Path integrals over the group manifolds

Particle motion in the SU(2) manifold is quantized by path integrals. It is shown that the Poschl-Teller, Wood-Saxon, and Rosen-Morse potentials are solved by relating their propagators to the path integrations over the SU(2) manifold. Examples with some other groups are mentioned.     

Path integrals on symmetric spaces and group manifolds

Invariant path integrals on symmetric and group spaces are defined in terms of a sum over the paths formed by broken geodesic segments. Their evaluation proceeds by using the mean value properties of functions over the geodesic and complex radius spheres. It is shown that on symmetric spaces the invariant path integral gives a kernel of the Schrödinger equation in terms of the spectral resolution of the zonal functions of the space. On compact group spaces the invariant path integral reduces to a sum over powers of Gaussian-type integrals which, for a free particle, yields the standard Van Vleck-Pauli propagator. Explicit calculations are performed for the case ofSU(2) andU(N) group spaces.     

An accurate power series expansion for path integrals on curved manifolds

A general and simple framework for treating path integrals on curved manifolds is presented. The crucial point is expanding the exponent of the propagator of general diffusion processes in a power series in time. The expansion coefficients are determined by recursive relations and can be analytically evaluated to any desired level of accuracy int. The treatment is both theoretically and numerically advantageous with respect to the other path integral methods known in the literature. Its power is illustrated on two exactly solvable models. The propagator obtained is shown to be much more accurate over a broad range oft than the standard short time approximation. In view of its numerical application this means significant reducing the number of time steps that are required to evaluate a path integral.     

Path integrals on half-line

In the framework of path integrals we present a solution to the Schrödinger equation for a free particle confined to the half-linex > 0. A solution in question corresponds to the boundary condition (/x) (0,t)= (0,t) where is a real constant.     

Path integrals and parastatistics

The propagator and corresponding path integral for a system of identical particles obeying parastatistics are derived. It is found that the statistical weights of topological sectors of the path integral for parafermions and parabosons are simply related through multiplication by the parity of the permutation of the final positions of the particles. Appropriate generalizations of statistics are proposed obeying unitarity and factorizability (strong cluster decomposition). The realization of simple maximal occupancy (Gentile) statistics is shown to require ghost states.     

Graphic Bernstein results in curved pseudo-Riemannian manifolds

We generalize a Bernstein-type result due to Albujer and Alías, for maximal surfaces in a curved Lorentzian product 3-manifold of the form , to higher dimension and codimension. We consider M a complete spacelike graphic submanifold with parallel mean curvature, defined by a map f:Σ₁→Σ₂ between two Riemannian manifolds and of sectional curvatures K₁ and K₂, respectively. We take on Σ₁×Σ₂ the pseudo-Riemannian product metric g₁−g₂. Under the curvature conditions, and K₁≥K₂, we prove that, if the second fundamental form of M satisfies an integrability condition, then M is totally geodesic, and it is a slice if at some point. For bounded K₁, K₂ and hyperbolic angle θ, we conclude that M must be maximal. If M is a maximal surface and , we show M is totally geodesic with no need for further assumptions. Furthermore, M is a slice if at some point pΣ₁, K₁(p)>0, and if Σ₁ is flat and K₂<0 at some point f(p), then the image of f lies on a geodesic of Σ₂.     

On biharmonic submanifolds in non-positively curved manifolds

In the biharmonic submanifolds theory there is a generalized Chen’s conjecture which states that biharmonic submanifolds in a Riemannian manifold with non-positive sectional curvature must be minimal. This conjecture turned out false by a counter example of Y.L. Ou and L. Tang in Ou and Tang (2012). However it remains interesting to find out sufficient conditions which guarantee this conjecture to be true. In this note we prove that:1. Any complete biharmonic submanifold (resp. hypersurface) $(M,g)$ in a Riemannian manifold $(N,h)$ with non-positive sectional curvature (resp. Ricci curvature) which satisfies an integral condition: for some $p\in (0,+\infty )$, ${\int }_M{\left|\overrightarrow{H}\right|}^{p}d{\mu }_g<+\infty$, where $\overrightarrow{H}$ is the mean curvature vector field of $M\hookrightarrow N$, must be minimal. This generalizes the recent results due to N. Nakauchi and H. Urakawa in Nakauchi and Urakawa (2013, 2011).2. Any complete biharmonic submanifold (resp. hypersurface) in a Riemannian manifold of at most polynomial volume growth whose sectional curvature (resp. Ricci curvature) is non-positive must be minimal.3. Any complete biharmonic submanifold (resp. hypersurface) in a non-positively curved manifold whose sectional curvature (resp. Ricci curvature) is smaller than $-ϵ$ for some $ϵ>0$ which satisfies that ${\int }_{B_{\rho }\left(x_0\right)}{\left|\overrightarrow{H}\right|}^{p+2}d{\mu }_g(p\ge 0)$ is of at most polynomial growth of $\rho$, must be minimal.We also consider $\epsilon$-superbiharmonic submanifolds defined recently in Wheeler (2013) by G. Wheeler and prove similar results for $\epsilon$-superbiharmonic submanifolds, which generalize the result in Wheeler (2013).     

