(A.3)
e
−
e
∆(x − y) = −i
(2π)3 2Ep
P
p+m
and we have made use of the equalities u† = ūγ 0 ,
uα (p, s) ūβ (p, s) = /2m
αβ
s
P
p−m
/
and took into consideration that only terms
vα (p, s) v̄β (p, s) = 2m
and
αβ
s
ˆ contribute.
involving {b̂, b̂ } and {dˆ† , d}
†
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 pointing 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,
Conventions and side calculations
56
1996)
∆(0, x) = 0 ,
∂0 ∆(x0 , x)x =0 = −δ (3) (x) ,
0
∂i ∆(x0 , x)x0 =0 = 0 .
(A.4a)
(A.4b)
(A.4c)
Now we may proceed by multiplying both sides of the above equation
n
o
Ψ(x, t), Ψ† (y, t′ ) = (i∂/x + m) γ 0 i∆(x − y) ,
(A.5)
by the projection operator Λ+ from left and right
n
o
Λ+ Ψ(x, t), Ψ† (y, t′ ) Λ+ = Λ+ (i∂/x + m) γ 0 i∆(x − y)Λ+ .
|
{z
}
|
{z
}
RHS
(A.6)
LHS
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(x, t)
|
{z
}
(Λ+ Ψ(y,t′ ))†
†
†
= ψ + (x, x+ )ψ + (y, y + ) + ψ + (y, y + )ψ + (x, x+ )
n
o
†
= ψ + (x, x+ ), ψ + (y, y + ) ,
(A.7)
RHS : Λ+ (i∂/x + m)αβ γ 0 i∆(x − y)Λ+
= Λ+ (iγ 0 ∂0 + iγ i ∂i + m)αβ γ 0 i∆(x − y)Λ+
= Λ+ (−γ 0 ∂0 − γ i ∂i + im)αβ γ 0 ∆(x − y)Λ+ x0 =y0




= Λ+ −γ 0 ∂0 ∆(x − y) −γ i ∂i ∆(x − y) +im ∆(x − y)
|
{z
}
{z
}
| {z }
|
=0
−δ 3 (x−y)
=0
= Λ+ (γ 0 )2 δ 3 (x − y) Λ+
| {z }
γ 0 Λ+ x0 =y0
αβ
=1
+ 2 3
= (Λ ) δ (x − y)
= Λ+ δ 3 (x − y) .
(A.8)
Thus, we are left with the following anti-commutation relation
n
o
†
ψ + (x), ψ + (y) + + = Λ+ δ 3 (x − y) ,
x =y
(A.9)
Side calculations
57
or, in other words, we have
n
o
†
ψ + (x), ψ + (y)
= Λ+ δ(x− − y − ) δ 2 (x⊥ − y⊥ ) .
x+ =y +
(A.10)
A.2.2 The generators of Poincaré algebra for a free fermion
field
Using the Equations (3.15), (3.19) and (3.20), the generators of Poincaré algebra
for a free fermion field in (1+1) dimensions can primarily be written as follows
P+ =
P− =
1
2
1
2
K− =
1
2
=
1
4
Z
ZΣ
dx− T ++ ,
(A.11a)
dx− T +− ,
(A.11b)
Σ
+−
M
Z
Σ
dx− x− T ++ − x+ T +− .
(A.11c)
Rewriting the Lagrangian density of the free fermion field as below
L=
i
i
Ψγ + ∂ − Ψ + Ψγ − ∂ + Ψ − mΨΨ ,
2
2
(A.12)
we may start to calculate T ++ and T +− . Hence, we have
T +− = ∂ − Ψ
∂L
∂
|
+
1 −
2∂ Ψ
{z
=0
= iΨγ + ∂ − Ψ
}
∂L
∂
1 −
2∂ Ψ
∂ − Ψ − g +− L
| {z }
=0
= iΨ† γ 0 γ + ∂ − Ψ
| {z }
2Λ+
= 2iΨ† Λ+ ∂ − Ψ
2
= 2iΨ† Λ+ ∂ − Ψ
†
= 2i Λ+ Ψ ∂ − Λ+ Ψ
†
= 2iψ + ∂ − ψ +
2
† m
= 2ψ +
ψ+ .
+
i∂
(A.13)
58
Conventions and side calculations
Similarly, for T ++ we have
T ++ = ∂ + Ψ
∂L
∂
|
{z
=0
= iΨγ + ∂ + Ψ
†
+
1 −
2∂ Ψ
= 2iψ + ∂ − ψ + .
