Membranes and Consistent Quantization of Nambu Dynamics

The dynamics of even topological open membranes relies on Nambu Brackets. Consequently, such 2p-branes can be quantized through the consistent quantization of the underlying Nambu dynamical structures. This is a summary construction relying on the methods detailed in refs [1, 2]. The classical motion of topological open membranes is controlled by Nambu Brackets, the multilinear generalization of Poisson Brackets [3]. Without loss of generality, consider first an illustrative Nambu Bracket (NB) dynamical law in phase space for a particle with two degrees of freedom. Time-evolution is specified by a phase-space Jacobian, df dt = ∂(f, L1, L2, L3) ∂(x, px, y, py) ≡ {f, L1, L2, L3} . (1) For an arbitrary function f of phase-space variables, df = ∂lf dz , where z ≡ (x, px, y, py). Thus, this phase-space Jacobian is usually written symbolically as a set of Nambu 4-Brackets, ż = {z, L1, L2, L3}. (2) The Lis are arbitrary independent functions of the 4-d phase-space variables, and play the role of three “Hamiltonians”, as required in Nambu dynamics. They are manifestly time-invariant by the complete antisymmetry of all arguments in the Jacobian. In what action principle does this motion arise? The action for this evolution is given by the analog of the Hamilton-Poincaré symplectic 2-form (dω1 = dx ∧ dpx − dH ∧ dt), now extended to an exact 4-form [4, 5], dω3 = dx ∧ dpx ∧ dy ∧ dpy − dL1 ∧ dL2 ∧ dL3 ∧ dt = (dx− {x, L1, L2, L3}dt) ∧ (dpx − {px, L1, L2, L3}dt) ∧ (dy − {y, L1, L2, L3}dt) ∧ (dpy − {py, L1, L2, L3}dt) . (3) The 4-integral of this form on an open 4-surface yields a 3-form action evaluated on the 3-boundary of that surface, S = ∫