Functional Integral Representation Research Articles

Single scale cluster expansions with applications to many Boson and unbounded spin systems

We develop a cluster expansion to show exponential decay of correlations for quite general single scale spin systems, as they arise in lattice quantum field theory and discretized functional integral representations for observables of quantum statistical mechanics. We apply our results to the small field approximation to the coherent state correlation functions of the grand canonical Bose gas at negative chemical potential, constructed by Balaban et al. [Ann. Henri Poincaré 11, 151–350 (2010c)], and to N component unbounded spin systems with repulsive two body interaction and massive, possibly complex, covariance. Our cluster expansion is derived by a single application of the Brydges-Kennedy-Abdesselam-Rivasseau interpolation formula.

Nonperturbative quantization of the electroweak model’s electrodynamic sector

Consider the Euclidean functional integral representation of any physical process in the electroweak model. Integrating out the fermion degrees of freedom introduces twenty-four fermion determinants. These multiply the Gaussian functional measures of the Maxwell, $Z$, $W$ and Higgs fields to give an effective functional measure. Suppose the functional integral over the Maxwell field is attempted first. This paper is concerned with the large amplitude behavior of the Maxwell effective measure. It is assumed that the large amplitude variation of this measure is insensitive to the presence of the $Z$, $W$ and $H$ fields; they are assumed to be a subdominant perturbation of the large amplitude Maxwell sector. Accordingly, we need only examine the large amplitude variation of a single QED fermion determinant. To facilitate this the Schwinger proper time representation of this determinant is decomposed into a sum of three terms. The advantage of this is that the separate terms can be non-perturbatively estimated for a measurable class of large amplitude random fields in four dimensions. It is found that the QED fermion determinant grows faster than $\exp \left[ce^2\int\mathrm d^4x\, F_{\mu\nu}^2\right]$, $c>0$, in the absence of zero mode supporting random background potentials. This raises doubt on whether the QED fermion determinant is integrable with any Gaussian measure whose support does not include zero mode supporting potentials. Including zero mode supporting background potentials can result in a decaying exponential growth of the fermion determinant. This is \textit{prima facie} evidence that Maxwellian zero modes are necessary for the non-perturbative quantization of QED and, by implication, for the non-perturbative quantization of the electroweak model.

Two-particle irreducible functional renormalization group schemes—a comparative study

We derive functional renormalization group (fRG) schemes for Fermi systems which are based on the two-particle irreducible approach to the quantum many-body problem. In a first step, the cutoff is introduced in the non-interacting propagator as it is commonly done in fRG based on one-particle irreducible vertex functions. The most natural truncation of the resulting infinite hierarchy of flow equations is shown to be fully equivalent to self-consistent perturbation theory. An earlier suggested alternative truncation strategy is considered as well. In a second step, the cutoff is introduced in the two-particle interaction. Again two truncation procedures are investigated, one of which was derived before. In the latter, the mean-field solution of the many-body problem is considered as the starting point of the renormalization group flow. We compare the performance and the required numerical resources for solving the coupled flow equations for all the approximate schemes by applying them to the problem of the quantum anharmonic oscillator. In a functional integral representation, this model has a formal similarity to the quantum many-body problem. The perspectives for applying the derived two-particle irreducible fRG approaches to zero- and one-dimensional systems of correlated fermions are discussed.

Statistical dynamics of classical systems: a self-consistent field approach.

We develop a self-consistent field theory for particle dynamics by extremizing the functional integral representation of a microscopic Langevin equation with respect to the collective fields. Although our approach is general, here we formulate it in the context of polymer dynamics to highlight satisfying formal analogies with equilibrium self-consistent field theory. An exact treatment of the dynamics of a single chain in a mean force field emerges naturally via a functional Smoluchowski equation, while the time-dependent monomer density and mean force field are determined self-consistently. As a simple initial demonstration of the theory, leaving an application to polymer dynamics for future work, we examine the dynamics of trapped interacting Brownian particles. For binary particle mixtures, we observe the kinetics of phase separation.

Spin-boson model through a Poisson-driven stochastic process

We give a functional integral representation of the semigroup generated by the spin-boson Hamiltonian by making use of a Poisson point process and a Euclidean field. We present a method of constructing Gibbs path measures indexed by the full real line which can be applied also to more general stochastic processes with jump discontinuities. Using these tools we then show existence and uniqueness of the ground state of the spin-boson, and analyze ground state properties. In particular, we prove super-exponential decay of the number of bosons, Gaussian decay of the field operators, derive expressions for the positive integer, fractional and exponential moments of the field operator, and discuss the field fluctuations in the ground state.

