Scattering in Time-dependent Basis Light-Front Quantization

Scattering in Time-dependent Basis Light-Front Quantization

Abstract

We introduce a nonperturbative, first principles numerical approach for solving time-dependent problems in quantum field theory, using light-front quantization. As a first application we consider QED in a strong background field, and the process of non-linear Compton scattering in which an electron is excited by the background and emits a photon. We track the evolution of the quantum state as a function of time. Observables, such as the invariant mass of the electron-photon pair, are first checked against results from perturbation theory, for suitable parameters. We then proceed to a test case in the strong background field regime and discuss the various nonperturbative effects revealed by the approach.

pacs:
11.10.Ef, 11.15.Tk, 12.20.Ds

I Introduction

Treating quantum field theory in the nonperturbative regime remains a significant challenge. “Basis Light-Front Quantization” (BLFQ) Vary:2009gt (), which adopts light-front quantization and the Hamiltonian formalism, offers a first-principles approach to nonperturbative quantum field theory (QFT) Vary:2009gt (); Honkanen:2010rc (). Diagonalization of the full Hamiltonian of the quantum field theory yields the physical eigenvalues and eigenvectors of the mass eigenstates. This approach offers new insights into bound state properties and scattering processes Brodsky:1997de () as well as opportunities to address many outstanding puzzles in nuclear and particle physics Brodsky:2011vc (); Brodsky:2012rw ().

The BLFQ approach is real-time (as opposed to imaginary-time, as normally used in lattice-QFT, see though Hebenstreit:2013qxa ()) and therefore naturally applicable to time-dependent problems. There is currently much interest in gauge theories with an explicit time dependence introduced by a background field, in particular QED in ultra-intense laser fields Heinzl:2008an (); DiPiazza:2011tq () and QCD in strong magnetic fields Chernodub:2011mc (); Bali:2011qj (); Basar:2011by (); Tuchin:2012mf (). In both cases, the greatest interest lies in the case for which the fields are strong enough to require a nonperturbative treatment and this motivates the approach we present here.

In this paper we introduce time-dependent Basis Light-Front Quantization (tBLFQ), which is an extension of BLFQ to time-dependent problems in quantum field theory. In this approach, BLFQ provides the eigenstates of the time-independent part of the Hamiltonian. We then solve for the time evolution of a chosen initial state under the influence of an applied background field, which is introduced through explicitly time-dependent interaction terms in the Hamiltonian. Although we treat a specific application in the present work, the method is more generally applicable to time evolution even in the absence of external fields where one is simply following the evolution of a chosen non-stationary state of the system.

In this paper we will apply tBLFQ to “strong field QED”, in which the background field models the high-intensity fields of modern laser systems. Such light sources now routinely reach intensities of W/cm, and there is ongoing research into using intense lasers to investigate previously unmeasured effects such as vacuum birefringence Heinzl:2006xc (); King () and Schwinger pair production Dunne:2008kc (). Within this research field, the use of large-scale numerical codes, based on kinetic models, is becoming increasingly popular Fedotov:2010ja (); Sokolov:2010am (); Elkina:2010up (); King:2013zw (). The two main advantages of such approaches are that they are real-time, and that huge numbers of particles can be treated via particle-in-cell (PIC) simulations. However, there exists no first-principles derivation of the required kinetic equations from QED. Consequently, this approach is based on a forced welding of classical and quantum theories, in which particles and photons are treated as classical ballistic objects, and QED cross sections are added by hand to model instantaneous collisions. This leads to problems with double-counting and the inclusion of higher-order processes.

Here, we consider an alternative approach. We restrict ourselves to low numbers of particles, but we perform a fully quantum and real-time calculation within QED. Specifically, we will study “non-linear Compton scattering” (nCs), in which an electron is excited by a background field and emits a photon Nikishov:1963 (); Nikishov:1964a (). This is one of the simplest background field processes, as there are no thresholds to overcome as in, say, pair creation. We note that light-front quantization is the natural setting for this investigation Neville:1971zk (); Ilderton:2012qe (), since lasers have inherently “light-front” properties: all photons propagate on the light-front.

This paper is organized as follows. We provide the background to our approach in Sec. II, followed by details of the BLFQ method in Sec. III. We then introduce tBLFQ in Sec. IV and provide illustrative numerical results for our first application to non-linear Compton scattering in Sec. V. We present our conclusions and outlook in Sec. VI. The Appendices contain a number of useful details.

Ii Background

Our approach is based on light-front quantization, and on a previously developed method called BLFQ Vary:2009gt (); Honkanen:2010rc (). We begin here with a brief review of relevant aspects of the light-front formalism Brodsky:1997de (); Heinzl:2000ht (), and an outline, in terms of textbook methods, of the calculation which we wish to perform.

Physical processes in light-front dynamics are described in terms of light-front coordinates (, , , ), in which =+ plays the role of time. Hence, quantization surfaces are null hyperplanes given by =constant, and on which initial conditions are specified. = is the “longitudinal” direction, and the remaining two spatial directions are called “transverse”, ={, }. The evolution of quantum states is governed as usual by the Schrödigner equation, which in light-front quantization takes the form

(1)

where is the (Schrödinger picture) state at light-front time and is the light-front Hamiltonian. Our Hamiltonian will contain two parts; which is the full light-front Hamiltonian of QED, and which contains interaction terms introduced by a background field, so

(2)

contains, in general, an explicit time dependence. It is therefore natural to use an interaction picture, but we must immediately stress two things: first, we are not using the usual “free + interacting” split of the Hamiltonian and, second, we are not working in perturbation theory. Instead, the full QED Hamiltonian replaces the customary “free” Hamiltonian, and is naturally the interaction term. The interaction picture states are then defined by

(3)

(since is time-independent), and obey

(4)

in which , “the interaction Hamiltonian in the interaction picture”, is

(5)

