Light-cone quantization of (3+1)-dimensional electrodynamics is discussed,
using discretization as an infrared regulator and paying careful attention to
the interplay between gauge choice and boundary conditions. In the zero
longitudinal momentum sector of the theory a general gauge fixing is performed
and the corresponding relations that determine the constrained modes of the
gauge field are obtained. The constraints are solved perturbatively and the
structure of the theory is studied to lowest nontrivial order. (Talk presented
at ``Theory of Hadrons and Light-Front QCD,'' Polana Zgorzelisko, Poland,
August 1994.)Comment: 6 pages, LaTeX, OSU-NT-94-0

Robertson, David G.

English

arXiv

Light-cone quantization of (3+1)-dimensional electrodynamics is discussed, using discretization as an infrared regulator and paying careful attention to the interplay between gauge choice and boundary conditions. In the zero longitudinal momentum sector of the theory a general gauge fixing is performed and the corresponding relations that determine the constrained modes of the gauge field are obtained. The constraints are solved perturbatively and the structure of the theory is studied to lowest nontrivial order

Otterbein University

Otterbein University Digital Commons @ Otterbein Physics Faculty Scholarship Physics 8-15-1994 Light Cone Quantization of Electrodynamics David G. Robertson Otterbein University Follow this and additional works at: https://digitalcommons.otterbein.edu/phys_fac  Part of the Physics Commons Repository Citation Robertson, David G., "Light Cone Quantization of Electrodynamics" (1994). Physics Faculty Scholarship. 6. https://digitalcommons.otterbein.edu/phys_fac/6 This Conference Proceeding is brought to you for free and open access by the Physics at Digital Commons @ Otterbein. It has been accepted for inclusion in Physics Faculty Scholarship by an authorized administrator of Digital Commons @ Otterbein. For more information, please contact digitalcommons07@otterbein.edu. OSU–NT–94–07Light-Cone Quantization of Electrodynamics1David G. RobertsonDepartment of Physics, The Ohio State University, Columbus, OH 43210AbstractLight-cone quantization of (3+1)-dimensional electrodynamics is discussed, using dis-cretization as an infrared regulator and paying careful attention to the interplay betweengauge choice and boundary conditions. In the zero longitudinal momentum sector of thetheory a general gauge fixing is performed and the corresponding relations that deter-mine the constrained modes of the gauge field are obtained. The constraints are solvedperturbatively and the structure of the theory is studied to lowest nontrivial order.1. IntroductionThe primary motivation for considering light-cone quantization when attempting toderive a constituent picture of hadrons from QCD is that a single cutoff, removing states ofsmall longitudinal momentum, renders the vacuum of the theory completely trivial. Thisimmediately results in a picture in which all partons in a physical hadronic state are tiedto the hadron, instead of being disconnected excitations in a complicated vacuum. Thesituation is quite different when quantizing on a spacelike surface, where no simple cutoffcan remove the states that are kinematically allowed to mix with the bare Fock vacuum.In this case a constituent description would necessarily be in terms of quasiparticle states,i.e., collective excitations above a complicated ground state.One calculational scheme which accomplishes (in most cases) this small-k+ cutoff is“discretized” light-cone quantization (DLCQ). In this approach one imposes periodicityconditions on the fields in the x− direction, leading to a discrete set of allowed longitudinalmomenta. For Fermi fields it is permissible to choose antiperiodic boundary conditionsso that there is no zero-momentum mode. For bosonic fields one must generally chooseperiodic boundary conditions, but the zero mode is in most cases not an independentfield; rather, it is a constrained functional of the other, dynamical fields in the theory.1 Talk presented at “Theory of Hadrons and Light-Front QCD,” Polana Zgorzelisko, Poland, August1994.1The bare vacuum is thus the only state in the theory with k+ = 0, and so is an exacteigenstate of the full Hamiltonian. The price we pay for this simplification is that wemust solve the constraints that determine the zero modes.The exception to this is gauge theories. Here one finds that the zero modes of Ai andA− are constrained fields, but certain of the zero modes of A+ can in fact be dynamical.Furthermore, these cannot be removed from the theory by a gauge choice. Thus in thiscase there are k+ = 0 particle states that can in principle mix with the bare vacuum togive a nontrivial physical vacuum state, although there are many fewer such states thantypically occur in equal-time quantization. In this case the nontrivial vacuum must eitherbe confronted or further ad hoc truncations must be made in order to obtain a trivialvacuum.In this talk I summarize recent work with A. Kalloniatis on the DLCQ of Abeliangauge theory with fermions [1]. The central problem is disentangling the constrained anddynamical zero modes in the context of a particular gauge fixing of the theory. Afterthis has been done, we then face the problem of solving the relations that determine theconstrained zero modes so that the Hamiltonian can be written down. I shall describehere the formulation of the theory to lowest order in perturbation theory.2. Gauge