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We show how background field inhomogeneities modify the non-perturbative structure of the effective action. The simple Borel poles of the Euler-Heisenberg effective action become branch points, and new branch points also appear, indicating new non-perturbative effects. This information is resurgently encoded in the perturbative weak field expansion, and becomes physically significant for strongly inhomogeneous fields. We also show that resurgent extrapolation methods permit the decoding of a surprising amount of non-perturbative information from a relatively modest amount of perturbative input, enabling accurate analytic continuations from weak field to strong field, and of a spatially dependent magnetic background to a time dependent electric background. These extrapolations are far superior to standard WKB and locally constant field approximations., Comment: 26 pages, 10 figures

Motivated by the search for the QCD critical point, we discuss how to obtain the singular behavior of a thermodynamic system near a critical point, namely the Lee-Yang singularities, from a limited amount of local data generated in a different region of the phase diagram. We show that by using a limited number of Taylor series coefficients, it is possible to reconstruct the equation of state past the radius of convergence, in particular in the critical region. Furthermore we also show that it is possible to extend this reconstruction to go from a crossover region to the first-order transition region in the phase diagram, using a uniformizing map to pass between Riemann sheets. We illustrate these ideas via the Chiral Random Matrix Model and the Ising Model., Comment: 23 pages, 16 figures

We determine the 1=N2 f and 1=N3 f contributions to the QED beta function stemming from the closed set of nested diagrams. At the order of 1=N2 f, we discover a new logarithmic branch cut closer to the origin when compared to the 1=Nf results. The same singularity location appears at 1=N3 f, and these correspond to a UV renormalon singularity in the finite part of the photon two-point function.

We consider quantum field theoretic systems subject to a time-dependent perturbation, and discuss the question of defining a time dependent particle number not just at asymptotic early and late times, but also during the perturbation. Naively, this is not a well-defined notion for such a non-equilibrium process, as the particle number at intermediate times depends on a basis choice of reference states with respect to which particles and anti-particles are defined, even though the final late-time particle number is independent of this basis choice. The basis choice is associated with a particular truncation of the adiabatic expansion. The adiabatic expansion is divergent, and we show that if this divergent expansion is truncated at its optimal order, a universal time dependence is obtained, confirming a general result of Dingle and Berry. This optimally truncated particle number provides a clear picture of quantum interference effects for perturbations with non-trivial temporal sub-structure. We illustrate these results using several equivalent definitions of adiabatic particle number: the Bogoliubov, Riccati, Spectral Function and Schrodinger picture approaches. In each approach, the particle number may be expressed in terms of the tiny deviations between the exact and adiabatic solutions of the Ermakov-Milne equation for the associated time-dependent oscillators.

We consider the time evolution of the adiabatic particle number in both time-dependent electric fields and in de Sitter spaces, and define a super-adiabatic particle number in which the (divergent) adiabatic expansion is truncated at optimal order. In this super-adiabatic basis, the particle number evolves smoothly in time, according to Berry's universal adiabatic smoothing of the Stokes phenomenon. This super-adiabatic basis also illustrates clearly the quantum interference effects associated with particle production, in particular for sequences of time-dependent electric field pulses, and in eternal de Sitter space where there is constructive interference in even dimensions, and destructive interference in odd dimensions., Comment: 25 pages; 17 figures

We study non-self-dual classical solutions in the ${\mathbb{C}\mathbb{P}}^{N\ensuremath{-}1}$ model with ${\mathbb{Z}}_{N}$ twisted boundary conditions on the spatially compactified cylinder. These solutions have finite, and fractional, classical action and topological charge, and are ``unstable'' in the sense that the corresponding fluctuation operator has negative modes. We propose a physical interpretation of these solutions as saddle point configurations whose contributions to a resurgent semi-classical analysis of the quantum path integral are imaginary nonperturbative terms that must be cancelled by infrared renormalon terms generated in the perturbative sector.