The formal solution to (4) is

(6)

where is light-front time ordering. Now let us imagine that we could “solve” QED and identify the eigenstates and eigenvalues of the theory. Call these and respectively, so

(7)

Having these covariant solutions, we would then be interested in the transitions between such states introduced by the background field interactions contained in . We choose the external field (modeling an intense laser) to vanish , , prior to . At , we expand a chosen initial state as a sum over QED eigenstates:

(8)

where is the initial data such that

(9)

We then expand a solution of the interaction picture state at later times,

(10)

in which the coefficients characterize the nontrivial part of the state’s time evolution induced by the external field. Plugging (10) into (4) yields an equation for the :

(11)

(Summation notation in the second line.) This is an intractable infinite-dimensional system of coupled differential equations, and it is at this point that one would normally switch to perturbation theory in the interaction . However, the background fields we wish to treat are strong and therefore not amenable to perturbation theory. We therefore write down the formal solution to (II), which is, regarding as a column vector and as a matrix, both with infinite dimensions,

(12)

In our approach, BLFQ provides finite dimensional approximate solutions for the eigenstates . In tBLFQ, the time evolution in (12) is performed numerically, beginning with the initial vector , to find the vector . The coefficients can then be read off, allowing one to reconstruct the evolved state itself from the overlap

(13)

In this way we solve equation (II) with initial conditions (9).

Let us compare the above to the usual calculation of scattering amplitudes in QED. Such amplitudes are based on the split of the QED Hamiltonian into a free particle Hamiltonian, , and an interaction. For the application here, this split produces an interaction that would be the sum of the QED interaction terms, call them , and the additional interaction terms introduced by the background, .

A scattering calculation would begin with an initial state which is a free particle state , prepared at . This state would be evolved through all time using the -matrix operator Weinberg (),

(14)

and projected onto a final state , describing free particles at . Thus, one obtains the -matrix element

(15)

We are also calculating “scattering amplitudes”, but there are two important differences between our approach and that based on the -matrix. First, we calculate transitions based upon the eigenstate basis of QED (for example physical electrons) rather than between free particle states. Second, and related, we calculate finite-time, rather than asymptotic, transitions between such states. For all times before and after the external field acts on our chosen state, we have, in principle, the full quantum amplitude expressed as a superposition of physical states (mass eigenstates of QED). A specific experimental setup will then project this full amplitude onto states to which that setup is sensitive.

ii.1 Application: Nonlinear Compton Scattering

In this paper we apply tBLFQ to the process of single photon emission from an electron accelerated by a background field. Taking the background to model an intense laser, this process often goes by the name “non-linear Compton scattering” and is well-studied in plane wave backgrounds Boca:2009zz (); Heinzl:2009nd (); Seipt:2010ya (); Mackenroth:2010jr (). An appropriate experimental setup would see the (almost head on) collision of an electron with the laser, and the subsequent measurement of either the emitted photon Harvey:2012ie () or electron Boca:2012pz () spectra.

We begin with an electron at light-front time =0 when it first encounters the laser field. The electron may be both accelerated (invariant mass unchanged but 4-vector altered) and excited (invariant mass changed) by the laser field. Excitation produces electron–photon final states. After time the background field switches off and no further acceleration or excitation may occur. This setup is sketched in Fig. 1 for two of the four dimensions in the problem. The natural question to ask is how the quantum states of the electron and (emitted) photon fields evolve with light-front time , and this will indeed be studied below.

While, in principle, there is nothing to stop us including arbitrarily complex background fields, as a first step we consider a simple model. The background is turned on only for finite light-front time , during which it is independent of but inhomogeneous in ,

(16)

where is the electron charge and is the electron mass. We have written out the exponential form of cosine to highlight that the field both “pushes” and “pulls” particles in the longitudinal direction. This field has periodic structure in the longitudinal direction with frequency and the dimensionless parameter measures the field strength in relativistic units, . ( corresponds to an intensity of W/cm at optical frequency Heinzl:2008an ().) It is uniform in the transverse plane, as for plane waves, but unlike plane waves is longitudinally polarized. The profile (16) describes, in the lab frame, a beam of finite duration propagating along the direction. Classically, such a field accelerates charges in the () direction as time () evolves. The accelerated charges subsequently radiate, see Fig. 1, and it is the quantum version of this radiation which we will investigate below.

Note that (16) does not obey Maxwell’s equations in vacuum. This is not an issue for us since we are interested here not in phenomenology but in a first demonstration of the framework of tBLFQ. Whether the background obeys Maxwell or not has no impact on our methods. With future developments of our formalism in mind, we note that a simple background field model obeying Maxwell would be a plane wave. However, it is also common to consider time-dependent electric fields, which do not obey Maxwell, as models of the focus of counter-propagating pulses Dunne:2008kc (). Insisting on background field profiles which are both realistic (finite energy, pulsed in all four dimensions) and obey Maxwell’s equations is a challenge, as very few such solutions exist in closed form. An exception is given in IVAN (), and while there is nothing to stop us including such backgrounds in principle, doing so goes somewhat beyond the initial “proof-of-concept” presented here.

Figure 1: An illustration of non-linear Compton scattering. An electron enters a laser field, is accelerated, and emits a photon. After emission the electron can be further accelerated until it leaves the field.

Iii Basis Light-front Quantization (BLFQ)

We are interested in how eigenstates of the full QED Hamiltonian evolve due to interactions with a background field. (This is analogous to, but clearly not the same as, studying transitions between bare states induced by perturbative QED interactions.) To begin, we must therefore find the eigenstates of QED, for which we must adopt an approximation.

