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{{
<!--Introduction-->
[[File:World line.svg|thumb|right|alt=A light cone|The light cone of special relativity. Light-front quantization
uses light-front (or light-cone) coordinates to select an initial surface that is tangential to the light cone. Equal-time quantization uses an initial surface that is horizontal, labeled here as the "hypersurface of the present".]]
The '''light-front quantization'''<ref name="whitepaper">{{ cite journal |author1=B. L. G. Bakker |author2=A. Bassetto |author3=S. J. Brodsky |author4=W. Broniowski |author5=S. Dalley |author6=T. Frederico |author7=S. D. Glazek |author8=J. R. Hiller |
Line 14 ⟶ 12:
doi= 10.1016/j.nuclphysbps.2014.05.004
|volume=251–252
|bibcode=2014NuPhS.251..165B|s2cid=117029089 |display-authors=etal}}</ref><ref name="Burkardt">{{ Cite book |
last=Burkardt |first=Matthias |chapter=Light Front Quantization |
journal=[[Advances in Nuclear Physics]] | volume= 23 | pages= 1–74 | year=1996 |
doi= 10.1007/0-306-47067-5_1|arxiv=hep-ph/9505259 |
isbn= 978-0-306-45220-8 |s2cid=19024989 }}</ref><ref name="PhysRep">{{ cite journal |author1=S.J. Brodsky |author2=H.-C. Pauli |author3=S.S. Pinsky |
title= Quantum chromodynamics and other field theories on the light cone |
journal=[[Physics Reports]] | volume= 301 |issue=4–6 | pages= 299–486 | year=1998 |
doi= 10.1016/S0370-1573(97)00089-6 | bibcode=1998PhR...301..299B|arxiv = hep-ph/9705477 |s2cid=118978680 }}</ref>
of [[Quantum field theory|quantum field theories]]
provides a useful alternative to ordinary equal-time
Line 28 ⟶ 25:
particular, it can lead to a [[Special relativity|relativistic]] description of [[Bound state|bound systems]]
in terms of [[Quantum mechanics|quantum-mechanical]] [[wave function]]s. The quantization is
based on the choice of [[light-front coordinates]],<ref name="Dirac">{{ cite journal |
author= P. A. M. Dirac |
title= Forms of Relativistic Dynamics |
Line 34 ⟶ 31:
issue= 3 | pages= 392–399 | year=1949 |
doi= 10.1103/RevModPhys.21.392 | bibcode=1949RvMP...21..392D|doi-access= free }}</ref>
where <math>x^+\equiv ct+z</math> plays the role of time and the corresponding spatial coordinate is <math>x^-\equiv ct-z</math>. Here, <math>t</math> is the ordinary time, <math>z</math>
is one [[Cartesian coordinate system|Cartesian coordinate]], and <math>c</math> is the speed of light. The other two Cartesian coordinates, <math>x</math> and <math>y</math>, are untouched and often called transverse or perpendicular, denoted by symbols of the type
<math>\vec x_\perp = (x,y)</math>. The choice of the [[frame of reference]] where the time
<math>t</math> and <math>z</math>-axis are defined can be left unspecified in an exactly soluble relativistic theory, but in practical calculations some choices may be more suitable than others.
== Overview ==
In practice, virtually all measurements are made at fixed light-front time. For example, when an [[electron]] [[scattering|scatters]] on a [[proton]] as in the famous [[SLAC]] experiments that discovered the [[quark]] structure of [[hadrons]], the interaction with the constituents occurs at a single light-front time. When one takes a flash photograph, the recorded image shows the object as the front of the [[Electromagnetic radiation|light wave]] from the flash crosses the object. Thus [[Paul Dirac|Dirac]] used the terminology "light-front" and "front form" in contrast to ordinary instant time and "instant form".<ref name="Dirac" />
Light waves traveling in the negative <math>z</math> direction continue to propagate in <math>x^-</math> at a single light-front time <math>x^+</math>.
As emphasized by Dirac, [[Lorentz boost#boost|Lorentz boosts]] of states at fixed light-front time are simple [[kinematic]] transformations. The description of physical systems in light-front coordinates is unchanged by light-front boosts to frames moving with respect to the one specified initially. This also means that there is a separation of external and internal coordinates (just as in nonrelativistic systems), and the internal wave functions are independent of the external coordinates, if there is no external force or field. In contrast, it is a difficult dynamical problem to calculate the effects of boosts of states defined at a fixed instant time <math>t</math>.
The description of a bound state in a quantum field theory, such as an atom in [[quantum electrodynamics]] (QED) or a hadron in [[quantum chromodynamics]] (QCD), generally requires multiple wave functions, because quantum field theories include processes which [[particle creation|create]] and [[annihilation|annihilate]] particles. The state of the system then does not have a definite number of particles, but is instead a quantum-mechanical linear combination of [[Fock state]]s, each with a definite particle number. Any single measurement of particle number will return a value with a probability determined by the [[probability amplitude|amplitude]] of the Fock state with that number of particles. These amplitudes are the light-front wave functions. The light-front wave functions are each frame-independent and independent of the total [[momentum]].
The wave functions are the solution of a field-theoretic analog of the [[Schrödinger equation]]
<math>H\psi=E\psi</math> of nonrelativistic quantum mechanics. In the nonrelativistic theory the
[[Hamiltonian (quantum mechanics)|Hamiltonian]] [[Operator (physics)|operator]]
<math>H</math> is just a kinetic piece <math>-\frac{\hbar^2}{2m}\nabla^2</math> and a [[potential energy|potential]] piece <math>V(\vec r)</math>. The wave function <math>\psi</math> is a function of the coordinate <math>\vec r</math>, and
<math>E</math> is the [[energy]]. In light-front quantization, the formulation is
usually written in terms of light-front momenta
Line 99 ⟶ 54:
<math>p_i^+\equiv\sqrt{p_i^2+m_i^2}+p_{iz}</math>,
<math>\vec p_{\perp i}=(p_{ix},p_{iy})</math>, and <math>m_i</math> the particle [[mass]], and light-front
energies <math>p_i^-\equiv\sqrt{p_i^2+m_i^2}-p_{iz}</math>. They satisfy the [[mass shell|mass-shell]] condition <math>m_i^2=p_i^+p_i^--\vec p_{\perp i}^2</math>
The analog of the nonrelativistic Hamiltonian <math>H</math> is the light-front operator <math>\mathcal{P}^-</math>, which generates [[Translation (physics)#Translations in physics|translations]] in light-front time. It is constructed from the [[Lagrangian (field theory)|Lagrangian]] for the chosen quantum field theory. The total light-front momentum of the system, <math>\underline{P}\equiv(P^+,\vec P_\perp)</math>, is the sum of the single-particle light-front momenta. The total light-front energy <math>P^-</math> is fixed by the mass-shell condition to be <math>(M^2+P_\perp^2)/P^+</math>, where <math>M</math> is the invariant mass of the system. The Schrödinger-like equation of light-front quantization is then <math>\mathcal{P}^-\psi=\frac{M^2+P_\perp^2}{P^+}\psi</math>. This provides a foundation for a [[nonperturbative]] analysis of quantum field theories that is quite distinct from the [[Lattice gauge theory|lattice]] approach.<ref name="Wilson">{{ cite journal |
author=K. G. Wilson |
title= Confinement of Quarks |
Line 122 ⟶ 62:
issue= 8 | pages= 2445–2459 | year=1974 |
doi= 10.1103/PhysRevD.10.2445
| bibcode=1974PhRvD..10.2445W }}</ref><ref name="Gattringer">{{ cite book |
first1=C. | last1=Gattringer |
first2=C.B. | last2=Lang |
Line 129 ⟶ 68:
publisher=Springer |
___location=Berlin |
date=2010 }}</ref><ref name="Rothe">{{ cite book |
first=H. | last=Rothe |
title= Lattice Gauge Theories: An Introduction 4e |
Line 137 ⟶ 75:
date=2012 }}</ref>
Quantization on the light-front provides the rigorous field-theoretical realization of the intuitive ideas of the [[Parton (particle physics)|parton model]] which is formulated at fixed <math>t</math> in the infinite-momentum frame.<ref name="Feynman:1969ej">{{ cite journal |
author= R. P. Feynman |
title= Very high-energy collisions of hadrons |
Line 147 ⟶ 81:
issue= 24 | pages= 1415–1417 | year=1969 |
doi= 10.1103/PhysRevLett.23.1415 | bibcode=1969PhRvL..23.1415F|
url= https://authors.library.caltech.edu/3871/1/FEYprl69.pdf }}</ref><ref name="KogutSusskind">{{ cite journal |author1=J. B. Kogut |author2=L. Susskind |
title=The parton picture of elementary particles |
journal=[[Physics Reports]] | volume= 8 |issue=2 | pages= 75–172 | year=1973 |
doi= 10.1016/0370-1573(73)90009-4|bibcode = 1973PhR.....8...75K }}</ref>
(see [[#Infinite momentum frame]]). The same results are obtained in the front form for any frame; e.g., the structure functions and other probabilistic parton distributions measured in [[deep inelastic scattering]] are obtained from the squares of the boost-invariant light-front wave functions,<ref name="Brodsky:2003pw">{{ cite journal |author1=S. J. Brodsky |author2=J. R. Hiller |author3=D. S. Hwang |author4=V. A. Karmanov |
title= The Covariant structure of light front wave functions and the behavior of hadronic form-factors |
journal=[[Physical Review D]] | volume= 69 |issue=7 | page= 076001 | year=2004 |
doi= 10.1103/PhysRevD.69.076001|arxiv = hep-ph/0311218 |bibcode = 2004PhRvD..69g6001B |s2cid=855584 }}</ref>
the eigensolution of the light-front Hamiltonian. The [[James Bjorken|Bjorken]] kinematic variable <math>x_{bj}</math> of deep inelastic scattering becomes identified with the light-front fraction at small
<math>x</math>. The Balitsky–Fadin–Kuraev–Lipatov
(BFKL)<ref name="Fadin:1998py">{{ cite journal |author1=V. S. Fadin |author2=L. N. Lipatov |
title= BFKL pomeron in the next-to-leading approximation |
journal=[[Physics Letters B]] | volume= 429 |issue=1–2 | pages= 127–134 | year=1998 |
doi= 10.1016/S0370-2693(98)00473-0|arxiv = hep-ph/9802290 |bibcode = 1998PhLB..429..127F |s2cid=15965017 }}</ref>
Regge behavior of structure functions can be demonstrated from the behavior of light-front wave functions at small <math>x</math>. The Dokshitzer–Gribov–Lipatov–Altarelli–Parisi ([[DGLAP]]) evolution<ref name="Salam:1999cn">{{ cite journal |
author= G. P. Salam |
title= An Introduction to leading and next-to-leading BFKL |
Line 179 ⟶ 102:
pages= 3679–3705 | year=1999 |
arxiv=hep-ph/9910492|bibcode = 1999AcPPB..30.3679S }}</ref>
of structure functions and the Efremov–Radyushkin–Brodsky–Lepage (ERBL) evolution<ref name="Lepage:1980fj">{{ cite journal |author1=G. P. Lepage |author2=S. J. Brodsky |
title= Exclusive Processes in Perturbative Quantum Chromodynamics |
journal=[[Physical Review D]] | volume= 22 |issue=9 | pages= 2157–2198 | year=1980 |
doi= 10.1103/PhysRevD.22.2157 | bibcode=1980PhRvD..22.2157L|osti=1445541 }}</ref><ref name="Efremov:1979qk">{{ cite journal |author1=A. V. Efremov |author2-link=Anatoly Radyushkin |author2=A. V. Radyushkin |
title= Factorization and Asymptotical Behavior of Pion Form-Factor in QCD |
journal=[[Physics Letters B]] | volume= 94 |issue=2 | pages= 245–250 | year=1980 |
doi= 10.1016/0370-2693(80)90869-2 | bibcode=1980PhLB...94..245E}}</ref>
of distribution amplitudes in <math>\log Q^2</math> are properties of the light-front wave functions at high transverse momentum.
