* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download Pair Production and the Light-front Vacuum
Atomic theory wikipedia , lookup
Double-slit experiment wikipedia , lookup
Quantum key distribution wikipedia , lookup
Many-worlds interpretation wikipedia , lookup
Bell's theorem wikipedia , lookup
Particle in a box wikipedia , lookup
Higgs mechanism wikipedia , lookup
Matter wave wikipedia , lookup
Orchestrated objective reduction wikipedia , lookup
Aharonov–Bohm effect wikipedia , lookup
Hydrogen atom wikipedia , lookup
Quantum state wikipedia , lookup
Coherent states wikipedia , lookup
Interpretations of quantum mechanics wikipedia , lookup
EPR paradox wikipedia , lookup
Dirac equation wikipedia , lookup
Quantum chromodynamics wikipedia , lookup
Symmetry in quantum mechanics wikipedia , lookup
Casimir effect wikipedia , lookup
Yang–Mills theory wikipedia , lookup
Path integral formulation wikipedia , lookup
Theoretical and experimental justification for the Schrödinger equation wikipedia , lookup
Topological quantum field theory wikipedia , lookup
Quantum field theory wikipedia , lookup
Wave–particle duality wikipedia , lookup
Quantum electrodynamics wikipedia , lookup
Renormalization group wikipedia , lookup
Renormalization wikipedia , lookup
Hidden variable theory wikipedia , lookup
Relativistic quantum mechanics wikipedia , lookup
History of quantum field theory wikipedia , lookup
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 }. 3 If ∀x, y ∈ X, ∃g ∈ G : gx = y, the action of group G on set X is called transitive (Isham, 1989). 4 If ∀a, b ∈ G1 there exist a bijection i : G1 → G2 for which i(ab) = i(a).i(b) , the map i is called an isomorphism of G1 with G2 , where · denotes the combination law in group G2 (Isham, 1989). 59 References Adler, S. L. 1971. Photon Splitting and Photon Dispersion in a Strong Magnetic Field. Annals of Physics, 67, 599–647. 5 Affleck, I. K., Alvarez, O., and Manton, N. S. 1982. Pair Production at Strong Coupling in Weak External Fields. Nuclear Physics B, 197(Apr), 509–519. 40, 48, 49 Aharony, O., and Berkooz, M. 1999. IR dynamics of d = 2, N = (4, 4) gauge theories and DLCQ of “little string theories”. Journal of High Energy Physics, 1999(Oct), 030. 35 Akhiezer, A. I., and Berestetskii, V. B. 1965. Quantum Electrodynamics. Interscience Monographs and Texts in Physics and Astronomy, V.11. New York: John Wiley & Sons. 22 Alexandrou, C., Rosenfelder, R., and Schreiber, A. W. 2000. Nonperturbative Mass Renormalization in Quenched QED from the Worldline Variational Approach. Phys. Rev. D, 62(Sep), 085009. 49 Alkofer, R., Hecht, M. B., Roberts, C. D., Schmidt, S. M., and Vinnik, D. V. 2001. Pair Creation and an X-Ray Free Electron Laser. Phys. Rev. Lett., 87(Oct), 193902. 7 Allor, D., Cohen, T. D., and McGady, D. A. 2008. Schwinger Mechanism and Graphene. Phys. Rev. D, 78(Nov), 096009. 7 Antonuccio, F., Pauli, H.-C., Pinsky, S., and Tsujimaru, S. 1998a. DLCQ bound states of N = (2, 2) super-Yang-Mills theory at finite and large N . Phys. Rev. D, 58(Nov), 125006. 35 Antonuccio, F., Lunin, O., Pinsky, S., Pauli, H.-C., and Tsujimaru, S. 1998b. DLCQ spectrum of N = (8, 8) super Yang-Mills theory. Phys. Rev. D, 58(Oct), 105024. 35 Antonuccio, F., Lunin, O., and Pinsky, S. 1998c. On exact supersymmetry in DLCQ. Physics Letters B, 442(1-4), 173–179. 35 Antonuccio, F., Filippov, I., Haney, P., Lunin, O., Pinsky, S., Trittmann, U., and Hiller, J. 1999. Supersymmetry and DLCQ. Pages 165–179 of: Strobel, G., and Mack, D. (eds), Proceedings of Workshop on Transition from low to high Q form factors. Athens, Georgia, Sep. 17, 1999. 35, 36 Archibald, J., Gleisberg, T., Höche, S., Krauss, F., Schönherr, M., Schumann, S., Siegert, F, and Winter, J.-C. 2008. Simulation of Photon-Photon Interactions in Hadron Collisions with Sherpa. Nuclear Physics B - Proceedings Supplements, 179–180(0), 218–225. Proceedings of the International Workshop on High-Energy Photon Collisions at the LHC. 4 60 61 References Arfken, G. B., and Weber, H. J. 2005. Mathematical Methods for Physicists. Massachusetts: Elsevier Science. 10 Artru, X., and Czyzewski, J. 1998. Transverse Polarization of Quark Pairs Created in String Fragmentation. Acta Physica Polonica B, 29(8), 2115–2127. 41 Bamber, C., Boege, S. J., Koffas, T., Kotseroglou, T., Melissinos, A. C., Meyerhofer, D. D., Reis, D. A., Ragg, W., Bula, C., McDonald, K. T., Prebys, E. J., Burke, D. L., Field, R. C., Horton-Smith, G., Spencer, J. E., Walz, D., Berridge, S. C., Bugg, W. M., Shmakov, K., and Weidemann, A. W. 1999. Studies of Nonlinear QED in Collisions of 46.6 GeV Electrons with Intense Laser Pulses. Phys. Rev. D, 60(Oct), 092004. 5 Banks, T., Fischler, W., Shenker, S. H., and Susskind, L. 1997. M theory as a matrix model: A conjecture. Phys. Rev. D, 55(Apr), 5112–5128. 35 Belgiorno, F., Cacciatori, S. L., Clerici, M., Gorini, V., Ortenzi, G., Rizzi, L., Rubino, E., Sala, V. G., and Faccio, D. 2010. Hawking Radiation from Ultrashort Laser Pulse Filaments. Phys. Rev. Lett., 105(Nov), 203901. 7 Belgiorno, F., Cacciatori, S. L., Clerici, M., Gorini, V., Ortenzi, G., Rizzi, L., Rubino, E., Sala, V. G., and Faccio, D. 2011. Belgiorno et al. Reply:. Phys. Rev. Lett., 107(Sep), 149402. 7 Bern, Z., and Kosower, D. A. 1992. The Computation of Loop Amplitudes in Gauge Theories. Nuclear Physics B, 379, 451–561. 