The method we use to construct the approximate eigenstates is Basis Light-front Quantization, or BLFQ Vary:2009gt (); Honkanen:2010rc (). This is a numerical method for calculating the spectrum of a Hamiltonian, using light-front quantization. The idea of finding, for example, the bound state spectrum via diagonalization of the Hamiltonian has a long history Brodsky:1997de (). One well-known approach is discretized light-cone quantization Brodsky:1997de (); Pauli:1985pv (); Pauli:1985ps (); Tang:1991rc (), on which BLFQ is in part based. The idea behind BLFQ, and its main advantage, is that its adopted basis should have the same symmetries as the full QED or QCD Hamiltonian. (BLFQ was initially designed for QCD Vary:2009gt () and is supported by successful anti-de Sitter QCD methods deTeramond:2008ht ().) This basis is therefore not the usual basis of momentum states. Usually, the more symmetries the basis captures, the less computational effort is needed for the solutions to reflect those symmetries. Because of this, BLFQ achieves an accurate representation of the Hamiltonian using available computational resources. The construction of the BLFQ basis therefore begins with symmetries of the light-front Hamiltonian.

iii.1 Basis construction

The derivation of the light-front QED Hamiltonian (in the presence of background fields) and a list of relevant mutually commuting operators, may be found in Appendix A. We do not need the detailed form of these operators in order to discuss the three symmetries directly encoded in the BLFQ basis. A fourth symmetry, transverse boost invariance, (also referred to as transverse Galilei invariance, Brodsky:1997de (); Heinzl:2008an ()) is discussed separately below as it is not encoded directly in the BLFQ basis but is easily accessible with the employed transverse basis.

The three directly encoded symmetries are 1) Translational symmetry in the longitudinal direction. The longitudinal momentum operator, , therefore commutes with the Hamiltonian , and total longitudinal momentum is conserved. 2) Rotational symmetry in the transverse plane. This means that the longitudinal projection of angular momentum is conserved, and the corresponding operator obeys . The operator can be decomposed into two parts for each particle species,

(17)

in which the subscript refers to the longitudinal projection of orbital angular momentum, while subscript refers to the longitudinal projection of the spin angular momentum. This defines the helicity of a particle in light-front dynamics. 3) Charge conservation , where is the charge operator with eigenvalue equal to the net fermion number .

The existence of these conserved quantities means that the QED eigenspace can be divided up into “segments”, which are groups of eigenstates with definite eigenvalues1 , and . The full spectrum of QED is the sum of all such segments.

The BLFQ basis is chosen to respect these symmetries. The essential point is that each basis state, call them , is an eigenstate of the three operators introduced above, with the eigenvalues,

(18)

Therefore, each state belongs to one and only one segment. As a consequence, the BLFQ basis divides into segments, and the QED Hamiltonian is accordingly block-diagonal in the BLFQ basis. As will be outlined below, this structure allows for a large reduction in (numerical) complexity in bound-state calculations.

The BLFQ basis states are built for each Fock-sector (of free-particle states) by allowing the particles to occupy orthonormalized modes of a single-particle basis that facilitates implementation of the full symmetries. The many-particle basis states in each Fock-sector are therefore direct products of single particle states, written , so in general. It clearly remains to specify the details of the single-particle states.

The single particle basis states are chosen to be two-dimensional harmonic oscillator (“2D-HO”) states in the transverse direction and discretized plane waves in the longitudinal direction. This is one choice (among many) that facilitates implementation of the symmetries mentioned above. We note in passing the contrast with treatments of the transverse degrees of freedom in a discretized two-dimensional plane wave basis where the orbital projection symmetry is lost. We also note the freedom to choose another orthonormal basis in the transverse space using cylindrical coordinates that may be better for some applications.

Each single particle state carries four quantum numbers,

(19)

The first quantum number, , labels the particle’s longitudinal momentum. For this degree of freedom we employ the usual plane-wave basis states, i.e. eigenstates of the free-field longitudinal momentum operator , see (60), with corresponding eigenvalues . In this paper, we compactify to a circle of length . We impose (anti) periodic boundary conditions on (fermions) bosons. As a result, the longitudinal momentum in our basis states takes the discrete values

(20)

where the dimensionless quantity =1, 2, 3,… for bosons (neglecting the zero mode) and for fermions. In particular, we have for the laser where is a natural number. For convenience, throughout this paper we take MeV so that can be interpreted as the longitudinal momentum in units of MeV.

The next two quantum numbers, and , label the degrees of freedom in the transverse directions. As mentioned above we take the transverse components of our single particle states to be eigenstates of a 2D-HO which is defined by two parameters, mass and frequency . (See below for the characteristic scale of the oscillator, which depends only on a combination of these parameters.) These eigenstates are labelled by the quanta of the radial excitation, , and the angular momentum quanta, . The eigenstate carrying these numbers has HO eigenenergy

(21)

Since they are not eigenstates of the transverse momentum operator , the BLFQ basis elements mix states with the same intrinsic motion but with different transverse center-of-mass momenta. This is the price we pay for employing the 2D-HO states as single particle basis states in the transverse plane. We may employ, when needed, a Lagrange multiplier technique to enforce factorization of the transverse center-of-mass component of the amplitude from the internal motion components following techniques used in non-relativistic nuclear physics Navratil:2000ww (); Navratil:2000gs (). We may also work with alternative coordinates chosen to achieve factorization ?.

The final quantum number, , labels the particle’s helicity, which is the eigenvalue of , see (17). The electron (photon) helicity takes values ().

We present only selected essentials of our method; more details of the basis states may be found in Appendix B. We note here that our transverse modes depend only on the combination (and not on and individually). This is a free parameter which must be chosen. Since our goal is to design a basis which matches as closely as possible the symmetries of the QED Hamiltonian, we note that there is only one mass scale in QED, and that is the physical electron mass . A sensible choice for our 2D-HO parameter is therefore2 , and we adopt this throughout.

Now, to see why this choice of basis is suited to light-front problems, we relate the single particle quantum numbers to the segment numbers of the states . So, consider a multi-particle state , which belongs to a particular segment and is an eigenvector of , , and with eigenvalues , and , respectively. If , , , and are the quantum numbers for, respectively, the longitudinal momentum, longitudinal projection of angular momentum, net fermion number and helicity of the particle in the state then, summing over particles , we have

(22)
(23)
(24)

(The single particle net fermion number is 1 for , -1 for and 0 for .) We see that the basis states are eigenstates of and individually, with eigenvalues and . Note, though, that it is the sum which is conserved by the light-front QED Hamiltonian.

