Wentzel's Path Integrals

Quantum mechanics can be arrived at in threeways, as Heisenberg, Schrodinger, and Feynman did,respectively. For the last way, an unknown (i.e.,forgotten) forerunner exists, which we have found in two papers by Gregor Wentzel, published before thefamous works by Heisenberg and Schrodinger, andcontemporary with the fundamental works of L. deBroglie. In those papers, one can find the basicinterpretation of the action integral as a phase together withthe concept of interference of all virtual paths —not only the classical solutions — and theinterpretation of the result of the interference as theamplitude of a transition probability, both given byFeynman 20 years later, following Dirac.     

Coherent-State Path Integrals and Brownian Motions

A stochastic approach to the rigorous foundation of the coherent-state (phase-space) path integral is given. Stochastic integrals and some generalizations of the Feynman–Kac theorem are used for this purpose. In this approach, quantum mechanics is described in terms of the Fock–Bargmann representation; a classical Hamiltonian is related to the corresponding quantum Hamiltonian on the Fock–Bargmann space, seen as a Hilbert subspace of L²(R²)L^{2}({\bf R}^{2}). The coherent-state path integral is realized as a conditional expectation of a stochastic process defined by the exponential of the Fisk–Stratonovich integral of the fundamental 1-form along a path of Brownian motion on the phase space R²{\bf R}^{2}.     

Path Integrals for Pseudo-Hermitian Hamiltonians

Inner products in pseudo-Hermitian quantum theories depend on the details of the Hamiltonians themselves, which makes them difficult to calculate. We shall see that, for some questions, the functional integrals for such theories can be calculated without needing to determine the inner product metric. The reason is that their derivation is based on the Heisenberg equations of motion and the canonical commutation relations, which are unchanged. In particular, this can greatly simplify the derivation of Hermitian theories that are equivalent to these pseudo-Hermitian systems.     

Path Integrals and Statistics of Identical Particles

We summarize the essential ingredients, which enabled us to derive the path-integral for a system of harmonically interacting spin-polarized identical particles in a parabolic confining potential, including both the statistics (Bose–Einstein or Fermi–Dirac) and the harmonic interaction between the particles. This quadratic model, giving rise to repetitive Gaussian integrals, allows to derive an analytical expression for the generating function of the partition function. The calculation of this generating function circumvents the constraints on the summation over the cycles of the permutation group. Moreover, it allows one to calculate the canonical partition function recursively for the system with harmonic two-body interactions. Also, static one-point and two-point correlation functions can be obtained using the same technique, which make the model a powerful trial system for further variational treatments of realistic interactions.     

Quantum Mechanics of H-Atom from Path Integrals

The quantum mechanical Coulomb problem in two and three dimensions is solved completely in terms of path integrals. We derive the integral representations for the Green's functions in configuration space and recover the wave functions from factorized residues.     

The Feynman Path Integrals and Everett's Universal Wave Function

We study here the properties of some quantum mechanical wave functions, which, in contrast to the regular quantum mechanical wave functions, can be predetermined with certainty (probability 1) by performing dense measurements (or continuous observations). These specific certain states are the junction points through which pass all the diverse paths that can proceed between each two such neighboring sure points. When we compare the properties of these points to the properties of the well-known universal wave functions of Everett we find a strong similarity between these two apparently uncorrelated entities, and in this way find the same similarity between the Feynman path integrals and Everett's universal wave functions.     

Homotopy and Path Integrals in the Time-dependent Aharonov-Bohm Effect

For time-independent fields the Aharonov-Bohm effect has been obtained by idealizing the coordinate space as multiply-connected and using representations of its fundamental homotopy group to provide information on what is physically identified as the magnetic flux. With a time-dependent field, multiple-connectedness introduces the same degree of ambiguity; by taking into account electromagnetic fields induced by the time dependence, full physical behavior is again recovered once a representation is selected. The selection depends on a single arbitrary time (hence the so-called holonomies are not unique), although no physical effects depend on the value of that particular time. These features can also be phrased in terms of the selection of self-adjoint extensions, thereby involving yet another question that has come up in this context, namely, boundary conditions for the wave function.     

Coherent State Path Integrals Without Resolutions of Unity

From the very beginning, coherent state path integrals have always relied on a coherent state resolution of unity for their construction. By choosing an inadmissible fiducial vector, a set of coherent states spans the same space but loses its resolution of unity, and for that reason has been called a set of weak coherent states. Despite having no resolution of unity, it is nevertheless shown how the propagator in such a basis may admit a phase-space path integral representation in essentially the same form as if it had a resolution of unity. Our examples are toy models of similar situations that arise in current studies of quantum gravity.     

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.     

Path Integrals for a Class of P-Adic Schrödinger Equations

The theme of doing quantum mechanics on all Abelian groups goes back to Schwinger and Weyl. If the group is a vector space of finite dimension over a non-Archimedean locally compact division ring, it is of interest to examine the structure of dynamical systems defined by Hamiltonians analogous to those encountered over the field of real numbers. In this Letter, a path integral formula for the imaginary time propagators of these Hamiltonians is derived.     

LETTER: Entropy Using Path Integrals for Quantum Black Hole Models

Canonical quantum gravity has been used in the search for eigenvalue equations that could describe black holes. In this paper we choose one of the simplest of these quantum equations to show how the usual Feynman's path integral approach can be applied to get the corresponding statistical properties. We get a logarithmic correction to the Bekenstein–Hawking entropy as already obtained by other authors by other means.     

Mathai–Quillen Formalism — From the Point of View of Path Integrals

We present here a path integral derivation of Mathai–Quillen formalism which gives a new interpretation of the Gauss–Bonnet–Chern theorem and Hopf–Poincaré theorem.     

Analysis of Path Integrals at Low Temperature: Box Formula,Occupation Time and Ergodic Approximation

We study the low temperature behavior of path integrals for a simple one-dimensional model. Starting from the Feynman–Kac formula, we derive a new functional representation of the density matrix at finite temperature, in terms of the occupation times for Brownian motions constrained to stay within boxes with finite sizes. From that representation, we infer a kind of ergodic approximation, which only involves double ordinary integrals. As shown by its applications to different potentials, the ergodic approximation turns out to be quite efficient, especially in the low-temperature regime where other usual approximations fail.     

Some New Developments in the Theory of Path Integrals,with Applications to Quantum Theory

A survey of recent developments concerning rigorously defined infinite dimensional integrals, mainly of the type of “Feynman path integrals,” is given. Both the theory and its applications, especially in quantum theory, are presented. As for the theory, general results are discussed including the case of polynomially growing phase functions, which are handled by exploiting the connection with probabilistic functional integrals. Also applications to continuous measurement theory and the stochastic Schrödinger equation are given. Other applications of probabilistic methods in non relativistic quantum theory and in quantum field theory, and their relations with statistical mechanics, are discussed.     

U(1/1)q q-Coherent States and Path Integrals for the q-Deformed Jaynes-Cummings Model

The q-coherent states for the quantum superalgebra U(1/1)_q, are introduced. Path integrals in the q-coherent state representations of the U(1/1)_q, are constructed and applied to the q-deformed Jaynes-Cummings model whose dynamical superalgebra is the quantum superalgebra U(1/1)_q.