Andrey Akhmeteli

No Drama Quantum Theory?

LTA Solid, Houston

Is it possible to offer a "no drama" quantum theory? Something as simple (in principle) as classical electrodynamics - a local realistic theory described by a system of partial differential equations in 3+1 dimensions, but reproducing unitary evolution of a quantum field theory in the configuration space (see details and references in Refs. [1-3])? Of course, the Bell inequalities cannot be violated in such a theory. But there are some reasons to believe these inequalities cannot be violated either in experiments or in quantum theory. Indeed, on the one hand, there is no loophole-free experimental evidence of violations of the Bell inequalities, on the other hand, to prove theoretically that the inequalities can be violated in quantum theory, one needs to use the projection postulate. The latter is, strictly speaking, in contradiction with the standard unitary evolution of the larger quantum system that includes the measured system and the measurement device, as such postulate introduces irreversibility and turns a superposition of states into their mixture. Therefore, mutually contradictory assumptions are required to prove the Bell theorem, so it is on shaky grounds both theoretically and experimentally and can be circumvented if, for instance, the projection postulate is rejected. The equations of scalar electrodynamics (the Klein-Gordon-Maxwell electrodynamics) in the unitary gauge are shown to describe independent dynamics of electromagnetic field in the following sense: if components of the 4-potential and their first derivatives with respect to time are known in the entire space at some time point, the equations yield the values of the second derivatives of these components with respect to time for the same time point, so integration yields the 4-potential for any value of time. In particular, the matter field can be naturally eliminated from the equations. There are reasons to believe that similar results can be obtained for spinor electrodynamics (the Dirac-Maxwell electrodynamics). In particular, it is shown that the Dirac equation is generally equivalent to a partial differential equation of the fourth order for one real function, so at least seven out of eight real components of the Dirac spinor field can be eliminated from the equations of spinor electrodynamics. The result for the scalar electrodynamics remains valid if conserved external currents are added to the right-hand side of the Maxwell equations. This is important for description of the hydrogen atom, the Aharonov-Bohm effect, and other quantum phenomena in terms of electromagnetic field only. This result can also be relevant to interpretation of quantum theory. For example, it allows an interesting modification of the de Broglie - Bohm interpretation. The results of this work suggest that electromagnetic field, rather than the matter field wave function, can be regarded as the guiding field. A second-quantized theory describing unitary evolution can be obtained from the resulting equations by a generalization of the Carleman linearization procedure. For a system of nonlinear partial differential equations this procedure generates a system of linear equations in the Hilbert space, which looks like a second-quantized theory and is equivalent to the original nonlinear system on the set of solutions of the latter.

[1] A. Akhmeteli, Int'l Journal of Quantum Information, Vol. 9, Suppl. (2011) 17-26, www.akhmeteli.org/akh-prepr-ws-ijqi2.pdf [2] A. Akhmeteli, quant-ph/1006.2578 [3] A. Akhmeteli, quant-ph/1008.4828