While each basis state belongs to one and only one segment, it is clear that the basis states themselves are not eigenstates of QED (written as ). These must still be constructed by diagonalizing in this basis. For example, the physical electron eigenstate can be expanded as

(25)

in which both the eigenstate on the left and the basis states on the right belong to the same segment. Diagonalizing the Hamiltonian in our basis would yield the coefficients , and hence the physical states . In order to do this, though we need to be able to implement our basis numerically, which requires some truncation. We turn to this now.

iii.2 Basis reduction

Since a quantum field theory contains an infinite number of degrees of freedom, reduction of the basis space is necessary in order for numerical calculations to be feasible. For us, this reduction takes place both in the basis states retained (exploiting symmetries) and in the Fock space itself (i.e. we retain only certain sectors and implement regulators).

The first type of reduction is called “pruning”, in which we exclude basis states which are not needed for desired observables. The pruning process is lossless, in that it does not lead to loss of accuracy in the desired observables. For example, in bound state problems, one is typically interested in states with definite and . Combining this with the longitudinal boost invariance inherent to light-front dynamics, one can choose based on the desired “resolution” for the longitudinal momentum partition among the basis particles Brodsky:1997de (). Thus, one only needs to work in a single segment of the QED eigenspace, neglecting the others, without loss of information. From here on we write “BLFQ basis” to mean the basis of a single segment.

Pruning alone is not enough to reduce the basis space to finite dimension, however, since even a single segment contains an infinite number of degrees of freedom. To further reduce the basis dimensionality we need to perform basis truncation, which unavoidably causes loss of accuracy in calculating observables. Basis truncation is implemented at two levels.

i) Fock-sector truncation. Consider the physical electron state. This has components in all Fock-sectors with , which we write schematically as

(26)

Included in this series are, for example, the bare electron and its photon-cloud dressing, , etc. Together, the bare fermion and its cloud of virtual particles comprise the observable, gauge invariant electron, as originally described by Dirac Dirac:1955uv (); Lavelle:1995ty (); Bagan:1999jf (). We implement basis truncation by assuming that higher Fock-sectors give (with an appropriate renormalization procedure implemented) decreasing contributions for the low-lying eigenstates in which we are mostly interested. (One motivation for this is the success of perturbation theory in QED). In this first paper, we make the simplest possible nontrivial truncation, which is to truncate our Fock-sectors to and . Thus, in this truncated basis, the physical electron state would be given by only the first two terms of (26). This is enough to calculate physical wavefunctions accurate up to the first-order of the electromagnetic coupling . Due to its simplicity this Fock sector truncation has been typical of light-front Hamiltonian approaches such as Refs. Honkanen:2010rc (); Chabysheva:2009ez () though an extension to include the 2-photon sector has been successfully implemented in solving for the electron’s anomalous magnetic moment Chabysheva:2009vm ().

ii) Truncation within Fock-sectors. Fock-sector truncation is still not enough to reduce the basis to finite dimension; each Fock particle has an infinite number of (momentum) degrees of freedom. In BLFQ, truncations of the longitudinal and transverse degrees of freedom are realized separately, and differently.

Truncation of the longitudinal basis space is realized through the finite size of the direction. By imposing (anti–) periodic boundary conditions, the longitudinal momentum for single particles can only take discrete values, see (20). Therefore, in a given segment with total longitudinal momentum , only a finite number of longitudinal momentum partitions is available for the particles in the basis states, since each particle’s momentum must obey and all the ’s must sum to . For segments with larger , more partitions of longitudinal momenta among particles are possible, allowing for a “finer” description of the longitudinal degrees of freedom. Thus, also regulates the longitudinal degrees of freedom; bases with larger have simultaneously higher ultra-violet (UV) and lower infra-red (IR) cutoffs in the longitudinal direction.

Now consider the transverse part. Recalling from above that the transverse states are eigenstates of a 2D-HO, with energies (21), we define the total transverse quantum number for multi-particle basis states as,

(27)

where the sum runs over all particles in the state. This number is used as the criterion for transverse basis truncation; all the retained basis states satisfy

(28)

for some chosen . Physically, this simply corresponds to restricting the total 2D-HO energy (summed over all particles). is specified globally across all Fock-sectors to ensure that the transverse motion in different Fock-sectors is truncated at the same energies. As shown in Appendix B.1, determines both the UV and IR cutoffs for the transverse basis space, see also Coon:2012ab (); Furnstahl:2012qg ().

This brings us to the end of our discussion on the BLFQ basis itself, so let us summarize the approach so far. The eigenspace of QED breaks up into segments, labelled by , and . The BLFQ basis is a basis of states for such a segment, with each basis element carrying the same three quantum numbers as the segment itself. The basis elements themselves are collections of Fock particle states. For each Fock particle, 2D-HO states/plane-waves are employed to represent the transverse/longitudinal degrees of freedom. The Fock particle states carry four quantum numbers, , , , , see above. A complete specification of a BLFQ basis requires 1) the segment numbers , 2) the parameters and pertaining to the transverse oscillator basis and length of the longitudinal direction, respectively, and 3) two truncation parameters, namely the choice of which Fock sectors to retain, and the transverse truncation parameter . (Recall, automatically serves as a longitudinal truncation parameter because is compact.)

Such a basis is finite dimensional. It is then a straightforward matter to diagonalize the QED Hamiltonian in the BLFQ basis. This yields, as well as the eigenvalues of the Hamiltonian, a representation of the physical states of QED in terms of the BLFQ basis, as in (25).