Dynamics of polymers: A mean-field theory

We derive a general mean-field theory of inhomogeneous polymer dynamics; a theory whose form has been speculated and widely applied, but not heretofore derived. Our approach involves a functional integral representation of a Martin-Siggia-Rose (MSR) type description of the exact many-chain dynamics. A saddle point approximation to the generating functional, involving conditions where the MSR action is stationary with respect to a collective density field ρ and a conjugate MSR response field ϕ, produces the desired dynamical mean-field theory. Besides clarifying the proper structure of mean-field theory out of equilibrium, our results have implications for numerical studies of polymer dynamics involving hybrid particle-field simulation techniques such as the single-chain in mean-field method.

Scaling dimensions in AdS/QCD and the gluon field strength propagator

We derive the scaling dimension of antisymmetric tensor operators in the boundary theory of the AdS/CFT correspondence using a functional integral representation of the boundary-to-boundary propagators of their dual fields in the bulk. We then apply this technique to AdS/QCD in which the bulk metric is warped, resulting in non-constant scaling dimensions. In particular, we compute the two-point correlation function of gluon field strength operators, for which it is prerequisite to know the ow of the anomalous scaling dimension under rescaling. The results are in very good agreement with quenched lattice QCD data, thus confirming the functional form of the scaling dimension.

FUNCTIONAL INTEGRAL REPRESENTATIONS AND GOLDEN–THOMPSON INEQUALITIES IN BOSON–FERMION SYSTEMS

For a general class of boson–fermion Hamiltonians H acting in the tensor product Hilbert space L2(ℝn) ⊗ ∧(ℂr) of L2(ℝn) and the fermion Fock space ∧(ℂr) over ℂr(n, r ∈ ℕ), we establish, in terms of an n-dimensional conditional oscillator measure, a functional integral representation for the trace Tr (F ⊗ zN f e-tH)(F ∈ L∞(ℝn), z ∈ ℂ∖{0}, t > 0), where N f is the fermion number operator on ∧(ℂr). We prove a Golden–Thompson type inequality for | Tr (F ⊗ zN f e-tH)|. Also we discuss applications to a model in supersymmetric quantum mechanics and present an improved version of the Golden–Thompson inequality in supersymmetric quantum mechanics given by Klimek and Lesniewski ([Lett. Math. Phys.21 (1991) 237–244]). An upper bound for the number of the supersymmetric states is given as well as a sufficient condition for the spontaneous supersymmetry breaking. Moreover, we derive a functional integral representation for the analytical index of a Dirac type operator on ℝn (Witten index) associated with the supersymmetric quantum mechanical model.

Investigations on the Theory of Riemann Zeta Function I: New Functional Equation, Integral Representation and Laurent Expansion for Riemann’s Zeta Function

We developed a new functional equation and a new integral representation for the Riemann zeta function.

Field-dependent BRST transformations in Yang–Mills theory

We find an explicit form for the Jacobian of arbitrary field-dependent BRST transformations in Yang–Mills theory. For the functional-integral representation of the (gauge-fixed) Yang–Mills vacuum functional, such transformations merely amount to a precise change in the gauge-fixing functional. This proves the independence of the vacuum functional under any field-dependent BRST transformation. We also give a formula for the transformation parameter functional which generates a prescribed change of gauge and evaluate it for connecting two arbitrary Rξ gauges.

Contour-ordered Green’s functions in stochastic field theory

We briefly review the functional formulation of the perturbation theory for various Green’s functions in quantum field theory. In particular, we discuss the contour-ordered representation of Green’s functions at a finite temperature. We show that the perturbation expansion of time-dependent Green’s functions at a finite temperature can be constructed using the standard Wick rules in the functional form without introducing complex time and evolution backward in time. We discuss the factorization problem for the corresponding functional integral. We construct the Green’s functions of the solution of stochastic differential equations in the Schwinger-Keldysh form with a functional-integral representation with explicitly intertwined physical and auxiliary fields.

The propagation kernel for strong coupling gravity

The strong coupling limit of Einstein gravity in d+1 dimensions gives rise to a quantum theory where after factorization of the conformal factor mode SL(d,R)/SO(d) nonlinear sigma-models are spatially coupled by the diffeomorphism constraint. A functional integral representation for the theoryʼs propagation kernel is derived in completions of the proper time gauge which manifestly invokes only physical gauge invariant degrees of freedom. In the weak field limit it reduces to the propagation kernel of massless and transversal-traceless free fields. For strong fields a covariant normal coordinate expansion is developed which covers the configuration manifold globally. Its leading order approximant resembles a semiclassical propagation kernel but without the need to solve the classical constraints. The results have implications for the ground state structure of quantum gravity.

