Statistical mechanics of a (1 + 1)-dimensional quantum field theory at finite density and temperature

The quantum field theory in one space + one time dimension described by the Lagrangian $\mathfrak{L} = \left(\frac{i}{2}\right){\phi }^{*}{\stackrel{}{\partial }}_0\phi -\left({\partial }_1{\phi }^{*}\right)\left({\partial }_1\phi \right)-c{\left|\phi \right|}^4$ is studied for systems with finite temperature and particle density. Using momentum-space techniques previously developed, a graphical procedure is obtained for calculating inner products of many-particle scattering state wave functions. The unitarity of the wave operator $U(0,-\infty )$ is demonstrated as a pattern of graphical cancellations. An operator formulation of statistical mechanics is derived in which partition functions are given in terms of matrix elements having the form of diagonal (forward) inner products. The importance of noncommutativity of the forward limit and the $i\epsilon \to 0$ limit is noted and traced to the presence of forward singular graphs in the inner product. Combining this observation with wave-operator unitarity, we obtain a graphical recipe for calculating $N$-body partition functions. The thermodynamics first described by Yang and Yang is obtained by summation of the fugacity series for the grand partition function.