We end this section with a few words on renormalization. Our focus in this paper is on an initial exploration of tBLFQ, and we neglect the necessary counter-terms when writing down our Hamiltonians. (Hence, we adopt physical values for the electron mass and charge.) Renormalization within the BLFQ framework is possible, via a sector-dependent scheme Karmanov:2008br (); Karmanov:2012aj (); Zhao:2013xx (); Chabysheva:2009ez (). For an application, see Zhao:2013xx (), in which the scheme is implemented for the QED Hamiltonian; the resulting electron anomalous magnetic moment agrees with the Schwinger value to within 1%.

Iv Time-dependent Basis Light-front Quantization (tBLFQ)

Now that we have the physical states of QED, we turn to the transitions between them as caused by an external field. Our Hamiltonian now consists of two terms,

(29)

in which the new term comprises the interactions introduced by the background (laser field), just as in (2). See Appendix A for the explicit form of the new interactions.

As discussed above, only a single segment of states is needed to address bound-state problems. The presence of background field terms means, in general, that the full Hamiltonian will not posses the symmetries associated with conservation of longitudinal momentum () and longitudinal projection of total angular momentum (). (Net fermion number is not affected, of course.) In other words, the background field can cause transitions between QED eigenstates in different segments. In order to account for this, the BLFQ basis must be extended to cover several segments. We refer to a collection of multiple BLFQ basis segments with different ’s and ’s as the “extended BLFQ basis”.

In fact, since our particular choice of background field (16) only adds longitudinal momentum (and lightfront energy) to the system, the transverse degrees of freedom remain untouched, and the symmetry associated with holds even with the laser field switched on. Therefore, for our current example, we only need to include segments with different total longitudinal momenta .

We therefore begin by applying BLFQ to in each segment, finding the physical states in that segment and representing them as in (25). The combination of all such eigenstates from all the segments forms the “tBLFQ basis”, which is a basis of physical eigenstates of QED. From here on we will write to represent the extended BLFQ basis, and to represent the tBLFQ basis of physical states. See Fig. 2 for an illustration of the two different bases and the relationship between them.

In constructing the tBLFQ basis, one needs to specify the number of the segments to be included. Larger, less-truncated, basis spaces yields more realistic and detailed descriptions of the underlying system. The price we pay for increased basis dimensionality is of course increased computational time.

We have reached the stage at which we have an (appropriate) set of physical eigenstates of QED. We now describe the preparation of the initial state, and its evolution in time under the Hamiltonian (29).

Figure 2: The BLFQ and tBLFQ bases. On the left, the extended BLFQ basis . This is a collection of bases in different segments, each segment labelled by , and . (Since nothing in our theory changes net fermion number, all segments of interest have fixed , in our case.) The states in each segment are bare states. Two such states, a bare electron and a bare electron + a photon, are illustrated. The BLFQ procedure diagonalizes the Hamiltonian in each segment. The basis states in are then rearranged into eigenstates of the QED Hamiltonian, shown on the right. These are the tBLFQ basis states.

iv.1 Initial state preparation

In perturbation theory, scattering calculations take initial states to be eigenstates of the free part of the Hamiltonian, following the usual assumption of asymptotic switching, see though Kulish:1970ut (); Horan:1999ba (). In our calculations, initial states are taken to be physical eigenstates of QED. For our nCs, process, for example, the initial state is a single physical electron with longitudinal momentum . This state can be identified as the “ground state” of the QED Hamiltonian in the segment , and = (since there is no other state in that segment with a lower energy). In the tBLFQ basis, which is just the set of eigenvectors of QED, this initial state is trivially defined.

iv.2 State evolution

Recalling the discussion in Section II, our initial state evolves, in the interaction picture, according to

(30)

in which is the initial state, equal to a chosen eigenstate of QED (or a superposition thereof). In general, the interaction operator will not commute with itself at different times. We decompose the time-evolution operator into many small steps in light-front time , introducing the step size ,

(31)

in which each square bracketed term is a matrix, and we let each of these matrices act on the initial state sequentially. Between each matrix multiplication we insert a (numerically truncated) resolution of the identity, so that the evaluation of (31) amounts to the repeated computation of the overlaps

(32)

in which the are the previously solved eigenergies of , and their presence follows from Eq. (7). In order to calculate the left hand side of (32) in our numerical scheme, it is simpler to first calculate the phase factor and then calculate the remaining overlap in terms of the (extended) BLFQ basis, as follows:

(33)

The resulting interaction picture matrix elements are the elementary building-blocks for evaluating all observables.

For our particular choice of background field, the structure of the matrix elements between BLFQ basis elements is simple. The interaction terms introduced by the chosen background do not contain the quantum gauge field (see Appendices A and E for details), and therefore do not directly connect different Fock sectors; matrix elements of the type are therefore all zero. (Physically, the only direct effect of the chosen background field is to either increase or decrease the longitudinal momentum of an electron by .) Matrix elements between the same Fock sectors (in our case and ), on the other hand, are nonzero. If and label two Fock electron states, then one finds for example

(34)

in which is the Kronecker delta. The magnitude of the matrix element is proportional to the field intensity . It is the sum of two terms, originating in the two exponentials in (16). Each term is the product of two Kronecker deltas. The first delta conserves all quantum numbers between the states except for the longitudinal momentum (since that is all that our background field alters). The second delta fixes the difference between the values of the basis elements to be ; this is simply the “conservation” of longitudinal momentum among the initial and final electrons, in that any added energy-momentum must come from the laser field.

iv.3 Numerical Scheme

A direct implementation of Eq. (31) leads to the so-called Euler scheme which relates the state at to that at ; this scheme is however not numerically stable (since it is not symmetric in time) and the norm of the state vector increases as time evolves, see Ref. Iitaka:1994 (). We therefore adopt the second order difference scheme MSD2 Askar:1978 (), which is a symmetrized version of the Euler scheme relating the state at to those at and via

(35)

It can be shown that the MSD2 scheme is stable, with the norm of the states conserved, provided that , where is the largest (by magnitude) eigenvalue of  Iitaka:1994 (). This requirement imposes an upper limit on the step size . Further limits on will be discussed below.

