Quantum Field Theory in Light-Front Coordinates

Transcription

1 Quantum Field Theory in Light-Front Coordinates Einstein's theory of special relativity requires that every physics law be covariant under inhomogeneous Lorentz transformations of coordinates, which include translations in space and time, rotations in space, and boosts along an arbitrary spatial direction. Questions that frequently arise concern an observer traveling almost with the speed of light: Can he formulate a consistent dynamical theory in his frame? Will his theory be equivalent to the theory in ordinary Lorentz frames? And is the new theory simpler to solve? This paper will comment on attempts made to answer these questions in the framework of quantum field theory, in particular, on efforts to solve non-perturbative Quantum Chromodynamics (QCD) using light-front coordinates. The reference frame with v ~ c was first used by Fubini and Furlan 1 2 to discuss current-algebra sum rules in particle physics, and was named the "infinite-momentum frame" (IMF). The Lorentz transformation into such a frame is obviously singular due to -y = 1/V l - (v!c) 2 ~ :xi. It was shown, however, that singularities of the transformation cancel in physical variables, such as the Poincaré generators. 3 The net result is a transformation to "lightfront coordinates," which were introduced by Dirac many years ago 4 : Comments N11c/. Part Phys 1993, Vol. 21, No. 2, pp Reprints available directly frorn the publisher Photocopying permitted by license only 1993 Gordon and Brcach. Science Publishers S.A. Printed in Singapol'c 123

2 - 1_ (xo 3) +X ' \IL X (1) x = (x 1 x 2 ) = x; _L ' ' where x + is the light-front time and x - is the light-front spatial coordinate. (Occasionally, "light-front" has been described as "lightcone"; the meaning is the same but the latter usuage is not accurate.) Therefore, it appears that a theory solved in light-front coordinates is equivalent to that solved in ordinary Lorentz coordinates if Lorentz invariance holds in the limiting procedure. However, as we shall discuss in Section II, this is only formally correct for a quantum field theory because ultraviolet divergences may interfere. A proof of equivalence becomes much more involved if the theory needs regularization and renormalization. Nonetheless, the study of field theories in light-front coordinates, or light-front quantization, has been attracting continuing interest because it has some important simplifications over theories in Lorentz frames. In particular, light-front QCD seems to provide a plausible picture of hadron physics. This paper is divided into four sections. The first section discusses canonicat light-front quantization and its consistency with special relativity. The second section deals with light-front perturbation theory and its equivalence to ordinary Feynman perturbation. The third section explains the simplifying features of lightfront theories and introduces non-perturbative methods, which aim to calculate the structure of bound states. The final section cites a few other applications of light-front coordinates to illustrate that these coordinates not only are convenient for solving field theories, but also have an intimate connection with physics at high energy. l. CAN WE FORMULATE A DYNAMICAL THEORY IN LIGHT-FRONT COORDINATES? In this section, we abandon the notion that the light-front system is a limiting case of Lorentz systems, and study the possibility of 124

3 directly constructing in it a dynamical theory consistent with special relativity. This discussion is useful because is avoids the limiting procedure to IMF, which is ill-defined when ultra-violet divergence is involved. Dirac 4 was the first to suggest that a relativistic dynamical theory can be formulated consistently in the light-front frame. This conclusion followed from a general investigation about the forms of Hamiltonian dynamics and the requirements of special relativity. According to Dirac, the necessary conditions for relativistic dynamics include (1) identification of a complete set of independent dynamical variables, (2) specification of a rule combining two dynamical variables to a third one, (3) the existence of ten fundamental dynamical variables which generate changes of all dynamical variables under inhomogeneous Lorentz transformations, and, finally, ( 4) these fundamental variables obey the Poincaré algebra. In the following, we use the example of QCD to examine these conditions in some detail. To formulate a dynamical theory, one must first identify all independent dynamical variables of the system. In ordinary dynamics, these variables live on the t = 0 initial-data plane of Minkowski space. Their values in other!-planes are determined by the Hamiltonian of the system which generates translations of the plane. In light-front coordinates, the analogous procedure is carried out at x+ = 0, where initial datais specified. However, it is not generally true that all the dynamical variables live in the initial x + = 0 plane. In fact, there is a well-known example which contradicts this. Consider the massless Klein-Gordon equation in two dimensions, 5 The most general solution of the equation is where f and g are arbitrary functions. Clearly, a specification of the solution at x + = 0 carries no information about f(x +). In this extreme case, the initial plane contains only half of the dynamical degrees of freedom of the system. In realistic field theories like QED or QCD, this problem may arise for massless gauge (2) (3) 125

