# Uniqueness of Segal quantization for oscillating systems

## TL;DR

This paper proves that the Segal quantization of decoupled harmonic oscillators is unique, as the one-particle Hilbert space is fully determined by natural symplectic and unitary evolution requirements.

## Contribution

It establishes the uniqueness of Segal quantization for oscillating systems based on symplectic and unitary constraints.

## Key findings

- Segal quantization is unique under natural conditions.
- The one-particle Hilbert space is fully determined by symplectic and unitary requirements.
- The result applies to arbitrary decoupled harmonic oscillators.

## Abstract

We show that the Segal quantization of an arbitrary system of decoupled harmonic oscillators is unique in the sense that the one particle Hilbert space is completely determined by the requests of being a naturally complex symplectic space carrying a unitary realization of the dynamical evolution of the considered system.