(Note that in order to provide sufficient initial conditions for the MSD2 scheme, we use the standard Euler scheme to evolve the initial state one half-step forward, generating . Then we use the MSD2 scheme to evolve an additional half-step forward, generating . With both and available the MSD2 scheme is ready to generate at subsequent times, in time steps of .)

This concludes our discussion of the principles behind, and the method of application, of BLFQ and tBLFQ. The reader interested in more details is referred to Appendix C for the (analytic) representation of states and operators in the BLFQ basis, and to Appendix D for a worked example of the construction of a small, simple BLFQ basis, diagonalization of the Hamiltonian and an example tBLFQ calculation.

In the next section we turn to the results of our calculation of the nCs process.

V Numerical Results

In this section we present numerical results for non-linear Compton scattering (nCs), computed in the tBLFQ framework. Since the laser matrix elements play an important role in the numerical results, we first check them against those from light-front perturbation theory, in Section V.1. We then perform a systematic study of nCs using the laser matrix elements obtained from BLFQ, in Section V.2. For interested readers we present the full details in the numerical calculation for the nCs process (in a “minimal” basis) in Appendix. D.

v.1 Comparison of laser matrix elements

The laser matrix elements

(36)

are calculated in the BLFQ framework from the wavefunctions of and found from diagonalizing . Due to the small value of the electromagnetic coupling these wavefunctions can also be calculated in perturbation theory, and we will use this to check the BLFQ procedure.

Let us begin with the perturbative calculation of the matrix element (36). The background field enters only as an operator sandwiched between the states. What we must do is to construct the QED eigenstates . This can be achieved using ordinary, time-independent perturbation theory. To be concrete we will take , the physical electron, and , the electron-photon scattering state. We will work to first order in the coupling. So, if is a complete set of eigenstates of the free light-front Hamiltonian , and the QED interaction linear in is (see the first line of (58)), then the physical electron can be written, to first order,

(37)

Similarly, the physical electron-photon state is

(38)

The matrix element (36) is therefore approximated in perturbation theory by

(39)

in which we have written a hat over the operator to distinguish it from the eigenvalues and . Note that and are eigenstates of , but in order to compare with the BLFQ calculation we need to evaluate the matrix elements (39), and hence the states, in the BLFQ basis . The calculation is uninstructive, so we simply present the result in Appendix E.

Now, how do we compare this with a BLFQ calculation? We begin by constructing a basis containing only two (-)segments, and , using the same parameters as in the perturbative calculation. In the segment we retain only the single electron (ground) state; this acts as the initial state. In the segment we retain only the electron-photon (excited) states. Such a basis, while heavily truncated, is all that is required for comparing the results of BLFQ and tBLFQ with the perturbative result (39).

In general we would expect that the two matrix elements will match in the case of small QED coupling . However, in our truncated Fock-space (with only the one-electron and one-photon-one-electron sectors) there is one further source of potential discrepancy, for the following reason: the perturbative matrix elements (39) are calculated in the (complete) momentum basis first and then projected onto the initial () and final () states in the BLFQ basis. The BLFQ matrix elements, on the other hand, are calculated in a truncated basis space throughout. In the language of perturbation theory, there exists extra truncation effects in the BLFQ matrix elements between the “propagator” and the QED vertices, cf. Eq. (39). Due to this “intermediate” basis truncation, exact agreement can only be expected in the continuum limit ().

Direct comparison as a function of is difficult, because as increases the spectrum of changes (more states appear in the spectrum) and it becomes difficult to keep track of the dependence for specific matrix elements. We will now look at a test case, the nCs process with perturbative and nonperturbative matrix elements as inputs, and compare their predictions for the population of various tBLFQ basis states as a function of .

We take , , and parameters , for the laser profile (16). Thus the laser can cause transitions between the and segments. In general, we expect the truncation error between perturbative and nonperburbative matrix elements to diminish as increases and more basis states are used. However, since we are neglecting various renormalization counter-terms, at sufficiently large , high order and divergent loop effects may lead to further discrepancies between the (leading order) perturbative and nonperburbative matrix elements. Contributions from these loop effects are proportional to (higher-than-leading) powers of and generally increase with the ultraviolet and/or infrared cutoff . Since at this stage we are interested only in verifying the decreasing truncation error as increases, we suppress here the contribution of loop effects by artificially reducing the coupling constant so that 1/13700.

Figure 3: (Color online) Time evolution of the electron system in the laser field (at =16). Upper, middle and lower panels correspond to exposure time =0, 100, 200 MeV respectively (the laser field is switched on at =0). Each dot on these plots corresponds to a tBLFQ basis state in =3.5 segment. Y-axis is the probability, , for each basis state and x-axis is the corresponding invariant mass, . Green (red) dots are results based on laser matrix elements evaluated nonperturbatively (perturbatively). Note that the electromagnetic coupling constant is reduced to 1/13700, see text for details.
Figure 4: (Color online) “Snapshots” of the system at MeV in bases of =8 (upper panel), 16 (middle panel) and 24 (lower panel). Each dot on these plots corresponds to a tBLFQ basis state in =3.5 segment. Y-axis is the probability, , (on a greatly expanded scale compared to Fig. 3) for each basis state and x-axis is the corresponding invariant mass, . Green (red) dots are results based on laser matrix elements evaluated nonperturbatively (perturbatively). Note that the electromagnetic coupling constant is reduced to 1/13700, see text for details.
Figure 5: (Color online) Time evolution of the average invariant mass of the electron system. Up, middle and lower panels are calculated in tBLFQ basis space with =8, 16, 24 respectively. Y-axis is the difference between the average invariant mass of the system at and that of a single electron . X-axis is the (lightfront) exposure time . Green (red) dots are results based on (non)perturbative laser matrix elements. Note that the electromagnetic coupling constant is reduced to 1/13700, see text for details.

