A path integral over the field configurations gives a representation of the thermodynamical partition function. The resulting Euclidean formulation of thermal field theory enables topologically nontrivial solutions for equations of motion, which in turn enables nonperturbative theories for the thermal ground state at high temperatures. Effective excitations in Yang-Mills theory requires a momentum space formulation to discern quantum fluctuations from thermal effects. The thermal ground state sets the scale of maximal resolution, so it needs to be expressly identified for a complete theory.
From the Minkowski metric, the path-integral representation mapping onto the space of field configurations itself includes a Gaussian integral, for which the expression can be normalized without paying too much attention to the resulting factor. Recover from that, the partition function Z.
This implies a periodic solution. It's a familiar equation, usually solved using something akin to variable conjugation in both time and space. From this arises the Matsubara frequency, and the Fourier transform of a differential operator D.
Extend the modified ideal gas law to these equations via the logarithm of the partition function.
The root term is in reference to the vacuum fluctuations. Free theories on flat, 4D spacetimes are defined to have no vacuum energy, so contributions with linear dependencies of vacuum fluctuations, and those without x-dependence, can be set to zero. For small a, the thermodynamical pressure P can be expanded to a completely time-dependent expression. Solving it requires a solution of the Riemann-Zeta function (for z > 1)
As the derivatives of I(a) with respect to a² diverge at a = 0, higher derivatives of P(a) than those of second order are not defined at a = 0. Instead, a resummed expansion should be used, though it modifies the integrand of I(a), which can then be split into an infinite sum and remainder. The final logarithmic contributions, and odd powers of a are the source of the nonanalycity of P in a² around a = 0.
The imaginary time in the finite-temperature field require integration over the Matsubara frequencies. This method makes quantum fluctuations impossible to identify. As mentioned, quantum and thermal effects are independent from one another, so the theory needs to be reformulated via the imaginary time propagator. The thermal topological field configurations contributing to the ground state can unfortunately only be expressed through imaginary time, while the formulation of thermodynamics of effective excitations on different time-integration contours in the action is inconsistent insofar that the partition function is only ever defined on a single spacetime over the entire theory.
The connection between imaginary and real-time formulations lies in the one-loop diagram. In imaginary time, it can be represented by a frequency sum of analytic functions. It may be set equal to the integral over a spatial momentum expression via a contour integral.
The factor in the integral has simple poles at p₀ = 2πnTi. C is assumed to be a set of closed circular curves with r < πT, and centered around the poles. The contours can be deformed into two lines parallel to the imaginary axis. The clockwise rotation of the contour of the sum at T = 0 to the real axis circumvents the poles. For T > 0, the contour must be closed by a semicircle over the infinite right half-plane.
The right side of the final equation is the real-time propagator of the scalar field in momentum space. It splits into temperature-independent parts and a part in which the thermal fluctuations are Bose-distributed on the mass shell. This description currently sees no momentum transfers in diagrams such daisy diagrams, so an additional formalism needs to be introduced, making sure that the contour for the integration is more precise/elaborate. The δ-function is replaced by a function describing the physical energy spectrum of intermediate states at fixed spatial momentum. This "spectral function" expresses the imaginary part of the self-energy, and erases the mathematical inconsistencies.
Comentários