Relativistic probability amplitudes I. Massive particles of any spin

TL;DR

This paper develops a relativistic quantum framework for massive particles of any spin, defining probability amplitudes, their transformation properties, and position operators, extending nonrelativistic concepts to relativistic quantum mechanics.

Contribution

It introduces a consistent relativistic formalism for probability amplitudes and position operators for particles of arbitrary spin, including transformation laws and symmetry considerations.

Findings

Probability amplitudes transform unitarily under Lorentz transformations.

Position and spin amplitudes are identified with proper transformation properties.

Relativistic position operators and their transformation laws are derived.

Abstract

We consider a massive particle of arbitrary spin and the basis vectors that carry the unitary, irreducible representations of the Poincar\'e group. From the complex coefficients in normalizable superpositions of these basis vectors, we identify momentum/spin-component probability amplitudes with the same interpretation as in the nonrelativistic theory. We find the relativistic transformations of these amplitudes, which are unitary in that they preserve the modulus-squared of scalar products from frame to frame. Space inversion and time reversal are also treated. We reconsider the Newton- Wigner construction of eigenvectors of position and the position operator. Position/spin-component probability amplitudes are also identified and their relativistic, unitary, transformations derived. Again, space inversion and time reversal are considered. For reference, we show how to construct positive energy solutions of the Klein-Gordon and Dirac equations in terms of probability amplitudes. We find the boost transformation of the position operator in the spinless case and present some results on the relativity of position measurements. We consider issues surrounding the classical concept of causality as it applies in quantum mechanics. We briefly examine the relevance of the results presented here for theories of interaction.