Time Evolution and Probability in Quantum Theory: The Central Role of Born's Rule

TL;DR

This paper introduces a time-dependent Generalized Born's Rule as the fundamental evolution equation in quantum theory, emphasizing quantum probability's central role and showing its invariance across different models, challenging traditional notions like wavefunction collapse.

Contribution

The paper proposes the Generalized Born's Rule as the core time evolution equation, independent of specific auxiliary equations like Schrödinger's, and clarifies its invariance across models such as Schrödinger and Heisenberg.

Findings

The Generalized Born's Rule is the unique fundamental evolution equation.

Schrödinger's equation is auxiliary, not fundamental, in quantum evolution.

Quantum phenomena like entanglement are explained without wavefunction collapse.

Abstract

In this treatise I introduce the time dependent Generalized Born's Rule for the probabilities of quantum events, including conditional and consecutive probabilities, as the unique fundamental time evolution equation of quantum theory. Then these probabilities, computed from states and events, are to be compared with relative frequencies of observations. Schrodinger's equation still is valid in one model of the axioms of quantum theory, which I call the Schrodinger model. However, the role of Schrodinger's equation is auxiliary, since it serves to help compute the continuous temporal evolution of the probabilities given by the Generalized Born's Rule. In other models, such as the Heisenberg model, the auxiliary equations are quite different, but the Generalized Born's Rule is the same formula (covariance) and gives the same results (invariance). Also some aspects of the Schrodinger model are not found in the isomorphic Heisenberg model, and they therefore do not have any physical significance. One example of this is the infamous collapse of the quantum state. Other quantum phenomena, such as entanglement, are easy to analyze in terms of the Generalized Born's Rule without any reference to the unnecessary concept of collapse. Finally, this leads to the possibility of quantum theory with other sorts of auxiliary equations instead of Schrodinger's equation, and examples of this are given. Throughout this treatise the leit motif is the central importance of quantum probability and most especially of the simplifying role of the time dependent Generalized Born's Rule in quantum theory.