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Redalyc.Gauge Theories on the Light-Front Brazilian Journal of Physics ISSN: 0103-9733 [email protected] Sociedade Brasileira de Física Brasil Brodsky, Stanley J. Gauge Theories on the Light-Front Brazilian Journal of Physics, vol. 34, núm. 1A, march, 2004, pp. 157-165 Sociedade Brasileira de Física Sâo Paulo, Brasil Available in: http://www.redalyc.org/articulo.oa?id=46400102 How to cite Complete issue Scientific Information System More information about this article Network of Scientific Journals from Latin America, the Caribbean, Spain and Portugal Journal's homepage in redalyc.org Non-profit academic project, developed under the open access initiative Brazilian Journal of Physics, vol. 34, no. 1A, March, 2004 157 Gauge Theories on the Light-Front Stanley J. Brodsky Stanford Linear Accelerator Center, Stanford University, Stanford, California 94309, USA Received on 10 February, 2003. The light-front quantization of gauge theories in light-cone gauge provides a frame-independent wavefunction representation of relativistic bound states, simple forms for current matrix elements, explicit unitary, and a trivial vacuum. The light-front Hamiltonian form of QCD provides an alternative to lattice gauge theory for the computation of nonperturbative quantities such as the hadronic spectrum and the corresponding eigenfunctions. In the case of the electroweak theory, spontaneous symmetry breaking is represented by the appearance of zero modes of the Higgs field. Light-front quantization then leads to an elegant ghost-free theory of massive gauge particles, automatically incorporating the Lorentz and ’t Hooft conditions, as well as the Goldstone boson equivalence theorem. + ¡ 2 1 Introduction The operator HLC = P P ¡P?; the “light-cone Hamilto- nian”, is frame-independent. This can in principle be solved One of the challenges of relativistic quantum field theory by diagonalizing the matrix hnjHLC jmi on the free Fock is to compute the wavefunctions of bound states such as basis: [2] the amplitudes which determine the quark and gluon sub- X 2 structure of hadrons in quantum chromodynamics. In light- hnjHLC jmi hmjÃi = M hnjªi : (2) front quantization [1], one fixes the initial boundary con- m ditions of a composite system as its constituents are inter- 2 0 cepted by a single light-wave evaluated on the hyperplane The eigenvalues fM g of HLC = HLC + VLC give the x+ = t + z=c. The light-front quantization of QCD pro- squared invariant masses of the bound and continuum spec- vides a frame-independent, quantum-mechanical represen- trum of the theory. The light-front Fock space is the eigen- tation of a hadron at the amplitude level, capable of en- states of the free light-front Hamiltonian; i.e., it is a Hilbert coding its multi-quark, hidden-color and gluon momentum, space of non-interacting quarks and gluons, each of which m2+k2 helicity, and flavor correlations in the form of universal 2 2 ¡ ? satisfy k = m and k = k+ ¸ 0: The projections process-independent and frame-independent hadron wave- fhnjªig of the eigensolution on the n-particle Fock states functions [2]. Remarkably, quantum fluctuations of the vac- provide the light-front wavefunctions. Thus solving a quan- uum are absent if one uses light-front time to quantize the tum field theory is equivalent to solving a coupled many- system, so that matrix elements such as the electromagnetic body quantum mechanical problem: form factors only depend on the currents of the constituents " # Z described by the light-cone wavefunctions. The degrees of Xn 2 2 X 2 m + k? 0 freedom associated with vacuum phenomena such as spon- M ¡ Ãn = hnjVLC jn i Ãn0 (3) xi taneous symmetry breaking in the Higgs model have their i=1 n0 counterpart in light-front k+ = 0 zero modes of the fields. where the convolution and sum is understood over the Fock In Dirac’s “Front Form” [1], the generator of light-front number, transverse momenta, plus momenta, and helicity of time translations is P ¡ = i @ : Boundary conditions are set @¿ the intermediate states. Light-front wavefunctions are also x x¡ = z ¡ ct on the transverse plane labelled by ? and . related to momentum-space Bethe-Salpeter wavefunctions P ¡ Given the Lagrangian of a quantum field theory, can be by integrating over the relative momenta k¡ = k0 ¡ kz constructed as an operator on the Fock basis, the eigenstates since this projects out the dynamics at x+ = 0: of the free theory. Since each particle in the Fock basis is The light-front quantization of gauge theory can be most k2 +m2 ¡ 0 3 ? + on its mass shell, k ´ k ¡ k = k+ ; and its energy conveniently carried out in the light-cone gauge A = 0 1 + ¡ 0 z ¡ k = 2 (k + k ) is positive, only particles with positive A + A = 0. In this gauge the A field becomes a depen- momenta k+ ´ k0 + k3 ¸ 0 can occur in the Fock basis. dent