may be written as expectation values in ordinary quantum field theory where the configuration is evolved by an imaginary time τ τ = i t ( 0 ≤ ≤ τ τ ≤ ≤ β β ) . One can therefore switch to a spacetime with Euclidean signature, where the above trace (Tr) leads to the requirement that all bosonic and fermionic fields be periodic and antiperiodic, respectively, with respect to the Euclidean time direction with periodicity β β = 1 / ( k T ) (we are assuming natural units ℏ ℏ = 1 ). This allows one to perform calculations with the same tools as in ordinary quantum field theory, such as functional integrals and Feynman diagrams, but with compact Euclidean time. Note that the definition of normal ordering has to be altered. In momentum space, this leads to the replacement of continuous frequencies by discrete imaginary (Matsubara) frequencies v n = n / β β and, through the de Broglie relation, to a discretized thermal energy spectrum E n = 2 n π π k T . This has been shown to be a useful tool in studying the behavior of quantum field theories at finite temperature. It has been generalized to theories with gauge invariance and was a central tool in the study of a conjectured deconfining phase transition of Yang–Mills theory. In this Euclidean field theory, real-time observables can be retrieved by analytic continuation. The Feynman rules for gauge theories in the Euclidean time formalism, were derived by C. W. Bernard.