Operator ordering schemes and covariant path integrals of quantum and stochastic processes in Curved space

A method is given for the derivation of covariant path integral solutions of quantum and stochastic processes in curved space.The correspondence between operator ordering schemes and ordinary functions in phase space is studied and applied to the explicit construction of a class of equivalent lattice representations of path integrals. It is shown that this class is uniquely related to a covariant functional integral. Some arguments forR/12 as the quantum mechanical curvature potential are given. Simple rules for nonlinear point transformations are stated. The connection with previous works is discussed.     

Zero momentum calculations for finite temperature field theory

Problems with calculations of Feynaman diagrams with no external leg carrying non-zero momentum for a finite temperature scalar field theory using a real time path integral approach are considered. The necessary extra rules for all such calculations are derived, covering all the usual methods of calculating the effective potential, and several problems with such calculations from the path integral viewpoint are resolved.     

Breakdown of the coherent state path integral: two simple examples

We show how the time-continuous coherent state path integral breaks down for both the single-site Bose-Hubbard model and the spin-path integral. Specifically, when the Hamiltonian is quadratic in a generator of the algebra used to construct coherent states, the path integral fails to produce correct results following from an operator approach. As suggested by previous authors, we note that the problems do not arise in the time-discretized version of the path integral.     

Power Series Expansion of Propagator for Path Integral and Its Applications

In this paper we obtain a propagator of path integral for a harmonic oscillator and a driven harmonic oscillator by using the power series expansion. It is shown that our result for the harmonic oscillator is more exact than the previous one obtained with other approximation methods. By using the same method, we obtain a propagator of path integral for the driven harmonic oscillator, which does not have any exact expansion. The more exact propagators may improve the path integral results for these systems.     

The quantum geometry of spin and statistics

Both, spin and statistics of a quantum system can be seen to arise from underlying (quantum) group symmetries. We show that the spin–statistics theorem is equivalent to a unification of these symmetries. Besides covering the Bose–Fermi case we classify the corresponding possibilities for anyonic spin and statistics. We incorporate the underlying extended concept of symmetry into quantum field theory in a generalised path integral formulation capable of handling general braid statistics. For bosons and fermions the different path integrals and Feynman rules naturally emerge without introducing Grassmann variables. We also consider the anyonic example of quons and obtain the path integral counterpart to the usual canonical approach.     

New framework for the Feynman path integral

The well-known Fourier integral solution of the free diffusion equation in an arbitrary Euclidean space is reduced to Feynmannian integrals using the method partly contained in the formulation of the Fresnelian integral. By replacing the standard Hilbert space underlying the present mathematical formulation of the Feynman path integral by a new Hilbert space, the space of classical paths on the tangent bundle to the Euclidean space (and more general to an arbitrary Riemannian manifold) equipped with a natural inner product, we show that our Feynmannian integral is in better agreement with the qualitative features of the original Feynman path integral than the previous formulations of the integral.     

Taming the cosmological constant in 2D causal quantum gravity with topology change

As shown in previous work, there is a well-defined nonperturbative gravitational path integral including an explicit sum over topologies in the setting of causal dynamical triangulations in two dimensions. In this paper we derive a complete analytical solution of the quantum continuum dynamics of this model, obtained uniquely by means of a double-scaling limit. We show that the presence of infinitesimal wormholes leads to a decrease in the effective cosmological constant, reminiscent of the suppression mechanism considered by Coleman and others in the four-dimensional Euclidean path integral. Remarkably, in the continuum limit we obtain a finite spacetime density of microscopic wormholes without assuming fundamental discreteness. This shows that one can in principle make sense of a gravitational path integral which includes a sum over topologies, provided suitable causality restrictions are imposed on the path integral histories.     

The Polyakov path integral over bordered surfaces

Following the general scheme of the convariant path integral quantization of gauge systems, two alternative formulations of the first quantized closed bosonic string in a position representation are presented. In both approaches the covariant path integral representations of the propagator and of the higher order off-shell amplitudes are constructed. For a wide class of gauges the explicit formulae for off-shell amplitudes are obtained. This paper is the continuation of our previous work where the corresponding problems in the open string case were considered [20].     

Path integral over reparametrizations: Lévy flights versus random walks

We investigate the properties of the path integral over reparametrizations (or the boundary value of the Liouville field in string theory). Discretizing the path integral, we apply the Metropolis–Hastings algorithm to numerical simulations of a proper (subordinator) stochastic process and find that typical trajectories are not Brownian but rather have discontinuities of the type of Lévy's flights. We study a fractal structure of these trajectories and show that their Hausdorff dimension is zero. We confirm thereby previous results on QCD scattering amplitudes by analytical and numerical calculations. We also perform Monte Carlo simulations of the path integral over reparametrization in the effective string ansatz for a circular Wilson loop and discuss their subtleties associated with the discretization of Douglas' functional.     

A Note on Calculation of Phase Space Path Integral by Means of Invariants

In this note we use the method of invariant to calculate the phase space path integral appearing in our previous article.     

