Hamiltonian finite-temperature quantum field theory from its vacuum on partially compactified space

The partition function of a relativistic invariant quantum field theory is expressed by its vacuum energy calculated on a spatial manifold with one dimension compactified to a 1-sphere $S^1(\beta )$, whose circumference $\beta$ represents the inverse temperature. Explicit expressions for the usual energy density and pressure in terms of the energy density on the partially compactified spatial manifold ${\mathbb{R}}^2\times S^1(\beta )$ are derived. To make the resulting expressions mathematically well defined a Poisson resummation of the Matsubara sums as well as an analytic continuation in the chemical potential are required. The new approach to finite-temperature quantum field theories is advantageous in a Hamilton formulation since it does not require the usual thermal averages with the density operator. Instead, the whole finite-temperature behavior is encoded in the vacuum wave functional on the spatial manifold ${\mathbb{R}}^2\times S^1(\beta )$. We illustrate this approach by calculating the pressure of a relativistic Bose and Fermi gas and reproduce the known results obtained from the usual grand canonical ensemble. As a first nontrivial application we calculate the pressure of Yang-Mills theory as a function of the temperature in a quasiparticle approximation motivated by variational calculations in Coulomb gauge.

Figure 1

The pressure of a gas of massive bosons as a function of the temperature calculated from Eq. (54) with $\mathcal{N}=1$ by including only the first $n=1$, 2 and 5 terms. The crosses give the result of the grand canonical ensemble (36).

Figure 2

The pressure $p$ of Yang-Mills theory calculated by the method of Sec. 2 for a gluon dispersion relation $\omega (p)=p+M^2/p$. Shown is the ratio $p/p_{SB}$ where $p_{SB}$ denotes the Stefan-Boltzmann limit (108). The crosses show the lattice data from Ref. [23].