This is the classical form of the theorem; quantum fluctuations are taken into account by replacing 2 k B T / ω ω with ℏ ℏ coth ⁡ ⁡ ( ℏ ℏ ω ω / 2 k B T ) (whose limit for ℏ ℏ → → 0 is 2 k B T / ω ω ). A proof can be found by means of the LSZ reduction, an identity from quantum field theory.