# The Pauli objection

## TL;DR

This paper challenges Pauli's argument against the existence of a general time operator in quantum mechanics by showing that when using an external clock, time emerges as a correlation rather than a conjugate variable.

## Contribution

It demonstrates that Pauli's objection does not hold when time is defined via correlations with an external clock, allowing for a consistent time operator in quantum systems.

## Key findings

- Time can be represented as a correlation with an external clock.
- The time operator is not conjugate to the Hamiltonian in this framework.
- Time eigenvalues satisfy the Schrödinger equation for arbitrary Hamiltonians.

## Abstract

Schroedinger's equation says that the Hamiltonian is the generator of time translations. This seems to imply that any reasonable definition of time operator must be conjugate to the Hamiltonian. Then both time and energy must have the same spectrum since conjugate operators are unitarily equivalent. Clearly this is not always true: normal Hamiltonians have lower bounded spectrum and often only have discrete eigenvalues, whereas we typically desire that time can take any real value. Pauli concluded that constructing a general a time operator is impossible (although clearly it can be done in specific cases). Here we show how the Pauli argument fails when one uses an external system (a "clock") to track time, so that time arises as correlations between the system and the clock (conditional probability amplitudes framework). In this case, the time operator is not conjugate to the system Hamiltonian, but its eigenvalues still satisfy the Schroedinger equation for arbitrary Hamiltonians.