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Lorentz Transformation in Quantum Mechanics

Marcello Baldo

TL;DR

This work examines whether standard quantum mechanics remains compatible with special relativity when measurement-induced wave-function reduction is included. It adopts a nonstandard-analysis-based stochastic reduction model on a space-time lattice and develops a relativistic extension to study how Lorentz transformations act on generic wave functions. A key finding is that Lorentz transformations cannot in general map quantum states at fixed times between frames; only the full evolution is covariant, while the reduction process itself is frame-dependent and localized to the detector frame where Born probabilities are defined. The results clarify how relativistic invariance manifests in measurement processes and offer a framework to interpret EPR/Bell correlations without signaling, highlighting a special role for the rest frame of the measuring apparatus in relativistic quantum mechanics.

Abstract

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.

Lorentz Transformation in Quantum Mechanics

TL;DR

This work examines whether standard quantum mechanics remains compatible with special relativity when measurement-induced wave-function reduction is included. It adopts a nonstandard-analysis-based stochastic reduction model on a space-time lattice and develops a relativistic extension to study how Lorentz transformations act on generic wave functions. A key finding is that Lorentz transformations cannot in general map quantum states at fixed times between frames; only the full evolution is covariant, while the reduction process itself is frame-dependent and localized to the detector frame where Born probabilities are defined. The results clarify how relativistic invariance manifests in measurement processes and offer a framework to interpret EPR/Bell correlations without signaling, highlighting a special role for the rest frame of the measuring apparatus in relativistic quantum mechanics.

Abstract

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.

Paper Structure

This paper contains 7 sections, 16 equations, 3 figures.

Table of Contents

  1. Introduction.
  2. Remarks on the Lorentz transformations.
  3. Lorentz transformations and the nonstandard sector.
  4. Lorentz transformations of a free wave packet.
  5. Measurement.
  6. Summary and conclusions
  7. Appendix.

Figures (3)

  • Figure 1: Schematic representation of the thought experiment illustrated by A. Einstein, as reported in ref. Bag.
  • Figure 2: Schematic representation of the thought experiment discussed in the text. The boxes indicate two detectors. The dashed circles represent pictorially the spherical wave packet of the relative motion of two emitted particles.
  • Figure 3: Schematic representation of the thought experiment discussed in the text. The boxes indicate two detectors. A particle at the position indicated by a black dot decays into two identical particles in opposite directions. A LT is performed to a reference frame with a velocity $V$ along the x-axis. As observed in this frame the arrivals of the two particles are not anymore simultaneous. As soon as the detector D2 is reached by a particle and the corresponding spin is measured, the spin of the other particle is fixed before it can reach D1.