Fluctuation Dissipation Theorem

The fluctuation dissipation theorem (Nyquist formula) is shown to be exact and a number of generalizations of it are given, including a four-dimensional formulation which is useful in the quantum theory of fields. The more general theorem can be used to calculate vacuum expectation values of field operators, and to deduce covariant commutation relations for the fields. For a four-potential field, the vacuum expectation values for operators at two space-time points x and x{'} are ${〈{A}_{{μ}}(x){A}_{{α}}({x}^{{'}})〉}₀={{}}{{π}}{∫}{0}^{{∞}}{{d}_{{μ}{α}}([x{-}{x}^{{'}}], {ω})}{{ω}}d{ω},$ where ${d}_{{μ}{α}}$ is a dissipation tensor.The covariant commutation relations are $[{A}_{{μ}}(x), {A}_{{α}}({x}^{{'}})]={{}}{{π}}{∫}{0}^{{∞}}{[{d}_{{μ}{α}}([x{-}{x}^{{'}}], {ω}){-}{d}_{{α}{μ}}([{x}^{{'}}{-}x], {ω})]}{{ω}}d{ω}.$A well-defined cut-off procedure is given for calculating observable fluctuations in cases where the theorem gives infinite results.For measurements with a linear device which has energy $E={}{{ω}}c$, the observable fluctuations of the vacuum electromagnetic fields are given by the exact expression $〈{V}²〉={{}}{{π}}{∫}{0}^{{{ω}}c}R({ω}){ω}d{ω}.$