Functional integral formulation of classical wave equations

It is shown in this paper that classical wave equations admit path integral formulations. For this, the evolution of the system is first set-up in terms of a fundamental solution or propagator. We choose this last name because it suggests a connection with functional integrals, which are exploited in this work. A functional integral in terms of non-singular functions is then proposed and shown to converge to the propagator in the appropriate limit for the case of scalar wave equations. One of the advantages of such formulation is that it provides an adequate framework for mesh-free numerical methods. This is demonstrated through a computational implementation that combines a simple second-degree polynomial local approximation of the continuous field and an approximate statement of the exact evolution equations. Numerical simulations of modal analysis and transient dynamics indicate the feasibility of the technique.     

Functional equations for Feynman integrals

New types of equations for Feynman integrals are found. It is shown that the latter satisfy functional equations that relate integrals with different kinematics. A regular method for obtaining such relations is proposed. A derivation of the functional equations for one-loop two-, three-, and four-point functions with arbitrary masses and external momenta is given. It is demonstrated that the functional equations can be used to analytically continue Feynman integrals to various kinematical domains.     

Renormalization of massless fields with hard-soft infrared decomposition

The decomposition of Feynman integrals with massless propagators into hard and soft contributions is systematically effected in renormalized field theory. It is shown that the decomposition leads to an elegant method of renormalizing massless field theories. Ultraviolet and infrared finite composite fields (normal products) are defined and renormalized field equations are derived. Exploiting a gauge principle, scalar ghosts arising in the hard-soft decomposition are eliminated and a renormalization group equation is derived to describe the effects of changes in the mass scale.     

Feynman integrals with tensorial structure in the negative dimensional integration scheme

The negative-dimensional integration method (NDIM) is revealing itself as a very useful technique for computing massless and/or massive Feynman integrals, covariant and noncovariant alike. Up until now, however, the illustrative calculations done using such method have been mostly covariant scalar integrals, without numerator factors. We show here how those integrals with tensorial structures also can be handled straightforwardly and easily. However, contrary to the absence of significant features in the usual approach, here the NDIM also allows us to come across surprising unsuspected bonuses. Toward this end, we present two alternative ways of working out the integrals and illustrate them by taking the easiest Feynman integrals in this category that emerge in the computation of a standard one-loop self-energy diagram. One of the novel and heretofore unsuspected bonuses is that there are degeneracies in the way one can express the final result for the referred Feynman integral. Received: 3 November 1998 /Published online: 3 August 1999     

Renormalization of QED in an external field

The Schwinger equations of QED are rewritten in three different ways as integral equations involving functional derivatives, which are called weak field, strong field, and SCF quantum electrodynamics. The perturbative solutions of these equations are given in terms of appropriate Feynman diagrams. The Green function that is used as an electron propagator in each case is discussed in detail. The general renormalization rules for each of the three equations are provided both in a non perturbative way (Dyson relations) and for Feynman diagrams.     

Calculating contracted tensor Feynman integrals

A recently derived approach to the tensor reduction of 5-point one-loop Feynman integrals expresses the tensor coefficients by scalar 1-point to 4-point Feynman integrals completely algebraically. In this Letter we derive extremely compact algebraic expressions for the contractions of the tensor integrals with external momenta. This is based on sums over signed minors weighted with scalar products of the external momenta. With these contractions one can construct the invariant amplitudes of the matrix elements under consideration, and the evaluation of one-loop contributions to massless and massive multi-particle production at high energy colliders like LHC and ILC is expected to be performed very efficiently.     

Three-loop vacuum integral with four-propagators using hypergeometry

A hypergeometric function is proposed to calculate the scalar integrals of Feynman diagrams.In this study,we verify the equivalence between the Feynman parametrization and the hypergeometric technique for the scalar integral of the three-loop vacuum diagram with four propagators.The result can be described in terms of generalized hypergeometric functions of triple variables.Based on the triple hypergeometric functions,we establish the systems of homogeneous linear partial differential equations(PDEs)satisfied by the scalar integral of three-loop vacuum diagram with four propagators.The continuation of the scalar integral from its convergent regions to entire kinematic domains can be achieved numerically through homogeneous linear PDEs by applying the element method.     