Computing hadronic matrix elements of currents is particularly simple on the light-front, since they can be obtained rigorously as overlaps of light-front wave functions as in the Drell–Yan–West formula.<ref name="Drell:1969km">{{ cite journal |author1=S. D. Drell |author2=T. -M. Yan |
title=Connection of Elastic Electromagnetic Nucleon Form-Factors at Large <math>Q^2</math> and Deep Inelastic Structure Functions Near Threshold |
journal=[[Physical Review Letters]] | volume= 24 |issue=4 | pages= 181–186 | year=1970 |
doi= 10.1103/PhysRevLett.24.181
| bibcode=1970PhRvL..24..181D|
author= G. B. West |
title= Phenomenological model for the electromagnetic structure of the proton |
journal=[[Physical Review Letters]] | volume= 24 |
issue= 21 | pages= 1206–1209 | year=1970 |
doi= 10.1103/PhysRevLett.24.1206 | bibcode=1970PhRvL..24.1206W}}</ref><ref name="Brodsky:1980zm">{{ cite journal |author1=S. J. Brodsky |author2=S. D. Drell |
title= The Anomalous Magnetic Moment and Limits on Fermion Substructure |
journal=[[Physical Review D]] | volume= 22 |issue=9 | pages= 2236–2243 | year=1980 |
doi= 10.1103/PhysRevD.22.2236 | bibcode=1980PhRvD..22.2236B|osti=1445649 }}</ref>
[[File:ComptonScattering.jpg|thumbnail|right|alt=Compton scattering|Compton scattering of a photon by an electron]]
The [[Gauge theory|gauge]]-invariant [[meson]] and [[baryon]] distribution amplitudes which control hard exclusive and direct reactions are the [[Quark model|valence]] light-front wave functions integrated over transverse momentum at fixed <math>x_i= {k^+ _i/ P^+}</math>. The "ERBL" evolution<ref name="Lepage:1980fj" /><ref name="Efremov:1979qk" /> of distribution amplitudes and the factorization theorems for hard exclusive processes can be derived most easily using light-front methods. Given the frame-independent light-front wave functions, one can compute a large range of hadronic observables including generalized parton distributions, Wigner distributions, etc. For example, the "handbag" contribution to the generalized parton distributions for deeply virtual [[Compton scattering]], which can be computed from the overlap of light-front wave functions, automatically satisfies the known [[Sum rule in quantum mechanics|sum rules]].
The light-front wave functions contain information about novel features of QCD. These include effects suggested from other approaches, such as [[color transparency]], hidden color, intrinsic [[Charm (quantum number)|charm]], [[Quark#Sea quarks|sea-quark]] symmetries, dijet diffraction, direct hard processes, and hadronic [[Spin (physics)|spin]] dynamics.
[[File:DIS parton model.svg|thumbnail|right|alt=Deep inelastic scattering|Deep inelastic electron-proton scattering]]
One can also prove fundamental theorems for relativistic quantum field theories using the front form, including:
(a) the [[cluster decomposition theorem]]<ref name="Brodsky:1985gs">{{ cite journal |author1=S. J. Brodsky |author2=C.-R. Ji |
title= Factorization Property of the Deuteron |
journal=[[Physical Review D]] | volume= 33 |issue=9 | pages= 2653–2659 | year=1986 |
doi= 10.1103/PhysRevD.33.2653|pmid=9956950 |bibcode = 1986PhRvD..33.2653B |osti=1447785 }}</ref>
and (b) the vanishing of the anomalous gravitomagnetic moment for any Fock state of a
hadron;<ref name="Brodsky:2000ii">{{ cite journal |author1=S. J. Brodsky |author2=D. S. Hwang |author3=B.-Q. Ma |author4=I. Schmidt |
title= Light cone representation of the spin and orbital angular momentum of relativistic composite systems |
journal=[[Nuclear Physics B]] | volume= 593 |issue=1–2 | pages= 311–335 | year=2001 |
doi= 10.1016/S0550-3213(00)00626-X|arxiv = hep-th/0003082 |bibcode = 2001NuPhB.593..311B |s2cid=7435760 }}</ref>
one also can show that a nonzero [[Anomalous magnetic dipole moment|anomalous magnetic moment]] of a bound state requires nonzero [[angular momentum]] of the constituents. The cluster properties<ref name="Antonuccio:1997tw">{{ cite journal |author1=F. Antonuccio |author2=S. J. Brodsky |author3=S. Dalley |
title= Light cone wave functions at small <math>x</math> |
journal=[[Physics Letters B]] | volume= 412 |issue=1–2 | pages= 104–110 | year=1997 |
doi= 10.1016/S0370-2693(97)01067-8|arxiv = hep-ph/9705413 |bibcode = 1997PhLB..412..104A |s2cid=118926903 }}</ref>
of light-front time-ordered [[Perturbation theory (quantum mechanics)|perturbation theory]], together with <math>J^z</math> conservation, can be used to elegantly derive the Parke–Taylor rules for multi-[[gluon]] scattering amplitudes.<ref name="Cruz-Santiago:2013vta">{{ cite journal |author1=C. A. Cruz-Santiago |author2=A. M. Stasto |
title= Recursion relations and scattering amplitudes in the light-front formalism |
journal=[[Nuclear Physics B]] | volume= 875 |issue=2 | pages= 368–387 | year=2013 |
doi= 10.1016/j.nuclphysb.2013.07.019|arxiv = 1308.1062 |bibcode = 2013NuPhB.875..368C |s2cid=119214902 }}</ref>
The counting-rule<ref name="Brodsky:1994kg">{{ cite journal |author1=S. J. Brodsky |last2=Burkardt |first2=Matthias |author3=I. Schmidt |
title= Perturbative QCD constraints on the shape of polarized quark and gluon distributions |
journal=[[Nuclear Physics B]] | volume= 441 |issue=1–2 | pages= 197–214 | year=1995 |
doi= 10.1016/0550-3213(95)00009-H|arxiv = hep-ph/9401328 |bibcode = 1995NuPhB.441..197B |s2cid=118969788 }}</ref>
behavior of structure functions at large <math>x</math> and Bloom–Gilman duality<ref name="BloomGilman1">{{ cite journal |author1=E. Bloom |author2=F. Gilman |
title= Scaling, Duality, and the Behavior of Resonances in Inelastic electron-Proton Scattering |
journal=[[Physical Review Letters]] | volume= 25 |issue=16 | pages= 1140–1143 | year=1970 |
doi= 10.1103/PhysRevLett.25.1140 | bibcode=1970PhRvL..25.1140B|citeseerx=10.1.1.412.3968 }}</ref><ref name="BloomGilman2">{{ cite journal |author1=E. Bloom |author2=F. Gilman |
title=Scaling and the Behavior of Nucleon Resonances in Inelastic electron-Nucleon Scattering |
journal=[[Physical Review D]] | volume= 4 |issue=9 | pages= 2901–2916 | year=1971 |
doi= 10.1103/PhysRevD.4.2901|bibcode = 1971PhRvD...4.2901B |citeseerx=10.1.1.412.5779 }}</ref>
have also been derived in light-front QCD (LFQCD). The existence of "lensing effects" at leading twist, such as the
<math>T</math>-odd "Sivers effect" in spin-dependent semi-inclusive deep-inelastic scattering, was first demonstrated using light-front methods.<ref name="Brodsky:2002cx">{{ cite journal |author1=S. J. Brodsky |author2=D. S. Hwang |author3=I. Schmidt |
title= Final state interactions and single spin asymmetries in semiinclusive deep inelastic scattering |
journal=[[Physics Letters B]] | volume= 530 |issue=1–4 | pages= 99–107 | year=2002 |
doi= 10.1016/S0370-2693(02)01320-5|arxiv = hep-ph/0201296 |bibcode = 2002PhLB..530...99B |s2cid=13446844 }}</ref>
Light-front quantization is thus the natural framework for the description of the nonperturbative relativistic bound-state structure of hadrons in quantum chromodynamics. The formalism is rigorous, relativistic, and frame-independent. However, there exist subtle problems in LFQCD that require thorough investigation. For example, the complexities of the [[Vacuum#Quantum mechanics|vacuum]] in the usual instant-time formulation, such as the [[Higgs mechanism]] and [[Vacuum expectation value|condensates]] in <math>\phi^4</math> theory, have their counterparts in [[zero modes]] or, possibly, in additional terms in the LFQCD Hamiltonian that are allowed by power counting.<ref name="Wilsonetal">{{ cite journal |author1=K. G. Wilson |author2=T. S. Walhout |author3=A. Harindranath |author4=W.-M. Zhang |author5=R. J. Perry |author6=S. D. Glazek |
title=Nonperturbative QCD: A Weak coupling treatment on the light front |
journal=[[Physical Review D]] | volume= 49 |issue=12 | pages= 6720–6766 | year=1994 |
doi= 10.1103/PhysRevD.49.6720|pmid=10016996 |arxiv = hep-th/9401153 |bibcode = 1994PhRvD..49.6720W |s2cid=119422380 }}</ref>
Light-front considerations of the vacuum as well as the problem of achieving full [[Lorentz covariance|covariance]] in LFQCD require close attention to the light-front [[Mathematical singularity|singularities]] and zero-mode contributions.<ref name="Nambu">{{ cite journal |author1=Y. Nambu |author2=G. Jona-Lasinio |
title=Dynamical model of elementary particles based on an analogy with auperconductivity |
journal=[[Physical Review]] | volume= 122 |issue=1 | pages= 345–358 | year=1961 |
doi= 10.1103/PhysRev.122.345|bibcode = 1961PhRv..122..345N |doi-access=free }}</ref><ref name="GOR">{{ cite journal |author1=M. Gell-Mann |author2=R. J. Oakes |author3=B. Renner |
title=Behavior of current divergences under SU(3) x SU(3) |
journal=[[Physical Review]] | volume= 175 |issue=5 | pages= 2195–2199 | year=1968 |
doi= 10.1103/PhysRev.175.2195|bibcode = 1968PhRv..175.2195G |url=https://authors.library.caltech.edu/3634/1/GELpr68.pdf }}</ref><ref name="tHooftVeltman">{{ cite journal |author1=G. 't Hooft |author2=M. Veltman |
title=Regularization and renormalization of gauge fields |
journal=[[Nuclear Physics B]] | volume= 44 |issue=1 | pages= 189–213 | year=1972 |
doi= 10.1016/0550-3213(72)90279-9|bibcode = 1972NuPhB..44..189T |hdl=1874/4845 |url=https://repositorio.unal.edu.co/handle/unal/81144 |hdl-access=free }}</ref><ref name="SVZ">{{ cite journal |author1=M. A. Shifman |author2=A.I. Vainshtein |author3=V. I. Zakharov |
title=QCD and Resonance Physics: Applications |
journal=[[Nuclear Physics B]] | volume= 147 |issue=5 | pages= 448–518 | year=1979 |
doi= 10.1016/0550-3213(79)90023-3|bibcode = 1979NuPhB.147..448S }}</ref><ref name="FeynmanQCD2">{{ cite journal |