49 Best, C., and Eisenberg, J. M. 1993. Pair creation in transport equations using the equal-time Wigner function. Phys. Rev. D, 47(May), 4639–4646. 40 Bethe, H. A. 1947. The Electromagnetic Shift of Energy Levels. Phys. Rev., 72(Aug), 339–341. viii Bialynicki-Birula, I., Górnicki, P., and Rafelski, J. 1991. Phase-space structure of the Dirac vacuum. Phys. Rev. D, 44(Sep), 1825–1835. 40 Bigatti, D., and Susskind, L. 1998. A Note on Discrete Light Cone Quantization. Physics Letters B, 425(3–4), 351–353. 36 Binosi, D., and Theußl, L. 2004. JaxoDraw: A graphical user interface for drawing Feynman diagrams. Computer Physics Communications, 161(1–2), 76–86. x Binosi, D., Collins, J., Kaufhold, C., and Theußl, L. 2009. JaxoDraw: A graphical user interface for drawing Feynman diagrams. Version 2.0 release notes. Computer Physics Communications, 180(9), 1709–1715. x Bjorken, J. D. 1969. Asymptotic Sum Rules at Infinite Momentum. Phys. Rev., 179(Mar), 1547–1553. 12 Bjorken, J. D., and Drell, S. D. 1965. Relativistic Quantum Fields. New York: McGraw-Hill Book Company. 41, 53, 55 Boca, M., and Florescu, V. 2009a. Non-linear Compton Scattering Effect with a Laser Pulse. Journal of Physics: Conference Series, 194, 032003. 5 Boca, M., and Florescu, V. 2009b. Nonlinear Compton Scattering with a Laser Pulse. Phys. Rev. A, 80(Nov), 053403. 5 Bordag, M., Mohideen, U., and Mostepanenko, V. M. 2001. New Developments in the Casimir Effect. Physics Reports, 353(Oct), 1–205. 3 62 References Brezin, E., and Itzykson, C. 1970. Pair Production in Vacuum by an Alternating Field. Phys. Rev. D, 2(Oct), 1191–1199. 40 Brodin, G., Marklund, M., and Stenflo, L. 2001. Proposal for Detection of QED Vacuum Nonlinearities in Maxwell’s Equations by the Use of Waveguides. Phys. Rev. Lett., 87(Oct), 171801. 4 Brodin, G., Stenflo, L., Anderson, D., Lisak, M., Marklund, M., and Johannisson, P. 2003. Light Bullets and Optical Collapse in Vacuum. Physics Letters A, 306(4), 206–210. 5 Brodin, G., Eriksson, D., and Marklund, M. 2006. Graviton Mediated PhotonPhoton Scattering in General Relativity. Phys. Rev. D, 74(Dec), 124028. 4 Brodsky, S. J. 1988 (Mar). Topics in Nuclear Chromodynamics: Color Transparency and Hadronization in the Nucleus. SLAC-PUB-4551. Invited talk presented at the Third Lake Louise Winter Institute on QCD: Theory and Experiment, Chateau Lake Louise, Alberta, Canada, March 6-12, 1988. 35 Brodsky, S. J. 1998. Light Cone Quantized QCD and Novel Hadron Phenomenology. Pages 1–64 of: C.-R. Ji, D.-P. Min (ed), QCD, light cone physics and hadron phenomenology: Proceedings. Singapore: World Scientific. 10th Summer School and Symposium on Nuclear Physics (NuSS 97), Seoul, Korea, Jun 23-28, 1997. 12, 36 Brodsky, S. J., and Pauli, H.-C. 1991. Light-Cone Quantization of Quantum Chromodynamics. Pages 51–121 of: Mitter, H., and Gausterer, H. (eds), Recent Aspects of Quantum Fields. Lecture Notes in Physics, vol. 396. Springer Berlin Heidelberg. 12, 35, 36 Brodsky, S. J., McCartor, G., Pauli, H.-C., and Pinsky, S. S. 1993. The challenge of light-cone quantization of gauge field theory. Particle World, 3, 109–124. 36 Brodsky, S. J., Pauli, H.-C., and Pinsky, S. S. 1998. Quantum Chromodynamics and other Field Theories on the Light Cone. Physics Reports, 301(Aug), 299–486. 17, 19, 35, 36 Buks, E., and Roukes, M. L. 2001. Stiction, Adhesion Energy, and the Casimir Effect in Micromechanical Systems. Phys. Rev. B, 63(Jan), 033402. 3 Bula, C., McDonald, K. T., Prebys, E. J., Bamber, C., Boege, S., Kotseroglou, T., Melissinos, A. C., Meyerhofer, D. D., Ragg, W., Burke, D. L., Field, R. C., Horton-Smith, G., Odian, A. C., Spencer, J. E., Walz, D., Berridge, S. C., Bugg, W. M., Shmakov, K., and Weidemann, A. W. 1996. Observation of Nonlinear Effects in Compton Scattering. Phys. Rev. Lett., 76(Apr), 3116–3119. 5 Bulanov, Sergei V., Esirkepov, Timur, and Tajima, Toshiki. 2003. Light Intensification towards the Schwinger Limit. Phys. Rev. Lett., 91(Aug), 085001. 2 Burkardt, M. 1989. The virial theorem and the structure of the deuteron in 1 + 1-dimensional QCD on the light cone. Nuclear Physics A, 504(Nov), 762–776. 35 Burkardt, M. 1993. Light-front quantization of the sine-Gordon model. Phys. Rev. D, 47(May), 4628–4633. 35 Burkardt, M. 2002. Light Front Quantization. Pages 1–74 of: Negele, J. W., and 63 References Vogt, E. (eds), Advances in Nuclear Physics. Advances in Nuclear Physics, vol. 23. Springer US. 17, 37 Carmichael, R. D. 1956. Introduction to the Theory of Groups of Finite Order. New York: Dover. 10 Casher, A. 1976. Gauge fields on the null plane. Phys. Rev. D, 14(Jul), 452–464. 36 Casher, A., Neuberger, H., and Nussinov, S. 1979. Chromoelectric-flux-tube model of particle production. Phys. Rev. D, 20(Jul), 179–188. 40 Casimir, H. B. G. 1948. On the Attraction Between Two Perfectly Conducting Plates. Proc. K. Ned. Akad. Wet., 51(North-Holland Publishing Company), 793– 795. 3 Casimir, H. B. G., and Polder, D. 1948. The Influence of Retardation on the London-van der Waals Forces. Phys. Rev., 73(Feb), 360–372. 3 Chabysheva, S. S., and Hiller, J. R. 2009. Zero Momentum Modes in Discrete Light-Cone Quantization. Phys. Rev. D, 79(May), 096012. 