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In Hamiltonian formulation of relativistic quantum field theory on Minkowski spacetime there are three essential choices of forms of hypersurfaces to foliate spacetime by [Dirac (1949)]:
(Fig. 1 from Brodsky, Pauli & Pinsky (1998))
The “instant form” on the left foliates spacetime by spacelike hypersurfaces (cf. Cauchy surface). This is the form considered for Hamiltonian mechanics in most of the literature (certainly in the case of quantum mechanics, cf. [Dirac (1949) §3]).
In contrast, light-cone quantization corresponds to the choice of foliation by lightlike wavefronts [Dirac (1949) §5].
The instant form has the advantage of being the one people are most familiar with, but I do not believe it is intrinsically any better for this reason. […]
The front form has the advantage that it requires only three Hamiltonians, instead of the four of the other forms. This makes it mathematically the most interesting form, and makes any problem of finding Hamiltonians substantially easier.
The front form has the further advantage that there is no square root in the Hamiltonians, which means that one can avoid negative energies for particles by suitably choosing the values of the dynamical variables in the front, without having to make a special convention about the sign of a square root.
(…)
For non-perturbative computations it turns out at least convenient (if not necessary) to in addition assume/impose periodic boundary conditions along the lightlike front, hence, in flat lightcone coordinates $x^{\pm}$, so that all fields $\phi$ satisfy
for some $L \in \mathbb{R}_{\gt 0}$ giving the circumference of a lightlike circle (which some authors take to be literally lightlike, of the kind shown eg. in Bär 2004, Exp. 2.21 (2), while others, like Seiberg 1997, consider to be the limit of boosts of a spacelike circle fiber).
Since, dually, this means that the corresponding lightcone momentum $p^+$ becomes quantized in units of the inverse radius $R$ of this circle
hence one speaks of discretized light-cone quantization or DLCQ, for short. This goes back to Maskawa & Yamakawi 1976; Casher 1976; Thorn 1977, 1978; Pauli & Brodsky (1985b), (1985a); Tang, Brodsky & Pauli 1991 (2.6); see also McCartor & Robertson 1994, §3; for review see Burkardt 1996, §5.1; Heinzl 2001, §3.4.
Often it turns out that negative values of $N$ in (1) can be neglected or integrated out, so that
becomes a natural number-parameter akin to that considered in the 't Hooft double line construction of gauge theories, and then the large N limit of the discrete light cone quantization becomes of interest.
In quantization of sigma models whose target space has a lightlike Killing vector, the strategy of light-cone gauge quantization is to gauge fix the metric and diffeomorphisms of the worldvolume such that also the worldvolume has a lightlike Killing vector field and such that this is mapped to the given one on target space.
This typically fixes most of the available gauge freedom, and the strategy is then to apply quantization to what remains. For more on this general idea see at quantization commutes with reduction.
Often this is considered for target space being Minkowski spacetime $\mathbb{R}^{d-1,1}$ and with $X^+ \coloneqq X^0 - X^1$ denoting one of its canonical lightlike coordinates. If $\tau$ denotes similarly a lightlike coordinate function on the worldvolume, then the condition of light-cone gauge reads
for some proportionality constant $p^+$, the light cone momentum. This is how light cone gauge appears in much of the string theory literature.
Light cone gauge quantization is the only method by which Green-Schwarz super p-brane sigma models have been quantized, to date.
Specifically, applying light-cone gauge quantization to the Green-Schwarz sigma model for the M2-brane on 11d Minkowski spacetime, combined with a certain regularization of the remaining light-cone Hamiltonian yields the BFSS matrix model.
Alternatively, applying the light cone gauge quantization of the Green-Schwarz sigma-model of the M2-brane not on Minkowski spacetime but, more generally, on 11d pp-wave spacetimes (which are Penrose limits of the near horizon geometry of the black M2-branes/M5-branes) yields the BMN matrix model.