author=R. P. Feynman |
title=The Qualitative Behavior of Yang-Mills Theory in (2+1)-Dimensions
journal=[[Nuclear Physics B]] | volume= 188 |
issue=3 | pages= 479–512 | year=1981 |
doi= 10.1016/0550-3213(81)90005-5|bibcode = 1981NuPhB.188..479F }}</ref><ref name="SSbreaking">{{ cite journal |
author=E. Witten |
title=Dynamical Breaking of Supersymmetry |
journal=[[Nuclear Physics B]] | volume= 188 |
issue=3 | pages= 513–554 | year=1981 |
doi= 10.1016/0550-3213(81)90006-7|bibcode = 1981NuPhB.188..513W }}</ref><ref name="GasserLeutwyler">{{ cite journal |author1=J. Gasser |author2=H. Leutwyler |
title=Chiral Perturbation Theory to One Loop |
journal=[[Annals of Physics]] | volume= 158 |issue=1 | pages= 142–210 | year=1984 |
doi= 10.1016/0003-4916(84)90242-2 |bibcode = 1984AnPhy.158..142G |url=
author=S. D. Glazek |
title=Light Front QCD in the Vacuum Background |
journal=[[Physical Review D]] | volume= 38 |
issue=10 | pages= 3277–3286 | year=1988 |
doi= 10.1103/PhysRevD.38.3277 |
pmid=9959077 |bibcode = 1988PhRvD..38.3277G }}</ref><ref name="Marisqq">{{ cite journal |author1=P. Maris |author2=C. D. Roberts |author3=P. C. Tandy |
title=Pion mass and decay constant |
journal=[[Physics Letters B]] | volume= 420 |issue=3–4 | pages= 267–273 | year=1998 |
doi= 10.1016/S0370-2693(97)01535-9|arxiv = nucl-th/9707003 |bibcode = 1998PhLB..420..267M |s2cid=16778465 }}</ref><ref name="Brodsky:2012ku">{{ cite journal |author1=S. J. Brodsky |author2=C. D. Roberts |author3=R. Shrock |author4=P. C. Tandy |
title= Confinement contains condensates |
journal=[[Physical Review C]] | volume= 85 |issue=6 | page= 065202 | year=2012 |
doi= 10.1103/PhysRevC.85.065202|arxiv = 1202.2376 |bibcode = 2012PhRvC..85f5202B |s2cid=118373670 }}</ref><ref name="CasherSusskind">{{ cite journal |author1=A. Casher |author2=L. Susskind |
title=Chiral magnetism (or magnetohadrochironics) |
journal=[[Physical Review D]] | volume= 9 |issue=2 | pages= 436–460 | year=1974 |
doi= 10.1103/PhysRevD.9.436|bibcode = 1974PhRvD...9..436C }}</ref>
The truncation of the light-front Fock-space calls for the introduction of effective quark and gluon degrees of freedom to overcome truncation effects. Introduction of such effective degrees of freedom is what one desires in seeking the dynamical connection between canonical (or current) quarks and effective (or constituent) quarks that Melosh sought, and [[Gell-Mann]] advocated, as a method for truncating QCD.
The light-front Hamiltonian formulation thus opens access to QCD at the amplitude level and is poised to become the foundation for a common treatment of [[Hadron spectroscopy|spectroscopy]] and the parton structure of hadrons in a single covariant formalism, providing a unifying connection between low-energy and high-energy experimental data that so far remain largely disconnected.
== Fundamentals ==
Front-form relativistic quantum mechanics was introduced by Paul Dirac in a 1949 paper published in Reviews of Modern Physics.<ref name="Dirac"/> Light-front quantum field theory is the front-form representation of local relativistic quantum field theory.
The relativistic invariance of a quantum theory means that the observables (probabilities, [[Expectation value (quantum mechanics)|expectation values]] and ensemble averages) have the same values in all [[inertial]] coordinate systems. Since different inertial coordinate systems are related by inhomogeneous [[Lorentz transformations]] ([[Henri Poincaré|Poincaré]] transformations), this requires that the Poincaré group is a symmetry group of the theory. [[Wigner]]<ref name="wigner:1939">{{ cite journal |
author=E.P. Wigner |
title=On unitary representations of the inhomogeneous Lorentz group |
Line 380 ⟶ 229:
doi= 10.2307/1968551|bibcode = 1939AnMat..40..149W |
jstor=1968551 |
s2cid=121773411 }}</ref>
and Bargmann<ref name="bargmann:1954">{{ cite journal |
author=V. Bargmann |
Line 389 ⟶ 238:
doi= 10.2307/1969831|
jstor=1969831 }}</ref>
showed that this symmetry must be realized by a unitary representation of the connected component of the Poincaré group on the Hilbert space of the quantum theory. The Poincaré symmetry is a dynamical symmetry because Poincaré transformations mix both space and time variables. The dynamical nature of this symmetry is most easily seen by noting that the Hamiltonian appears on the right-hand side of three of the [[commutators]] of the Poincaré generators,
<math>[K^j,P^k] = i\delta^{jk}H</math>, where <math>P^k</math> are components of the linear momentum and
<math>K^j</math> are components of rotation-less boost generators. If the Hamiltonian includes interactions, i.e. <math>H=H_0 +V</math>, then the commutation relations cannot be satisfied unless at least three of the Poincaré generators also include interactions.
Dirac's paper<ref name="Dirac" /> introduced three distinct ways to minimally include interactions in the [[Poincaré group|Poincaré Lie algebra]]. He referred to the different minimal choices as the "instant-form", "point-form" and "front-from" of the dynamics. Each "form of dynamics" is characterized by a different interaction-free (kinematic) subgroup of the Poincaré group. In Dirac's instant-form dynamics the kinematic subgroup is the three-dimensional Euclidean subgroup generated by spatial translations and rotations, in Dirac's point-form dynamics the kinematic subgroup is the Lorentz group and in Dirac's "light-front dynamics" the kinematic subgroup is the group of transformations that leave a three-dimensional hyperplane tangent to the [[light cone]] invariant.
A light front is a three-dimensional hyperplane defined by the condition:
Line 443 ⟶ 271:
</math>|{{EquationRef|3}}}}
In a front-form relativistic quantum theory the three interacting generators of the Poincaré group are <math>P^-:= H-\vec{P}\cdot \hat{n}</math>, the generator of translations normal to the light front, and <math>\vec{J}_{\perp}:= \vec{J} -\hat{n}(\hat{n} \cdot \vec{J})</math>, the generators of rotations transverse to the light-front. <math>P^-</math> is called the "light-front" Hamiltonian.
The kinematic generators, which generate transformations tangent to the light front, are free of interaction. These include <math>P^+:=
H+\vec{P}\cdot \hat{n}</math> and <math>\vec{P}_{\perp}:=
\vec{P} -\hat{n}( \hat{n} \cdot \vec{P})</math>,
which generate translations tangent to the light front,
<math>J_3:=\hat{n} \cdot \vec{J}</math> which generates rotations about the <math>\hat{n}</math> axis, and the generators
{{NumBlk|:|<math>
Line 477 ⟶ 295:
These properties have consequences that are useful in applications.
There is no loss of generality in using light-front relativistic quantum theories. For systems of a finite number of degrees of freedom there are explicit <math>S</math>-matrix-preserving unitary transformations that transform theories with light-front kinematic subgroups to equivalent theories with instant-form or point-form kinematic subgroups. One expects that this is true in quantum field theory, although establishing the equivalence requires a nonperturbative definition of the theories in different forms of dynamics.
=== Light-front Commutation Relations ===
[[Canonical commutation relations]] at equal time are the centerpiece of the [[canonical quantization]] method to quantized fields. In the standard quantization method (the "Instant Form" in Dirac's classification of relativistic dynamics<ref name="Dirac" />), the relations are, for example here for a spin-0 field <math> \phi </math> and its [[Conjugate_variables#Quantum_theory|canonical conjugate]] <math> \pi </math>:
<math display="block">{\rm Instant~Form:}~~[\phi(t, \vec x),\phi(t, \vec y)] = 0, \ \ [\pi(t, \vec x), \pi(t, \vec y)] = 0, \ \ [\phi(t, \vec x),\pi(t, \vec y)] = i\hbar \delta^3( \vec x- \vec y),</math>
where the relations are taken at equal time <math> t </math>, and <math> \vec x </math> and <math> \vec y </math> are the space variables. The equal-time requirement imposes that <math> \vec x - \vec y </math> is a [[Spacetime#Spacetime_interval|spacelike]] quantity. The non-zero value of the commutator <math>[\phi(t, \vec x),\pi(t, \vec y)]</math> expresses the fact that when <math> \phi </math> and <math> \pi </math> are separated by a spacelike distance, they cannot communicate with each other and thus commute, except when their separation <math> \vec x - \vec y \to 0</math>.<ref>{{cite book | last=Carroll | first=Sean | title=Spacetime and Geometry: An Introduction to General Relativity | publisher=Addison Wesley | year=2003 | isbn=0-8053-8732-3 | edition=Reprinted 2019 }}</ref>
In the Light-Front form however, fields at equal time <math> x^+ </math> are causally linked (i.e., they can communicate) since the Light-Front time <math> x^+ \equiv t-z </math> is along the light cone. Consequently, the Light-Front canonical commutation relations are different. For instance:<ref>{{cite book | last=Harindranath | first=A. | title=An Introduction to Light Front Dynamics for Pedestrians; In Light-Front Quantization and Non-Perturbative QCD | editor-last1=Vary | editor-first1=J.P. | editor-last2=Wolz | editor-first2=F. | publisher=International Institute of Theoretical and Applied Physics | ___location=Ames, IA | year=2000 | isbn=1-891815-00-8 | arxiv=hep-ph/9612244 }}</ref>
<math display="block">{\rm Light-Front~form:}~~[\phi(x^+, \vec x),\phi(x^+, \vec y)] = \frac{i}{4}\epsilon(x^- -y^-)\delta^2( \vec{x_\bot} - \vec{y_\bot}),</math>
where <math>\epsilon(x)=\theta(x)-\theta(-x)</math> is the antisymmetric [[Heaviside step function]].