36 Chaichian, M., and Demichev, A. 2001. Path Integrals in Physics: Volume I Stochastic Processes and Quantum Mechanics. Institute of physics series in mathematical and computational physics. IOP Publishing Ltd. 49 Chan, H. B., Aksyuk, V. A., Kleiman, R. N., Bishop, D. J., and Capasso, F. 2001. Quantum Mechanical Actuation of Microelectromechanical Systems by the Casimir Force. Science, 291(5510), 1941–1944. 3 Chang, S.-J., Root, R. G., and Yan, T.-M. 1973. Quantum Field Theories in the Infinite-Momentum Frame. I. Quantization of Scalar and Dirac Fields. Phys. Rev. D, 7(Feb), 1133–1146. 21, 22 Chernodub, M. N. 2012. Vacuum Superconductivity, Conventional Superconductivity and Schwinger Pair Production. International Journal of Modern Physics A, 27(15), 1260003. 6 Chervyakov, A., and Kleinert, H. 2009. Exact Pair Production Rate for a Smooth Potential Step. Phys. Rev. D, 80(Sep), 065010. 7 Chervyakov, A., and Kleinert, H. 2011 (Dec). On Electron-Positron Pair Production by a Spatially Nonuniform Electric Field. arXiv:1112.4120 [hep-th]. 7 Coester, F. 1992. Null-Plane Dynamics of Particles and Fields. Progress in Particle and Nuclear Physics, 29, 1–32. 12 Coleman, S. 1966. The Invariance of the Vacuum is the Invariance of the World. J. Math. Phys., 7(May), 787. 17 Collins, J. C. 1997 (May). Light-cone Variables, Rapidity and All That. arXiv:hep-ph/9705393. 15 Compton, A. H. 1923. A Quantum Theory of the Scattering of X-rays by Light Elements. Phys. Rev., 21(May), 483–502. 6 Corradini, O. 2012. Quantum Mechanical Path Integrals: from Transition Amplitudes to Worldline Formalism. School on Spinning Particles in Quantum Field Theory: Worldline Formalism, Higher Spins, and Conformal Geometry, Morelia, Michoacán, Mexico, Nov 19-23, 2012. http://indico.cern.ch/getFile. py/access?resId=10&materialId=4&confId=206621. 52 64 References Damour, T., and Ruffini, R. 1975. Quantum Electrodynamical Effects in KerrNewmann Geometries. Phys. Rev. Lett., 35(Aug), 463–466. 40 Dancoff, S. M. 1950. Non-Adiabatic Meson Theory of Nuclear Forces. Phys. Rev., 78(May), 382–385. 35 Dashen, R., and Gell-Mann, M. 1966. Representation of Local Current Algebra at Infinite Momentum. Phys. Rev. Lett., 17(Aug), 340–343. 12 Deriagin, B. V., and Abrikosova, I. I. 1957a. Direct measurement of the molecular attraction of solid bodies. 1. Statement of the problem and method of measuring forces by using negative feedback. Sov. Phys. JETP, 3, 819–829. 3 Deriagin, B. V., and Abrikosova, I. I. 1957b. Direct measurement of the molecular attraction of solid bodies. 2. Method for measuring the gap. Results of experiments. Sov. Phys. JETP, 4, 2–10. 3 Di Piazza, A. 2004. Pair Production at the Focus of Two Equal and Oppositely Directed Laser Beams: The Effect of the Pulse Shape. Phys. Rev. D, 70(Sep), 053013. 7 Di Piazza, A., Müller, C., Hatsagortsyan, K. Z., and Keitel, C. H. 2012a. Extremely High-Intensity Laser Interactions with Fundamental Quantum Systems. Rev. Mod. Phys., 84(Aug), 1177–1228. 2 Di Piazza, A., King, B., Hatsagortsyan, K., and Keitel, C. H. 2012b (Sep). Testing Strong-Field CED and QED with Intense Laser Fields. http://www.mpq.mpg. de/APS/Frontiers/vortragende/dipiazza.pdf. Invited talk at Frontiers in Intense Laser-Matter Interaction Theory Symposium, Garching, Germany, September 2012. 2 Dietrich, D. D., and Dunne, G. V. 2007. Gutzwiller’s Trace Formula and Vacuum Pair Production. Journal of Physics A: Mathematical and Theoretical, 40, F825– F830. 48, 49 Dirac, P. A. M. 1927. The Quantum Theory of the Emission and Absorption of Radiation. Proc. R. Soc. Lond. A, 114(Mar), 243–265. viii Dirac, P. A. M. 1945. On the Analogy Between Classical and Quantum Mechanics. Rev. Mod. Phys., 17(Apr), 195–199. 48 Dirac, P. A. M. 1949. Forms of Relativistic Dynamics. Rev. Mod. Phys., 21(Jul), 392–399. 11, 12 Dirac, P. A. M. 1950. Generalized Hamiltonian Dynamics. Canad. J. Math., 2(Feb), 129–148. 11, 12 Dumlu, C. K., and Dunne, G. V. 2011a. Complex Worldline Instantons and Quantum Interference in Vacuum Pair Production. Phys. Rev. D, 84(Dec), 125023. 48, 49 Dumlu, C. K., and Dunne, G. V. 2011b. Interference Effects in Schwinger Vacuum Pair Production for Time-dependent Laser Pulses. Phys. Rev. D, 83(Mar), 065028. 7 Dunne, G. V. 2009. New Strong-Field QED Effects at Extreme Light Infrastructure. The European Physical Journal D, 55, 327–340. 6 Dunne, G. V., and Schubert, C. 2005. Worldline Instantons and Pair Production in Inhomogeneous Fields. Phys. Rev. D, 72(Nov), 105004. 7, 48, 49, 51 65 References Dunne, G. V., and Schubert, C. 2006. Pair Creation in Inhomogeneous Fields from Worldline Instantons. AIP Conf. Proc., 857, 240–248. 48, 49, 51 Dunne, G. V., and Wang, Q.-h. 2006. Multidimensional Worldline Instantons. Phys. Rev. D, 74(Sep), 065015. 48, 49 Dunne, G. V., Wang, Q.-h., Gies, H., and Schubert, C. 2006. Worldline Instantons and the Fluctuation Prefactor. Phys. Rev. D, 73(Mar), 065028. 49 Dunne, G. V., Gies, H., and Schützhold, R. 2009. Catalysis of Schwinger Vacuum Pair Production. Phys. Rev. D, 80(Dec), 111301. 7 Elkina, N. V., Fedotov, A. M., Kostyukov, I. Yu., Legkov, M. V., Narozhny, N. B., Nerush, E. N., and Ruhl, H. 2011. QED Cascades Induced by Circularly Polarized Laser Fields. Phys. Rev. ST Accel. Beams, 14(May), 054401. 5 Eller, T., and Pauli, H.-C. 1989. Quantizing QED2 on the light-cone. Zeitschrift für Physik C Particles and Fields, 42(Mar), 59–67. 35 Eller, T., Pauli, H.-C., and Brodsky, S. J. 1987. Discretized light-cone quantization: The massless and the massive Schwinger model. Phys. Rev. D, 35(Feb), 1493– 1507. 