Fixing and the Zero Momentum ModesIn DLCQ the theory is defined in a light-cone box, with −L⊥ ≤ xi ≤ L⊥ and −L ≤x− ≤ L, and with some boundary conditions imposed on the fields. Because the gaugefield couples to a fermion bilinear, which is necessarily periodic in all coordinates, Aµmust be taken to be periodic in both x− and x⊥. We have more flexibility with the Fermifield, and it is convenient to choose this to be periodic in x⊥ and antiperiodic in x−. Thiseliminates the zero longitudinal momentum mode while still allowing an expansion of thefield in a complete set of basis functions.These functions are taken to be plane waves, and for periodic fields there will of coursebe zero-momentum modes. In previous work on the formulation of QED in DLCQ [2]the zero modes were discarded, and only the nonzero, or “normal,” modes were retained.This part of the theory is essentially unchanged. The “global” zero modes—those inde-pendent of all spatial coordinates—require special treatment, and the reader is referredto Appendix B of ref. [1] for more details. (It is shown there that they are irrelevant forthe calculations we shall present here.) The quantities that will be of central interest tous here are the “proper” zero modes, which are independent of x− but not independentof x⊥. Specifically, for a periodic quantity f , the proper zero mode is defined asf0(x⊥) ≡∫ L−Ldx−2Lf(x−, x⊥)−∫ L−Ldx−2L∫ L⊥−L⊥d2x⊥(2L⊥)2f(x−, x⊥) . (1)2Projecting the equation of motion ∂µFµν = gJν onto the proper zero mode sector gives− ∂2⊥A+0 = gJ+0 , (2)− 2(∂+)2A+0 − ∂2⊥A−0 − 2∂i∂+Ai0 = gJ−0 , (3)− ∂2⊥Ai0 + ∂i∂+A+0 + ∂i∂jAj0 = gJi0 . (4)We first observe that eq. (2) is a constraint which determines A+0 in terms of J+. Eqs.(3) and (4) then determine A−0 and Ai0. Thus all of the proper zero modes are constrainedfields.Eq. (2) is clearly incompatible with the light-cone gauge A+ = 0, which is most naturalin light-cone analyses of gauge theories. Here we encounter a common problem in treatingaxial gauges on compact spaces, which has nothing to do with light-cone quantization perse. The point is that the x−-independent part of A+ is in fact gauge invariant, since undera gauge transformationA+ → A+ + 2∂−ω , (5)where ω is a function periodic in all coordinates. Thus it is not possible to bring anarbitrary gauge field configuration to one satisfying A+ = 0 via a gauge transformation.We can (and will) set the normal mode part of A+ to zero, which is equivalent to∂−A+ = 0 . (6)This does not, however, completely fix the gauge—we are free to make arbitrary x−-independent gauge transformations without undoing eq. (6). We may therefore imposefurther conditions on Aµ in the zero mode sector of the theory.To see what might be useful in this regard, let us consider solving eq. (4) for thetransverse zero modes. By making use of current conservation we can rewrite eq. (4) as− ∂2⊥(δij −∂i∂j∂2⊥)Aj0 = g(δij −∂i∂j∂2⊥)Jj0 , (7)the general solution of which isAi0 = −g1∂2⊥J i0 + ∂iϕ(x+, x⊥) . (8)Here ϕ must be independent of x− but is otherwise arbitrary, and reflects the residualgauge invariance. Imposing a condition on, say, ∂iAi0 will uniquely determine this function.Now different choices for ϕ merely correspond to different gauge choices in the zeromode sector, so we expect that physical quantities should be independent of the specificϕ we choose. This is in fact the case. It turns out, however, that the unique choice whichallows simple (free-field) commutation relations among the fields is ϕ = 0. The easiest3way to see this is to note that for ϕ 6= 0 the kinematical operators P i do not have theirfree-field forms [1]. Thus it would be impossible to realize the Heisenberg relation[ψ+, Pi] = −i∂iψ+ (9)with the usual anticommutation relation between ψ+ and ψ†+. The complicated anticom-mutator required could in principle be determined, but this is not really necessary. It isfar simpler to choose the gauge ϕ = 0 and take the usual canonical commutation relationsamong the fields, and this is what we shall do.3. Perturbative FormulationWe shall now construct a perturbative solution of the constraints and study the struc-ture of the theory to lowest nontrivial order. This requires constructing the Hamiltonianthrough terms of O(g2), which in turn corresponds to taking the O(g) solutions for thezero modes Ai0 and A+0 . We obtain Ai0 at this order simply by setting g = 0 in the currentJ i0. A+0 is given exactly by inverting the ∂2⊥ in eq. (2).With Ai0 and A+0 in hand it is simple to construct the Hamiltonian. The contributionsfrom the normal mode part of the theory may be found in ref. [2].2 To this we must addthe contributions from the proper zero modes, which reduce to [1]P−Z =g22∫dx−d2x⊥( ∂i∂2⊥Jµ0)( ∂i∂2⊥J0µ), (10)where the currents are evaluated with g = 0. It is a straightforward (if tedious) exerciseto express P−Z in the Fock representation. We obtain various four-fermion operators, aswell as fermion bilinears (“self-induced inertias”) which arise when the four-point termsare brought into normal order.We can now compute with this Hamiltonian and study the effects of the zero mode-induced interactions (10). We shall here describe the calculation of the fermion self-energy(eigenvalue of P−) at O(g2). This is not the only quantity to which P−Z contributes, ofcourse. The four-fermion operators in P−Z will give rise to divergent contributions tothe e+e−γ vertex, and hence to the charge renormalization at this order. They will alsocontribute to tree-level scattering