On the other hand, the commutation relations for the [[creation and annihilation operators]] are similar for both the Instant and Light-Front forms:
<math display="block">{\rm Instant~Form:}~~[a(t, \vec k),a(t, \vec l)] = 0, \ \ [a^\dagger(t, \vec k),a^\dagger(t, \vec l)] = 0, \ \ [a(t, \vec k),a^\dagger(t, \vec l)]= \hbar \delta^3( \vec k- \vec l).</math>
<math display="block">{\rm Light-Front~form:}~~[a(x^+, \vec k),a(x^+, \vec l)] = 0, \ \ [a^\dagger(x^+, \vec k),a^\dagger(x^+, \vec l)] = 0, \ \ [a(x^+, \vec k),a^\dagger(x^+, \vec l)]= \hbar \delta(k^+-l^+) \delta^2( \vec{k_\bot}- \vec{l_\bot}).</math>
where <math> \vec k</math> and <math> \vec l</math> are the [[Wave vector|wavevectors]] of the fields, <math> k^+ = k_0 + k_3 </math> and <math> l^+ = l_0 + l_3 </math>.
=== Light-front boosts ===
In general if one multiplies a Lorentz boost on the right by a momentum-dependent rotation, which leaves the rest vector unchanged, the result is a different type of boost. In principle there are as many different kinds of boosts as there are momentum-dependent rotations. The most common choices are rotation-less boosts, [[Helicity (particle physics)|helicity]] boosts, and light-front boosts. The light-front boost ({{EquationNote|4}}) is a Lorentz boost that leaves the light front invariant.
The light-front boosts are not only members of the light-front kinematic subgroup, but they also form a closed three-parameter subgroup. This has two consequences. First, because the boosts do not involve interactions, the unitary representations of light-front boosts of an interacting system of particles are tensor products of single-particle representations of light-front boosts. Second, because these boosts form a subgroup, arbitrary sequences of light-front boosts that return to the starting frame do not generate Wigner rotations.
The spin of a particle in a relativistic quantum theory is the angular momentum of the particle in its [[rest frame]]. Spin observables are defined by boosting the particle's [[angular momentum tensor]] to the particle's rest frame
{{NumBlk|:|<math>
Line 530 ⟶ 343:
preserving boost, ({{EquationNote|4}}).
The light-front components of the spin are the components of the spin measured in the particle's rest frame after transforming the particle to its rest frame with the light-front preserving boost ({{EquationNote|4}}).
The light-front spin is invariant with respect to light-front preserving-boosts because these boosts do not generate Wigner rotations. The component of this spin along the <math>\hat{n}</math>
direction is called the light-front helicity. In addition to being invariant, it is also a kinematic observable, i.e. free of interactions. It is called a helicity because the spin quantization axis is determined by the orientation of the light front. It differs from the Jacob–Wick helicity, where the quantization axis is determined by the direction of the momentum.
These properties simplify the computation of current matrix elements because (1) initial and final states in different frames are related by kinematic Lorentz transformations, (2) the one-body contributions to the current matrix, which are important for hard scattering, do not mix with the interaction-dependent parts of the current under light front boosts and (3) the light-front helicities remain invariant with respect to the light-front boosts. Thus, light-front helicity is conserved by every interaction at every vertex.
Because of these properties, front-form quantum theory is the only form of relativistic dynamics that has true "frame-independent" impulse approximations, in the sense that one-body current operators remain one-body operators in all frames related by light-front boosts and the momentum transferred to the system is identical to the momentum transferred to the constituent particles. Dynamical constraints, which follow from rotational covariance and current covariance, relate matrix elements with different magnetic [[quantum numbers]]. This means that consistent impulse approximations can only be applied to linearly independent current matrix elements.
=== Spectral condition ===
A second unique feature of light-front quantum theory follows because the operator <math>P^+</math> is non-negative and kinematic. The kinematic feature means that the generator <math>P^+</math> is the sum of the non-negative single-particle <math>P_i^+</math> generators, (<math>P^+= \sum_i P_i^+)</math>. It follows that if <math>P^+</math> is zero on a state, then each of the individual <math>P_i^+</math> must also vanish on the state.
In perturbative light-front quantum field theory this property leads to a suppression of a large class of diagrams, including all vacuum diagrams, which have zero internal <math>P^+</math>. The condition <math>P^+=0</math>
corresponds to infinite momentum <math>(-P^3\to H)</math>. Many of the simplifications of light-front quantum field theory are realized in the infinite momentum limit<ref name="fubini:1965">{{ cite journal |author1=S. Fubini |title=Renormalization effects for partially conserved currents |author2=G. Furlan |
journal=Physics Physique Fizika | volume= 1 |issue=4 | page= 229 | year=1965 |doi=10.1103/PhysicsPhysiqueFizika.1.229 |doi-access=free }}</ref><ref name="weinberg:1966">{{ cite journal |
author= S. Weinberg |
title= Dynamics at infinite momentum |
Line 588 ⟶ 365:
of ordinary canonical field theory (see [[#Infinite momentum frame]]).
An important consequence of the spectral condition on <math>P^+</math> and the subsequent suppression of the vacuum diagrams in perturbative field theory is that the perturbative vacuum is the same as the free-field vacuum. This results in one of the great simplifications of light-front quantum field theory, but it also leads to some puzzles with regard the formulation of theories with [[spontaneous symmetry breaking|spontaneously broken symmetries]].
=== Equivalence of forms of dynamics ===
Sokolov<ref name="sokolov:1978">{{ cite journal |author1=S. N. Sokolov |author2=A. N. Shatini |
journal=Theoreticheskya I Matematicheskaya Fizika | volume= 37 | page= 291 | year=1978 }}</ref><ref name="polyzou:2010">{{ cite journal |
author= W. N. Polyzou |
title=Examining the equivalence of Bakamjian-Thomas mass operators in different forms of dynamics |
journal=[[Physical Review C]] | volume= 82 |
issue=6 | page= 064001 | year=2010 |
doi= 10.1103/PhysRevC.82.064001|arxiv = 1008.5222 |bibcode = 2010PhRvC..82f4001P
s2cid=26711947 }}</ref>
demonstrated that relativistic quantum theories based on different forms of dynamics are related by <math>S</math>-matrix-preserving unitary transformations. The equivalence in field theories is more complicated because the definition of the field theory requires a redefinition of the ill-defined local operator products that appear in the dynamical generators. This is achieved through renormalization. At the perturbative level, the ultraviolet divergences of a canonical field theory are replaced by a mixture of ultraviolet and infrared <math>(P^+=0)</math>
divergences in light-front field theory. These have to be renormalized in a manner that recovers the full rotational covariance and maintains the <math>S</math>-matrix equivalence. The [[renormalization]] of light front field theories is discussed in [[Light-front computational methods#Renormalization group]].
=== Classical vs quantum ===
One of the properties of the classical wave equation is that the light-front is a characteristic surface for the initial value problem. This means the data on the light front is insufficient to generate a unique evolution off of the light front. If one thinks in purely classical terms one might anticipate that this problem could lead to an ill-defined quantum theory upon quantization.
In the quantum case the problem is to find a set of ten self-adjoint operators that satisfy the Poincaré Lie algebra. In the absence of interactions, Stone's theorem applied to tensor products of known unitary irreducible representations of the Poincaré group gives a set of self-adjoint light-front generators with all of the required properties. The problem of adding interactions is no different<ref name="kato:1966">{{ cite book |
first=T. | last=Kato |
title=Perturbation Theory for Linear Operators |
Line 642 ⟶ 391:
date=1966 |
page=theorem 4.3 }}</ref>
than it is in non-relativistic quantum mechanics, except that the added interactions also need to preserve the commutation relations.
There are, however, some related observations. One is that if one takes seriously the classical picture of evolution off of surfaces with different values of <math>x^+</math>, one finds that the surfaces with
<math>x^+\not=0</math> are only invariant under a six parameter subgroup. This means that if one chooses a quantization surface with a fixed non-zero value of <math>x^+</math>, the resulting quantum theory would require a fourth interacting generator. This does not happen in light-front quantum mechanics; all seven kinematic generators remain kinematic. The reason is that the choice of light front is more closely related to the choice of kinematic subgroup, than the choice of an initial value surface.
In quantum field theory, the vacuum expectation value of two fields restricted to the light front are not well-defined distributions on test functions restricted to the light front. They only become well defined distributions on functions of four space time variables.<ref name="leutwyler:1970">{{ cite journal |author1=H. Leutwyler |author2=J.R. Klauder |author3=L. Streit |
title=Quantum field theory on lightlike slabs |
journal=[[Nuovo Cimento]] | volume= A66 |issue=3 | pages= 536–554 | year=1970 |
doi= 10.1007/BF02826338|bibcode = 1970NCimA..66..536L |s2cid=124546775 }}</ref><ref name="ullrich:2006">{{ cite journal |author1=P. Ullrich |author2=E. Werner |
title=On the problem of mass-dependence of the two-point function of the real scalar free massive field on the light cone |journal=[[Journal of Physics A]] | volume= 39 |issue=20 | pages= 6057–6068 | year=2006 |
doi= 10.1088/0305-4470/39/20/029|arxiv = hep-th/0503176 |bibcode = 2006JPhA...39.6057U |s2cid=32919998 }}</ref>
=== Rotational invariance ===
Line 685 ⟶ 417:
the theory recovered?
Given a dynamical unitary representation of rotations, <math>U(R)</math>, the product <math>U_0(R) U^{\dagger}(R)</math> of a kinematic rotation with the inverse of the corresponding dynamical rotation is a unitary operator that (1) preserves the <math>S</math>-matrix and (2) changes the kinematic subgroup to a kinematic subgroup with a rotated light front,
<math>\hat{n}'= R\hat{n}</math>. Conversely, if the <math>S</math>-matrix is invariant with respect to changing the orientation of the light-front, then the dynamical unitary representation of rotations,
<math>U(R)</math>, can be constructed using the generalized wave operators for different orientations of the light front<ref name="fuda:1990a">{{ cite journal |
author=M. Fuda |
title=A new picture for light front dynamics |
journal=[[Annals of Physics]] | volume= 197 |
issue=2 | pages= 265–299 | year=1990 |
doi= 10.1016/0003-4916(90)90212-7|bibcode = 1990AnPhy.197..265F }}</ref><ref name="fuda:1990b">{{ cite journal |
author= M. Fuda |
title=Poincaré invariant Lee model |
journal=[[Physical Review D]] | volume= 41 |
issue=2 | pages= 534–549 | year=1990 |
doi= 10.1103/PhysRevD.41.534 |
pmid=10012359 |bibcode = 1990PhRvD..41..534F }}</ref><ref name="fuda:1991">{{ cite journal |
author= M. Fuda |
title=Angular momentum and light front scattering theory |
journal=[[Physical Review D]] | volume= 44 |
issue=6 | pages= 1880–1890 | year=1991 |
doi= 10.1103/PhysRevD.44.1880|
pmid=10014068 |bibcode = 1991PhRvD..44.1880F }}</ref><ref name="fuda:1994">{{ cite journal |
author= M. Fuda |
title=A new picture for light front dynamics. 2 |
journal=[[Annals of Physics]] | volume= 231 |
issue=1 | pages= 1–40 | year=1994 |
doi= 10.1006/aphy.1994.1031|bibcode = 1994AnPhy.231....1F }}</ref><ref name="polyzou:1999">{{ cite journal |
author= W. N. Polyzou |
title=Left Coset Invariance and Relativistic Invariance |
journal=[[Few Body Systems]] | volume= 27 |
issue=2 | pages= 57–72 | year=1999 |
doi=10.1007/s006010050122|bibcode = 1999FBS....27...57P
s2cid=120699006 }}</ref>
and the kinematic representation of rotations
Line 732 ⟶ 455:
</math>|{{EquationRef|6}}}}
Because the dynamical input to the <math>S</math>-matrix is <math>P^-</math>, the invariance of the <math>S</math>-matrix with respect to changing the orientation of the light front implies the existence of a consistent dynamical rotation generator without the need to explicitly construct that generator. The success or failure of this approach is related to ensuring the correct rotational properties of the asymptotic states used to construct the wave operators, which in turn requires that the subsystem bound states transform irreducibly with respect to <math>SU(2)</math>.