35 Eriksson, D., Brodin, G., Marklund, M., and Stenflo, L. 2004. Possibility to Measure Elastic Photon-Photon Scattering in Vacuum. Phys. Rev. A, 70(Jul), 013808. 4 Faddeev, L., and Jackiw, R. 1988. Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett., 60(Apr), 1692–1694. 19, 20 Fedotov, A. M., and Narozhny, N. B. 2007. Generation of Harmonics by a Focused Laser Beam in the Vacuum. Physics Letters A, 362, 1–5. 4 Fedotov, A. M., Narozhny, N. B., Mourou, G., and Korn, G. 2010a. Erratum: Limitations on the Attainable Intensity of High Power Lasers [Phys. Rev. Lett. 105, 080402 (2010)]. Phys. Rev. Lett., 105(Nov), 199901. 5 Fedotov, A. M., Narozhny, N. B., Mourou, G., and Korn, G. 2010b. Limitations on the Attainable Intensity of High Power Lasers. Phys. Rev. Lett., 105(Aug), 080402. 5 Feynman, R. P. 1948. Space-Time Approach to Non-Relativistic Quantum Mechanics. Rev. Mod. Phys., 20(Apr), 367–387. 48 Feynman, R. P. 1950. Mathematical Formulation of the Quantum Theory of Electromagnetic Interaction. Phys. Rev., 80(Nov), 440–457. 48, 49 Feynman, R. P. 1951. An Operator Calculus Having Applications in Quantum Electrodynamics. Phys. Rev., 84(Oct), 108–128. 48, 49 Feynman, R. P. 1972. Photon-Hadron Interactions. California: Addison-Wesley Publishing Company, Inc. 12 Feynman, R. P., and Hibbs, A. R. 1965. Quantum Mechanics and Path Integrals. New York: McGraw-Hill Companies, Inc. 49 Fleming, G. N. 1991. The Vacuum on Null Planes. Pages 111–135 of: Saunders, S. W., and Brown, H. R. (eds), The Philosophy of Vacuum. UK: Clarendon Press. 11, 12, 15, 17 Fliegner, D., Haberl, P., Schmidt, M. G., and Schubert, C. 1998. The Higher Derivative Expansion of the Effective Action by the String Inspired Method, II. Annals of Physics, 264, 51–74. 49 66 References Fock, V. A. 1932. Konfigurationsraum und zweite Quantelung. Zeitschrift für Physik, 75, 622–647. 20 Foley, H. M., and Kusch, P. 1948. On the Intrinsic Moment of the Electron. Phys. Rev., 73(Feb), 412–412. viii Fradkin, E. S., Gitman, D. M., and Shvartsman, S. M. 1991. Quantum Electrodynamics with Unstable Vacuum. Berlin: Springer-Verlag. 40 Franke, V. A., Paston, S. A., and Prokhvatilov, E. V. 2006. QED(1+1) on the light front and its implications for semiphenomenological methods in QCD(3+1). In: Proceedings of Quarks 2006. 14th International Seminar on High Energy Physics: Quarks 2006, St. Petersburg, Russia, 19-25 May, 2006. 36 Fubini, S., and Furlan, G. 1965. Renormalization Effects for Partially Conserved Currents. Physics, 1, 229–247. 12, 14 Gabrielse, G., Hanneke, D., Kinoshita, T., Nio, M., and Odom, B. 2006. New Determination of the Fine Structure Constant from the Electron g Value and QED. Phys. Rev. Lett., 97(Jul), 030802. viii Gabrielse, G., Hanneke, D., Kinoshita, T., Nio, M., and Odom, B. 2007. Erratum: New Determination of the Fine Structure Constant from the Electron g Value and QED [Phys. Rev. Lett. 97, 030802 (2006)]. Phys. Rev. Lett., 99(Jul), 039902. viii Garriga, J., Kanno, S., Sasaki, M., Soda, J., and Vilenkin, A. 2012. Observer Dependence of Bubble Nucleation and Schwinger Pair Production. Journal of High Energy Physics, 2012, 006. 6 Gavrilov, S. P., and Gitman, D. M. 1996. Vacuum instability in external fields. Phys. Rev. D, 53(Jun), 7162–7175. 40 Gies, H. 2008. External Fields as a Probe for Fundamental Physics. J. Phys. A: Math. Theor., 41, 164039. 2 Gies, H. 2009. Strong Laser Fields as a Probe for Fundamental Physics. The European Physical Journal D, 55, 311–317. 2 Gies, H., and Klingmüller, K. 2005. Pair Production in Inhomogeneous Fields. Phys. Rev. D, 72(Sep), 065001. 49, 50 Gies, H., and Langfeld, K. 2001. Quantum Diffusion of Magnetic Fields in a Numerical Worldline Approach. Nuclear Physics B, 613, 353–365. 49 Gonoskov, A., Gonoskov, I., Harvey, C., Ilderton, A., Kim, A., Marklund, M., Mourou, G., and Sergeev, A. M. 2013 (Feb). Probing Nonperturbative QED with Optimally Focused Laser Pulses. arXiv:1302.4653 [hep-ph]. 7 Greiner, W., and Reinhardt, J. 1996. Field Quantization. Berlin [etc.]: SpringerVerlag. 55 Greiner, W., Müller, B., and Rafelski, J. 1985. Quantum Electrodynamics of Strong Fields. Berlin: Springer-Verlag. 40 Harindranath, A. 1997. An Introduction to Light Front Dynamics for Pedestrians. In: J. P. Vary, F. Wolz (ed), Light-Front Quantization and Non-Perturbative QCD. International Institute of Theoretical and Applied Physics, ISU, Ames, IA 50011, U.S.A., 6 May-2 Jun, 1996. 10, 19, 21, 36 67 References Harindranath, A. 2000. An Introduction to Light Front Field Theory and Light Front QCD. Lecture Notes, Saha Institute of Nuclear Physics, Calcutta. http://www. public.iastate.edu/~aharindr/lect.html. 15, 20, 22 Harindranath, A., and Vary, J. P. 1987. Solving two-dimensional ϕ4 theory by discretized light-front quantization. Phys. Rev. D, 36(Aug), 1141–1147. 35 Harindranath, A., and Vary, J. P. 1988a. Light-front Hamiltonian approach to relativistic two- and three-body bound-state problems in 1+1 dimensions. Phys. Rev. D, 37(Feb), 1064–1069. 35 Harindranath, A., and Vary, J. P. 1988b. Stability of the vacuum in scalar field models in 1+1 dimensions. Phys. Rev. D, 37(Feb), 1076–1078. 35 Harindranath, A., and Vary, J. P. 1988c. Variational calculation of the spectrum of two-dimensional ϕ4 theory in light-front field theory. Phys. Rev. D, 37(May), 3010–3013. 