The concept originates with
reviewed in
and was rediscovered several times, such as in the guise of “infinite momentum frame” field theory:
see also Chang & Ma 1969 and Kogut & Soper 1970
Early review:
Shau-Jin Chang, Robert G. Root, and Tung-Mow Yan, Quantum Field Theories in the Infinite-Momentum Frame. I. Quantization of Scalar and Dirac Fields, Phys. Rev. D 7, (1973) 1133 [doi:10.1103/PhysRevD.7.1133]
Stanley J. Brodsky, Gary McCartor, Hans-Christian Pauli, Stephen S. Pinsky, The challenge of light-cone quantization of gauge field theory, Particle World 3 3 (1993) 109-124 [cds:240388, inspire:335247]
Wei-Min Zhang, Light-Front Dynamics and Light-Front QCD, Chin. J. Phys. 32 (1994) 717-808 [arXiv:hep-ph/9412244]
Matthias Burkardt, Light Front Quantization, Adv. Nucl. Phys. 23 (1996) 1-74 [arXiv:hep-ph/9505259, doi:10.1007/0-306-47067-5_1]
Further review:
Guillaume Beuf, Meijian Li, Tolga Altinoluk, Fabio Dominguez, Course on Light-Cone Techniques applied to QCD (2022) [indico:1203236]
Stanley Brodsky, Thomas Heinzl et al. Light-Front Quantum Chromodynamics: A framework for the analysis of hadron physics, White Paper of International Light Cone Advisory Committee, Nuclear Physics B - Proceedings Supplements 251, 252 (2014) 165-174 [doi:10.1016/j.nuclphysbps.2014.05.004, arXiv:1309.6333]
J. Vary et al. Basis Light-Front Quantization: Recent Progress and Future Prospects, Few-Body Syst 57 (2016) 695–702 [doi:10.1007/s00601-016-1117-x]
Wikipedia, Light front quantization
Wikipedia, Light-front computational methods
See also:
The notion of lightlike circle compactification (“discretized light-cone quantization”):
Toshihide Maskawa, Koichi Yamakawi, The Problem of $P^+ = 0$ Mode in the Null-Plane Field Theory and Dirac’s Method of Quantization, Progress of Theoretical Physics 56 1 (1976) 270–283 [doi:10.1143/PTP.56.270]
Aharon Casher, Gauge fields on the null plane, Phys. Rev. D 14 (1976) 452 [doi:10.1103/PhysRevD.14.452]
Charles B. Thorn, On the derivation of dual models from field theory, Physics Letters B 70 1 (1977) 85-87 [doi:10.1016/0370-2693(77)90351-3]
Charles B. Thorn, Derivation of dual models from field theory. II, Phys. Rev. D 17 (1978) 1073 [doi:10.1103/PhysRevD.17.1073]
Hans-Christian Pauli, Stanley J. Brodsky, Solving field theory in one space and one time dimension, Phys. Rev. D 32 (1985) 1993 [doi:10.1103/PhysRevD.32.1993]
Hans-Christian Pauli, Discretized light-cone quantization and the effective interaction in hadrons, AIP Conf. Proc. 494 (1999) 80-139 [arXiv:hep-ph/9910203, doi:10.1063/1.1301662]
and on the issue of excluding $p^+ = 0$:
See also:
Lecture notes:
More on the case of gauge fields:
In relation to spontaneous symmetry breaking:
In relation to instant-time quantization:
Philip D. Mannheim, Light-front quantization is instant-time quantization [arXiv:1909.03548]
Philip D. Mannheim, Peter Lowdon, Stanley Brodsky, Comparing light-front quantization with instant-time quantization, Physics Reports 891 (2021) 1-65 [arXiv:2005.00109, doi:10.1016/j.physrep.2020.09.001]
Discussion of (reduced) phase spaces in light-cone coordinates:
Prem P. Srivastava, Constraints and Hamiltonian in light-front quantized field theory, Nuovo Cim. A 107 (1994) 549-558 [arXiv:hep-th/9308046, doi:10.1007/BF02768789]
Kianoosh Kargar, Ahmad Shirzad, Majid Monemzadeh, Dynamical structure of fields in light cone coordinates, Phys. Rev. D 99 (2019) 045019 [arXiv:1608.03255, doi:10.1103/PhysRevD.99.045019]
Application to quantum electrodynamics:
Shau-Jin Chang, Shang-Keng Ma, Feynman Rules and Quantum Electrodynamics at Infinite Momentum, Phys. Rev. 180 (1969) 1506 [doi:10.1103/PhysRev.180.1506]
John B. Kogut and Davison E. Soper, Quantum Electrodynamics in the Infinite-Momentum Frame, Phys. Rev. D 1 (1970) 2901 [doi:10.1103/PhysRevD.1.2901]
Andrew C. Tang, Discretized light cone quantization: Application to quantum electrodynamics, PhD thesis, Stanford (1990) [spire:296776, pdf, pdf]
Andrew C. Tang, Stanley J. Brodsky, Hans-Christian Pauli, Discretized light-cone quantization: Formalism for quantum electrodynamics, Phys. Rev. D 44 (1991) 1842 [doi:10.1103/PhysRevD.44.1842]