4 particles with zero transverse momenta. The number of degrees of freedom excluded from the x + = 0 plane is a set of measure zero, therefore presumably not as significant as in the massless two-dimensional case. At present, we do not fully understand the raie played by these missing degrees of freedom (only speculations are available), and generally it is assumed that they do not critically alter the dynamics of light-front theories. To discover rules combining two independent dynamical variables, we study the commutation relations between field variables at x + = O. For QCD in the light-front gauge (A _ = 0) and in light-front coordinates, the Lagrange density is expressed as + iïî(iq>.l - m)ljl, (4) where we have used the notation, ljl± = P ±lj! and P ± = (t)'y"'-y±. Since the fields ljl+ and Ai contain the evolution derivative (a+), which replaces the role of time in the usual scheme, they are the only dynamical degrees of freedom of the system. The other variables, ljl_ and A+, are not dynamically independent. They are related to ljl + and A; by constraint equations, which can be derived from the Lagrangian equations for these variables: (5) The Lagrangian in ( 4) is linear in the first-order evolution derivative of dynamical variables. If we use the canonical quantization procedure, i.e., calculate the canonical momenta conjugate to the field variables and postulate canonical commutation relations between them, we will find that the commutation relations 126

5 lead to incorrect dynamical equations of motion. The problem has nothing to do with the light-front coordinates that we are working in, rather it is related to the general question of how to quantize systems with Lagrangians having only linear dependence on the first-order time derivative. Answers to this question are well-known in the particular examples of non-relativistic field theory or the Dirac theory and the general solution to the problem is aiso available. 6 The correct commutation relations in our case are (6) where ô 3 (x - y) = ô(x - - y-)ô 2 (x J_ - y J_ ). [The commutation relation for the gluon potentials in Eq. (6) differs by a factor of t from that obtained from the canonical quantization procedure.] From the Lorentz invariance of the QCD Lagrangian, we can construct the following conserved currents: (7) with aµ.tµ.v = 0 and aamaµ.v = O. Here TIL" is the symmetric energymomentum tensor. Maµ.v is the angular momentum density and is antisymmetric in µ and v. The corresponding conserved charges are defined as, pµ. = J dx- dxj.t+µ. and}il'' = J dx- dxj.m+µ.... Using the light-front commutation relations (6), we can prove that pµ. and Jµ.v generate correct Lorentz transformations for the quark and gluon fields. Thus they constitute the ten fundamental dynamical variables of the system. 1t is easy to show, with the use of the constraint equations and partial integrations, that seven of these generators, P _, P;, 1 12, J + _ and J -i are independent of the dynamics of the system, and they leave the initial-value plane invariant. The three other generators, P + and J + ;, depend on the interactions in the Lagrangian, and they generate changes of the 127

6 plane. Clearly, a theory in the light-front coordinate system has one more kinematic generator than that in Lorentz frames, as first noticed by Dirac. 4 It is tedious but easy to show that the ten generators of Lorentz transformations obey the commutation relations of the Poincaré algebra. This completes our discussion that light-front QCD forms a consistent dynamical system. This line of argument can also be applied to other Lorentz invariant Lagrangian field theories in light-front coordinates, and the conclusion remains the same. II. ARE LIGHT-FRONT THEORIES THE SAME AS THEORIES IN LORENTZ FRAMES? If the Iight-front version of a theory can be sensibly defined, is it equivalent to the corresponding theory in Lorentz frames? This question can be answered unambiguously in the context of perturbation theory, as we can compare, order by order, the lightfront and Feynman perturbation series. Unfortunately, in field theory, terms in the perturbative expansion are usually divergent, and a comparison only makes sense after both series are regularized and renormalized. However, it seems difficult to find a regularization scheme which suits both expansions. To obtain the light-front perturbation theory (LFPTh) for QCD, we first need to construct the light-front Hamiltonian which is a sum of a free term H 0 and a perturbative interaction term H 1. This can be done simply by substituting the constraint equations (5) into the Poincaré generator P + constructed from (7), eliminating ail the dynamical-dependent fields. The resulting Hamiltonian contains only fields which admit a free-field expansion in terms of Fock space quanta. In light-front gauge, these quanta include physical quarks (b:j, antiquarks (d:j, and gluons (a:j with different colors, helicities, and momenta. The difference in appearance between the light-front and ordinary Hamiltonians lies in some extra terms in the former, which are regarded as instantaneous interactions. These interactions arise from Z-diagrams in Feynman perturbation theory and summarize the effects of backward-moving particles. 7 An explicit expression for the light-front QCD Hamiltonian can be found in Ref. 8. The rules for LFPTh can be obtained 128