35 Harvey, C., Heinzl, T., and Ilderton, A. 2009. Signatures of high-intensity Compton scattering. Phys. Rev. A, 79(Jun), 063407. 5 Hawking, S. W. 1974. Black Hole Explosions? Nature, 248(March), 30–31. 7 Hawking, S. W. 1975. Particle Creation by Black Holes. Communications in Mathematical Physics, 43, 199–220. 7 Hebenstreit, F., Alkofer, R., and Gies, H. 2008. Pair Production Beyond the Schwinger Formula in Time-Dependent Electric Fields. Phys. Rev. D, 78(Sep), 061701. 7 Hebenstreit, F., Alkofer, R., Dunne, G. V., and Gies, H. 2009. Momentum Signatures for Schwinger Pair Production in Short Laser Pulses with a Subcycle Structure. Phys. Rev. Lett., 102(Apr), 150404. 7 Hebenstreit, F., Ilderton, A., and Marklund, M. 2011a. Pair Production: The View From the Lightfront. Phys. Rev. D, 84(Dec), 125022. 7 Hebenstreit, F., Ilderton, A., Marklund, M., and Zamanian, J. 2011b. Strong Field Effects in Laser Pulses: The Wigner Formalism. Phys. Rev. D, 83(Mar), 065007. 7 Heinzl, T. 1998 (Dec). Light-Cone Dynamics of Particles and Fields. Habilitation thesis, Faculty of Physics, University of Regensburg, Regensburg, Germany. arXiv:hep-th/9812190. 9 Heinzl, T. 2001. Light-Cone Quantization: Foundations and Applications. Pages 55–142 of: Latal, H., and Schweiger, W. (eds), Methods of Quantization. Lecture Notes in Physics, vol. 572. Springer Berlin Heidelberg. 9, 10, 11, 12, 19, 20, 22, 29 Heinzl, T., and Ilderton, A. 2008 (Jul). Extreme Field Physics and QED. arXiv:0809.3348 [hep-ph]. Presented at Vulcan 10 PW upgrade meeting, Steventon, UK, July 16, 2008. 2 Heinzl, T., and Ilderton, A. 2009. Exploring high-intensity QED at ELI. The European Physical Journal D, 55(2), 359–364. 3, 7 Heinzl, T., and Schröder, O. 2006. Large Orders in Strong-Field QED. J. Phys. A: Math. Gen., 39(18), 11623. 3 68 References Heinzl, T., Seipt, D., and Kämpfer, B. 2010a. Beam-Shape Effects in Nonlinear Compton and Thomson Scattering. Phys. Rev. A, 81(Feb), 022125. 5 Heinzl, T., Ilderton, A., and Marklund, M. 2010b. Finite Size Effects in Stimulated Laser Pair Production. Physics Letters B, 692, 250–256. 5 Heisenberg, W., and Euler, H. 1936. Folgerungen aus der Diracschen Theorie des Positrons. Zeitschrift für Physik, 98(11-12), 714–732. 1, 49 Heyl, J. S., and Hernquist, L. 1997. Birefringence and Dichroism of the QED Vacuum. J. Phys. A: Math. Gen., 30(18), 6485. 3 Hiller, J. R. 2000. Calculations with DLCQ. Nuclear Physics B - Proceedings Supplements, 90(Dec), 170–174. Non-Perturbative QCD and Hadron Phenomenology. 36 Hoffman, K., and Kunze, R. 1971. Linear Algebra. 2nd edn. Englewood Cliffs, New Jersey: Prentice-Hall, Inc. 4 Hornbostel, K. 1988. The Application of Light-Cone Quantization to Quantum Chromodynamics in One-Plus-One Dimensions. Ph.D. thesis, SLAC report SLAC-PUB-333. 35 Hornbostel, K., Brodsky, S. J., and Pauli, H.-C. 1990. Light-Cone-Quantized QCD in 1+1 Dimensions. Phys. Rev. D, 41(Jun), 3814–3821. 35 Hyun, S. 1998. The background geometry of DLCQ supergravity. Physics Letters B, 441(1–4), 116–122. 35 Hyun, S., and Kiem, Y. 1998. Background geometry of DLCQ M theory on a p-torus and holography. Phys. Rev. D, 59(Dec), 026003. 35 Hyun, S., Kiem, Y., and Shin, H. 1998. Effective action for membrane dynamics in DLCQ M theory on a two-torus. Phys. Rev. D, 59(Dec), 021901. 35 Ilderton, A. 2011. Trident Pair Production in Strong Laser Pulses. Phys. Rev. Lett., 106(Jan), 020404. 5 Ilderton, A. 2012. QED Processes in Intense Laser Fields. International Journal of Modern Physics: Conference Series, 14, 394–402. 2, 3, 5 Ilderton, A., Johansson, P., and Marklund, M. 2011. Pair Annihilation in Laser Pulses: Optical Versus X-ray Free-Electron Laser Regimes. Phys. Rev. A, 84(Sep), 032119. 7 Isham, C. J. 1989. Lectures on Groups and Vector Spaces for Physicists. World Scientific Lecture Notes in Physics. Singapore: World Scientific. 59 Itzykson, C., and Zuber, J.-B. 1980. Quantum Field Theory. New York: McGrawHill Book Company. 49 Jackiw, R. 1993. (Constrained) Quantization Without Tears. Pages 367–381 of: Colomo, F., Lusanna, L., and Marmo, G. (eds), Constraint Theory and Quantization Methods: From Relativistic Particles to Field Theory and General Relativity: proceedings. River Edge, N.J.: World Scientific. 2nd Workshop on Constraint Theory and Quantization Methods, Montepulciano, Italy, 28 Jun-1 Jul, 1993. 19, 20 Kaku, M. 1993. Quantum Field Theory: A Modern Introduction. Oxford University Press. 10 69 References Kalloniatis, A. C. 1995. A paradigm for solving the QCD zero mode problem in DLCQ. Pages 101–105 of: Glazek, S. D. (ed), Theory of Hadrons and Light Front QCD: Proceedings. River Edge, N.J.: World Scientific. 4th International Workshop on Light-Front Quantization and Non-Perturbative Dynamics, Polana Zgorselisko, Poland, Aug 15-25, 1994. 12 Kim, S. P., and Page, D. N. 2002. Schwinger Pair Production via Instantons in Strong Electric Fields. Phys. Rev. D, 65(Apr), 105002. 48 Kim, S. P., and Page, D. N. 2006. Schwinger Pair Production in Electric and Magnetic Fields. Phys. Rev. D, 73(Mar), 065020. 7 Kim, S. P., and Page, D. N. 2008. Schwinger Pair Production in dS2 and AdS2 . Phys. Rev. D, 78(Nov), 103517. 6 Kim, S. P., Lee, H. W., and Ruffini, R. 2012 (Jul). Schwinger Pair Production in Pulsed Electric Fields. arXiv:1207.5213 [hep-th]. 