These observations make it clear that the rotational covariance of the theory is encoded in the choice of light-front Hamiltonian. Karmanov<ref name="karmanov:1976">{{ cite journal |
author=V.A. Karmanov |
title=Wave Functions of Relativistic Bound Systems |
journal=[[Journal of Experimental and Theoretical Physics]] | volume= 44 | page= 210 | year=1976 |bibcode = 1976JETP...44..210K }}</ref><ref name="karmanov:1982a">{{ cite journal |
author= V.A. Karmanov |
title=Angular Condition Imposed On The State Vector Of A Compound System For A Light Front |
journal=[[Soviet Physics JETP Letters]] | volume= 35 | page= 276 | year=1982 }}</ref><ref name="karmanov:1982b">{{ cite journal |
author= V.A. Karmanov |
title=Complete System Of Equations For The State Vector Of A Relativistic Composite System On The Light Front |
journal=[[Journal of Experimental and Theoretical Physics]] | volume= 56 | page= 1 | year=1982
issue=1 |
bibcode=1982JETP...56....1K }}</ref>
introduced a covariant formulation of light-front quantum theory, where the orientation of the light front is treated as a degree of freedom. This formalism can be used to identify observables that do not depend on the orientation, <math>\hat{n}</math>, of the light front (see [[#Covariant formulation]]).
While the light-front components of the spin are invariant under light-front boosts, they Wigner rotate under rotation-less boosts and ordinary rotations. Under rotations the light-front components of the single-particle spins of different particles experience different Wigner rotations. This means that the light-front spin components cannot be directly coupled using the standard rules of angular momentum addition. Instead, they must first be transformed to the more standard canonical spin components, which have the property that the Wigner rotation of a rotation is the rotation. The spins can then be added using the standard rules of angular momentum addition and the resulting composite canonical spin components can be transformed back to the light-front composite spin components. The transformations between the different types of spin components are called Melosh rotations.<ref name="Melosh">{{ cite journal |
author=H. J. Melosh|
journal=[[Physical Review D]] | volume= 9 |
Line 781 ⟶ 477:
doi= 10.1103/PhysRevD.9.1095
| title=Quarks: Currents and constituents|bibcode = 1974PhRvD...9.1095M |
url=https://thesis.library.caltech.edu/4807/1/Melosh_hj_IV_1973.pdf }}</ref><ref name="keister:1991">{{ cite journal |author1=B. D. Keister |author2=W. N. Polyzou |
title=Relativistic Hamiltonian dynamics in nuclear and particle physics |
journal=[[Advances in Nuclear Physics]] |
volume=20 |
year=1991}}</ref>
They are the momentum-dependent rotations constructed by multiplying a light-front boost followed by the inverse of the corresponding rotation-less boost. In order to also add the relative orbital angular momenta, the relative orbital angular momenta of each particle must also be converted to a representation where they Wigner rotate with the spins.
While the problem of adding spins and internal orbital angular momenta is more complicated,<ref name="glockle:2013">{{ cite journal |author1=W. N. Polyzou |author2=W. Glockle |author3=H. Witala |
title=Spin in relativistic quantum theory |
journal=[[Few Body Systems]] | volume= 54 |issue=11 | pages= 1667–1704 | year=2013 |
doi= 10.1007/s00601-012-0526-8 |arxiv = 1208.5840 |bibcode = 2013FBS....54.1667P |s2cid=42925952 }}</ref>
it is only total angular
momentum that requires interactions; the total spin does not necessarily require an interaction dependence. Where the interaction dependence explicitly appears is in the relation between the total spin and the total angular momentum<ref name="keister:1991" /><ref name="leutwyler:1977b">{{ cite journal |author1=H. Leutwyler |author2=J. Stern |
title=Covariant Quantum Mechanics on a Null Plane |
journal=[[Physics Letters B]] | volume= 69 |issue=2 | pages= 207–210 | year=1977 |
Line 819 ⟶ 502:
</math>|{{EquationRef|1}}}}
where here <math>P^-</math> and <math>M</math> contain interactions. The transverse components of the light-front spin, <math>\vec{j}_{\perp}</math> may or may not have an interaction dependence; however, if one also demands cluster properties,<ref name="keister:2012">{{ cite journal |author1=B. D. Keister |author2=W. N. Polyzou |
title=Model Tests of Cluster Separability In Relativistic Quantum Mechanics |
journal=[[Physical Review C]] | volume= 86 |issue=1 | page= 014002 | year=2012 |
doi= 10.1103/PhysRevC.86.014002|arxiv = 1109.6575 |bibcode = 2012PhRvC..86a4002K |s2cid=41960696 }}</ref>
then the transverse components of
total spin necessarily have an interaction dependence. The result is that by choosing the light front components of the spin to be kinematic it is possible to realize full rotational invariance at the expense of cluster properties. Alternatively it is easy to realize cluster properties at the expense of full rotational symmetry. For models of a finite number of degrees of freedom there are constructions that realize both full rotational covariance and cluster properties;<ref name="coester:1982">{{ cite journal |author1=F. Coester |author2=W. N. Polyzou |
title=Relativistic Quantum Mechanics Of Particles With Direct Interactions |
journal=[[Physical Review D]] | volume= 26 |issue=6 | pages= 1348–1367 | year=1982 |
doi= 10.1103/PhysRevD.26.1348 |bibcode = 1982PhRvD..26.1348C }}</ref>
these realizations all have additional [[many-body problem|many-body]] interactions in the generators that are functions of fewer-body interactions.
The dynamical nature of the rotation generators means that tensor and spinor operators, whose commutation relations with the rotation generators are linear in the components of these operators, impose dynamical constraints that relate different components of these operators.
=== Nonperturbative dynamics ===
The strategy for performing nonperturbative calculations in light-front field theory is similar to the strategy used in lattice calculations. In both cases a nonperturbative regularization and renormalization are used to try to construct effective theories of a finite number of degrees of freedom that are insensitive to the eliminated degrees of freedom. In both cases the success of the renormalization program requires that the theory has a fixed point of the renormalization group; however, the details of the two approaches differ. The renormalization methods used in light-front field theory are discussed in [[Light-front computational methods#Renormalization group]]. In the lattice case the computation of observables in the [[effective theory]] involves the evaluation of large-dimensional integrals, while in the case of light-front field theory solutions of the effective theory involve solving large systems of linear equations. In both cases multi-dimensional integrals and linear systems are sufficiently well understood to formally estimate numerical errors. In practice such calculations can only be performed for the simplest systems. Light-front calculations have the special advantage that the calculations are all in [[Minkowski space]] and the results are wave
functions and scattering amplitudes.
== Relativistic quantum mechanics ==
While most applications of light-front quantum mechanics are to the light-front formulation of quantum field theory, it is also possible to formulate relativistic quantum mechanics of finite systems of directly interacting particles with a light-front kinematic subgroup. Light-front relativistic quantum mechanics is formulated on the direct sum of tensor products of single-particle Hilbert spaces. The kinematic representation <math>U_0(\Lambda ,a)</math> of the Poincaré group on this space is the direct sum of tensor products of the single-particle unitary irreducible representations of the Poincaré group. A front-form dynamics on this space is defined by a dynamical representation of the Poincaré group <math>U(\Lambda ,a)</math> on this space where <math>U(g) = U_0(g)</math> when <math>g</math> is in the kinematic subgroup of the Poincare group.
One of the advantages of light-front quantum mechanics is that it is possible to realize exact rotational covariance for system of a finite number of degrees of freedom. The way that this is done is to start with the non-interacting generators of the full Poincaré group, which are sums of single-particle generators, construct the kinematic invariant mass operator, the three kinematic generators of translations tangent to the light-front, the three kinematic light-front boost generators and the three components of the light-front spin operator. The generators are well-defined functions of these operators<ref name="leutwyler:1977b" /><ref name="leutwyler:1977a">{{ cite journal |author1=H. Leutwyler |author2=J. Stern |
title=Relativistic Dynamics on a Null Plane |
journal=[[Annals of Physics]] | volume= 112 |issue=1 | pages= 94–164 | year=1978 |
doi= 10.1016/0003-4916(78)90082-9|bibcode = 1978AnPhy.112...94L }}</ref>
given by ({{EquationNote|1}})
and <math> P^- = (\vec{P}_{\perp}^2 + M^2)/P^+</math>. Interactions that commute with all of these operators except the kinematic mass are added to the kinematic mass operator to construct a dynamical mass operator. Using this mass operator in ({{EquationNote|1}}) and the expression for <math>P^-</math> gives a set of dynamical Poincare generators with a light-front kinematic subgroup.<ref name="coester:1982" />
A complete set of irreducible eigenstates can be found by diagonalizing the interacting mass operator in a basis of simultaneous eigenstates of the light-front components of the kinematic momenta, the kinematic mass, the kinematic spin and the projection of the kinematic spin on the <math>{\hat{n}}</math> axis. This is equivalent to solving the center-of-mass Schrödinger equation in non-relativistic quantum mechanics. The resulting mass eigenstates transform irreducibly under the action of the Poincare group. These irreducible representations define the dynamical representation of the Poincare group on the Hilbert space.
This representation fails to satisfy cluster properties,<ref name="keister:2012" /> but this can be restored using a front-form generalization<ref name="keister:1991" /><ref name="coester:1982" /> of the recursive construction given by Sokolov.<ref name="sokolov:1978" />
== Infinite momentum frame ==
{{
The infinite momentum frame (IMF) was originally introduced<ref name="fubini:1965"/><ref name="weinberg:1966" /> to provide a physical interpretationof the Bjorken variable <math>x_{bj} = \frac{Q^2}{2 M\nu}</math> measured in deep inelastic [[lepton]]-proton scattering <math>\ell p \to \ell^\prime X</math> in Feynman's parton model. (Here <math>Q^2=-q^2</math> is the square of the spacelike momentum transfer imparted by the lepton and <math>\nu =E_\ell-E_{\ell^\prime}</math> is the energy transferred in the proton's rest frame.) If one considers a hypothetical Lorentz frame where the observer is moving at infinite momentum, <math>P \to \infty</math>, in the negative <math>\hat z</math> direction, then <math>x_{bj} </math> can be interpreted as the longitudinal momentum fraction <math>x = \frac{k^z}{P^z}</math> carried by the struck quark (or "parton") in the incoming fast moving proton. The structure function of the proton measured in the experiment is then given by the square of its instant-form wave function boosted to infinite momentum.