7 with the use of the ordinary time-independent perturbation theory. They have also been listed in Ref. 8. A forma[ proof of the equivalence of the Feynman and the lightfront perturbations can be made in the following way. Consider the unregularized Feynman perturbative expansion, make a coordinate change to the light-front system in momentum space, and then integrate over ail the internai k- coordinates. The result thus obtained is exactly the same as the above LFPTh series. 9 Another method of proof is to start with the ordinary time-independent perturbation theory in Lorentz frames and boost each diagram to the IMF. This was first discussed by Weinberg for scalar field theory 10 and then generalized to theories with non-zero spin particles by Dreil, Levy and Yan 11 and Brodsky, Roskies and Suaya. 7 The arguments presented above are incomplete bccause in physical field theories like QED or QCD integrals in the Feynman perturbation series diverge. To make them well defined, we have to regularize. Unfortunately, for most of the regularization schemes in use, the k- integration cannot be simply done after regularization, with the exception of Pauli-Villars regularization with a sufficient number of ghost particles. 12 Here "sufficient" assures convergence not only for covariant integrations in the Feynman perturbation series, but also for the individual spatial integrals after performing the k- integration. On the other hand, if one formally integrates out k - without any regularization, the result is not Lorentz covariant due to end-point contributions. It will remain so after introducing regularization. One needs to introduce noncovariant counterterms to cancel these non-covariant end-point contributions. A further complication arises from the so-called spurious light-front singularities, which should be separately regularized and cancelled when different diagrams with the same topology are combined. After the non-covariant renormalization, about which we do not yet have much knowledge, it is not clear that we recover a finite, covariant perturbation series coinciding with ordinary Feynman perturbation. On the other hand, we can check bath expansions explicitly in the lowest few orders. For instance, in QED, Brodsky, Burkardt and Langnau have developed a systematic way to calculate g - 2 in LFPTh with a minimum number of Pauli-Villars subtractions The non-covariant contributions appear because of the 129

8 aforementioned light-front singularity. It can be taken care of by certain empirical methods like the alternating denominator method and the tensor method. They found that the result coïncides with that obtained from Feynman perturbative calculations. In QCD, Perry, Harindranath and Zhang have calculated the renormalization constant for the quark-gluon vertex at one loop with eut-off regularization. 14 Despite the serious light-front singularities, the final result reproduces the famous asymptotic freedom. Nevertheless, a proof that LFPTh can be systematically regularized and renormalizec at ail orders, and that the result is the same as the covariant perturbation, is far from complete. This is due to the explicit breaking of Lorentz invariance and gauge invariance of LFPTh. In particular, there are now two different distance scales in the problem: the longitudinal scale x- and the transverse scale x 1. The counterterms required for renormalizing the theory have much more variety than the terms appearing in the naive Hamiltonian. Power counting suggests the existence of counterterm fonctions, the forms of which are arbitrary. 15 The Lorentz symmetry and gauge invariance may eventually be used as constraints on the forms of counterterms. The establishment of the equivalence or of a comparison between LFPTh and Feynman perturbation can offer insights into the structure of light-front theories. For instance, the vacuum diagrams in LFPTh would apparently vanish due to lack of backwardmoving particles. However, there are delta-fonction-like contributions coming from the point k+ = 0 in LFPTh which can be easily missed. 5 Thus, it appears that the vacuum in the light-front coordinates contains only virtual particles with k+ = 0, and thus its fonction may be characterized by a few constants. Compared with the vacuum in Lorentz frames, this achieves a great simplification. In the non-perturbative approaches to light-front theory discussed in the next section, most calculations done for two dimensional theories reproduce the results of Lorentz-frame calculations when available. This provides us with confidence about those nonperturbative methods and, at the same time, shows the identity of the physics in the light-front coordinates and Lorentz frames. Of course, a rigorous non-perturbative proof of this for theories like QCD is unavailable. 130