7 Klein, O. 1929. Die Reflexion von Elektronen an einem Potentialsprung nach der relativistischen Dynamik von Dirac. Zeitschrift für Physik, 53(3-4), 157–165. 1 Kleinert, H. 2004. Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets. World Scientific. 49 Kleinert, H., Ruffini, R., and Xue, S.-S. 2008. Electron-Positron Pair Production in Space- or Time-Dependent Electric Fields. Phys. Rev. D, 78(Jul), 025011. 7 Kluger, Y., Eisenberg, J. M., Svetitsky, B., Cooper, F., and Mottola, E. 1992. Fermion pair production in a strong electric field. Phys. Rev. D, 45(Jun), 4659– 4671. 17, 40 Kluger, Y., Mottola, E., and Eisenberg, J. M. 1998. Quantum Vlasov equation and its Markov limit. Phys. Rev. D, 58(Nov), 125015. 40 Kogut, J. B., and Soper, D. E. 1970. Quantum Electrodynamics in the InfiniteMomentum Frame. Phys. Rev. D, 1(May), 2901–2914. 22, 45 Kohlfürst, C., Mitter, M., von Winckel, G., Hebenstreit, F., and Alkofer, R. 2012 (Dec). Optimizing the Pulse Shape for Schwinger Pair Production. arXiv:1212.1385 [hep-ph]. 7 Lamb, W. E., and Retherford, R. C. 1947. Fine Structure of the Hydrogen Atom by a Microwave Method. Phys. Rev., 72(Aug), 241–243. viii Lenz, F., Thies, M., Levit, S., and Yazaki, K. 1991. Hamiltonian Formulation of Two-Dimensional Gauge Theories on the Light Cone. Annals of Physics, 208, 1–89. 17 Leutwyler, H., and Stern, J. 1978. Relativistic Dynamics on a Null Plane. Annals of Physics, 112, 94–164. 11 Leutwyler, H., Klauder, J. R., and Streit, L. 1970. Quantum Field Theory on Lightlike Slabs. Il Nuovo Cimento A, 66, 536–554. 20 Lifschytz, G. 1998. DLCQ-M(atrix) description of string theory, and supergravity. Nuclear Physics B, 534(1–2), 83–95. 35 Lifshitz, E. M. 1956. The Theory of Molecular Attractive Forces Between Solids. JETP USSR, 2(Jan), 73–83. 3 Lipkin, H. J., and Meshkov, S. 1966. Spin Independence, W Spin, Parity, and SU(6) Symmetry. Phys. Rev., 143(Mar), 1269–1274. 12 70 References Lorentz, H. A. 1909. The Theory of Electrons. Leipzig: B. G. Teubner. 32 Lundin, J. 2010. QED and Collective Effects in Vacuum and Plasmas. Ph.D. thesis, Department of Physics, Umeå University, Umeå, Sweden. 2 Lundström, E., Brodin, G., Lundin, J., Marklund, M., Bingham, R., Collier, J., Mendonça, J. T., and Norreys, P. 2006. Using High-Power Lasers for Detection of Elastic Photon-Photon Scattering. Phys. Rev. Lett., 96(Mar), 083602. 4 Lunin, O., and Pinsky, S. S. 1999. SDLCQ: Supersymmetric discrete lightcone quantization. Pages 140–218 of: Ji, C.-R., and Min, D.-P. (eds), New Directions in Quantum Chromodynamics. AIP Conference Proceedings 494, Melville, NY. 35 Mackenroth, F., and Di Piazza, A. 2011. Nonlinear Compton Scattering in Ultrashort Laser Pulses. Phys. Rev. A, 83(Mar), 032106. 5 Mandl, F., and Shaw, G. 2010. Quantum Field Theory. A Wiley-Interscience publication. John Wiley & Sons. 2 Marklund, M. 2010. Fundamental Optical Physics: Probing the Quantum Vacuum. Nature Photonics, 4(Feb), 72–74. 2 Marklund, M., and Lundin, J. 2009. Quantum Vacuum Experiments Using High Intensity Lasers. The European Physical Journal D, 55(2), 319–326. 2, 5, 6 Marklund, M., and Shukla, P. K. 2006. Nonlinear collective effects in photon-photon and photon-plasma interactions. Rev. Mod. Phys., 78(May), 591–640. 2 Marklund, M., Brodin, G., and Stenflo, L. 2003. Electromagnetic Wave Collapse in a Radiation Background. Phys. Rev. Lett., 91(Oct), 163601. 5 Marklund, M., Eliasson, B., Shukla, P. K., Stenflo, L., Dieckmann, M. E., and Parviainen, M. 2006. Electrostatic Pair Creation and Recombination in Quantum Plasmas. Journal of Experimental and Theoretical Physics Letters, 83, 313–317. 6 Marklund, M., Ilderton, A., and Lundin, J. 2011. Probing New Physics Using HighIntensity Laser Systems. Proc. SPIE, 8080(Jun), 80801H. Diode-Pumped High Energy and High Power Lasers; ELI: Ultrarelativistic Laser-Matter Interactions and Petawatt Photonics; and HiPER: the European Pathway to Laser Energy. 2 Maskawa, T., and Yamawaki, K. 1976. The Problem of P + = 0 Mode in the NullPlane Field Theory and Dirac’s Method of Quantization. Progress of Theoretical Physics, 56(1), 270–283. 35, 36 Mattuck, R. D. 1976. A Guide to Feynman Diagrams in the Many-Body Problem. New York: McGraw-Hill International Book Company. 1 McCartor, G. 1988. Light-Cone Quantization for Massless Fields. Zeitschrift für Physik C Particles and Fields, 41(Jun), 271–275. 35, 40 McCartor, G. 1991. Light-cone gauge Schwinger model. Zeitschrift für Physik C Particles and Fields, 52(Dec), 611–625. 35 McCartor, G. 1994. Schwinger model in the light-cone representation. Zeitschrift für Physik C Particles and Fields, 64(Jun), 349–353. 35 Melissinos, A. C. 2009. Tests of QED with Intense Lasers. Pages 497–518 of: Brabec, T. (ed), Strong Field Laser Physics. Springer Series in Optical Sciences, vol. 134. Springer New York. 2 71 References Mendonça, J. T. 2001. Theory of Photon Acceleration. Series in Plasma Physics. Bristol and Philadelphia: Institute of Physics Publishing. 5 Mendonça, J. T., Shukla, P. K., and Bingham, R. 1998. Photon Acceleration by Gravitational Waves. Physics Letters A, 250, 144–148. 5 Mendonça, J. T., Marklund, M., Shukla, P. K., and Brodin, G. 2006. Photon Acceleration in Vacuum. Physics Letters A, 359(6), 700–704. 6 Milonni, P. W. 1994. The Quantum Vacuum : An Introduction to Quantum Electrodynamics. Academic Press. 2 Milonni, P. W., and Shih, M.-L. 1992. Casimir Forces. Contemporary Physics, 33(5), 313–322. 