# A no-go result for the quantum damped harmonic oscillator

## TL;DR

This paper demonstrates that canonical quantization of the quantum damped harmonic oscillator using the Bateman Lagrangian is fundamentally flawed, as no proper vacuum state exists within the standard framework, highlighting the need for more rigorous approaches.

## Contribution

It proves the non-existence of a proper vacuum state in the canonical quantization of the damped harmonic oscillator with the Bateman Lagrangian, challenging previous assumptions.

## Key findings

- No square integrable vacuum exists for the system's ladder operators.
- Vacua can only be represented as distributions, not proper states.
- Previous formal approaches lack rigorous mathematical foundation.

## Abstract

In this letter we show that it is not possible to set up a canonical quantization for the damped harmonic oscillator using the Bateman lagrangian. In particular, we prove that no square integrable vacuum exists for the {\em natural} ladder operators of the system, and that the only vacua can be found as distributions. This implies that the procedure proposed by some authors is only formally correct, and requires a much deeper analysis to be made rigorous.