Multi-loop Feynman integrals and conformal quantum mechanics

New algebraic approach to analytical calculations of D-dimensional integrals for multi-loop Feynman diagrams is proposed. We show that the known analytical methods of evaluation of multi-loop Feynman integrals, such as integration by parts and star-triangle relation methods, can be drastically simplified by using this algebraic approach. To demonstrate the advantages of the algebraic method of analytical evaluation of multi-loop Feynman diagrams, we calculate ladder diagrams for the massless φ³ theory. Using our algebraic approach we show that the problem of evaluation of special classes of Feynman diagrams reduces to the calculation of the Green functions for specific quantum mechanical problems. In particular, the integrals for ladder massless diagrams in the φ³ scalar field theory are given by the Green function for the conformal quantum mechanics.     

Path Integrals and Alternative Effective Dynamics in Loop Quantum Cosmology

The alternative dynamics of loop quantum cosmology is examined by the path integral formulation.We consider the spatially flat FRW models with a massless scalar field,where the alternative quantizations inherit more features from full loop quantum gravity.The path integrals can be formulated in both timeless and deparameterized frameworks.It turns out that the effective Hamiltonians derived from the two different viewpoints are equivalent to each other.Moreover,the first-order modified Friedmann equations are derived and predict quantum bounces for contracting universe,which coincide with those obtained in canonical theory.     

Renormalization and scaling behavior of non-Abelian gauge fields in curved spacetime

In this article we discuss the one loop renormalization and scaling behavior of non-Abelian gauge field theories in a general curved spacetime. A generating functional is constructed which forms the basis for both the perturbation expansion and the Ward identities. Local momentum space representations for the vector and ghost particles are developed and used to extract the divergent parts of Feynman integrals. The one loop diagram for the ghost propagator and the vector-ghost vertex are shown to have no divergences not present in Minkowski space. The Ward identities insure that this is true for the vector propagator as well. It is shown that the above renormalizations render the three- and four-vector vertices finite. Finally, a renormalization group equation valid in curved spacetimes is derived. Its solution is given and the theory is shown to be asymptotically free as in Minkowski space.     

Quadratic Lagrangians and Topology in Gauge Theory Gravity

We consider topological contributions to the action integral in a gauge theory formulation of gravity. Two topological invariants are found and are shown to arise from the scalar and pseudoscalar parts of a single integral. Neither of these action integrals contribute to the classical field equations. An identity is found for the invariants that is valid for non-symmetric Riemann tensors, generalizing the usual GR expression for the topological invariants. The link with Yang–Mills instantons in Euclidean gravity is also explored. Ten independent quadratic terms are constructed from the Riemann tensor, and the topological invariants reduce these to eight possible independent terms for a quadratic Lagrangian. The resulting field equations for the parity non-violating terms are presented. Our derivations of these results are considerably simpler than those found in the literature.     

The Hadamard Construction of Green's Functions on a Curved Space-Time with Symmetries

The Hadamard constituents of Green's functions for a ζ-parametrized generalization of the massless scalar d'Alembert equation to a curved space-time including the conformally invariant wave equation: the world function of space-time, the transport scalar, and the tail-term coefficients, being simultaneously coefficients in the Schwinger-DeWitt expansion of the Feynman propagator for the corresponding invariant Klein-Gordon equation, are considered on a general static spherically symmetric and (2,2)-decomposable metric. The construction equations determining the Hadamard building elements are cast into a symmetry-adapted form and used to obtain, on a specific model metric, exact explicit solutions.     

New Integrable and Linearizable Nonlinear Difference Equations

A systematic investigation to derive nonlinear lattice equations governed by partial difference equations (PΔΔE) admitting specific Lax representation is presented. Further it is shown that for a specific value of the parameter the derived nonlinear PΔΔE's can be transformed into a linear PΔΔE's under a global transformation. Also it is demonstrated how to derive higher order ordinary difference equations (OΔE) or mappings in general and linearizable ones in particular from the obtained nonlinear PΔΔE's through periodic reduction. The question of measure preserving property of the obtained OΔE's and the construction of more than one integrals (or invariants) of them is examined wherever possible.     

