A path-integral formulation of the Pauli equation using two real spin coordinates

Feynman's path-integral quantum-mechanical formulation is generalised for particles of spin 1/2. In the one-particle case, the path-integral formulation uses paths in a Euclidean real five-dimensional space, two coordinates (u, v) being reserved for spin. The path integral is proven to correspond exactly to the Pauli equation. A canonical density-matrix formulation is also dealt with. Basic ideas are to start with differential spin operators instead of the Pauli matrices and apply them to functions ψ=ψ ₁(r,t)u +gy ₂(r,t)v whereψ ₁,ψ ₂ are the Pauli wave functions. Then a ‘nilpotent’ spin ‘kinetic-energy’ term is added to the Hamiltonian. This enables us to find a non-matrix spin-dependent Lagrangian which is used as usual in the action of a path integral of the Feynman type. Integral relations are derived from which the path integral can be transformed into components of the Pauli matrix Green's function (propagator) or the canonical density matrix. As an example, a path-integral calculation of the normal Zeeman splitting is carried out.     

A Rigorous Path Integral for Supersymmetic Quantum Mechanics and the Heat Kernel

In a rigorous construction of the path integral for supersymmetric quantum mechanics on a Riemann manifold, based on Bär and Pfäffle’s use of piecewise geodesic paths, the kernel of the time evolution operator is the heat kernel for the Laplacian on forms. The path integral is approximated by the integral of a form on the space of piecewise geodesic paths which is the pullback by a natural section of Mathai and Quillen’s Thom form of a bundle over this space. In the case of closed paths, the bundle is the tangent space to the space of geodesic paths, and the integral of this form passes in the limit to the supertrace of the heat kernel.     

Some comments on Chern-Simons gauge theory

Following M. F. Atiyah and R. Bott [AB] and E. Witten [W], we consider the space of flat connections on the trivialSU(2) bundle over a surfaceM, modulo the space of gauge transformations. We describe on this quotient space a natural hermitian line-bundle with connection and prove that if the surfaceM is now endowed with a complex structure, this line bundle is isomorphic to the determinant bundle. We show heuristically how path-integral quantisation of the Chern-Simons action yields holomorphic sections of this bundle.I.M.S. and T.R.R. supported by DOE grant DE-FG02-88ER 25066. J.W. supported by NSF Mathematical Sciences post-doctoral research scholarship 8807291     

The Anomaly Line Bundle of the Self-Dual Field Theory

In this work, we determine explicitly the anomaly line bundle of the abelian self-dual field theory over the space of metrics modulo diffeomorphisms, including its torsion part. Inspired by the work of Belov and Moore, we propose a non-covariant action principle for a pair of Euclidean self-dual fields on a generic oriented Riemannian manifold. The corresponding path integral allows one to study the global properties of the partition function over the space of metrics modulo diffeomorphisms. We show that the anomaly bundle for a pair of self-dual fields differs from the determinant bundle of the Dirac operator coupled to chiral spinors by a flat bundle that is not trivial if the underlying manifold has middle-degree cohomology, and whose holonomies are determined explicitly. We briefly sketch the relevance of this result for the computation of the global gravitational anomaly of the self-dual field theory, that will appear in another paper.     

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.     

General procedure of gauge fixing based on BRS invariance principle

We propose a simple gauge-fixing procedure based on the BRS invariance principle. It does not refer to the path integral at all and is applicable to the cases for which the standard path-integral method of Faddeev and Popov does not work.     

On the geometrical representation of the path integral reduction Jacobian: The case of dependent variables in a description of reduced motion

The geometrical representation of the path integral reduction Jacobian obtained in the problem of the path integral quantization of a scalar particle motion on a smooth compact Riemannian manifold with the given free isometric action of the compact semisimple Lie group has been found for the case when the local reduced motion is described by means of dependent coordinates. The result is based on the scalar curvature formula for the original manifold which is viewed as a total space of the principal fiber bundle.     

Superfield formulation of the phase space path integral

We give a superfield formulation of the path integral on an arbitrary curved phase space, with or without first class constraints. Canonical tranformations and BRST transformations enter in a unified manner. The superpartners of the original phase space variables precisely conspire to produce the correct path integral measure, as Pfaffian ghosts. When extended to the case of second-class constraints, the correct path integral measure is again reproduced after integrating over the superpartners. These results suggest that the superfield formulation is of first-principle nature.     

Supersymmetry in stochastic processes with higher-order time derivatives

A supersymmetric path-integral representation is developed for stochastic processes whose Langevin equation contains any number N of time derivatives, thus generalizing the presently available treatment of first-order Langevin equations by Parisi and Sourlas [Phys. Rev. Lett. 43 (1979) 744; Nucl. Phys. B 206 (1982) 321] to systems with inertia (Kramers' process) and beyond. The supersymmetric action contains N fermion fields with first-order time derivatives whose path integral is evaluated for fermionless asymptotic states.     

Galilei invariance and superfluidity II

Continuing the (heuristic) analysis of the mathematical structure of the Landau excitations, we find that inone dimension they may be described by a vector bundle over the base space of the boosts. The total space is a direct integral of all irreducible representations (of a given class) of the Galilei group. The existence of an energy-momentum spectrum requires the action of the boosts to be non-linear. This action can also be formulated as a superselection rule.     