Formally, there is a simple connection between the Hamiltonian formulation of quantum field theories quantized at fixed time <math>t</math> (the "instant form" ) where the observer is moving at infinite momentum and light-front Hamiltonian theory quantized at fixed light-front time <math>\tau= t+z/c</math> (the "front form"). A typical energy denominator in the instant-form is <math>{1/ [E_{initial} - E_{intermediate}+ i \epsilon]}</math>
where <math>E_{intermediate} = \sum_j E_j = \sum_j\sqrt{m^2+ {\vec k}^2_j}</math>
is the sum of energies of the particles in the intermediate state. In the IMF, where the observer moves at high momentum <math>P</math> in the negative <math>\hat z</math> direction, the leading terms in
<math>P</math> cancel, and the energy denominator becomes <math>2P / [ \mathcal{M}^2-
\sum_j \big[{k^2_\perp + \frac{m^2}{x_i}}\big]_j + i \epsilon] </math> where
<math>\mathcal{M}^2</math> is invariant mass squared of the initial state. Thus, by keeping the terms in <math>\frac{1}{P}</math> in the instant form, one recovers the energy denominator which appears in light-front Hamiltonian theory. This correspondence has a physical meaning: measurements made by an observer moving at infinite momentum is analogous to making observations approaching the speed of light—thus matching to the front form where measurements are made along the front of a light wave. An example of an application to quantum electrodynamics
can be found in the work of Brodsky, Roskies and
Suaya.<ref name="IMF-QED">{{ cite journal |author1=S. J. Brodsky |author2=R. Roskies |author3=R. Suaya |
title=Quantum Electrodynamics and Renormalization Theory in the Infinite Momentum Frame |
journal=[[Physical Review D]] | volume= 8 |issue=12 | pages= 4574–4594 | year=1973 |
doi= 10.1103/PhysRevD.8.4574|bibcode = 1973PhRvD...8.4574B |osti=1442551 }}</ref>
The vacuum state in the instant form defined at fixed <math>t</math> is acausal and infinitely complicated. For example, in quantum electrodynamics, bubble graphs of all orders, starting with the <math>e^+ e^- \gamma</math>
intermediate state, appear in the ground state vacuum; however, as shown by Weinberg,<ref name="weinberg:1966" /> such vacuum graphs are frame-dependent and formally vanish by powers of <math>1/ P^2</math> as the observer moves at <math>P \to \infty</math>. Thus, one can again match the instant form to the front-form formulation where such vacuum loop diagrams do not appear in the QED ground state. This is because the <math>+</math> momentum of each constituent is positive, but must sum to zero in the vacuum state since the <math>+</math>momenta are conserved. However, unlike the instant form, no dynamical boosts are required, and the front form formulation is causal and frame-independent. The infinite momentum frame formalism is useful as an intuitive tool; however, the limit
<math>P\to \infty</math> is not a rigorous limit, and the need to boost the instant-form wave function introduces complexities.
== Covariant formulation ==
In light-front coordinates, <math>x^+=ct+z</math>, <math>x^-=ct-z</math>, the spatial coordinates <math>x,y,z</math>
do not enter symmetrically: the coordinate <math>z</math> is distinguished, whereas <math>x</math> and <math>y</math> do not appear at all. This non-covariant definition destroys the spatial symmetry that, in its turn, results in a few difficulties related to the fact that some transformation of the reference frame may change the orientation of the light-front plane. That is, the transformations of the reference frame and variation of orientation of the light-front plane are not decoupled from each other. Since the wave function depends dynamically on the orientation of the plane where it is defined, under these transformations the light-front wave function is transformed by dynamical operators (depending on the interaction). Therefore, in general, one should know the interaction to go from given reference frame to the new one. The loss of symmetry between the coordinates <math>z</math> and <math>x,y</math>
complicates also the construction of the states with definite angular momentum since the latter is just a property of the wave function relative to the rotations which affects all the coordinates <math>x,y,z</math>.
To overcome this inconvenience, there was developed the explicitly covariant version<ref name="karmanov:1976"/><ref name="karmanov:1982a" /><ref name="karmanov:1982b" /> of
light-front quantization (reviewed by Carbonell
et al.<ref name="cdkm">{{ cite journal |author1=J. Carbonell |author2=B. Desplanques |author3=V.A. Karmanov |author4=J.F. Mathiot |
title=Explicitly covariant light front dynamics and relativistic few body systems |
journal=[[Physics Reports]] | volume= 300 |issue=5–6 | pages= 215–347 | year=1998 |
doi= 10.1016/S0370-1573(97)00090-2 |arxiv = nucl-th/9804029 |bibcode = 1998PhR...300..215C |s2cid=119329870 }}</ref>),
in which the state vector is defined on the light-front plane of
general orientation:
Line 1,034 ⟶ 581:
by the wave function dependence on the four-vector <math>\omega</math>.
There were formulated the rules of graph techniques which, for a given Lagrangian, allow to calculate the perturbative decomposition of the state vector evolving in the light-front time <math>\sigma=\omega\cdot x</math> (in contrast to the evolution in the direction <math>x^+</math> or <math>t</math>). For the instant form of dynamics,
these rules were first developed by Kadyshevsky.<ref name="kadyshevsky:1964">{{ cite journal |
author=V.G. Kadyshevsky |
journal=[[Soviet JETP]] | volume= 19 | year=1964 | page=443 }}</ref><ref name="kadyshevsky:1968">{{ cite journal |
author=V.G. Kadyshevsky |
title=Quasipotential type equation for the relativistic scattering amplitude |
Line 1,048 ⟶ 590:
issue=2 | year=1968 | pages=125–148 |
doi= 10.1016/0550-3213(68)90274-5|bibcode = 1968NuPhB...6..125K }}</ref>
By these rules, the light-front amplitudes are represented as the integrals over the momenta of particles in intermediate states. These integrals are three-dimensional, and all the four-momenta <math>k_i</math>
are on the corresponding mass shells <math>k_i^2=m_i^2</math>,
in contrast to the Feynman rules containing four-dimensional integrals over the off-mass-shell momenta. However, the calculated light-front amplitudes, being on the mass shell, are in general the off-energy-shell amplitudes. This means that the on-mass-shell four-momenta, which these amplitudes depend on, are not conserved in the direction <math>x^-</math>
(or, in general, in the direction <math>\omega</math>).
The off-energy shell amplitudes do not coincide with the Feynman amplitudes, and they depend on the orientation of the light-front plane. In the covariant formulation, this dependence is explicit:
the amplitudes are functions of <math>\omega</math>. This allows one to apply to them in full measure the well known techniques developed for the covariant [[Feynman amplitudes]] (constructing the invariant variables, similar to the Mandelstam variables, on which the amplitudes depend; the decompositions, in the case of particles with spins, in invariant amplitudes; extracting electromagnetic form factors; etc.). The irreducible off-energy-shell amplitudes serve as the kernels of equations for the light-front wave functions. The latter ones are found from these equations and used to analyze hadrons and nuclei.
For spinless particles, and in the particular case of <math>\omega=(1/c,0,0,-1/c)</math>, the amplitudes found by the rules of covariant graph techniques, after replacement of variables, are reduced to the amplitudes given by the Weinberg rules<ref name="weinberg:1966" /> in the [[#Infinite momentum frame|infinite momentum frame]]. The dependence on orientation of the light-front plane manifests itself in the dependence of the off-energy-shell Weinberg amplitudes on the variables <math>\vec{k}_{\perp i}, x_i</math> taken separately but not in some particular combinations like the Mandelstam variables <math>s,t</math>.
On the energy shell, the amplitudes do not depend on the four-vector <math>\omega</math> determining orientation of the corresponding light-front plane. These on-energy-shell amplitudes coincide with the on-mass-shell amplitudes given by the Feynman rules. However, the dependence on <math>\omega</math> can survive because of approximations.
== Angular momentum ==
The covariant formulation is especially useful for constructing the states with definite angular momentum. In this construction, the four-vector <math>\omega</math> participates on equal footing with other four-momenta, and, therefore, the main part of this problem is reduced to the well known one. For example, as is well known, the wave function of a non-relativistic system, consisting of two spinless particles with the relative momentum <math>\vec{k}</math>
and with total angular momentum <math>l</math>, is proportional to the spherical function <math>Y_{lm}(\hat{\vec{k}})</math>: <math>\psi_{lm}(\vec{k})=f(k)Y_{lm}(\hat{k})</math>,
where <math>\hat{k}=\vec{k}/k</math> and <math>f(k)</math> is a function depending on the modulus <math>k=|\vec{k}|</math>. The angular momentum operator reads: <math>\vec{J}=-i[\vec{k}\times \partial \vec{k}]</math>.
Then the wave function of a relativistic system in the covariant formulation of light-front dynamics obtains the similar form:
{{NumBlk|:|<math>
Line 1,102 ⟶ 612:
</math>|{{EquationRef|7}}}}
where <math>\hat{n}=\vec{\omega}/|\vec{\omega}|</math> and <math>f_{1,2}(k,\vec{k}\cdot\hat{n})</math> are functions depending, in addition to <math>k</math>, on the scalar product <math>\vec{k}\cdot\hat{n}</math>.
The variables <math>k</math>, <math>\vec{k}\cdot\hat{n}</math> are invariant not only under rotations
of the vectors <math>\vec{k}</math>, <math>\hat{n}</math> but also under rotations and the Lorentz
transformations of initial four-vectors <math>k</math>, <math>\omega</math>. The second contribution <math>\propto Y_{lm}(\hat{n})</math> means that the operator of the total angular momentum in explicitly covariant light-front dynamics obtains an additional term: <math>\vec{J}=-i[\vec{k}\times \partial \vec{k}] -i[\hat{n}\times \partial \hat{n}]</math>.
For non-zero spin particles this operator obtains the contribution of the spin operators:<ref name="fuda:1990a" /><ref name="fuda:1990b" /><ref name="fuda:1991" /><ref name="fuda:1994" /><ref name="karmanov:1979">{{ cite journal |
author=V.A. Karmanov | title=Wave function with spin on a light front |
journal=[[Journal of Experimental and Theoretical Physics]] | volume= 49 | page= 954 | date=June 1979 |bibcode = 1979JETP...49..954K }}</ref><ref name="karmanov:1981">{{ cite journal |
Line 1,132 ⟶ 631:
The fact that the transformations changing the orientation of the light-front plane are dynamical (the corresponding generators of the Poincare group contain interaction) manifests itself in the dependence of the coefficients <math>f_{1,2}</math> on the scalar product <math>\vec{k}\cdot\hat{n}</math> varying when the orientation of the unit vector <math>\hat{n}</math> changes (for fixed <math>\vec{k}</math>). This dependence (together with the dependence on <math>k</math>) is found from the dynamical equation for the wave function.
