# Born's Rule From the Principle of Unitary Equivalence

## TL;DR

This paper derives Born's rule by interpreting quantum observables as unitary representations of abelian Lie groups and enforcing invariance under second quantization, linking probabilities to invariant frequency averages.

## Contribution

It introduces a novel formalism connecting quantum observables with unitary group representations and derives Born's rule from the principle of unitary equivalence.

## Key findings

- Born's rule is derived from invariance principles.
- Quantum observables are represented as unitary group representations.
- Frequency interpretation of probability underpins the derivation.

## Abstract

Complex phase factors are viewed not only as redundancies of the quantum formalism but instead as remnants of unitary transformations under which the probabilistic properties of observables are invariant. It is postulated that a quantum observable corresponds to a unitary representation of an abelian Lie group, the irreducible subrepresentations of which correspond to the observable's outcomes. It is shown that this identification agrees with the conventional identification as self-adjoint operators. The upshot of this formalism is that one may 'second quantize' the representation to which an observable corresponds, thus obtaining the corresponding Fock space representation. This Fock space representation is then also identifiable as an observable in the same sense, the outcomes of which are naturally interpretable as ensembles of outcomes of the corresponding non-second quantized observable. The frequency interpretation of probability is adopted, i.e. probability as the average occurrence, from which Born's rule is deduced by enforcing the notion of 'average' to such that are invariant under the second quantized unitary representation which defines the quantum observable to which the initial state is an outcome. The enforcement of this invariance is an application the principle referred to as the 'principle of unitary equivalence'.