}
∂L
∂
1 −
2∂ Ψ
∂ + Ψ − g ++ L
| {z }
=0
(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).
Notes
Chapter 2
1 A set G which has a group structure and is also a n-dimensional ‘differentiable
manifold’ 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 Gx0 of G at the point x0 is
technically defined as (Isham, 1989)
Gx0 := {g ∈ G | gx0 = x0 }.
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4 If ∀a, b ∈ G1 there exist a bijection i : G1 → G2 for which
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Subject index
γ matrices, 24
light-front, 53
Majorana-Weyl representation, 24, 53
N, 36
action, 24
classical, 49
effective, 50
one-loop effective, 50, 52
worldline, 51
algebra
Lie, 10
Poincaré, 10, 21, 22, 57
anomalous magnetic moment, viii
anti-commutation relation, 38, 44, 53
antisymmetric Green function, 20
approach
numerical, 49
semiclassical, 49
string-inspired, 49
worldline, 52
approximation
semiclassical, 51
weak field, 50
bijection, 59
boost, 10, 15, 45
generator, 21, 23
boost and Galilei invariance, 18
bound states, 12
boundary condition, 1, 3, 35, 49, 52
anti-periodic, 35
periodic, 35, 36
canonical
momentum, 45
cascades, 5
Casimir effect, 3, 8
dynamical∼, 3
causality, 9
Clifford algebra, 24, 53
coherent laser beams, 6
Coleman’s theorem, 17
commutation relation
equal time, 20
equal-x+ , 20
Compton wavelength, 6, 9
conducting plates, 3
constituent picture of the hadron, 12
77
constraint
mass-shell, 15
correspondence principle, 11
covariant
theory, 12
cross section, 12
cut-off, 36
Dirac
spinor, 53
Dirac equation, 33, 40, 42, 47
Dirac’s
delta function, 29, 38
forms, 10
DLCQ, vii, 17, 35
eigenoperator, 44
electron-positron pair, 40
virtual, 5, 6
electron-positron pairs
virtual, 4
energy levels of hydrogen, viii
energy-momentum tensor, 10, 22
equation
∼ of motion, 25
Dirac, 33, 40, 42, 47
Euler-Lagrange, 25
Klein-Gordon, 49
equation of motion, 22
Euler-Lagrange equation, 19
excitation, 35
excited states, 15
existence, 9
expectation value, 41
external disturbance, 1, 8
Faddeev and Jackiw method, 19, 20
Fermi-Dirac statistics, 52
Feynman slash notation, 21
field
background, vii, 5, 15, 24, 32, 40, 49–51
componen, 25
constrained, 23, 25, 33
critical, 6
dynamical, 23, 25, 33
electric, 6, 30
electromagnetic, 6, 7
external, 1, 6, 8, 40, 49
fermion, 28, 33, 36, 40
Subject index
78
dynamical, 22, 23
free fermion, 20, 21, 57
free scalar, 20
gauge, 40
homogeneous, 40
inhomogeneous, 50
intensity, 7
non-constraint, 36
quantization, vii
real scalar, 19
strong, 2
tensor, 30
field theory
classical, 31
quantum, 1, 9, 15, 35, 40
conventional, 12
null-plane, 17
Fock
representation, 21
space expansion, 20, 22
foliation
of space-time, 9, 11, 14, 18
form
front, vii, viii, 11, 12, 14, 18, 22, 29, 36
instant, 10–12, 15, 18, 19, 22, 29
point, 11, 12, 18
Fourier
decomposition, 36
space, 44
transform, 26, 44
functional integral, 48, 52
gamma matrices, 41, 54
gauge
anti-lightcone, 24, 33, 42, 52
conventions, 53
covariant derivative, 24
fixing, 9
generator
dynamical, 11
kinematical, 11
ground state, 12
group
action
transitive, 11, 59
combination law, 59
Euclidean, 12
generator, 10
isomorphic, 59
isospin, 10
Lie, 59
Lorentz, 10
Poincaré, 10
generator, 11, 12, 21
stability, 11, 12, 59
theory, 10
Hamiltonian
dynamics, vii, 12, 15, 18, 36
normal-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, 11
equal-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, 2
light and matter interaction, viii
light bullets formation, 5
light-cone
coordinates, 14, 29, 32, 36, 53
dynamics, 15
energy, 14
framework, 15
momentum, 14, 17
quantization, 36
space, 14, 40
time, 14, 30, 33, 40
velocity, 14
light-fron
framework, 52
light-front, vii, 48
γ matrices, 53
