nLab light-cone gauge quantization

Contents

Context

Physics

Idea
Applications

Quantization of Green-Schwarz super $p$ -brane sigma models
BFSS matrix model
BMN matrix model

Related concepts
References

General
Quantization of the M2-brane sigma-model to a matrix model

Idea

What is called light-cone gauge quantization is one approach to the quantization of sigma models whose target space has a lightlike Killing vector. The strategy of this approach 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

X^+ = p^+ \tau

for some proportionality constant $p^+$ , the light cone momentum This is how light cone gauge appears in much of the physics literature.

If in addition, one assumes that the coordinate function $X^+$ on spacetime is periodic, hence that it actually runs along a circle fiber (which some authors take to be literally lightlike, while others consider the limit of boosts of a spacelike circle fiber), then the lightcone momentum $p^+$ becomes quantized in units of the inverse radius $R$ of this circle

(1)

p^+ \;=\; N/R \phantom{AAA} N \in \mathbb{Z} \,.

The quantization of a sigma model in this situation is hence called the discrete light cone quantization or DLCQ, for short.

Often it turns out that negative values of $N$ in (1) can be neglected or integrated out, so that

N \in \mathbb{N}

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.

Applications

Quantization of Green-Schwarz super $p$ -brane sigma models

Light cone gauge quantization is the only method by which Green-Schwarz super p-brane sigma models have been quantized, to date.

BFSS matrix model

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 l9ight-cone Hamiltonian yields the BFSS matrix model.

BMN 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.

Related concepts

References

General

For QCD:

Stanley Brodsky, H-C Pauli, S. Pinsky, Quantum Chromodynamics and Other Field Theories on the Light Cone, Phys. Rept. 301:299-486, 1998 (arXiv:hep-ph/9705477)

Review

Badis Ydri, Section 3.9 of: Review of M(atrix)-Theory, Type IIB Matrix Model and Matrix String Theory (arXiv:1708.00734), published as: Matrix Models of String Theory, IOP 2018 (ISBN:978-0-7503-1726-9)

All the standard introductory texts on string theory have sections devoted to light-cone quantization. For instance

Michael Green, John Schwarz, Edward Witten, section 5.2.1 of: Superstring theory, 3 vols. Cambridge Monographs on Mathematical Physics

Quantization of the M2-brane sigma-model to a matrix model

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:

Jens Hoppe, Quantum theory of a massless relativistic surface and a two-dimensional bound state problem, MIT 1982 (dspace:1721.1/15717, pdf)

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:

Bernard de Wit, Jens Hoppe, Hermann Nicolai, On the Quantum Mechanics of Supermembranes, Nucl. Phys. B305 (1988) 545. (pdf, pdf, spire:261702)

Observation that the spectrum is continuous:

Bernard de Wit, W. Lüscher, Hermann Nicolai, The supermembrane is unstable, Nucl. Phys. B320 (1989) 135 (spire:266584, doi:10.1016/0550-3213(89)90214-9)

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)