A peculiarity of this construction is in the fact that there exists the operator <math>A=(\hat{n}\cdot \vec{J})^2</math> which commutes both with the Hamiltonian and with <math>\vec{J}^2, J_z</math>. Then the states are labeled also by the eigenvalue <math>a</math> of the operator <math>A</math>: <math>\psi=\psi_{lma}(\vec{k},\hat{n})</math>.
For given angular momentum <math>l</math>, there are <math>l+1</math> such the states. All of them are
degenerate, i.e. belong to the same mass (if we do not make an approximation). However, the wave function should also satisfy the so-called angular condition<ref name="karmanov:1982a" /><ref name="karmanov:1982b" /><ref name="CarlsonJi">{{ cite journal |author1=C. Carlson |author2=C.-R. Ji |
title=Angular conditions, relations between Breit and light front frames, and subleading power corrections |
journal=[[Physical Review D]] | volume= 67 |issue=11 | page= 116002 | year=2003 |
doi= 10.1103/PhysRevD.67.116002|arxiv = hep-ph/0301213 |bibcode = 2003PhRvD..67k6002C |s2cid=7978843 }}</ref><ref name="AC-spin1">{{ cite journal |author1=B. L. G. Bakker |author2=C.-R. Ji |
title=Frame dependence of spin one angular conditions in light front dynamics |
journal=[[Physical Review D]] | volume= 65 |issue=7 | page= 073002 | year=2002 |
doi= 10.1103/PhysRevD.65.073002|arxiv = hep-ph/0109005 |bibcode = 2002PhRvD..65g3002B |s2cid=17967473 }}</ref><ref name="AC-spin2">{{ cite journal |
author=B. L. G. Bakker, H.-M.Choi and C.-R. Ji |
title=The vector meson form-factor analysis in light front dynamics |
Line 1,162 ⟶ 647:
issue=11 | page= 116001 | year=2002 |
doi= 10.1103/PhysRevD.65.116001
|arxiv = hep-ph/0202217 |bibcode = 2002PhRvD..65k6001B
s2cid=55018990 }}</ref>
After satisfying it, the solution obtains the form of a unique superposition of the states <math>\psi_{lma}(\vec{k},\hat{n})</math> with different eigenvalues
<math>a</math>.<ref name="karmanov:1982b" /><ref name="cdkm" />
The extra contribution <math>-i[\hat{n}\times \partial \hat{n}]</math> in the light-front angular momentum operator increases the number of spin components in the light-front wave function. For example, the non-relativistic [[deuteron]] wave function is determined by two components (<math>S</math>- and <math>D</math>-waves). Whereas, the relativistic light-front deuteron wave function is determined by six components.<ref name="karmanov:1979" /><ref name="karmanov:1981" />
These components were calculated in the one-boson exchange model.<ref name="carb-karm:1995">{{ cite journal |author1=J. Carbonell |author2=V.A. Karmanov |
title=Relativistic deuteron wave function in the light front dynamics |
journal=[[Nuclear Physics A]] | volume= 581 |issue=3–4 | pages= 625–653 | year=1995 |
Line 1,181 ⟶ 660:
== Goals and prospects ==
The central issue for light-front quantization is the rigorous description of hadrons, nuclei, and systems thereof from first principles in QCD. The main goals of the research using light-front dynamics are:
* Evaluation of masses and wave functions of hadrons using the light-front Hamiltonian of QCD.
Line 1,195 ⟶ 671:
The nonperturbative analysis of light-front QCD requires the following:
* Continue testing the light-front Hamiltonian approach in simple theories in order to improve our understanding of its peculiarities and treacherous points vis a vis manifestly-covariant quantization methods. This will include work on theories such as Yukawa theory and QED and on theories with unbroken supersymmetry, in order to understand the strengths and limitations of different methods. Much progress has already been made along these lines.
* Construct symmetry-preserving regularization and renormalization schemes for light-front QCD, to include the
title=Comparison of quantum field perturbation theory for the light front with the theory in lorentz coordinates |
journal=[[Theoretical and Mathematical Physics]] | volume= 112 |issue=3 | pages= 1117–1130 | year=1997 |
doi= 10.1007/BF02583044|arxiv = hep-th/9901110 |bibcode = 1997TMP...112.1117P |s2cid=5441075 }}</ref><ref name="StPete2">{{ cite journal |author1=S.A. Paston |author2=V.A. Franke |author3=E.V. Prokhvatilov |
title=Constructing the light-front QCD Hamiltonian |
journal=[[Theoretical and Mathematical Physics]] | volume= 120 |issue=3 | pages= 1164–1181 | year=1999 |
doi= 10.1007/BF02557241|bibcode = 1999TMP...120.1164P |arxiv=hep-th/0002062 |s2cid=119099826 }}</ref> Glazek-Wilson similarity renormalization-group procedure for Hamiltonians,<ref name="Glazek-Wilson1">{{ cite journal |author1=S. D. Glazek |author2=K. G. Wilson |
title=Renormalization of Hamiltonians |
journal=[[Physical Review D]] | volume= 48 |issue=12 | pages= 5863–5872 | year=1993 |
doi= 10.1103/PhysRevD.48.5863|pmid=10016252 |bibcode = 1993PhRvD..48.5863G |arxiv=hep-th/9706149 |s2cid=39086918 }}</ref><ref name="Glazek-Wilson2">{{ cite journal |author1=S. D. Glazek |author2=K. G. Wilson |
journal=[[Physical Review D]] | volume= 49 |issue=8 | pages= 4214–4218 | year=1994 |
doi= 10.1103/PhysRevD.49.4214
| title=Perturbative renormalization group for Hamiltonians|pmid=10017426 |bibcode = 1994PhRvD..49.4214G }}</ref><ref name="Glazek-Wilson3">{{ cite journal |author1=S. D. Glazek |author2=K. G. Wilson |
journal=[[Physical Review D]] | volume= 57 |issue=6 | pages= 3558–3566 | year=1998 |
doi= 10.1103/PhysRevD.57.3558
| title=Asymptotic freedom and bound states in Hamiltonian dynamics|arxiv = hep-th/9707028 |bibcode = 1998PhRvD..57.3558G |s2cid=16805417 }}</ref> Mathiot-Grange test functions,<ref name="Mathiot-Grange">{{ cite journal |author1=P. Grange |author2=J.-F. Mathiot |author3=B. Mutet |author4=
title=Taylor-Lagrange renormalization scheme, Pauli-Villars subtraction, and light-front dynamics |
journal=[[Physical Review D]] | volume= 82 |issue=2 | page= 025012 | year=2010 |
doi= 10.1103/PhysRevD.82.025012|arxiv = 1006.5282 |bibcode = 2010PhRvD..82b5012G |s2cid=118513433 }}</ref> Karmanov-Mathiot-Smirnov<ref name="kms2012">{{ cite journal |author1=V.A. Karmanov |author2=J.-F. Mathiot |author3=A.V. Smirnov |
title=Ab initio nonperturbative calculation of physical observables in light-front dynamics. Application to the Yukawa model |
journal=[[Physical Review D]] | volume= 86 |issue=8 | page= 085006 | year=2012 |
doi= 10.1103/PhysRevD.86.085006
| bibcode=2012PhRvD..86h5006K|arxiv = 1204.3257 |s2cid=119000243 }}</ref> realization of sector-dependent renormalization, and determine how to incorporate symmetry breaking in light-front quantization;<ref name="Bender">{{ cite journal |author1=C. M. Bender |author2=S. S. Pinsky |author3=B. van de Sande |
title=Spontaneous symmetry breaking of <math>\phi^4</math> in (1+1)-dimensions in light front field theory |
journal=[[Physical Review D]] | volume= 48 |issue=2 | pages= 816–821 | year=1993 |
doi= 10.1103/PhysRevD.48.816|pmid=10016310 |arxiv = hep-th/9212009 |bibcode = 1993PhRvD..48..816B |s2cid=14265514 }}</ref><ref name="Pinsky2">{{ cite journal |author1=S. S. Pinsky |author2=B. van de Sande |
title=Spontaneous symmetry breaking of (1+1)-dimensional <math>\phi^4</math> theory in light front field theory. 2 |
journal=[[Physical Review D]] | volume= 49 |issue=4 | pages= 2001–2013 | year=1994 |
doi= 10.1103/PhysRevD.49.2001|pmid=10017185 |arxiv = hep-ph/9309240 |bibcode = 1994PhRvD..49.2001P |s2cid=17165941 }}</ref><ref name="Pinsky3">{{ cite journal |author1=S. S. Pinsky |author2=B. van de Sande |author3=J.R. Hiller |
title=Spontaneous symmetry breaking of (1+1)-dimensional <math>\phi^4</math> theory in light front field theory. 3 |
journal=[[Physical Review D]] | volume= 51 |issue=2 | pages= 726–733 | year=1995 |
doi= 10.1103/PhysRevD.51.726|pmid=10018525 |arxiv = hep-th/9409019 |bibcode = 1995PhRvD..51..726P |s2cid=15291034 }}</ref><ref name="Rozowsky:2000gy">{{ cite journal |author1=J. S. Rozowsky |author2=C. B. Thorn |
title= Spontaneous symmetry breaking at infinite momentum without P+ zero modes |
journal=[[Physical Review Letters]] | volume= 85 |issue=8 | pages= 1614–1617 | year=2000 |
doi= 10.1103/PhysRevLett.85.1614 | bibcode=2000PhRvL..85.1614R|arxiv = hep-th/0003301 | pmid=10970571|s2cid=17968437 }}</ref><ref name="Chakrabarti:2003tc">{{ cite journal |author1=D. Chakrabarti |author2=A. Harindranath |author3=L. Martinovic |author4=G. B. Pivovarov |author5=J. P. Vary |
title= Ab initio results for the broken phase of scalar light front field theory |
journal=[[Physics Letters B]] | volume= 617 |issue=1–2 | pages= 92–98 | year=2005 |
doi= 10.1016/j.physletb.2005.05.012|arxiv = hep-th/0310290 |bibcode = 2005PhLB..617...92C |s2cid=119370407 }}</ref><ref name="Kim:2003ha">{{ cite journal |author1=V. T. Kim |author2=G. B. Pivovarov |author3=J. P. Vary |
title= Phase transition in light front <math>\phi^4_{1+1}</math> |
journal=[[Physical Review D]] | volume= 69 |issue=8 | page= 085008 | year=2004 |
doi= 10.1103/PhysRevD.69.085008|arxiv = hep-th/0310216 |bibcode = 2004PhRvD..69h5008K |s2cid=119524638 }}</ref><ref name="Dskulsh">{{ cite journal |author1=U. Kulshreshtha |author2=D. S. Kulshreshtha |author3=J. P. Vary|title= Hamiltonian, Path Integral and BRST Formulations of Large N Scalar $QCD_{2}$ on the Light-Front and Spontaneous Symmetry Breaking |journal=[[Eur. Phys. J. C]]| volume= 75|issue= 4 |page= 174 | year=2015| doi=10.1140/epjc/s10052-015-3377-x|arxiv=1503.06177 | bibcode=2015EPJC...75..174K|s2cid=119102254 }}</ref> this is likely to require an analysis of zero modes and in-hadron condensates.<ref name="Wilson" /><ref name="Nambu" /><ref name="GOR" /><ref name="tHooftVeltman" /><ref name="SVZ" /><ref name="FeynmanQCD2" /><ref name="SSbreaking" /><ref name="GasserLeutwyler" /><ref name="condensates" /><ref name="Marisqq" /><ref name="Brodsky:2012ku" /><ref name="CasherSusskind" />
* Develop computer codes which implement the regularization and renormalization schemes. Provide a platform-independent, well-documented core of routines that allow investigators to implement different numerical approximations to field-theoretic eigenvalue problems, including the light-front coupled-cluster method<ref name="LFCC}. Consider various quadrature schemes and basis sets, including Discretized Light-Cone Quantization (DLCQ),<ref name="DLCQ1">{{ cite journal |author1=H.-C. Pauli |author2=S. J. Brodsky |
title=Solving field theory in one space one time dimension |
journal=[[Physical Review D]] | volume= 32 |issue=8 | pages= 1993–2000 | year=1985 |
doi= 10.1103/PhysRevD.32.1993|pmid=9956373 |bibcode = 1985PhRvD..32.1993P }}</ref>
<ref name="DLCQ2">{{ cite journal |author1=H.-C. Pauli |author2=S. J. Brodsky |
title=Discretized light cone quantization: Solution to a field theory in one space one time dimensions |
journal=[[Physical Review D]] | volume= 32 |issue=8 | pages= 2001–2013 | year=1985 |
doi= 10.1103/PhysRevD.32.2001|pmid=9956374 |bibcode = 1985PhRvD..32.2001P }}</ref>
finite elements, function expansions,<ref name="Vary:2009gt">{{ cite journal |author1=J. P. Vary |author2=H. Honkanen |author3=J. Li |author4=P. Maris |author5=S. J. Brodsky |author6=A. Harindranath |author7=G. F. de Teramond |author8=P. Sternberg |
title= Hamiltonian light-front field theory in a basis function approach |
journal=[[Physical Review C]] | volume= 81 |issue=3 | page= 035205 | year=2010 |
doi= 10.1103/PhysRevC.81.035205|arxiv = 0905.1411 |bibcode = 2010PhRvC..81c5205V |s2cid=33206182 }}</ref>
and the complete orthonormal wave functions obtained from AdS/QCD. This will build on the Lanczos-based MPI code developed for nonrelativistic nuclear physics applications and similar codes for Yukawa theory and lower-dimensional supersymmetric Yang—Mills theories.
