# Standard Quantum Mechanics representation as a constrained 2 + 1 space-time (membrane) theory?

Could a particular Standard Quantum Mechanics representation be a constrained 2 + 1 space-time theory (membrane theory) ?

(i) This question is motivated by a possible (approximative) analogy with Matrix theory for M-Theory (see this reference and these blogs articles 1 and 2 )

(ii) More precisely, standard quantum mechanics is a (infinite) matrix theory (or operator theory), which basics degrees of freedom as represented as $O_{lm}(t)$, where $O$ is some operator. An observable $O(t)$ is an hermitian operator. We could so consider $lm$ as a (maybe non-trivial) discretization of some compact 2-dimensions. So it is expected that there exists a correspondance between $O_{lm}(t)$ and some function $O(\sigma_1, \sigma_2, t)$ defined on a 2D manifold (like a plan, a sphere, a torus, or maybe a Klein Bottle)

(iii) If we restrict the infinite operators of Quantum mechanics to a $N*N$ matrix, we will find for observables, $N*N$ hermitian matrix, so living in the adjoint representation of $SU(n)$. So we could expect, as $N$ goes to infinity, that this is representing some diffeomorphism for the 2D manifold.

(iv) It will be interesting to see, if it is only the topology of the 2D manifold which is important, or/and the shape and size (the geometry).

(v) Here, we don't want at all to say that Quantum mechanics is replaced by a non-sense classical mechanics, it's only, if it is possible, a different representation of Quantum Mechanics. In particular, it will be very interesting to see how quantum constraints (Heinsenberg inequalities) are expressed in this representation

-