Reconstruction theorem for groupoids and principal fiber bundles

The loop space formulation of 3+1 canonical quantum gravity premises that all physical information is contained within the holonomy loop functionals. This assumption is the result of the reconstruction theorem for a principla fiber bundle on a base loop space. The gauge connection for interacting gauge theories is more appropriately and readily reconstructed on a path space as opposed to a loop space. We generalize the reconstruction theorem to a base path space. Employing a holonomy groupoid map and a path connection, we trivially construct an abstract Lie groupoid from which a principal fiber bundle and gauge connection can be derived as distinctive examples. The groupoid reconstruction theorem is valid on both connected and nonconnected base manifolds, unlike the holonomy group reconstruction theorem, which can only be utilized for connected manifolds.     

Topological Chern—Simons vortices in the O (3) σ -model

Based on the phase-space path integral (functional integral) for a system with a regular or singular Lagrangian, the generalized Ward identities for phase space generating functional under the global transformation in phase space are derived respectively. The canonical Noether theorem at the quantum level is also established. It is pointed out that the connection between the symmetries and conservation laws in classical theories, in general,is no longer preserved in quantum theories. The advantage of our formulation is that we do not need to carry out the integration over the canonical momenta as usually performed. Applying the present formulation to Yang-Mills theory, the quantal BRS conserved quantity and Ward-Takahashi identity for BRS tranformation are derived; the Ward identities for gaugeghost proper vertices and new quantal conserved quantity are also found. In comparison of quantal conservation laws with those one deriving from configuration-space path integral using the Faddeev-Popov(F-P) trick is discussed. A precise study of path-integral quantisation for a nonlinear sigma model with Hopf and Chern-Simons (CS) terms is reexamined. It has been shown that the angular momentum at the quantum level is equal to classical (Noether ) one. Applying our formulation to non-Abelian CS theory, the quantal conserved angular momentum of this system is obtained which differs from classical one in that one needs to take into account the contribution of angular momenta of ghost fields.     

Path integrals and degrees of freedom in many-body systems and relativistic field theories

The identification of physical degrees of freedom is sometimes obscured in the path-integral formalism, and this makes it difficult to impose some constraints or to do some approximations. I review a number of cases where the difficulty is overcome by deriving the path integral from the operator form of the partition function after such identification has been made.     

Proof of factorization of fragmentation function in non-equilibrium QCD

In this paper, we prove factorization of fragmentation function in non-equilibrium QCD by using Schwinger-Keldysh closed-time path integral formalism. We use the background field method of QCD in a pure gauge in path integral approach to prove factorization of fragmentation function in non-equilibrium QCD. Our proof is valid in any arbitrary gauge fixing parameter α. This may be relevant to study hadron production from quark-gluon plasma at high energy heavy-ion colliders at RHIC and LHC.     

On reproducing kernels and quantization of states

Quantization of a mechanical system with the phase space a Kähler manifold is studied. It is shown that the calculation of the Feynman path integral for such a system is equivalent to finding the reproducing kernel function. The proposed approach is applied to a scalar massive conformal particle interacting with an external field which is described by deformation of a Hermitian line bundle structure.     

Path Integral Formulation of Fractionally Perturbed Lagrangian Oscillators on Fractal

Fractional path integration and particles trajectories in fractional dimensional space are motivating issues in quantum mechanics and kinetics. In this paper, a fractional path integral characterized by a fractional propagator is developed based on the framework of the fractional action-like variational approach. A fractional generalization of the free particle problem is found, the corresponding fractional Schrödinger equation is derived and a fractional path integral formulation of harmonic oscillators characterized by a perturbed Lagrangian is constructed after reducing the fractional action to an integral action on fractal. The new fractal-like path integral offers a number of motivating features which are discussed and analyzed. The main outcome is connected to the possibility of constructing on a fractal a path integral for the oscillators characterized by modified ground energy. In particular for low-temperature case, the fractional perturbed oscillator is characterized by a free energy larger than the standard value \( E_{0} = {{\hbar \omega } \mathord{\left/ {\vphantom {{\hbar \omega } 2}} \right. \kern-0pt} 2}.\) Such an increase in the ground energy generalizes the uncertainty principle without involving differentiable paths or even invoking new phenomenological theories based on deformed algebra.     

Quantum Yang-Mills on the two-sphere

We obtain the quantum expectations of gauge-invariant functions of the connection on a principalG=SU(N) bundle overS ². We show that the spaceA/g _m of connections modulo gauge transformations which are the identity at one point is itself a principal bundle over G, based loops in the symmetry group. The fiber inA/g _m is an affine linear space. Quantum expectations are iterated path integrals first over this fiber then over G, each with respect to the push-forward toA/g _m of the measure s^-S(A) DA.S(A) denotes the Yang-Mills action onA. There is a global section ofA/g _m on which the first integral is a Gaussian. The resulting measure on G is the conditional Wiener measure. We explicitly compute the expectations of a special class of Wilson loops.