9 III. ARE LIGHT-FRONT THEORIES SIMPLER TO SOLVE? The primary reason for the interest in light-front quantization is that it may offer a simpler approach to field theory problems, in particular, the bound states. One of the main advantages of the light-front frame is, as we have just mentioned, the absence of k+ > 0 quanta in the physical vacuum. Because of this, the vacuum behaves like its perturbative counterpart except for a k+ = O particle condensation. This feature was first discovered by Weinberg, 10 and based on this, Brodsky et al.7 argue that the LFPTh has practical calculational advantage over the ordinary time-dependent perturbation, for many of the Iight-front time-ordered diagrams vanish. 7 Another related property of the light-front theory is that its Fock space is somewhat "smaller" than the ordinary Fock space due to the absence of k+ < 0 particles. We shall explain this in the course of our discussion. In the Hamiltonian approach to QCD, hadrons are described by Fock space wave fonctions. This remains truc in the Jight-front coordinate system. For instance, a pion of momentum P = (P_, P _]_) can be described by the following light-front expansion 8 : (8) Here ln:x;p _, x;p j_ + k j_j, '/..-;) represents the Fock basis with n quarks and gluons, each of which has momentum (k; = x;p _, x;p j_ + k_j_;) and polarization À.;, and their color indices have been coupled properly into singlet. (Here X; is a light-cone fraction, not the transverse coordinates introduced earlier.) The expansion coefficient ljl,,(xh k_j_;, À.;) is the wave fonction amplitude for finding the pion in the Fock state. The nove) feature of the expansion in momentum space is its independence of any specific frame. In fact, ail X; and kj_j here are relative variables. The usefolness of the expansion in light-front coordinates lies in the speculation that a few terms in (8) might be sufficient to 131

10 approximate the structure of a hadron. This is supported first by the simplicity of the vacuum structure, which implies that every Fock state in the expansion represents the deviation of the hadron state from the vacuum. It may not be an unreasonable assumption that hadron structure can be described by a few Fock states building upon the vacuum state. On the other hand, in the Fock space representation in an ordinary frame, one already needs a complicated Fock expansion to mode! the nontrivial vacuum, let alone to expand the hadron states. For a given number of Fock particles, the number of ways to build a hadron with light-front momentum P + is constrained by "Z;k t = P + and kt > O. Clearly, without the positivity constraint, the number of solutions of the constraint equation increases dramatically. In the extreme case of two dimensions and with fields confined in a box, the number of solutions stays finite with the constraint and is infinite without. Two non-perturbative methods have been proposed in recent years to calculate the wave function amplitudes. The first method is the Discrete Light-Cone Quantization (DLCQ) suggested by Brodsky and Pauli, 16 and the second method is the Light-Front Tamm-Dancoff (LFTD) first suggested by Weinberg 10 and implemented by Perry, Harindranath and Wilson.17 Bath methods have been used to salve various two-dimensional models as well as a few higher dimensional ones. While they are not yet effective enough to solve QCD meaningfully, the success of bath approaches is noticeable in simple examples. The DLCQ truncates the Fock space by discretizing momentum space and limiting the value of harmonie resolution K, which is essentially the light-front momentum of the hadron. The continuum limit is believed to be reached by letting K ~ x. For each finite K, the Fock space has a finite dimension in two-dimensional space-time, and at higher dimensions, additional truncations in transverse momenta are needed to keep the space finite. The Hamiltonian becomes a matrix in the truncated Fock space and the eigenvalues and eigenvectors are masses and wave functions of bare states. The renormalizations can be done by fitting the masses of the lowest states to that of observed elementary particles, and the mass spectrum of the physical states are subsequently determined. This method has been used to salve QCD and other theories in two dimensions

11 The LFTD truncates the number of particles in the Fock states. The equations for the few-particle wave amplitudes can be derived by taking the matrix elements of the light-front mass eigen-equation between the truncated basis. These equations must be renormalized properly before being attacked with analytical or numerical methods. The renormalization technique in this case has not been systematically understood. The Ohio State group has studied Yukawa theory with truncations up to one boson, first in 1+1 dimensions and then in dimensions. 19 Their results are encouraging. There are presently a few prominent obstacles that must be overcome before directly attacking light-front QCD. First, the problem must be truncated to a calculable size and yet still maintain the essential physics. In DLCQ, the large dimensionality of the Hamiltonian matrix is difficult to handle. One must find a better way to choose the basis. In LFTD, the equations are often very complicated even with a few Fock particles. In fact, most investigations along this direction are limited to one particle beyond the minimum. Secondly, one must better understand non-perturbative regularization and renormalization, and in the process, the restoration of gauge invariance and Lorentz invariance. 20 And finally, the structure of the vacuum deserves proper attention. In the Fock expansion approach, the complication of the physical vacuum is neglected, and yet meaningful solutions have been obtained. This presumably is due to the absence of k + f. 0 quanta in the vacuum. However, one must understand in this picture how spontaneous symmetry breaking occurs and how certain operators develop their vacuum expectation values. It would be very interesting if one can demonstrate the speculation that the k + = 0 quanta, or zero modes, provide a solution to this problem. 21 IV. OTHER USES OF LIGHT-FRONT COORDINATES Light-front coordinates not only provide new insights into nonperturbative field theory, but also a natural coordinate system to discuss certain high-energy phenomena. Here we mention two applications of light-front coordinates in high-energy physics. The first is the derivation of fixed-mass sum rules through Iight-cone 133

