Geometrizing the Quantum—A Toy Model

Abstract

It is shown that the equations of relativistic Bohmian mechanics for multiple bosonic particles have a dual description in terms of a classical theory of conformally "curved" space-time. This shows that it is possible to formulate quantum mechanics as a purely classical geometrical theory. The results are further generalized to interactions with an external electromagnetic field.PACS numbers: 04.62.+v, 03.65.Ta A. IntroductionIn standard quantum mechanics observables and the corresponding uncertainty are promoted to a fundamental principle. But it was shown by David Bohm that this does not necessarily have to be the case [1]. In the de Broglie-Bohm (dBB) interpretation it is explained that the uncertainty "principle" and the description by means of operators can be understood in terms of uncontrollable initial conditions and non-local interactions between an additional field (the "pilot wave") and the measuring apparatus. This theory was further generalized to relativistic quantum mechanics and quantum field theory with bosonic and fermionic fields [2][3][4][5]. It is well known that (due to its non-locality) the dBB theory is not in contradiction to the Bell inequalities [6]. Due to its contextuality, the dBB theory is also not affected by the Kochen-Specker theorem [7].One mayor drawback of the dBB theory is that the pilot wave and the corresponding "quantum potential" have to be imposed by hand without further justification. In a previous work [17] it was shown that the relativistic dBB theory for a single particle is dual to a scalar theory of curved spacetime. In this dual theory the ominous "pilot wave" can be readily interpreted as a well known physical quantity, namely a space-time dependent conformal factor of the metric. This work on the single particle has many features in common with other publications on the subject [9][10][11][12][13][14]. However, having a duality for the single particle case is not enough, because the dBB interpretation only is a consistent quantum theory when it also has the many particle case. The many particle theory is for example crucial for understanding the quantum uncertainty and for evading the non existence theorems [6,7]. Therefore, we will generalize the previous results and present a dual for the relativistic many particle dBB theory. B. Relativistic dBB for many particlesIn this section we shortly list the ingredients for the interpretation of the many particle quantum Klein-Gordon equation in terms of Bohmian trajectories. For a detailed description of subsequent topics in the dBB theory like particle creation, the theory of quantum measurement, many particle states, and quantum field theory the reader is referred to [5]. Let |0 be the vacuum and |n be an arbitrary n-particle state. The corresponding n-particle wave function is [3] ψ(x 1 ; . . . ; x n ) = Ps √ n! 0|Φ(x 1 ) . . .Φ(x n )|n , where theΦ(x j ) are scalar Klein-Gordon field operators and the symbol P s denotes symmetrization over all positions x j . For free fields, the wave function satisfies the equat...

Geometric description of the Schrödinger equation in (3n+1)-dimensional configuration space

“…we find that the geometrical equation (18) is identical to the quantum equation (9). Please note that in relativistic case the dual theory was developed in 4n dimensions [19] while here equation (18) is obtained from the geometric theory developed in (1 + 3n) dimensions.…”

“…Using the language of [21,22], it was shown that such trajectories naturally arise in the configuration space for the complex Klein Gordon equation. It was further found that the evolution equation for those trajectories can be cast in the form of a geodesics equation in a conformally rescaled configuration space [2,3,7,9,18,19]. Thus, the relativistic Klein Gordon equation can be rewritten in a geometric language with non-trivial trajectories in configuration space.…”

“…It is thus natural that there are numerous attempts to reformulate quantum mechanics in a more geometrical way [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20].…”

On the Noncommutative Eikonal

“…Geometric treatments of quantum mechanics have also been studied in depth; for a sample see, e.g. [2,[20][21][22][23][24][25][26][27].…”

Two particle entanglement and its geometric duals

“…Thus it naturally led many physicists to the attempt to reformulate quantum mechanics in a geometric language, like GR. References [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19] are a few such efforts towards the geometrical rewriting of quantum laws.…”