Universal Form of Stochastic Evolution for Slow Variables in Equilibrium Systems

Nonlinear, multiplicative Langevin equations for a complete set of slow variables in equilibrium systems are generally derived on the basis of the separation of time scales. The form of the equations is universal and equivalent to that obtained by Green. An equation with a nonlinear friction term for Brownian motion turns out to be an example of the general results. A key method in our derivation is to use different discretization schemes in a path integral formulation and the corresponding Langevin equation, which also leads to a consistent understanding of apparently different expressions for the path integral in previous studies.     

Quantum Field Formalism for the Electromagnetic Interaction of Composite Particles in a Nonrelativistic Gauge Model

A classical nonrelativistic U(1) × U(1) gauge field model for the electromagnetic interaction of composite particles is proposed and the quantum formalism is constructed. This gauge model containing a Chern–Simons U(1) field and the electromagnetic U(1) field can be coupled to both a bosonic or a fermionic matter field. We explicitly consider the second case, a composite fermion system in the presence of an electromagnetic field, and we carry out the canonical quantization by the Dirac method. The path integral approach is developed and the Feynman rules are established. A simplified model is considered. As an alternative path integral method, the BRST formalism for this gauge model is also treated.     

Nuclear field theory

By using path integral techniques nuclear field theory (NFT) is developed for Fermi systems interacting via a general two-body force. The NFT Lagrangian is strictly derived. As a by-product, the corresponding graphical rules are obtained. The relation between the NFT and the conventional Feynman diagrammatic many-body perturbation theory is established for processes connecting initial and final states, too.     

Coadjoint orbits,spin and dequantization

In this Letter we propose two path integral approaches to describe the classical mechanics of spinning particles. We show how these formulations can be derived from the associated quantum ones via a sort of geometrical dequantization procedure proposed in a previous paper.     

Quantum Field Formalism for the Electromagnetic Interaction of Composite Particles in a Nonrelativistic Gauge Model III

A classical nonrelativistic U(1)×U(1) gauge field model that describes the topologically massive electromagnetic interaction of composite particles in (2+1) dimensions is proposed. The model, generalization of a previously postulated one, contains a Chern-Simons U(1) field and the topologically massive electromagnetic U(1) field, and it uses both a composite boson system or a composite fermion one. The second case is considered explicitly. By using the Dirac Hamiltonian method for constrained systems, the canonical quantization is carried out. By means of the Faddeev-Senjanovic formalism, the path integral quantization is developed. Consequently, the Feynman rules are established and the diagrammatic structure is treated. The application of the Becchi-Rouet-Stora-Tyutin algorithm is discussed. The present and previous models are compared.     

A new approach of solving Green’s function for wave propagation in an inhomogeneous absorbing medium

A new approach is developed to solve the Green's function that satisfies the Hehmholtz equation with complex refractive index. Especially, the Green's function for the Helmholtz equation can be expressed in terms of a one-dimensional integral, which can convert a Helmholtz equation into a Schr?dinger equation with complex potential. And the Schr?dinger equation can be solved by Feynman path integral. The result is in excellent agreement with the previous work.     

Factored coset models: a unifying approach to different bosonization schemes

We discuss various bosonization schemes from a path integral perspective. Our analysis shows that the existence of different bosonization schemes, such as abelian bosonization of non-abelian models and non-abelian bosonization of fermions with colour and flavour indices, can be understood as different ways of factoring out a dynamically trivial coset which contains the fermions. From this perspective follows the importance of the coset model in ensuring the correct superselection rules on the bosonic level.     

Quantum methods in field theories with singular higher derivative lagrangians

The path integral quantization for higher derivative Chern-Simons theories in (2+1) dimensions coupled to fermions is treated. The diagrammatic and the Feynman rules are constructed and the regularization and renormalization of this higher derivative model are analyzed in the framework of the perturbation theory. Finally, the unitarity problem related to the possible appearance of ghost states with negative norm is also discussed.     

Classical paths and the WKB approximation

Recently, new connection formulas for the WKB method have been proposed, without justification, for quantum tunneling problems. We show that these formulas can be associated with diagrammatic rules within the complex time framework of the path integral formalism and then we express the relevant Green functions in terms of a sum of contributions coming from (easily interpreted) classical paths. The method is applied to barrier penetration and the double well. Received: 6 June 2000 / Published online: 27 October 2000     

Geometro-Differential Conception of Extended Particles and the Semigroup of Trajectories in Minkowski Space-Time

The semigroup of trajectories in Minkowski space-time and its induced representations are constructed as a generalization of the Galilei case. They describe relativistic pointlike particles and yield the free propagator as a path integral in the space of trajectories parametrized by a fifth parameter. This non physical propagator in a five-dimensional space is integrated over the fifth parameter to yield the physical propagator in Minkowski space. Thereafter, this notion is applied to a model of extended particles with internal Poincaré symmetry and moving in an external Minkowski space. The geometrical structure is of Hilbert bundles and the interaction is introduced as a connection. The propagator is a path integral with respect to either the internal and external trajectories and reduces to a product of an internal and an external propagator when the interaction is ignored, just as has been found in a previous work with representations of the group rather than those of the semigroup.