degree of freedom, and it can be eliminated from the + P + Since the total plus momentum P = n kn is conserved, Hamiltonian in favor of a set of specific instantaneous light- the light-cone vacuum cannot have any particle content. front time interactions. In fact in QCD(1 + 1) theory, this The Heisenberg equation on the light-front is instantaneous interaction provides the confining linear x¡ interaction between quarks. In 3 + 1 dimensions, the trans- 2 ? HLC j ªi = M j ªi : (1) verse field A propagates massless spin-one gluon quanta 158 Stanley J. Brodsky with polarization vectors [3] which satisfy both the gauge orbital angular momenta + condition ²¸ = 0 and the Lorentz condition k ¢ ² = 0. à ! LF quantization is especially useful for quantum chro- @ @ lz = ¡i k1 ¡ k2 (5) modynamics, since it provides a rigorous extension of j j @k2 j @k1 many-body quantum mechanics to relativistic bound states: j j the quark, and gluon momenta and spin correlations of a derives from the n ¡ 1 relative momenta. This excludes hadron become encoded in the form of universal process- the contribution to the orbital angular momentum due to the independent, Lorentz-invariant wavefunctions [4]. For ex- motion of the center of mass, which is not an intrinsic prop- ample, the eigensolution of a meson in QCD, projected on erty of the hadron. The numerator structure of the light- the eigenstates fj nig of the free Hamiltonian HQCD(g = LC front wavefunctions is in large part determined by the angu- 0) ¿ = t ¡ z=c at fixed with the same global quantum num- lar momentum constraints. Thus wavefunctions generated bers, has the expansion: by perturbation theory provide a template for the numerator ¯ E structure of nonperturbative light-front wavefunctions. ¯ + ~ ¯ªM ; P ; P?; ¸ = Dae Sung Hwang, Bo-Qiang Ma, Ivan Schmidt, and I X Z 2 [5] have shown that the light-front wavefunctions generated n d k?idxi ¦i=1 p 3 by the radiative corrections to the electron in QED provide xi16¼ n¸2;¸i a simple system for understanding the spin and angular mo- 0 1 à ! mentum decomposition of relativistic systems. This pertur- Xn Xn 3 (2) ~ bative model also illustrates the interconnections between £16¼ ± @1 ¡ xjA ± k?` (4) Fock states of different particle number. The model is pat- j ` terned after the quantum structure which occurs in the one- ¯ E ¯ + ~ ~ ~ loop Schwinger ®=2¼ correction to the electron magnetic £ ¯n; xiP ; xiP? + k?i; ¸i Ãn=M (xi; k?i; ¸i): 1 moment [6]. In effect, we can represent a spin- 2 system 1 as a composite of a spin- 2 fermion and spin-one vector The set of light-front Fock state wavefunctions fÃn=M g rep- boson. A similar model has been used to illustrate the ma- resent the ensemble of quark and gluon states possible when trix elements and evolution of light-front helicity and orbital the meson is intercepted at the light-front. The light-front angular momentum operators [7]. This representation of a momentum fractions x = k+=P + = (k0 + kz)=(P 0 + P z) P i i ¼P i composite system is particularly useful because it is based n ~ n ~ ~ with i=1 xi = 1 and k?i with i=1 k?i = 0? represent on two constituents but yet is totally relativistic. We can then 2 2 the relative momentum coordinates of the QCD constituents explicitly compute the form factors F1(q ) and F2(q ) of and are independent of the total momentum of the state. the electromagnetic current and the various contributions to Remarkably, the scalar light-front wavefunctions the form factors A(q2) and B(q2) of the energy-momentum ~ Ãn=p(xi; k?i; ¸i) are independent of the proton’s momen- tensor. + 0 z tum P = P + P , and P?. (The light-cone spinors and Recently Ji, Ma, and Yuan [8] have derived perturbative polarization vectors multiplying Ãn=p are functions of the QCD counting rules for light-front wavefunctions with gen- absolute coordinates.) Thus once one has solved for the eral values of orbital angular momentum which constrain light-front wavefunctions, one can compute hadron matrix their form at large transverse momentum. elements of currents between hadronic states of arbitrary momentum. The actual physical transverse momenta are ~ ~ z ~p?i = xiP? + k?i: The ¸i label the light-front spin S pro- 3 Applications of Light-Front Wave- jections of the quarks and gluons along the quantization z direction. The spinors of the light-front formalism automat- functions ically incorporate the Melosh-Wigner rotation. The physical gluon polarization vectors ²¹(k; ¸ = §1) are specified in Matrix elements of spacelike currents such as spacelike elec- light-cone gauge by the conditions k ¢² = 0; ´ ¢² = ²+ = 0: tromagnetic form factors have an exact representation in The parton degrees of freedom are thus all physical; there terms of simple overlaps of the light-front wavefunctions in are no ghost or negative metric states.
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