* Address the problem of computing rigorous bounds on truncation errors, particularly for energy scales where QCD is strongly coupled.
Line 1,280 ⟶ 734:
errors.
* Solve for hadronic masses and wave functions. Use these wave functions to compute form factors, generalized parton distributions, scattering amplitudes, and decay rates. Compare with perturbation theory, lattice QCD, and model calculations, using insights from AdS/QCD, where possible. Study the transition to nuclear degrees of freedom, beginning with light nuclei.
* Classify the spectrum with respect to total angular momentum. In equal-time quantization, the three generators of rotations are kinematic, and the analysis of total angular momentum is relatively simple. In light-front quantization, only the generator of rotations around the <math>z</math>-axis is kinematic; the other two, of rotations about axes
<math>x</math> and <math>y</math>, are dynamical. To solve the angular momentum classification problem, the eigenstates and spectra of the sum of squares of these generators must be constructed. This is the price to pay for having more kinematical generators than in equal-time quantization, where all three boosts are dynamical. In light-front quantization, the boost along <math>z</math> is kinematic, and this greatly simplifies the calculation of matrix elements that involve boosts, such as the ones needed to calculate form factors. The relation to covariant Bethe–Salpeter approaches projected on the light-front may help in understanding the angular momentum issue and its relationship to the Fock-space truncation of the light-front Hamiltonian. Model-independent constraints from the general angular condition, which must be satisfied by the light-front helicity amplitudes, should also be explored. The contribution from the zero mode appears necessary for the hadron form factors to satisfy angular momentum conservation, as expressed by the angular condition. The relation to light-front quantum mechanics, where it is possible to exactly realize full rotational covariance and construct explicit representations of the dynamical rotation generators, should also be investigated.
* Explore the [[AdS/QCD correspondence]] and [[light front holography]].<ref name="deTeramond:2005su">{{ cite journal |author1=G. F. de Teramond |author2=S. J. Brodsky |
title= Hadronic spectrum of a holographic dual of QCD |
journal=[[Physical Review Letters]] | volume= 94 |issue=20 | page= 201601 | year=2005 |
doi= 10.1103/PhysRevLett.94.201601 | bibcode=2005PhRvL..94t1601D|arxiv = hep-th/0501022 | pmid=16090235|s2cid=11006078 }}</ref><ref name="deTeramond:2008ht">{{ cite journal |author1=G. F. de Teramond |author2=S. J. Brodsky |
title= Light-Front Holography: A First Approximation to QCD |
journal=[[Physical Review Letters]] | volume= 102 | page= 081601 | year=2009 |
doi= 10.1103/PhysRevLett.102.081601 | pmid=19257731 | issue=8|arxiv = 0809.4899 |bibcode = 2009PhRvL.102h1601D |s2cid=33855116 }}</ref><ref name="Brodsky:2011sk">{{ cite journal |author1=S. J. Brodsky |author2=F. -G. Cao |author3=G. F. de Teramond |
title= AdS/QCD and Applications of Light-Front Holography |
journal= [[Communications in Theoretical Physics]] | volume= 57 |issue=4 | pages= 641–664 | year=2012 |
doi= 10.1088/0253-6102/57/4/21 | bibcode=2012CoTPh..57..641S|arxiv=1108.5718 |s2cid=73629251 }}</ref><ref name="Forkel">{{ cite journal |author1=H. Forkel |author2=M. Beyer |author3=T. Frederico |
title= Linear square-mass trajectories of radially and orbitally excited hadrons in holographic QCD |
journal= [[JHEP]] | volume= 0707 |issue=7 | page= 077 | year= 2007 |
doi= 10.1088/1126-6708/2007/07/077|arxiv = 0705.1857 |bibcode = 2007JHEP...07..077F |s2cid=5282022 }}</ref><ref name="Gutsche:2012wb">{{ cite journal |author1=T. Gutsche |author2=V. E. Lyubovitskij |author3=I. Schmidt |author4=A. Vega |
title= Nucleon resonances in AdS/QCD |
journal=[[Physical Review D]] | volume= 87 |issue=1 | page= 016017 | year=2013 |
doi= 10.1103/PhysRevD.87.016017 | bibcode=2013PhRvD..87a6017G|arxiv = 1212.6252 |s2cid=118685470 }}</ref><ref name="Gutsche:2012ez">{{ cite journal |author1=T. Gutsche |author2=V. E. Lyubovitskij |author3=I. Schmidt |author4=A. Vega |
title= Chiral Symmetry Breaking and Meson Wave Functions in Soft-Wall AdS/QCD |
journal=[[Physical Review D]] | volume= 87 |issue=5 | page= 056001 | year=2013 |
doi= 10.1103/PhysRevD.87.056001 | bibcode=2013PhRvD..87e6001G|arxiv = 1212.5196 |s2cid=118377538 }}</ref>
The approximate duality in the limit of massless quarks motivates few-body analyses of meson and baryon spectra based on a one-dimensional light-front Schrödinger equation in terms of the modified transverse coordinate <math>\zeta</math>. Models that extend the approach to massive quarks have been proposed, but a more fundamental understanding within QCD is needed. The nonzero quark masses introduce a non-trivial dependence on the longitudinal momentum, and thereby highlight the need to understand the representation of rotational symmetry within the formalism. Exploring AdS/QCD wave functions as part of a physically motivated Fock-space basis set to diagonalize the LFQCD Hamiltonian should shed light on both issues. The complementary Ehrenfest
interpretation<ref name="Ehrenfest">{{ cite journal |author1=S. D. Glazek |author2=A. P. Trawinski |
title= Model of the AdS/QFT duality |
journal=[[Physical Review D]] | volume= 88 |issue=10 | page= 105025 | year=2013 |
doi= 10.1103/PhysRevD.88.105025|arxiv = 1307.2059 |bibcode = 2013PhRvD..88j5025G |s2cid=118455480 }}</ref>
can be used to introduce effective
degrees of freedom such as diquarks in
baryons.
* Develop numerical methods/computer codes to directly evaluate the partition function (viz. thermodynamic potential) as the basic thermodynamic quantity. Compare to lattice QCD, where applicable, and focus on a finite chemical potential, where reliable lattice QCD results are presently available only at very small (net) quark densities. There is also an opportunity for use of light-front AdS/QCD to explore non-equilibrium phenomena such as transport properties during the very early state of a heavy ion collision. Light-front AdS/QCD opens the possibility to investigate hadron formation in such a non-equilibrated strongly coupled quark-gluon plasma.
* Develop a light-front approach to the [[neutrino oscillation]] experiments possible at [[Fermilab]] and elsewhere, with the goal of reducing the energy spread of the neutrino-generating hadronic sources, so that the three-energy-slits interference picture of the oscillation pattern<ref name="Glazek:2012wp">{{ cite journal |author1=S. D. Glazek |author2=A. P. Trawinski |
title= Neutrino oscillations in the front form of Hamiltonian dynamics |
journal=[[Physical Review D]] | volume= 87 |issue=2 | page= 025002 | year=2013 |
doi= 10.1103/PhysRevD.87.025002|arxiv = 1208.5255 |bibcode = 2013PhRvD..87b5002G |s2cid=119206502 }}</ref> can be resolved and the front form of Hamiltonian dynamics utilized in providing the foundation for qualitatively new (treating the vacuum differently) studies of neutrino mass generation mechanisms.
* If the renormalization group procedure for effective particles (RGPEP)<ref name="RGPEPrecent1">{{ cite journal |
Line 1,388 ⟶ 779:
journal=[[Acta Physica Polonica B]] | volume= 43 |
issue=9 | page= 1843 | year=2012 |
doi= 10.5506/APhysPolB.43.1843|doi-access=free }}</ref><ref name="RGPEPrecent2">{{ cite journal |
author=S. D. Glazek |
title=Fermion mass mixing and vacuum triviality in the renormalization group procedure for effective particles |
journal=[[Physical Review D]] | volume= 87 |
issue=12 | page= 125032 | year=2013 |
doi= 10.1103/PhysRevD.87.125032 |arxiv = 1305.3702 |bibcode = 2013PhRvD..87l5032G |
s2cid=119222650 }}</ref> does allow one to study intrinsic charm, bottom, and glue in a systematically renormalized and convergent light-front Fock-space expansion, one might consider a host of new experimental studies of production processes using the intrinsic components that are not included in the calculations based on gluon and quark splitting functions. == See also ==
* [[Light-front computational methods]]
* [[Light-front quantization applications]]
* [[Quantum field theories]]
* [[Quantum chromodynamics]]
* [[Quantum electrodynamics]]
* [[Light front holography|Light-front holography]]
== References ==
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