The Moment Problem in Quantum Field Theory

Andrew Eberhardt, Michael Kopp, Alvaro Zamora +1 more

We introduce a numerical method and python package, chimes, that simulates quantum systems initially well approximated by mean field theory using a second order extension of the classical field approach. We call this the field moment expansion method. In this way, we can accurately approximate the evolution of first and second field moments beyond where the mean field theory breaks down. This allows us to estimate the quantum break time of a classical approximation without any calculations external to the theory. We investigate the accuracy of the field moment expansion using a number of well studied quantum test problems. Interacting bosonic systems similar to scalar field dark matter are chosen as test problems. We find that successful application of this method depends on two conditions: the quantum system must initially be well described by the classical theory, and the growth of the higher order moments must be hierarchical.

CHRISTOF WETTERICH

Sara Murciano, Giuseppe Di Giulio, Pasquale Calabrese

Abstract We present a thorough analysis of the entanglement entropies related to different symmetry sectors of free quantum field theories (QFT) with an internal U(1) symmetry. We provide explicit analytic computations for the charged moments of Dirac and complex scalar fields in two spacetime dimensions, both in the massive and massless cases, using two different approaches. The first one is based on the replica trick, the computation of the partition function on Riemann surfaces with the insertion of a flux α, and the introduction of properly modified twist fields, whose two-point function directly gives the scaling limit of the charged moments. With the second method, the diagonalisation in replica space maps the problem to the computation of a partition function on a cut plane, that can be written exactly in terms of the solutions of non-linear differential equations of the Painlevé V type. Within this approach, we also derive an asymptotic expansion for the short and long distance behaviour of the charged moments. Finally, the Fourier transform provides the desired symmetry resolved entropies: at the leading order, they satisfy entanglement equipartition and we identify the subleading terms that break it. Our analytical findings are tested against exact numerical calculations in lattice models.

Jayant Vishnu Narlikar, Τ. Padmanabhan

H.-J. Unger

H. B. Thacker

The quantum-mechanical implications of the inverse-scattering-transform method and its relationship to the structure of Bethe's ansatz are discussed in the context of the nonlinear Schr\"odinger equation (many-body problem) associated with the classical (quantum) field theory $\mathcal{L}=(\frac{i}{2}){\ensuremath{\varphi}}^{*}{\stackrel{}{\ensuremath{\partial}}}_{0}\ensuremath{\varphi}\ensuremath{-}{|{\ensuremath{\partial}}_{1}\ensuremath{\varphi}|}^{2}\ensuremath{-}c{|\ensuremath{\varphi}|}^{4}$. We review the transformation of the classical problem to action and angle variables and the derivation of an infinite number of polynomial conservation laws. The values of the conserved constants are given by the moments of the classical action variable. It is suggested that there exists a corresponding set of conserved polynomial operators in the quantum field theory and that they reflect the conservation of velocity content which characterizes the solution of the many-body scattering problem (Bethe's ansatz). This implies that the quantized action variable is just the occupation-number density operator in the asymptotic momentum-(velocity-) parameter space of Bethe's ansatz, and that Bethe's wave functions are eigenstates of all conserved operators with eigenvalues given by the moments of the $N$-particle distribution in asymptotic momentum space. These statements are verified for the first four operators, including one which has not previously been studied.