# Contents

## Idea

The next higher dimensional generalization of the concept of a relativistic particle and a relativistic string is called the relativistic membrane.

The concept of a bosonic membrane was considered already by Dirac 62 as a hypothetical model for the electron, then abandoned and eventually re-incarnated as the “super-membrane in 11d” due to Bergshoeff-Sezgin-Townsend 87 (whose “m” became the “M” in “M-theory”), namely the Green-Schwarz sigma-model for super p-branes corresponding to the brane scan-entry with $p=2$, $D=11$, now commonly known as the M2-brane.

In fact, according to the brane scan, a super 2-brane sigma model exists on superspacetimes of dimension 4, 5, 7, and 11:

## References

### Pre-history

Early speculations trying to unify the electron and the muon as two excitations of a single relativistic membrane:

• Paul Dirac, An Extensible Model of the Electron, Proc. Roy. Soc. A268, (1962) 57-67 (jstor:2414316)

(also proposing the Dirac-Born-Infeld action)

• Paul Dirac, The motion of an Extended Particle in the Gravitational Field, in Relativistic Theories of Gravitation, Proceedings of a Conference held in Warsaw and Jablonna, July 1962, ed. L. Infeld, P. W. N. Publishers, 1964, Warsaw, 163-171; discussion 171-175 (spire:1623740)

• Paul Dirac, Particles of Finite Size in the Gravitational Field, Proc. Roy. Soc. A270, (1962) 354-356 (doi:10.1098/rspa.1962.0228)

### Super-membrane/M2-brane as a sigma model

The Green-Schwarz sigma-model-type formulation of the super-membrane in 11d (as in the brane scan and in contrast to the black brane-solutions of 11d supergravity) first appears in:

The equations of motion of the super membrane are derived via the superembedding approach in

and the Lagrangian density for the super membrane is derived via the superembedding approach in

Discussion from the point of view of Green-Schwarz action functional-∞-Wess-Zumino-Witten theory is in

The double dimensional reduction of the M2-brane to the Green-Schwarz superstring was observed in

The interpretation of the super-membrane as an object related to string theory via double dimensional reduction, hence as the M2-brane was proposed in

around the time when M-theory became accepted due to

### 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:

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:

The generalization to pp-wave spacetimes (leading to the BMN matrix model):