12 commutators. And the second is the quark and gluon distributions in the nucleon probed through hard scattering processes. A set of fixed-mass sum rules were first derived by Dashen, Fubini and Gell-Mann using the equal-time current algebra and IMF technique. 2 These fixed mass sum rules include the celebrated Adler sum rule and Cabibbo-Radicati sum rule. 2 Later, Jackiw and his collaborntors 2 12 studied the Iight-cone current algebracommutation relations between physical currents at an equal lightfront time-from the light-front quantized field theory. They showed that these algebras can be simply established through the lightcone commutation relations, such as (6) for QCD, and they can be used to derive ail the fixed-mass sum rules. The light-cone algebra they obtained contain the so-called Schwinger terms which cannot be correctly given through the IMF procedure. Those Schwinger terms can be tested through the sum rules that involve the current with dynamically dependent fields. A second popular use of light-front coordinates is made in highenergy scattering processes and related parton distributions. Cheng and Wu 23 are perhaps the first to discuss the high-energy scattering in the Iight-front frame and later Bjorken, Kogut and Soper 24 gave a very thorough discussion of this in the context of QED. The rationale for using this coordinate system here is that high-energy particles travel almost with the speed of light and for these particles the light-front coordinates are more natural, just as Galilean coordinates are more suitable for non-relativistic particles. For composite particles participating in high-energy scattering, their structures are described by parton distributions, à la Feynman. ln QCD, the quark distribution in the nucleon is 1 f dà. - f 1 (x) = - - e'h.x(pit)j(ü)-y+tji(àn)ip) 2p+ 21T (9) where n is a four vector along the x - direction and IP) represents the nucleon state with momentum P. Due to the high-energy nature of the scattering, the relevant structure fonction is a light-front correlation! 22 The natural choice of the coordinate system for calculating/ 1 (x) is the light-front system. In this sytem, the light-front correlation becomes a "static" correlation and the dynamics is completely 134

13 included in the light-front wave fonction of the nucleon. 25 In Lorentz frames, however, we are faced with a dynamical correlation which is much more difficult to calculate because we have to translate the field from one time to another with correct dynamics, and knowing the ground state is not sufficient. The calculntion off 1 (x) in the light-front frame also preserves the boo. t invariance along the 3-direction, a property very difficult to majntain in L remzframe modeling of the nucleon. The light-frame was first introduced in Its practical use came 16 years later. After QCD was invented in 1973, light-front investigations again waned until very recently. Now perturbative QCD has achieved a great success, and most of the non-perturbative QCD methods, including mode! building and lattice calculation, have been pushed to their limits. Light-front QCD provides a fresh look at QCD that has a clear physical picture and a close connection with the fondamental theory. It deserves to be further explored. Acknowledgments I thank R. Jackiw for suggesting this paper and discussions on the quantization procedure. I am indebted to T. D. Lee and z. M. Qiu for their hospitality al China Center of Advanced Science and Technology where part of this work was done. Discussion with M. Burkardt on perturbative renormalization and comments from R. Perry and S. Pinsky are also acknowledged. This work was supported in part by funds provided by the U.S. Department of Energy (D.O.E.) under contract #DE-AC02-76ER XIANGDONG JI Center for Theoretica/ Physics, Laboratory for Nuclear Science and Department of Physics, Massachuserts lnstitute of Technology, Cambridge, Massachusetts and China Center of Advanced Science and Techno/ogy, The A cademy of Sciences of China, Beijing , PRC References 1. S. Fubini and G. Furlan, Physics 229 (1965). 2. V. DeAlfaro, S. Fubini, G. Furlan and C. Rossetti, Currents in Hadron Physics (North-Holland/American Elsevier, Amsterdam/New York, 1973). 135

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