Feynman Rules and Generalized Ward Identities in Phase Space Functional Integral

Based on the phase-space generating functional of Green function, the generalizedcanonical Ward identities are derived It is point out that one can deduce Feynmanrules in tree approximation without carring out explicit integration over canonicalmomenta in phase-space generating functional. If one adds a four-dimensionaldivergence term to a Lagrangian of the field, then, the propagator of the field can bechanged.     

Theory of Atom Optics: Feynman’s Path Integral Approach

The present theory of atom optics is established mainly on the Schrödinger equations or the matrix mechanics equation. The authors present a new theoretical formulation of atom optics: Feynman’s path integral theory. Its advantage is that one can describe the diffraction and interference of atoms passing through slits (or grating), apertures, and standing wave laser field in Earth’s gravitational field by using a type of wave function and calculation is simple. For this reason, we derive the wave functions of particles in the following configurations: single slit (and slit with the van der Waals interaction), double slit, N slit, rectangular aperture, circular aperture, the Mach-Zehnder-type interferometer, the interferometer with the Raman beams, the Sagnac effect, the Aharonov-Casher effect, the Kapitza-Dirac diffraction effect, and the Aharonov-Bohm effect. The authors give a wave function of the state of particles on the screen in abovementioned configurations. Our formulas show good agreement with present experimental measurements.     

Quantum Measurement and Extended Feynman Path Integral

Quantum measurement problem has existed many years and inspired a large of literature in both physics and philosophy, but there is still no conclusion and consensus on it. We show it can be subsumed into the quantum theory if we extend the Feynman path integral by considering the relativistic effect of Feynman paths. According to this extended theory, we deduce not only the Klein--Gordon equation, but also the wave-function-collapse equation. It is shown that the stochastic and instantaneous collapse of the quantum measurement is due to the “potential noise” of the apparatus or environment and “inner correlation” of wave function respectively. Therefore, the definite-status of the macroscopic matter is due to itself and this does not disobey the quantum mechanics. This work will give a new recognition for the measurement problem.     

A Note on Feynman Path Integral for Electromagnetic External Fields

We propose a Fresnel stochastic white noise framework to analyze the nature of the Feynman paths entering on the Feynman Path Integral expression for the Feynman Propagator of a particle quantum mechanically moving under an external electromagnetic time-independent potential.     

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.     

A Mathematical Construction of the Non-Abelian Chern-Simons Functional Integral

We construct rigorously an infinite dimensional distribution which corresponds to the Chern-Simons (CS) functional integral associated with a principal fiber bundle over R ³ with structure group a compact connected Lie group. We determine the ‘moments’ of the CS distribution and show that these coincide with those used in informal studies of the CS integral. A locality property of the CS distribution is proven. The complexified theory of Fr?hlich and King is also discussed within our framework. Received: 2 April 1996 / Accepted: 15 August 1996     

On the Formulation of the Feynman Path Integral Through Broken Line Paths

We study the formulation of the Feynman path integral through broken line paths in non-relativistic quantum mechanics. This formulation is very familiar to us and well known to be useful. But its rigorous meaning is given little except for special cases. In the present paper, using the ideas in the theory of difference methods and the theory of pseudo-differential operators, we show rigorously for some class of potentials that this formulation is well defined and that this Feynman path integral gives the probability amplitude, i.e., the solution of the Schr?dinger equation. Received: 21 August 1996 / Accepted: 13 February 1997     

相空间路径积分中的Feynman规则和广义Ward恒等式

基于Green函数的相空间生成泛函,导出了广义正则Ward恒等式.指出无须作出相空间生成泛函中对正则动量的路径积分,即可求得树图近似下的Feynman规则.对场的拉氏量添加一个四维散度项后,场的传播子发生了改变.     

On the Feynman Path Integral for the Dirac Equation in the General Dimensional Spacetime

The Feynman path integral for the Dirac equation in the general dimensional spacetime is determined mathematically in the form of the sum-over-histories satisfying the superposition principle, i.e., the “sum” of the probability amplitudes with a common weight over all possible paths that go in any direction at any speed forward and backward in time. It is noted that our Feynman path integral is determined not in configuration space, but in phase space.     

