Preferred frame and two meanings of time: diagonal form of the 'Lorentz-boost' transformation matrix

TL;DR

This paper integrates quantum mechanics with special relativity by analyzing the Lorentz-boost transformation matrix, revealing a preferred frame and distinguishing between vital and frozen time, leading to a generalized uncertainty principle.

Contribution

It introduces a diagonal form of the Lorentz-boost matrix within a quantum framework, connecting relativity principles with quantum mechanics and identifying a preferred frame.

Findings

Diagonal Lorentz-boost matrix relates relativity to quantum mechanics.

Covariant description is a preferred frame with frozen time.

Generalized Heisenberg uncertainty principle consistent with quantum analysis.

Abstract

The main purpose of this paper is to rethink the relativity issue within the framework of the fundamental postulates of quantum mechanics. The aspect of so-called ``double special relativity'' (DSR) is a starting point in our discussion. The three elementary ideas were involved to show that special relativity may be treated as an integral part of quantum mechanics. These ideas (or observations) are: (1) the necessity of distinguishing the two time meanings, namely: (i) the vital one referring to description of system evolution, and (ii) the frozen one, referring to energy measure by means of inverse time units; (2) the existence of the energy-momentum (and time-distance) comparison scale in relativistic description; and (3) a possibility of introduction of mass by means of a light-cone frame description. The resulting quantum-mechanical analysis allows us to find diagonal form of the Lorentz-boost transformation matrix and thus to relate the interval invariant relativity principle with the principles of quantum mechanics. The manner, in which the diagonal form of the transformation matrix was found, shows that covariant description itself is a preferred frame description and that the time that undergoes relativistic transformation rules is the frozen time, whereas the vital time is the Lorentz invariant. A generalized form of the Heisenberg uncertainty principle proposed by Witten (Phys. Today, Apriel 1996), is derived. It turns out, that this form is equivalent to the one know from the analysis of the covariant harmonic oscillator given by Kim and Noz (the book, 1986). As a by-product of this analysis one finds that special relativity itself preserves the Planck length, however, a particle cannot be seen any longer as a material point, but rather as an extended quantum object.