3 Moore, G. T. 1970. Quantum Theory of the Electromagnetic Field in a VariableLength One-Dimensional Cavity. J. Math. Phys., 11(Sep), 2679. 3 Narozhny, N. B., Bulanov, S. S., Mur, V. D., and Popov, V. S. 2004. e+ e− –Pair Production by a Focused Laser Pulse in Vacuum. Physics Letters A, 330, 1–6. 7 Neville, R. A., and Rohrlich, F. 1971. Quantum Field Theory off Null Planes. Il Nuovo Cimento A Series 11, 1(4), 625–644. 12, 40 Nikishov, A. I. 1970. Pair Production by a Constant External Field. Sov. Phys. JETP, 30(Apr), 660–662. 50 Nikishov, A. I., and Ritus, V. I. 1964. Quantum Processes in the Field of a Plane Electromagnetic Wave and in a Constant Field. I. Sov. Phys. JETP, 19, 529–541. 5 Odom, B., Hanneke, D., D’Urso, B., and Gabrielse, G. 2006. New Measurement of the Electron Magnetic Moment Using a One-Electron Quantum Cyclotron. Phys. Rev. Lett., 97(Jul), 030801. viii Ortı́n, T. 2004. Gravity and Strings. Cambridge, UK: Cambridge University Press. 24, 54 Palasantzas, G., and De Hosson, J. Th. M. 2005. Pull-in Characteristics of Electromechanical Switches in the Presence of Casimir Forces: Influence of Self-Affine Surface Roughness. Phys. Rev. B, 72(Sep), 115426. 3 Pauli, H. C. 2000. A Compendium of Light-Cone Quantization. Nuclear Physics B - Proceedings Supplements, 90(Dec), 259–272. NON-PERTURBATIVE QCD AND HADRON PHENOMENOLOGY. 11, 12 Pauli, H.-C., and Brodsky, S. J. 1985a. Discretized light-cone quantization: Solution to a field theory in one space and one time dimension. Phys. Rev. D, 32(Oct), 2001–2013. 35, 36 Pauli, H.-C., and Brodsky, S. J. 1985b. Solving field theory in one space and one time dimension. Phys. Rev. D, 32(Oct), 1993–2000. 35 Perry, R. J., Harindranath, A., and Wilson, K. G. 1990. Light-front Tamm-Dancoff field theory. Phys. Rev. Lett., 65(Dec), 2959–2962. 12, 36 Peskin, M. E., and Schroeder, D. V. 1995. An Introduction to Quantum Field Theory. Frontiers in physics. Perseus Books Publishing, L.L.C. 19 Pioline, B., and Troost, J. 2005. Schwinger Pair Production in AdS2 . Journal of High Energy Physics, 2005, 043. 6 72 References Polyakov, A. M. 1987. Gauge Fields and Strings. Contemporary concepts in physics. Chur, Switzerland: Harwood Academic Publishers. 49 Remington, B. A. 2005. High Energy Density Laboratory Astrophysics. Plasma Physics and Controlled Fusion, 47(May), 191. 2 Reuter, M., Schmidt, M. G., and Schubert, C. 1997. Constant External Fields in Gauge Theory and the Spin 0, 12 , 1 Path Integrals. Annals of Physics, 259, 313–365. 49 Roberts, C. D., Schmidt, S. M., and Vinnik, D. V. 2002. Quantum Effects with an X-Ray Free-Electron Laser. Phys. Rev. Lett., 89(Sep), 153901. 6, 7 Rohrlich, F. 1971. Null Plane Field Theory. Pages 277–322 of: Urban, P. (ed), Concepts in Hadron Physics. Few-Body Systems, vol. 8/1971. Springer Vienna. 21, 40 Rozanov, N. N. 1998. Self-Action of Intense Electromagnetic Radiation in an Electron-Positron Vacuum. Journal of Experimental and Theoretical Physics, 86(2), 284–288. 5 Ruffini, R., Vereshchagin, G., and Xue, S.-S. 2010. Electron–Positron Pairs in Physics and Astrophysics: From Heavy Nuclei to Black Holes. Physics Reports, 487(1–4), 1–140. 6 Ryder, L. H. 1985. Quantum Field Theory. Cambridge, UK: Cambridge University Press. 11 Sauter, F. 1931. Über das Verhalten eines Elektrons im homogenen elektrischen Feld nach der relativistischen Theorie Diracs. Zeitschrift für Physik, 69(11-12), 742–764. 1, 6, 15, 49 Schubert, C. 1996. An Introduction to the Worldline Technique for Quantum Field Theory Calculations. Acta Phys. Polon., B27, 3965–4001. 49 Schubert, C. 2001. Perturbative Quantum Field Theory in the String-Inspired Formalism. Physics Reports, 355, 73–234. 49 Schubert, C. 2007. QED in the Worldline Representation. AIP Conference Proceedings, 917, 178–194. 49 Schubert, C. 2012. Lectures on the Worldline Formalism. School on Spinning Particles in Quantum Field Theory: Worldline Formalism, Higher Spins, and Conformal Geometry, Morelia, Michoacán, Mexico, Nov 19-23, 2012. http://indico. cern.ch/getFile.py/access?resId=0&materialId=5&confId=206621. 49, 52 Schützhold, R., and Unruh, W. G. 2011. Comment on “Hawking Radiation from Ultrashort Laser Pulse Filaments”. Phys. Rev. Lett., 107(Sep), 149401. 7 Schützhold, R., Gies, H., and Dunne, G. 2008. Dynamically Assisted Schwinger Mechanism. Phys. Rev. Lett., 101(Sep), 130404. 7 Schwinger, J. 1951a. On Gauge Invariance and Vacuum Polarization. Phys. Rev., 82(Jun), 664–679. 6, 15, 30, 40, 50 Schwinger, J. 1951b. The Theory of Quantized Fields. I. Phys. Rev., 82(Jun), 914–927. 19 Schwinger, J. 1953a. CXXIV. A Note on the Quantum Dynamical Principle. Philosophical Magazine Series 7, 44(357), 1171–1179. 19 73 References Schwinger, J. 1953b. The Theory of Quantized Fields. II. Phys. Rev., 91(Aug), 713–728. 19 Seipt, D., and Kämpfer, B. 2011. Nonlinear Compton Scattering of Ultrashort Intense Laser Pulses. Phys. Rev. A, 83(Feb), 022101. 5 Serry, F. M., Walliser, D., and Maclay, G. J. 1998. The Role of the Casimir Effect in the Static Deflection and Stiction of Membrane Strips in Microelectromechanical Systems (MEMS). J. Appl. Phys., 84(Sep), 2501. 3 Shen, B., and Yu, M. Y. 2002. High-Intensity Laser-Field Amplification between Two Foils. Phys. Rev. Lett., 89(Dec), 275004. 