and to pure electromagnetism:
Martina M. Brisudova, Electromagnetic duality and light-front coordinates, Phys. Rev. D 59 (1999) 087702 [arXiv:hep-th/9806196, doi:10.1103/PhysRevD.59.087702]
Sucheta Majumdar, Residual gauge symmetry in light-cone electromagnetism, J. High Energ. Phys. 2023 215 (2023) [arXiv:2212.10637, doi:10.1007/JHEP02(2023)215]
Application of (discretized) light cone quantization to to QCD:
G. Peter Lepage, Stanley J. Brodsky, Exclusive processes in perturbative quantum chromodynamics, Phys. Rev. D 22 (1980) 2157 [doi:10.1103/PhysRevD.22.2157]
Hans-Christian Pauli, Stanley J. Brodsky, Discretized light-cone quantization: Solution to a field theory in one space and one time dimension, Phys. Rev. D 32 (1985) 2001 [doi:10.1103/PhysRevD.32.2001]
Stanley Brodsky, Hans-Christian Pauli, Stephen S. Pinsky, Quantum Chromodynamics and Other Field Theories on the Light Cone, Phys. Rept. 301 (1998) 299-486 [arXiv:hep-ph/9705477, doi:10.1016/S0370-1573(97)00089-6]
Review in the broader context of non-perturbative quantum field theory:
A light-cone QCD-Lagrangian density adapted to MHV amplitudes:
Light cone quantization of the string sigma-model originates with
see also
All the standard introductory texts on string theory have sections devoted to light-cone quantization. For instance:
In the context of the BFSS matrix model as a discrete light-cone formulation of M-theory:
Nathan Seiberg, Why is the Matrix Model Correct?, Phys. Rev. Lett. 79 (1997) 3577-3580 [arXiv:hep-th/9710009, doi:10.1103/PhysRevLett.79.3577]
Adel Bilal, DLCQ of M-Theory as the Light-Like Limit, Phys. Lett. B 435 (1998) 312-318 [arXiv:hep-th/9805070, doi:10.1016/S0370-2693(98)00811-9]
Review:
The Poisson bracket-formulation of the classical light-cone gauge Hamiltonian for the bosonic relativistic membrane and the corresponding matrix commutator regularization is due to:
Some exact solutions:
Jens Hoppe, Exact algebraic M(em)brane solutions $[$arXiv:2107.00569$]$
Jens Hoppe, The fast non-commutative sharp drop $[$arXiv:2302.13146$]$
On the regularized light-cone gauge quantization of the Green-Schwarz sigma model for the M2-brane on (super) Minkowski spacetime, yielding the BFSS matrix model:
Original articles:
Observation that the spectrum is continuous:
Review:
Hermann Nicolai, Robert C. Helling, Supermembranes and M(atrix) Theory, In Trieste 1998, Nonperturbative aspects of strings, branes and supersymmetry 29-74 (arXiv:hep-th/9809103, spire:476366)
Jens Hoppe, Membranes and Matrix Models (arXiv:hep-th/0206192)
Arundhati Dasgupta, Hermann Nicolai, Jan Plefka, An Introduction to the Quantum Supermembrane, Grav. Cosmol. 8:1, 2002; Rev. Mex. Fis. 49S1:1-10, 2003 (arXiv:hep-th/0201182)
Gijs van den Oord, On Matrix Regularisation of Supermembranes, 2006 (pdf)
Meer Ashwinkumar, Lennart Schmidt, Meng-Chwan Tan, Section 2 of: Matrix Regularization of Classical Nambu Brackets and Super $p$-Branes (arXiv:2103.06666)
The generalization to pp-wave spacetimes (leading to the BMN matrix model):
Keshav Dasgupta, Mohammad Sheikh-Jabbari, Mark Van Raamsdonk, Section 2 of: Matrix Perturbation Theory For M-theory On a PP-Wave, JHEP 0205:056, 2002 (arXiv:hep-th/0205185)
Keshav Dasgupta, Mohammad Sheikh-Jabbari, Mark Van Raamsdonk, Section 2 of: Matrix Perturbation Theory For M-theory On a PP-Wave, JHEP 0205:056, 2002 (arXiv:hep-th/0205185)
See also
Mike Duff, T. Inami, Christopher Pope, Ergin Sezgin, Kellogg Stelle, Semiclassical Quantization of the Supermembrane, Nucl.Phys. B297 (1988) 515-538 (spire:247064)
Daniel Kabat, Washington Taylor, section 2 of: Spherical membranes in Matrix theory, Adv. Theor. Math. Phys. 2: 181-206, 1998 (arXiv:hep-th/9711078)
Nathan Berkovits, Towards Covariant Quantization of the Supermembrane (arXiv:hep-th/0201151)
Qiang Jia, On matrix description of D-branes (arXiv:1907.00142)
A new kind of perturbation series for the quantized super-membrane:
Relation to the string dilaton under double dimensional reduction:
Last revised on January 29, 2024 at 06:16:14. See the history of this page for a list of all contributions to it.