We study the formation of vacuum condensates in $2+1$ dimensional QED in the presence of inhomogeneous background magnetic fields. For a large class of magnetic fields, the condensate is shown to be proportional to the inhomogeneous magnetic field, in the large flux limit. This may be viewed as a {\it local} form of the {\it integrated} degeneracy-flux relation of Aharonov and Casher., 13 pp, LaTeX, no figures; to appear in Phys. Rev. D

We introduce a nonperturbative continuum framework to study the dynamics of quantum field theory (QFT), applied here to the $CP(N\ensuremath{-}1)$ model. We show that the ambiguities in perturbation theory due to infrared renormalons are exactly canceled by corresponding ambiguities in the nonperturbative sector coming from amplitudes of certain nonperturbative objects: neutral bions and bion-antibions. This provides an explicit weak-coupling interpretation of the IR-renormalons. We use \'Ecalle's theory of resurgent trans-series and the physical principle of continuity to continuously connect QFT to quantum mechanics, while preventing all intervening rapid crossovers or phase transitions. The quantum mechanics contains the germ of all nonperturbative data, e.g., mass gap, of the QFT, all of which are calculable. For $CP(N\ensuremath{-}1)$, the results obtained at arbitrary $N$ are consistent with lattice and large-$N$ results. The trans-series expansion, in which perturbative and nonperturbative effects are intertwined, encapsulates the multilength-scale nature of the theory, and eliminates all perturbative and nonperturbative ambiguities under consistent analytic continuation of the coupling. A theorem by Pham et al. implies that the mass gap is a resurgent function, for which resummation of the semiclassical expansion yields finite exact results in the weakly coupled domain.

We present a simple unifying gauge theoretical formulation of gravitational theories in two dimensional spacetime. This formulation includes the effects of a novel matter-gravity coupling which leads to an extended de Sitter symmetry algebra on which the gauge theory is based. Contractions of this theory encompass previously studied cases., 19pp, no figs., CTP 2228, UCONN-93-6

We estimate the splitting between the spatial correlation lengths in the scalar and pseudoscalar channels in QCD at high temperature. The splitting is due to the contribution of the instanton/anti-instanton chains in the thermal ensemble, even though instanton contributions to thermodynamic quantities are suppressed. The splitting vanishes at asymptotically high temperatures as $\ensuremath{\Delta}M/M\ensuremath{\propto}({\ensuremath{\Lambda}}_{\mathrm{QCD}}/T{)}^{b}$, where $b$ is the beta function coefficient.

We consider a model of nonrelativistic charged matter interacting with a massive Abelian gauge field and a neutral scalar field in (2+1)-dimensional space-time. As in the model considered by Jackiw and Pi with a pure Chern-Simons interaction, the model admits classical static self-dual solutions. These solutions are nontopological solitons which may be threaded by a number of vortices. Furthermore, these self-dual classical solutions are zero-energy configurations.

The computation of the one-loop effective action in a radially symmetric background can be reduced to a sum over partial-wave contributions, each of which is the logarithm of an appropriate one-dimensional radial determinant. While these individual radial determinants can be evaluated simply and efficiently using the Gel'fand-Yaglom method, the sum over all partial-wave contributions diverges. A renormalization procedure is needed to unambiguously define the finite renormalized effective action. Here we use a combination of the Schwinger proper-time method, and a resummed uniform DeWitt expansion. This provides a more elegant technique for extracting the large partial-wave contribution, compared to the higher order radial WKB approach which had been used in previous work. We illustrate the general method with a complete analysis of the scalar one-loop effective action in a class of radially separable SU(2) Yang-Mills background fields. We also show that this method can be applied to the case where the background gauge fields have asymptotic limits appropriate to uniform field strengths, such as for example in the Minkowski solution, which describes an instanton immersed in a constant background. Detailed numerical results will be presented in a sequel., Comment: 35 pages

We extend the worldline instanton technique to compute the vacuum pair production rate for spatially inhomogeneous electric background fields, with the spatial inhomogeneity being genuinely two or three dimensional, both for the magnitude and direction of the electric field. Other techniques, such as WKB, have not been applied to such higher dimensional problems. Our method exploits the instanton dominance of the worldline path integral expression for the effective action., 22 pages, 13 figures

We analyze the large-order behavior of the perturbative weak-field expansion of the effective Lagrangian density of a massive scalar in de Sitter and anti de Sitter space, and show that this perturbative information is not sufficient to describe the non-perturbative behavior of these theories, in contrast to the analogous situation for the Euler-Heisenberg effective Lagrangian density for charged scalars in constant electric and magnetic background fields. For example, in even dimensional de Sitter space there is particle production, but the effective Lagrangian density is nevertheless real, even though its weak-field expansion is a divergent non-alternating series whose formal imaginary part corresponds to the correct particle production rate. This apparent puzzle is resolved by considering the full non-perturbative structure of the relevant Feynman propagators, and cannot be resolved solely from the perturbative expansion., 18 pages