Green functions in stochastic field theory

Functional representations are reviewed for the generating function of Green functions of stochastic problems stated either with the use of the Fokker-Planck equation or the master equation. Both cases are treated in a unified manner based on the operator approach similar to quantum mechanics. Solution of a second-order stochastic differential equation in the framework of stochastic field theory is constructed. Ambiguities in the mathematical formulation of stochastic field theory are discussed. The Schwinger-Keldysh representation is constructed for the Green functions of the stochastic field theory which yields a functional-integral representation with local action but without the explicit functional Jacobi determinant or ghost fields.

Generalizing the ADM computation to quantum field theory

The absence of recognizable, low energy quantum gravitational effects requires that some asymptotic series expansion be wonderfully accurate, but the correct expansion might involve logarithms or fractional powers of Newton’s constant. That would explain why conventional perturbation theory shows uncontrollable ultraviolet divergences. We explore this possibility in the context of the mass of a charged, gravitating scalar. The classical limit of this system was solved exactly in 1960 by Arnowitt, Deser and Misner, and their solution does exhibit nonanalytic dependence on Newton’s constant. We derive an exact functional integral representation for the mass of the quantum field theoretic system, and then develop an alternate expansion for it based on a correct implementation of the method of stationary phase. The new expansion entails adding an infinite class of new diagrams to each order and subtracting them from higher orders. The zeroth-order term of the new expansion has the physical interpretation of a first quantized Klein–Gordon scalar which forms a bound state in the gravitational and electromagnetic potentials sourced by its own probability current. We show that such bound states exist and we obtain numerical results for their masses.

FUNCTIONAL-INTEGRAL SOLUTION FOR THE SCHRÖDINGER EQUATION WITH POLYNOMIAL POTENTIAL: A WHITE NOISE APPROACH

A functional integral representation for the weak solution of the Schrödinger equation with polynomially growing potentials is proposed in terms of a white noise functional.

Coulomb blockade of Majorana-fermion-induced transport

We study Coulomb charging effects for transport through a topologically nontrivial superconducting wire, where Majorana bound states are present at the interface to normal-conducting leads. We construct the general Keldysh functional integral representation, and provide detailed results for the nonlinear current-voltage relation under weak Coulomb blockade conditions.

Anomalous scaling in the random-force-driven Burgers' equation: a Monte Carlo study

We present a new approach to determine numerically the statistical behavior of small-scale structures in hydrodynamic turbulence. Starting from the functional integral representation of the random-force-driven Burgers' equation we show that Monte Carlo simulations allow us to determine the anomalous scaling of high-order moments of velocity differences. Given the general applicability of Monte Carlo methods, this opens up the possibility of also addressing other systems relevant to the turbulence within this framework.

Extended self-energy functional approach for strongly correlated lattice bosons in the superfluid phase

Among the various numerical techniques to study the physics of strongly correlated quantum many-body systems, the self-energy functional approach (SFA) has become increasingly important. In its previous form, however, SFA is not applicable to Bose-Einstein condensation or superfluidity. In this paper we show how to overcome this shortcoming. To this end we identify an appropriate quantity, which we term $D$, that represents the correlation correction of the condensate order parameter, as it does the self-energy for the Green's function. An appropriate functional is derived, which is stationary at the exact physical realizations of $D$ and of the self-energy. Its derivation is based on a functional-integral representation of the grand potential followed by an appropriate sequence of Legendre transformations. The approach is not perturbative and therefore applicable to a wide range of models with local interactions. We show that the variational cluster approach based on the extended self-energy functional is equivalent to the "pseudoparticle" approach introduced in Phys. Rev. B, 83, 134507 (2011). We present results for the superfluid density in the two-dimensional Bose-Hubbard model, which show a remarkable agreement with those of Quantum-Monte-Carlo calculations.

The Ground State Energy of the Massless Spin-Boson Model

We provide an explicit combinatorial expansion for the ground state energy of the massless spin-Boson model as a power series in the coupling parameter. Our method uses the technique of cluster expansion in constructive quantum field theory and takes as a starting point the functional integral representation and its reduction to an Ising model on the real line with long range interactions. We prove the analyticity of our expansion and provide an explicit lower bound on the radius of convergence. We do not need multiscale nor renormalization group analysis. A connection to the loop-erased random walk is indicated.

Integral estimates for differentiable functions on irregular domains

Integral representations for functions and their partial derivatives in terms of some fixed system of partial derivatives are constructed on irregular domains in a Euclidean space. Embedding theorems for Sobolev-type spaces into a Lebesgue space are established and the norms of the derivatives are estimated.Bibliography: 17 titles.