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Journal of Physics Conference Series

Quantum mechanics and local Lorentz symmetry violation

Petr Jizba B, Fabio Scardigli

Published May 10, 2012• 5 citations
Publisher

Abstract

We show how a Brownian motion on a short scale can originate a relativistic motion on scales larger than the particle's Compton wavelength. Thus, Lorentz symmetry appears to be not a primitive concept, but rather it statistically emerges when a coarse graining average over distances of order, or longer than the Compton wavelength, is taken. We also present the generalizations needed to accommodate in our scheme the doubly special relativistic dynamics. In this way, a previously unsuspected, common stochastic origin of the two frameworks is revealed for the first time. Issues such as generalized commutation relations are also discussed.

DOI
10.1088/1742-6596/361/1/012026

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We show how a Brownian motion on a short scale can originate a relativistic motion on scales larger than the particle's Compton wavelength. Thus, Lorentz symmetry appears to be not a primitive concept, but rather it statistically emerges when a coarse graining average over distances of order, or longer than the Compton wavelength, is taken. We also present the generalizations needed to accommodate in our scheme the doubly special relativistic dynamics. In this way, a previously unsuspected, common stochastic origin of the two frameworks is revealed for the first time. Issues such as generalized commutation relations are also discussed.
There is a theoretical evidence that relativistically invariant quantum dynamics at (enough) large space-time scales can result from a cooperative process of two inter-correlated non-relativistic stochastic dynamics, operating at different energy scales. We show that the Euclidean transition amplitude for a relativistic particle is identical to the transition probability of a Brownian particle propagating in a granular space. We discuss the issue of the robustness of the special-relativistic quantum mechanics thus obtained under small changes in the granular-space distribution. Experimental implications for early Universe cosmology are also briefly outlined.
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For two classical Brownian particles an analog of continuous-variable quantum entanglement is presented: The common probability distribution of the two coordinates and the corresponding coarse-grained velocities cannot always be prepared via mixing of any factorized distributions referring to the two particles separately. This is possible for particles which have interacted in the past, but do not interact at present. Three factors are crucial for the effect: (1) separation of time scales of coordinate and momentum which motivates the definition of coarse-grained velocities; (2) the resulting uncertainty relations between the coordinate of the Brownian particle and the change of its coarse-grained velocity; (3) the fact that the coarse-grained velocity, though pertaining to a single Brownian particle, is defined on a common context of two particles. The Brownian entanglement is a consequence of a coarse-grained description and disappears for a finer resolution of the Brownian motion. Analogies with the quantum situation are discussed, as well as possibilities of experimental realization of the effect in examples of macroscopic Brownian motion.
The compatibility of special relativity and Quantum Mechanics has been questioned by several authors. The origin of this tension can be traced back mainly to the introduction of the measurement processes and the corresponding wave function reduction, which play a crucial role in Quantum Mechanics. We approach this problem with the help of a recent proposal for a model of Quantum Mechanics, where the measurement is explicitly described as a specific stochastic process. This implements ordinary Quantum Mechanics, where measurement and reduction are treated as phenomenological events of unknown origin without any physical justification. To state clearly the question in general, we first discuss and establish the effect of a Lorentz transformation on a generic wave function in space-time. Alongside the analysis we formulate the relativistic version of the model. We then consider few thought experiments in order to analyze to what extent Quantum Mechanics follows relativistic invariance and find the specific critical points where non invariance possibly occurs. The analysis can shade light to the interpretation of the existing experimental observations.
The analogy between quantum 'paths' and random walks of a Brownian particle is extended to the relativistic regime. This yields a coherent picture of the fine structure of quantum paths: these can be viewed as analogies of the trajectory of a random walker. The fractal dimension of quantum paths is df=2 at scales large compared with the Compton wavelength Lambda but small compared with the de Broglie wavelength lambda and becomes df=1 at scales smaller than Lambda . The transition between the two regimes is exhibited explicitly and is analogous to the cross-over from ballistic to diffusive motion of Brownian particles. The analogue of the collisions in Brownian motion is electron-positron pair creations and annihilations, and the unit thermodynamic free energy kT corresponds to the rest mass mc2 of the particle.