In a previous paper [1], it was shown that the worldline expression for the nonperturbative imaginary part of the QED effective action can be approximated by the contribution of a special closed classical path in Euclidean spacetime, known as a worldline instanton. Here we extend this formalism to compute also the prefactor arising from quantum fluctuations about this classical closed path. We present a direct numerical approach for determining this prefactor, and we find a simple explicit formula for the prefactor in the cases where the inhomogeneous electric field is a function of just one spacetime coordinate. We find excellent agreement between our semiclassical approximation, conventional WKB, and recent numerical results using numerical worldline loops., 28 pages, 1 figure; v2 references added, version in PRD

We compute the finite temperature correction to the induced fermion number for fermions coupled to a static SU(2) chiral background, using the derivative expansion technique. At zero temperature the induced fermion number is topological, being the winding number of the chiral background. At finite temperature however, higher order terms in the derivative expansion give nontopological corrections to the winding number. We use this result to show that the standard cancellation of the fractional parts of the fermion number inside and outside the bag in an SU(2) x SU(2) hybrid chiral bag model of the nucleon does not occur at nonzero temperature., 9 pages, no figures

We compute the finite temperature induced fermion number for fermions coupled to a static nonlinear sigma model background in (2+1) dimensions, in the derivative expansion limit. While the zero temperature induced fermion number is well known to be topological (it is the winding number of the background), at finite temperature there is a temperature dependent correction that is nontopological -- this finite T correction is sensitive to the detailed shape of the background. At low temperature we resum the derivative expansion to all orders, and we consider explicit forms of the background as a CP^1 instanton or as a baby skyrmion.

We demonstrate by explicit construction that while the untwisted Harrington-Shepard caloron A{sub {mu}} is manifestly periodic in Euclidean time, with a period {beta}=1/T, when transformed to the Weyl (A{sub 0}=0) gauge, the caloron gauge field A{sub i} is periodic only up to a large gauge transformation, with the winding number equal to the caloron's topological charge. This helps clarify the tunneling interpretation of these solutions, and their relation to Chern-Simons numbers and winding numbers.

We apply the resolvent technique to the computation of the QED effective action in time dependent electric field backgrounds. The effective action has both real and imaginary parts, and the imaginary part is related to the pair production probability in such a background. The resolvent technique has been applied previously to spatially inhomogeneous magnetic backgrounds, for which the effective action is real. We explain how dispersion relations connect these two cases, the magnetic case which is essentially perturbative in nature, and the electric case where the imaginary part is nonperturbative. Finally, we use a uniform semiclassical approximation to find an expression for very general time dependence for the background field. This expression is remarkably similar in form to Schwinger's classic result for the constant electric background.

In this talk we consider the Heisenberg equations of motion {dot q} = {minus}i(q, H), {dot p} = {minus}i(p, H), for the quantum-mechanical Hamiltonian H(p, q) having one degree of freedom. It is a commonly held belief that such operator differential equations are intractable. However, a technique is presented here that allows one to obtain exact, closed-form solutions for huge classes of Hamiltonians. This technique, which is a generalization of the classical action-angle variable methods, allows us to solve, albeit formally and implicitly, the operator differential equations of two anharmonic oscillators whose Hamiltonians are H = p{sup 2}/2 + q{sup 4}/4 and H = p{sup 4}/4 + q{sup 4}/4.

In a previous paper we introduced a method for obtaining exact solutions to the operator differential equations of quantum mechanics. In that paper we showed how to solve some simple quantum-mechanical models and we suggested that the method could be used to obtain exact solutions to the operator differential equations of more complicated models, such as the anharmonic oscillator whose Hamiltonian is $H=\frac{1}{2}{p}^{2}+\frac{1}{4}{q}^{4}$. In this paper we further sharpen the formalism and introduce the concept of a minimal solution. We then obtain the exact minimal solution to the operator differential equations arising from two different anharmonic-oscillator models whose Hamiltonians are $H=\frac{1}{2}{p}^{2}+\frac{1}{4}{q}^{4}$ and $H=\frac{1}{4}{p}^{4}+\frac{1}{4}{q}^{4}$.