2 Shukla, P. K., and Eliasson, B. 2004. Modulational and Filamentational Instabilities of Intense Photon Pulses and Their Dynamics in a Photon Gas. Phys. Rev. Lett., 92(Feb), 073601. 5 Sokolov, I. V., Naumova, N. M., Nees, J. A., and Mourou, G. A. 2010. Pair Creation in QED-Strong Pulsed Laser Fields Interacting with Electron Beams. Phys. Rev. Lett., 105(Nov), 195005. 5 Soljačić, M., and Segev, M. 2000. Self-Trapping of Electromagnetic Beams in Vacuum Supported by QED Nonlinear Effects. Phys. Rev. A, 62(Sep), 043817. 5 Soper, D. E. 1971 (Sep). Field Theories in the Infinite Momentum Frame. SLACR-137. Prepared for the U.S. Atomic Energy Commission. 14 Sparnaay, M. J. 1958. Measurements of Attractive Forces Between Flat Plates. Physica, 24, 751–764. 3 Srednicki, M. 2007. Quantum Field Theory. Cambridge, UK: Cambridge University Press. 9 Srinivasan, K., and Padmanabhan, T. 1999a. A Novel Approach to Particle Production in an Uniform Electric Field. gr-qc/9911022. 41 Srinivasan, K., and Padmanabhan, T. 1999b. Particle Production and Complex Path Analysis. Phys. Rev. D, 60(Jun), 024007. 41 Srivastava, P. P. 1998. Light front quantized field theory: Some new results. Lectures given at the 9th Brazilian School of Cosmology and Gravitation (BSCG 98), Centro Brasileiro de Pesquisas Fı́sicas, Rio de Janeiro, Brazil, 27 Jul - 7 Aug, 1998. 17 Stamatescu, I.-O., and Schmidt, M. G. 2003. Matter Determinants in Background Fields Using Random Walk World Line Loops on the Lattice. Modern Physics Letters A, 18, 1499–1515. 49 Strassler, M. J. 1992. Field Theory without Feynman Diagrams: One-Loop Effective Actions. Nuclear Physics B, 385, 145–184. 49 Strickland, D., and Mourou, G. 1985. Compression of Amplified Chirped Optical Pulses. Optics Communications, 55, 447–449. 2 Susskind, L. 1997 (Apr). Another Conjecture about M(atrix) Theory. arXiv:hep-th/9704080. 35 Tajima, T., and Mourou, G. 2002. Zettawatt-Exawatt Lasers and their Applications in Ultrastrong-Field Physics. Phys. Rev. ST Accel. Beams, 5(Mar), 031301. 2 Tamm, I. 1945. Relativistic Interaction of Elementary Particles. J. Phys. (USSR), 9, 449–460. 35 74 References Thorn, C. B. 1978. Derivation of dual models from field theory. II. Phys. Rev. D, 17(Feb), 1073–1085. 36 Toll, J. S. 1952. The Dispersion Relation for Light and its Application to Problems Involving Electron Pairs. Ph.D. thesis (unpublished), Princeton. 3 Tomaras, T. N., Tsamis, N. C., and Woodard, R. P. 2000. Back reaction in light cone QED. Phys. Rev. D, 62(Nov), 125005. 40, 41 Tomaras, T. N., Tsamis, N. C., and Woodard, R. P. 2001. Pair creation and axial anomaly in light-cone QED2 . Journal of High Energy Physics, 2001(11), 008. 40, 41, 42, 43, 44, 45, 46 Tong, D. 2006. Quantum Field Theory. Lecture Notes, Department of Applied Mathematics and Theoretical Physics, University of Cambridge. http://www. damtp.cam.ac.uk/user/tong/qft.html. 9 Troup, G. J., and Perlman, H. S. 1972. Pair Production in a Vacuum by an Alternating Field. Phys. Rev. D, 6(Oct), 2299–2299. 40 Unruh, W. G. 1976. Notes on Black-Hole Evaporation. Phys. Rev. D, 14(Aug), 870–892. 7 van Blokland, P. H. G. M., and Overbeek, J. T. G. 1978. van der Waals Forces Between Objects Covered with a Chromium Layer. J. Chem. Soc., Faraday Trans. 1, 74, 2637–2651. 3 Weinberg, S. 1966. Dynamics at Infinite Momentum. Phys. Rev., 150(Oct), 1313– 1318. 12, 14, 36 Weinberg, S. 1967. Dynamics at Infinite Momentum. Phys. Rev., 158(Jun), 1638– 1638. 12, 14, 36 Weinberg, S. 1995. The Quantum Theory of Fields, Vol. 1: Foundations. Cambridge, UK: Cambridge University Press. 11 Wilczek, F. 1999. Quantum Field Theory. Rev. Mod. Phys., 71(Mar), S85–S95. 9 Wilson, C. M., Johansson, G., Pourkabirian, A., Simoen, M., Johansson, J. R., Duty, T., Nori, F., and Delsing, P. 2011. Observation of the Dynamical Casimir Effect in a Superconducting Circuit. Nature, 479(Nov), 376–379. 3 Wilson, K. G. 1990. Ab initio Quantum Chemistry: A Source of Ideas for Lattice Gauge Theorists. Nuclear Physics B - Proceedings Supplements, 17(Sep), 82–92. 12 Wilson, K. G., Walhout, T. S., Harindranath, A., Zhang, W.-M., Perry, R. J., and Glazek, S. D. 1994. Nonperturbative QCD: A Weak-Coupling Treatment on the Light Front. Phys. Rev. D, 49(Jun), 6720–6766. 12, 35 Witten, E. 1983. Global Aspects of Current Algebra. Nuclear Physics B, 223, 422–432. 12 Witten, E. 1984. Non-Abelian Bosonization in Two Dimensions. Communications in Mathematical Physics, 92, 455–472. 12 Woodard, R. P. 2001. Light Cone QED in a Homogeneous Electric Background. In: Fried, H. M., Gabellini, Y., and Müller, B. (eds), Non-Perturbative QCD: Proceedings. Singapore: World Scientific. 6th Workshop on Nonperturbative Quantum Chromodynamics, Paris, France, Jun 5-9, 2001. 44, 45 75 References Woodard, R. P. 2002. Resolving the p+ = 0 Ambiguity in a Homogeneous Electric Background. Nuclear Physics B - Proceedings Supplements, 108(Apr), 165–169. LIGHT-CONE PHYSICS: PARTICLES AND STRINGS Proceedings of the International Workshop TRENTO 2001. 45 Yamawaki, K. 1998. Zero mode problem on the light front. Pages 116–199 of: C.-R. Ji, D.-P. Min (ed), QCD, light cone physics and hadron phenomenology: Proceedings. Singapore: World Scientific. 10th Summer School and Symposium on Nuclear Physics (NuSS 97), Seoul, Korea, Jun 23-28, 1997. 36 Zwiebach, B. 2004. A First Course in String Theory. Cambridge, UK: Cambridge University Press. 9 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