# A Theory for the Time Arrow

## TL;DR

This paper proposes a law stating that entropy in quantum systems generally increases over time, establishing an arrow of time and limiting reversible quantum states beyond traditional conservation laws.

## Contribution

It introduces a new entropy-based law for quantum dynamics that defines a direction of time and restricts certain state evolutions, extending the understanding of quantum irreversibility.

## Key findings

- Entropy of coherent states tends to increase over time.
- States with constant entropy are stationary eigenstates.
- Quantum theory is not time-reversible except for stationary states.

## Abstract

Physical laws for elementary particles can be described by the quantum dynamics equation given a Hamiltonian. The solution are probability amplitudes in Hilbert space that evolve over time. A probability density function over position and time is given as the magnitude square of such probability amplitude. An entropy can be associated with these probability densities characterizing the position information of a particle. Coherent states are localized wave packets and may describe the spatial distribution for some particle states. We show that due to a dispersion property of Hamiltonians in quantum physics, the entropy of coherent states increases over time. We investigate a partition of the Hilbert space into four sets based on whether the entropy is (i) increasing but not a constant, (ii) decreasing but not a constant, (iii) a constant, (iv) oscillating. We propose a law that entropy (weakly) increases in time in quantum physics. Thus, states in set (ii) are disallowed, and the states in set (iii) cannot complete an oscillation period. Then, according to this law, quantum theory is not time-reversible unless the state is in set (iii), i.e., it is a stationary state (an eigenstate of the Hamiltonian.). This additional law in quantum theory limits physical scenarios beyond conservation laws, providing causality reasoning by defining an arrow of time.