At =0 we switch on the laser field and evolve the initial single electron state according to Eq. (IV.3). The population of tBLFQ basis states (the probabilities ) in the segment, as a function of light-front time, are shown in Fig. 3, along with the corresponding perturbative results. Different tBLFQ basis states are distinguished by their respective invariant masses, , as defined in Eq. (83). As time evolves the probability for the single electron state (with invariant mass 0.511MeV) drops and various electron-photon states (with invariant mass above 0.6MeV) in the =3.5 segment are gradually populated. At and  MeV, a peak structure is seen around the invariant mass of 0.74 MeV. This can be understood as follows: because our laser profile (16) is only trivially dependent on light-front time, in that it switches on and off but is otherwise constant, in the infinite time limit only transitions between basis states with the same light-front energy can accumulate (the transition amplitudes between states with unequal energies oscillate with a period inversely proportional to their energy difference). In this case the light-front energy of the initial (single electron) state is  MeV, basis states in the segment with invariant mass around  MeV will thus accumulate and form a peak. The full peak develops over longer times and is located at approximately 0.8 MeV, independent of .

In order to study the convergence between the perturbative and nonperturbative laser matrix elements as a function of , we consider snapshots of the system at a fixed exposure time ( MeV) calculated in bases with increasing in Fig. 4. As expected the overall agreement between the results from perturbative and nonperturbative laser matrix elements indeed improves systematically as the basis increases.

As a measure of the energy transfer between the system and the laser field, we calculate the evolution of the average invariant mass of the system as a function of the exposure time . The numerical results calculated in bases with =8,16,24 are compared in Fig. 5.

As the exposure time increases, the laser field pumps energy into the system, and the invariant mass of the system increases accordingly, as seen from Fig. 5. Again, as expected the agreement between the results from perturbative and nonperturbative laser matrix elements improves as the basis size () increases.

In the next subsection we will study the nCs process systematically in a larger basis space using laser matrix elements from the BLFQ approach.

v.2 Numerical results for nCs

With the laser matrix elements checked, we now turn to nCs in a larger basis. This basis consists of three segments with . In each segment we retain both the single electron (ground) and electron-photon (excited) state(s). The initial state for the nCs process is a single (ground state) electron in the segment. This basis allows for the ground state to be excited twice by the background (from the segment with = through to segment with +2). In this calculation, we take and , with and . We present the evolution of the electron system in Fig. 6, at increasing (top to bottom) lightfront time.

Figure 6: (Color online) Time evolution of the single electron system in the laser field. From top to bottom, the panels in each row successively correspond to lightfront-time =0, 0.2, 0.4, 0.6MeV (the laser field is switched on at =0). Each dot on these plots stands for a tBLFQ basis state. Y-axis is the probability for the tBLFQ basis state and x-axis is its corresponding invariant mass . The panels on the left (with y-axis up to 1.1) illustrate the evolution of the single electron (ground) states in =1.5, 3.5, 5.5 segments respectively and the panels on the right with y-axis “zoomed-in” show the evolution of various electron-photon (excited) states. The electromagnetic coupling constant is 1/137.

The initial system is shown in the top panel of Fig. 6; the only populated basis state is the single electron (ground) state in the segment. As time evolves, the background causes transitions from the ground state to states in the segment. Both the single electron state and electron-photon states are populated; the former represent the acceleration of the electron by the background, while the later represent the process of radiation. At times  MeV, the single electron state3 in becomes populated while the probability for finding the initial state begins to drop. In the right hand panel, the populated electron-photon states begin forming a peak structure. The location of the peak is around the invariant mass of 0.8 MeV, roughly consistent with the expected value of  MeV, cf. the discussion in Section V.1.

Once the basis states in become populated, “second” transitions to the segment become possible. This can be seen in the third row of Fig. 6, at  MeV. In the left hand panel, one sees that the probability of the electron to remain in its ground state () is further decreased, the probability of it being accelerated (to ) is increased, and that the single electron state becomes populated. In the right hand panel, the electron-photon states in the segment also become populated as a result of the second transitions. A second peak arises here at the invariant mass of around  MeV (distinct from at that  MeV, above, formed by the electron-photon states from the first transitions). The peak in the segment is at a larger invariant mass than that in the segment simply because the basis states in the segment follow from the initial state being excited twice by the background field, and thus receive more energy than states in the segment.

As time evolves further, the probability of finding a single electron exceeds that of finding a segment electron, see the bottom left panels in Fig. 6. At this time,  MeV, the system is most likely be found in the single electron state, with probability 0.6. The probability for finding the single electron state is around  and the initial electron state is almost completely depleted. In the right hand panel, we see that the probability for finding =5.5 electron-photon states increases with time. One also notices that at later times, the probability for =3.5 electron-photon states also begin to drop. This is because (like the =3.5 single electron) the electron-photon states are coupled to the single electron; as the probability of the =5.5 single electron state increases, it “absorbs” both the single electron and the electron-photon states in the =3.5 segment. At =0.6MeV we terminate the evolution process, as the system is already dominated by the single electron state in the maximum -segment. Further evolution without artifacts would require bases with segments of .

This calculation, although performed in a basis of limited size, illustrates the basic elements of the tBLFQ framework. The acceleration of the single electron state and the radiation of a photon are treated coherently within the same Hilbert space.

Since the states encode all the information of the system, they can be employed to construct other observables. As an example, in Fig. 7 we present the evolution of the average invariant mass of the system as a function of time. The increase of the invariant mass with time reflects the fact that energy is pumped into the electron-photon system by the laser field. This invariant mass can be accessed experimentally by measuring the momenta of both the final electron, , and photon, in an nCs experiment. The invariant mass can be compared with the expectation value of measured over many repetitions of the nCs experiment.