Numerical evaluation of multi-loop integrals by sector decomposition

In a recent paper [Nucl. Phys. B 585 (2000) 741] we have presented an automated subtraction method for divergent multi-loop/leg integrals in dimensional regularisation which allows for their numerical evaluation, and applied it to diagrams with massless internal lines. Here we show how to extend this algorithm to Feynman diagrams with massive propagators and arbitrary propagator powers. As applications, we present numerical results for the master 2-loop 4-point topologies with massive internal lines occurring in Bhabha scattering at two loops, and for the master integrals of planar and non-planar massless double box graphs with two off-shell legs. We also evaluate numerically some two-point functions up to 5 loops relevant for beta-function calculations, and a 3-loop 4-point function, the massless on-shell planar triple box. Whereas the 4-point functions are evaluated in non-physical kinematic regions, the results for the propagator functions are valid for arbitrary kinematics.     

Which Green functions does the path integral for quasi-Hermitian Hamiltonians represent?

We explain how Feynman diagrams and the functional integral for quasi-Hermitian theories “know” about the metric η. The answer turns out be that their derivation is based fundamentally on the Heisenberg equations of motion and the canonical equal-time commutation relations, which only take their standard form when matrix elements are evaluated using η.     

Dimensionally regularized one-loop tensor integralswith massless internal particles

A set of one-loop vertex and box tensor integrals with massless internal particles has been obtained directly without any reduction method to scalar integrals. The results with one or two massive external lines for the vertex integral and zero or one massive external lines for the box integral are shown in this report. Dimensional regularization is employed to treat any soft and collinear (IR) divergence. A series expansion of tensor integrals with respect to an extra space-time dimension for the dimensional regularization is also given. The results are expressed by very short formulas in a manner suitable for a numerical calculation. Arrival of the final proofs: 25 November 2005     

Path integral and effective Hamiltonian in loop quantum cosmology

We study the path integral formulation of Friedmann universe filled with a massless scalar field in loop quantum cosmology. All the isotropic models of $k=0,+1,-1$ are considered. To construct the path integrals in the timeless framework, a multiple group-averaging approach is proposed. Meanwhile, since the transition amplitude in the deparameterized framework can be expressed in terms of group-averaging, the path integrals can be formulated for both deparameterized and timeless frameworks. Their relation is clarified. It turns out that the effective Hamiltonian derived from the path integral in deparameterized framework is equivalent to the effective Hamiltonian constraint derived from the path integral in timeless framework, since they lead to same equations of motion. Moreover, the effective Hamiltonian constraints of above models derived in canonical theory are confirmed by the path integral formulation.     

Invariant Forms of the Lorentz Group

We study trilinear and multilinear invariant forms for the homogeneous Lorentz group. The residues of these trilinear forms generate particular trilinear forms themselves. They appear also if we sum Taylor expansions partially into a series of expressions each of which is covariant under infinitesimal Lorentz transformations. Multilinear invariant forms are submitted to harmonic analysis in different channels. They are thus expressed by invariant functions. Invariant functions for different channels are related by integral equations involving 6χ-symbols, 9χ-symbols etc. as “crossing kernels”. It is shown by construction that all invariant functions and nχ-symbols can be represented as finite sums of Barnes type integrals. As example we analyze explicitly the four-point Schwinger function of the massless Euclidean Thirring field with arbitrary spin and dimension.     

Operator approach to analytical evaluation of Feynman diagrams

The operator approach to analytical evaluation of multiloop Feynman diagrams is proposed. We show that the known analytical methods of evaluation of massless Feynman integrals, such as the integration-by-parts method and the method of “uniqueness” (which is based on the star-triangle relation), can be drastically simplified by using this operator approach. To demonstrate the advantages of the operator method of analytical evaluation of multiloop Feynman diagrams, we calculate ladder diagrams for the massless ϕ ³ theory (analytical results for these diagrams are expressed in terms of multiple polylogarithms). It is shown how operator formalism can be applied to calculation of certain massive Feynman diagrams and investigation of the Lipatov integrable chain model. The text was submitted by the authors in English.