Path integrals in Riemannian spaces

The covariant path integral for a free particle in curved space will be evaluated by means of a spectral analysis of smooth paths. No discretization rule will be required to put the action on a lattice. The connection between the resulting quantum hamiltonian and the Onsager-Machlup lagrangian for diffusion processes willbe discussed. The present treatment corrects an earlier version.     

Conformal anomaly for free scalar propagation on curved bounded manifolds

The trace anomaly for free propagation in the context of a conformally invariant scalar field theory defined on a curved manifold of positive constant curvature with boundary is evaluated through use of an asymptotic heat kernel expansion. In addition to their direct physical significance the results are also of relevance to the holographic principle and to Quantum Cosmology.     

Some connectedness problems in positively curved Finsler manifolds

This paper studies some connectedness problems under the positivity hypothesis of various curvatures (k$k$-Ricci and flag curvature). Our approach uses Morse Theory for general end conditions (see [Ioan Radu Peter, The Morse index theorem where the ends are submanifolds in Finsler geometry, Houston J. Math. 32 (4) (2006) 995–1009]). Some previous results related to the flag curvature were obtained in [Ioan Radu Peter, A connectedness principle in positively curved Finsler manifolds, in: H. Shimada, S. Sabau (Eds.), Advanced Studies in Pure Mathematics, Finsler Geometry, Sapporo 2005-In Memory of Makoto Matsumoto, Mathematical Society of Japan, 2007]. Some results from Riemannian geometry are extended to the Finsler category also. The Finsler setting is much more complicated and the difference between Finsler and Riemann settings will be emphasized during the paper.     

Path integrals with kinetic coupling potentials

Path integral solutions with kinetic coupling potentials p ₁ p ₂ are evaluated. As examples a Morse oscillator, i.e., a model in molecular physics, and the double pendulum in the harmonic approximation are given. The former one is solved by some well-known path integral techniques, whereas the latter one by an affine transformation.     

Path integrals and linear canonical transformations

For a system withN bosonic or fermionic degrees of freedom I calculate the coherent state propagator, i.e. the matrix element between coherent states of the evolution operator, for a general quadratic Hamiltonian plus a source term, using the holomorphic form of the path integral. The analysis and the result obtained are used to discuss the transformation properties of the path integral for linear canonical transformations (Bogoliubov-Valatin trfs), a preliminary to the formulation of a geometric theory of path integral quantization.

     

Path integrals in chemical kinetics I

Proceeding from the approach of Zel'dovich and Ovchinnikov we construct, in terms of path integrals, the formal solution of the master equation for chemical systems with diffusion. The analogue of the action functional is derived and the properties of extremal paths are discussed. As an illustration, an example of a diffusion-controlled chemical reaction is analysed.     

Zeta function regularization of path integrals in curved spacetime

This paper describes a technique for regularizing quadratic path integrals on a curved background spacetime. One forms a generalized zeta function from the eigenvalues of the differential operator that appears in the action integral. The zeta function is a meromorphic function and its gradient at the origin is defined to be the determinant of the operator. This technique agrees with dimensional regularization where one generalises ton dimensions by adding extra flat dimensions. The generalized zeta function can be expressed as a Mellin transform of the kernel of the heat equation which describes diffusion over the four dimensional spacetime manifold in a fith dimension of parameter time. Using the asymptotic expansion for the heat kernel, one can deduce the behaviour of the path integral under scale transformations of the background metric. This suggests that there may be a natural cut off in the integral over all black hole background metrics. By functionally differentiating the path integral one obtains an energy momentum tensor which is finite even on the horizon of a black hole. This energy momentum tensor has an anomalous trace.