Semi-classical Locality for the Non-relativistic Path Integral in Configuration Space

In an accompanying paper Gomes (arXiv:1504.02818, 2015), we have put forward an interpretation of quantum mechanics based on a non-relativistic, Lagrangian 3+1 formalism of a closed Universe M, existing on timeless configuration space \(\mathcal {Q}\) of some field over M. However, not much was said there about the role of locality, which was not assumed. This paper is an attempt to fill that gap. Locality in full can only emerge dynamically, and is not postulated. This new understanding of locality is based solely on the properties of extremal paths in configuration space. I do not demand locality from the start, as it is usually done, but showed conditions under which certain systems exhibit it spontaneously. In this way we recover semi-classical local behavior when regions dynamically decouple from each other, a notion more appropriate for extension into quantum mechanics. The dynamics of a sub-region O within the closed manifold M is independent of its complement, \(M-O\), if the projection of extremal curves on \(\mathcal {Q}\) onto the space of extremal curves intrinsic to O is a surjective map. This roughly corresponds to \(e^{i\hat{H}t}\circ \mathsf {pr}_{\mathrm{O}}= \mathsf {pr}_{\mathrm{O}}\circ e^{i\hat{H}t}\), where \(\mathsf {pr}_{\mathrm{O}}:\mathcal {Q}\rightarrow \mathcal {Q}_O^{\partial O}\) is a linear projection. This criterion for locality can be made approximate—an impossible feat had it been already postulated—and it can be applied for theories which do not have hyperbolic equations of motion, and/or no fixed causal structure. When two regions are mutually independent according to the criterion proposed here, the semi-classical path integral kernel factorizes, showing cluster decomposition which is the ultimate aim of a definition of locality.     

Integral representation for the dimensionally renormalized Feynman amplitude

A compact convergent integral representation for dimensionally renormalized Feynman amplitudes is explicitly constructed. The subtracted integrand is expressed as a distribution in the Schwinger -parametric space, and is obtained by applying upon the bare integrand a new subtraction operatorR' which respects Zimmermann's forest structure.     

Covariant Hamiltonian Field Theory: Path Integral Quantization

The Hamiltonian counterpart of classical Lagrangian field theory is covariant Hamiltonian field theory where momenta correspond to derivatives of fields with respect to all world coordinates. In particular, classical Lagrangian and covariant Hamiltonian field theories are equivalent in the case of a hyperregular Lagrangian, and they are quasi-equivalent if a Lagrangian is almost-regular. In order to quantize covariant Hamiltonian field theory, one usually attempts to construct and quantize a multisymplectic generalization of the Poisson bracket. In the present work, the path integral quantization of covariant Hamiltonian field theory is suggested. We use the fact that a covariant Hamiltonian field system is equivalent to a certain Lagrangian system on a phase space which is quantized in the framework of perturbative quantum field theory. We show that, in the case of almost-regular quadratic Lagrangians, path integral quantizations of associated Lagrangian and Hamiltonian field systems are equivalent.     

Global Phase Time and Path Integral for the Kantowski-Sachs Anisotropic Universe

The action functional of the anisotropic Kantowski-Sachs cosmological model is turned into that of an ordinary gauge system. Then a global phase time is identified for the model by imposing canonical gauge conditions, and the quantum transition amplitude is obtained by means of the usual path integral procedure of Fadeev and Popov.     

