In semiclassical treatments of transition amplitudes, the determinant arising from fluctuations around a stationary solution needs to be evaluated through a correct limiting procedure. The relation between the determinant and an associated initial-value problem is clarified for coherent-state and Slater-determinant functional integrals.
We consider the slave-boson method within the coherent-state functional integral representation of the partition function, and show how to deal with the continuum imaginary-time limit required by the very definition of the functional integral. We find that a correct treatment of the continuum limit modifies the free energy when fluctuation (1/N) corrections beyond the mean-field solution are considered. Numerical results are presented for a two-level single-site model system (with an infinite Hubbard repulsion), for which the additional terms in the free energy introduced by the correct continuum limit act to validate the 1/N expansion. Our analysis calls for a revision of several outcomes of the slave-boson method with the inclusion of fluctuation corrections.
Recently, doubts have been cast on the validity of the continuous-time coherent state path integral. This has led to controversies regarding the correct way of performing calculations with path integrals, and to several alternative definitions of what should be their continuous limit. Furthermore, the issue of a supposedly proper ordering of the Hamiltonian operator, entangled with the continuous-time limit, has led to considerable confusion in the literature. Since coherent state path integrals are at the basis of the modern formulation of many-body quantum theory, it should be laid on solid foundations. Here, we show that the standard coherent state path integral is fine as it is, and does not need any modification. We show that the issues raised above are coming from illegitimate use of the (standard) rules of calculus, which are not necessarily valid in path integrals. This is well known in the context of stochastic equations, in particular in their path integral formulation. This insight allows for solving these issues, and addressing the correspondence between the various orderings at the level of the path integral. We also use this opportunity to address the proper calculation of a functional determinant in the presence of a Hubbard-Stratonovich field, which shares in the controversies.
The massless harmonic oscillator is a rare example of a system whose Feynman path integral can be explicitly computed and receives its main contributions from regions of the functional space that are far from the classical and semiclassical configurations near the stationary point of the action. The functional average ⟨qm2⟩ of the square of the coordinate at a time tm which is intermediate between the initial and final time gives a measure of the amplitude of quantum fluctuations with respect to the classical path. This average, or "transition element", is divergent in the massless limit, signaling a quantum runaway. We show that the divergence is not due to the continuum limit and formulate the conjecture that the divergent contributions come from regions where the action S is constant and therefore the interference factor e−iS/ℏ does not oscillate. For most systems these regions have zero functional measure and thus give a null contribution to the path integral, but this is not the case for the massless oscillator. We study the simplest functional subspace with constant action, namely the one with S=0, which is connected to the classical solutions but extends to infinity, like an hyperplane through the origin; this subspace turns out to be infinite-dimensional. Some possible applications and developments are mentioned.
The feasibility of using Monte Carlo methods in finite fermion systems has been investigated. The basic tool is a (recently suggested) functional integral representation of $Z=\mathrm{tr}[\mathrm{exp}(\ensuremath{-}\ensuremath{\beta}II)]$ using real coherent states of Slater determinants. Conditions for using the discretized (N-point lattice) version as a starting point for Monte Carlo integrations are studied. The limit of large $\ensuremath{\beta}$ is achieved only with large $\ensuremath{\surd}N$. For a two-fermion system numerical calculations of the ground-state energy agree farily well with the exact value.