amplitudes, etc.In fact, we shall consider only the part of the self-energy that is quadratically divergentin a transverse momentum cutoff. The contributions coming from the normal mode sectorof the theory may be taken from the work of Tang, et al. [2] (subject to the caveat in2 Note that the Hamiltonian given in ref. [2] should be symmetrized under charge conjugation (b↔ d).This is the effect of using the explicitly C-odd form of the current Jµ = 12[ψ, γµψ], or equivalently ofproperly symmetrizing products of noncommuting operators in the construction of P−.4footnote 2). It has the formδP− = cΛ2⊥[2 ln 2p+−1(p+)2L], (11)where p+ is the momentum of the fermion and c is a dimensionless constant.Two points should be made about this result. The first is that it represents a failure ofchiral symmetry. Corrections to the fermion mass are supposed to be proportional to m,so that they vanish in the chiral limit. Here, however, there is a correction to m [the firstterm in eq. (11)] that is completely independent of m. Furthermore, the second term ineq. (11) does not have the correct form to be interpreted as the redefinition of a parameterin the Lagrangian. It would have to be removed by a noncovariant counterterm.There is another quadratically divergent contribution to the self-energy, however, com-ing from the zero mode-induced interactions (10). It reduces toδP−zm = +cΛ2⊥[1(p+)2L], (12)so that the noncovariant part of the quadratic divergence is in fact canceled when thezero modes are retained. Thus inclusion of the zero modes renders the UV behavior ofthe theory more benign. The quadratic divergence proportional to 1p+survives, unlike ina continuum formulation (with a Lorentz-covariant regulator), but this may be removedby a redefinition of the fermion kinetic mass.4. DiscussionThe box length L is introduced to regulate the theory in the infrared, and in principleit should be taken to infinity at the end of the day. More concretely, one should calculatephysical quantities of interest, which may be functions of the parameter L, and studythem in the limit L → ∞. For the energy of the one-fermion state, for example, thismeans taking L→∞ with the p+ of the state held fixed. Thus the term in eq. (11) thatis canceled by the zero mode contribution is explicitly L-dependent, in addition to beingquadratically divergent in Λ⊥. It is in some sense “irrelevant”—it goes to zero as theinfrared cutoff is removed—but this is not the point we wish to emphasize. Rather it isthat the zero mode-induced interactions removed this L-dependence. This suggests thatat least some of the zero mode interactions can be thought of as infrared counterterms,that serve to remove dependence on the infrared cutoff L.In a theory like QED, in which we do not expect any vacuum structure (in the senseof, e.g., vacuum expectation values or a θ-vacuum), we therefore expect that inclusion ofthe zero mode interactions mainly helps to remove L-dependence from the solution of thetheory and so makes the infinite volume limit easier to reach. There is numerical evidence5that this is the case [3]. A useful analogy might be with improved actions for latticetheories, in which dependence on the lattice spacing a is explicitly removed through someorder. Calculations using the improved action can then be done at a larger value of a(fewer lattice points for a given physical volume) for a fixed numerical accuracy.It is clearly of interest to learn whether there are zero mode interactions that aremarginal or relevant in the renormalization group sense. This question can perhaps beaddressed systematically from the point of view of the light-cone power-counting analysisof Wilson [4].It is also important to develop more sophisticated, nonperturbative methods for solvingthe constraints. For QED with a realistic value of the electron charge, however, it ispossible that a perturbative treatment of the constraints could suffice; that is, that wecould use a perturbation theory to construct the Hamiltonian, and then diagonalize itnonperturbatively. This approach is similar in spirit to that advocated in ref. [4], wherethe idea is to use a perturbative realization of the renormalization group to construct aneffective Hamiltonian for QCD, which is then solved nonperturbatively.We have not addressed here the general issue of removing dependence on the transversecutoff Λ⊥ from the theory. This is perhaps the more difficult problem, and in fact it maymake its presence felt in calculations like those presented here. The constraint relationsare strictly speaking ill-defined due to transverse UV divergences, and how one regulatesand removes these presumably affects the treatment of the zero modes. This may be animportant issue for theories with nontrivial UV divergences.AcknowledgementsIt is a pleasure to thank the organizers of the conference for their hospitality. Thiswork was supported by the National Science Foundation under Grants Nos. PHY-9203145,PHY-9258270, and PHY-9207889.References1. A.C. Kalloniatis and D.G. Robertson, Heidelberg/OSU preprint MPIH–V10–1994,OSU–NT–94–03 (hep-th/9405176), to appear in Phys. Rev. D.2. A.C. Tang, S.J. Brodsky, and H.-C. Pauli, Phys. Rev. D44, 1842 (1991).3. J.J. Wivoda and J.R. Hiller, Phys. Rev. D47, 4647 (1993).4. K.G. Wilson, et al., Phys. Rev. D49, 6720 (1994).6

Light Cone Quantization of Electrodynamics

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Light-Cone Quantization of Electrodynamics

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