A general form of Wick rotation is the “Weyl’s unitary trick” or Unitarian trick which is related with groups like literally everything else in physics.

In quantum field theory, Wick rotation refers to a transformation from real time to imaginary time. This is done by replacing t → iτ, where τ is the new imaginary time. The goal of this transformation is often to simplify the mathematical treatment of quantum systems by turning oscillatory integrals (involving \(e^{-iEt / ℏ}\)) into exponentially decaying functions (like \(e^{-Eτ / ℏ}\)), which are easier to handle apperantly. So transformation looks like this:

\[\large e^{-iEt / \hbar} => e^{-E / k_B T}\]

Wick rotation consideres t in quantum mechanics as an imaginary time, so

\[\large t = -iτ\]

Here, we no longer have an oscillatory behavior but an exponentially decaying factor τ . And this is the connection to statistical mechanics by,

\[\large e^{-Eτ / \hbar} => e^{-E / k_B T}\]


In order to understand Wick rotation better, we can introduct a little bit of quantum field theory without resorting to complex branches of mathematics, especially groups. A field in physics defined as a physical quantity with components of scalars, vectors and tensors that has a value for each point in space and time. A classical field is a function of all space and time coordinates, like E(r, t), B(r, t). They have infinite amount of degrees of freedom since they are continuous. A field has to specify a value for each of these points, leading to infinitely many independent variables to describe its state. These independent variables correspond to the degrees of freedom.

Quantum Field Theory arised because quantum mechanics failed to explain: Relativity and creation and annihilation of particles.

To describe particles appearing, disappearing, or interacting, we need something that can exist everywhere in space and time—a field. To understand the further ideas of QFT, like particles as excitations of fields and relativistic spacetime symmetries, let’s understand these field manner in QFT.

When a classical field is quantized (e.g., in quantum field theory), the infinite degrees of freedom translate into an infinite number of quantum states or modes, often corresponding to particles or quanta of the field. Quantizing classical fields is the central idea of quantum field theory.