Figure 7: (Color online) Time evolution of the average invariant mass of the electron system calculated in tBLFQ basis space with =8. Y-axis is the difference between the average invariant mass of the system at and that of a single electron . X-axis is the (lightfront) exposure time . The electromagnetic coupling constant is 1/137.

Work in deriving other observables, such as the cross sections for specific electron-photon final states, is in progress.

In this section we have demonstrated a) the general procedure for treating processes nonperturbatively in tBLFQ, and b) the accessibility of the full configuration (wavefunction) of the system at finite time.

Vi Conclusions and Outlook

In this paper we constructed a nonperturbative framework for time-dependent problems in quantum field theory, referred to as time-dependent BLFQ (tBLFQ). This framework is based on the previously developed Basis Light-front Quantization (BLFQ) and adopts the light-front Hamiltonian formalism. Given the Hamiltonian and the initial configuration of a quantum field system as input, the system’s subsequent evolution is evaluated by solving the Schrödinger equation of light-front dynamics. The eigenstates of the time-independent part of the Hamiltonian, found by the BLFQ approach, provide the basis for the time-evolution process. Basis truncation and time-step discretization are the only approximations in this fully nonperturbative approach. (Note that the choice of background field is an input parameter; although a simple background is adopted in this work, the tBLFQ framework is in principle capable of dealing with realistic background fields with generic spatial and temporal dependence.) One feature of the tBLFQ framework is that the complete wavefunction of the quantum field system is accessible at any intermediate time during the evolution, which provides convenience for detailed studies of time-dependent processes.

As an initial application we have applied this framework to an external field problem. We have studied the process in which an electron absorbs energy-momentum from an intense background laser field, and emits a single photon. In contrast to current numerical approaches to strong laser physics, tBLFQ is fully quantum mechanical and allows us to see both the acceleration of the electron by the background and the creation of a photon, in real-time. Note that tBLFQ is also applicable to problems without external fields but in which nontrivial time-dependence arises from using an initial state which is a non-stationary superposition of mass eigenstates.

Future developments will be made in two directions. First, further improvement of tBLFQ itself. The initial step is to implement renormalization so that the BLFQ representation of the physical eigenspectrum of QED can be improved (and then used in tBLFQ calculations). Currently we are working on implementing a sector-dependent renormalization scheme within the BLFQ framework. The inclusion of higher Fock sectors in our calculation is also important, as it will not only result in more realistic representations of quantum states but will also allow for the description of a larger variety of processes, e.g., multi-photon emissions.

The second direction to be pursued is the extension of tBLFQ’s range of applications. In the field of intense laser physics, the inclusion of transverse (), longitudinal () and time () dependent structures to the background field will be used to more realistically model the focussed beams of next-generation laser facilities IVAN (). In addition to intense laser physics, we will also apply tBLFQ to relativistic heavy-ion physics, specifically the study of particle production in the strong (color)-electromagnetic fields of two colliding nuclei. Ultimately, the goal is to use tBLFQ to address strong scattering problems with hadrons in the initial and/or final states. As supercomputing technology continues to evolve, we envision that tBLFQ will become a powerful tool for exploring QCD dynamics.

Acknowledgements.
We acknowledge valuable discussions with K. Tuchin, H. Honkanen, S. J. Brodsky, P. Hoyer, P. Wiecki and Y. Li. This work was supported in part by the Department of Energy under Grant Nos. DE-FG02-87ER40371 and DESC0008485 (SciDAC-3/NUCLEI) and by the National Science Foundation under Grant No. PHY-0904782. A. I. is supported by the Swedish Research Council, contract 2011-4221. Diagrams created using JaxoDraw Binosi:2003yf (); Binosi:2008ig ().

Appendix A The light-front QED Hamiltonian

In this section we follow the derivation of the Hamiltonian in Brodsky:1997de (), but with an additional background field. The Lagrangian is

(40)

in which and is the sum of the background and quantum gauge fields respectively. Note that is calculated from alone, i.e. there is no kinetic term for the background. The equations of motion for the fields are

(41)

which defines the current , and

(42)

The background field appears in the equations of motion for the fermion, but not for the gauge field. We now analyze these equations in light-front coordinates (, and ). We work in light-front gauge, so that . The component of (41) does not contain time derivatives, and can be written

(43)

This is a constraint equation which relates the (non-dynamical) field to the transverse components and the fermion current. Similarly, if we multiply (42) by on the left, we find a constraint equation for the fermion field. Defining first the orthogonal field components

(44)

the constraint equation may be written

(45)

Hence, the field is non-dynamical and can be expressed in terms of the dynamical field . We now turn to the construction of the Hamiltonian. The conjugate momentaü are

(46)

and the Hamiltonian is then

(47)

in which the first line is the standard Legendre transformation, and in the second line we have used the equations of motion. It is convenient to add a total derivative to the Hamiltonian Brodsky:1997de (), the term , and again use the equations of motion to write

(48)

In order to complete the transition to the Hamiltonian picture we need to eliminate the light-front time derivatives of the fields in favour of the fields themselves, and their momenta. The gauge field terms are simplest. Let be transverse indices and define

(49)

The first line of (48) then becomes

(50)

using the constraint (43). The field is that which survives the limit , and is therefore referred to as a “free field”. Turning now to the spinor terms in (48), we have

(51)

and the spinor equations of motion (42) then give

(52)

The first line follows from , we used (45) in the second line and in the third line we commuted to the right. In analogy to , we introduce , defined by

(53)

Again, this is the field which survives the limit. Our final task is to insert (53) into (52) and rewrite this in terms of only the “tilde” variables. First, the -free terms of of (51) are:

(54)

Next, we have terms in (51) which are linear in :

(55)

using (53) in the second line. Note the tilde on in the third line. Finally, we have the terms quadratic in , which are

(56)

Now, we sum (52) (54), (55) and (56) to obtain the full Hamiltonian: we drop the “tilde” on all variables from now on, so that one must remember that