ambiguity, viii, 35
dynamics, 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, 9
Lorentz
force, 32
invariant, 12, 21
transformation, 10, 14, 45
manifold, 45, 47
differentiable, 59
many body system
Subject index
79
non-relativistic, 15
relativistic, 15
many-body theory, 9
mass-energy equivalence, 1
microscopic causality, 20
mode function, 41
momentum
angular, 10
canonical, 17, 27
conjugate, 19, 27
four-∼, 10
kinetic, 15, 17, 45
light-cone, 17
operator, 21
physical, 45
non-perturbative, 35
solution, 35
non-relativistic
limit, 11
many body system, 15
quantum mechanics, 48
normal modes, 3
on-shell, 27
one-loop effective, 50
operator
annihilation, 45
creation, 20, 45
Hamiltonian, 21, 23
Hermitian, 27
initial value, 40
momentum, 21
non-singular, 41
projection, 22, 24
oscillating integral, 49
pair creation, 6, 49
pair production, 40, 48, 50, 52
probability, 50, 51
rate, 50
spontaneous, 6
stimulated, 5
trident, 5
particle
accelerated, 32
anti∼, 9
identical, 9
on-mass shell, 15, 36
point, 9
relativistic, 32
virtual, 1, 50
path integral, 48, 50
path integral approach, 48
Pauli-Jordan function, 55
phenomenon
back reaction, 40
nonlinear vacuum ∼, vii
pair production, vii, 44, 50
photon acceleration, 5
photon splitting, 5
plane wave, 12
plasma oscillation, 5
predominant path, 50, 51
predominant trajectory, 50
propagator, 48, 49
proper time, 50
integral, 50, 51
QED, viii
non-perturbative, 2
strong-field∼, 8
QED(1+1) , 35
quantization
canonical, 20, 27
free field, 19
light-front, 23, 24, 40
non-standard, 23
null-plane, 17
quantum
electrodynamics, 49
system, 48
vacuum, viii
quantum field theory
abelian, 36
non-abelian, 36
quantum mechanics, 7, 9, 48
non-relativistic, 48
regularization
infrared, 36
relativistic
dispersion relation, 15
dynamics, viii, 16
invariance, 10
many body system, 15
many-body theory, 9
particle, 49
particle point, 9
scalar particle, 49
speed, 9
renormalization
perturbative, 36
representation
functional integral, 49
irreducible, 10
momentum, 35
path integral, 48, 50
rest-mass energy, 6
Sauter-Schwinger limit, vii, 6, 7
scattering
nonlinear Compton, 5
photon-photon, 4
Schrödinger picture, 48
Schwinger
mechanism, 7
pair production, vii, viii, 7, 8
Subject index
80
pairs, viii, 15, 17, 18, 29, 33
proper time parameter, 49
Schwinger’s
quantum action principle, 19
vacuum persistence amplitude, 44
self-lensing effects, 5
semiclassical approach, 48
singularity, 26, 34, 36
space
configuration, 48
Euclidean, 59
n-dimensional, 59
Fock, 35
momentum, 35
space-time, 36, 48
Euclidean, 49, 52
loop, 50
volume, 50
special relativity, viii, 7, 9
spectrum
discrete k− , 36
mass, 35
spin factor, 52
spinor loop, 52
squirming path, 50
string theory, 9
string/M-theory, 35
strong-field
processes, 2
theory, vii
structure
topological, 36
supergravity, 35
superposition, 4
supersymmetry, 35
symmetry, 10
Tamm-Dancoff method, 35
Tomaras-Tsamis-Woodard (TTW) solution,
vii, 40, 48, 52
alternative, 48
topological structure of space-time, 37
topology, 11
trajectory, 49
transition probability amplitude, 48
uncertainly principle, 1
uniqueness, 9
vacuum, 36, 48, 49
birefringence, 3
fluctuation, 1, 3–5, 8
free theory, 29
interacting theory, 29
light-front, vii, 15, 17, 19, 33, 38, 40, 52
nonlinear, 2, 8
nonlinear nature, 1
physical, 29
polarization, 6
quantum, vii, 2, 4, 5
structure, viii
stable, 17, 29
state, 17
trivial, 17, 29, 39
vacuum birefringence, 8
variational method, 49
velocity
group∼, 5
virtual
photon, 3
wave function, 35
worldline
action, 51
approach, 52
instanton, 49, 51
loop equation of motion, 51
path integral approach, 48
worldsheet, 9
Yukawa theory, 35
zero body problem, 1
zero-mode, vii, 36, 39
excitations, 17
issue, vii, 17, 29, 38, 39