Theory of Atom Optics: Feynman''s Path Integral Approach

The present theory of atom optics is established mainly on the Schr?dinger equations or the matrix mechanics equation. The authors present a new theoretical formulation of atom optics: Feynman’s path integral theory. Its advantage is that one can describe the diffraction and interference of atoms passing through slits (or grating), apertures, and standing wave laser field in Earth’s gravitational field by using a type of wave function and calculation is simple. For this reason, we derive the wave functions of particles in the following configurations: single slit (and slit with the van der Waals interaction), double slit, N slit, rectangular aperture, circular aperture, the Mach-Zehnder-type interferometer, the interferometer with the Raman beams, the Sagnac effect, the Aharonov-Casher effect, the Kapitza-Dirac diffraction effect, and the Aharonov-Bohm effect. The authors give a wave function of the state of particles on the screen in abovementioned configurations. Our formulas show good agreement with present experimental measurements.     

Phase Space Path Integral Method for Problem of Obtaining Propagator of a Particle with a Force Quadratic in Velocity

This article uses the phse space path integral method to find the propagator for a particle with a force quadratic in velocity. Two specific canonical transformations has been used for this purpose.     

Semiclassical Representation of the Scattering Matrix by a Feynman Integral

We study the scattering problem for one-dimensional Schr?dinger equations in the semiclassical limit when the energy level is close to the quadratic maxima of the potential. Starting from the formula of the scattering matrix obtained by the exact WKB method, we represent its principal term in a reduced form of the Feynman integral, an absolutely convergent sum of suitably defined probability amplitudes over countably many trajectories on ℝ generated by classical trajectories and tunneling effects. Received: 14 November 1997 / Accepted: 6 April 1998     

A Discrete, Deterministic Construction of the Phase in Feynman Paths

In the path-integral formulation of quantum mechanics, the phase factor e ^iS(x〈t〉) is associated with every path x〈t〉. Summing this factor over all paths yields Feynman's propagator as a sum-over-paths. In the original formulation, the complex phase was a mathematical device invoked to extract wave behaviour in a particle framework. In this paper we show that the continuous phase itself can have a discrete origin in time reversal and that the propagator can be drawn by a single deterministic path. ¹On leave from Department of Mathematics, Ryerson University, Toronto, Ontario Canada.     

Variational Method for the Calculation of Plasma Phase Diagrams in the Path Integral Representation

The formal general expression of thermodynamic potential of fermion system is obtained by representation of Coulomb interaction with functional integral and formal introduction of complex field. The variational method of the thermodynamic potential with respect to field cumulants is elaborated. The phase diagrams are constructed at first approximation of fermion propagator.     

Path Integral Solution for the Coulomb Potential in a Curved Space of Constant Positive Curvature

A new path integral treatment of a hydrogen-like atom in a uniformly curved space with a constant positive curvature is presented. By converting the radial path integral into a path integral for the modified Pöschl-Teller potential with the help of the space-time transformation technique, the radial Green’s function is expressed in closed form, from which the energy spectrum and the corresponding normalized wave functions of the bound states are extracted. In the limit of vanishing curvature, the Green’s function, the energy spectra and the correctly normalized wave functions of bound and scattering states for a standard hydrogen-like atom are found.     

The Feynman Propagator on Perturbations of Minkowski Space

The singular values squared of the random matrix product \({Y = {G_{r} G_{r-1}} \ldots G_{1} (G_{0} + A)}\), where each \({G_{j}}\) is a rectangular standard complex Gaussian matrix while A is non-random, are shown to be a determinantal point process with the correlation kernel given by a double contour integral. When all but finitely many eigenvalues of A*A are equal to bN, the kernel is shown to admit a well-defined hard edge scaling, in which case a critical value is established and a phase transition phenomenon is observed. More specifically, the limiting kernel in the subcritical regime of \({0 < b < 1}\) is independent of b, and is in fact the same as that known for the case b =  0 due to Kuijlaars and Zhang. The critical regime of b =  1 allows for a double scaling limit by choosing \({{b = (1 - \tau/\sqrt{N})^{-1}}}\), and for this the critical kernel and outlier phenomenon are established. In the simplest case r =  0, which is closely related to non-intersecting squared Bessel paths, a distribution corresponding to the finite shifted mean LUE is proven to be the scaling limit in the supercritical regime of \({b > 1}\) with two distinct scaling rates. Similar results also hold true for the random matrix product \({T_{r} T_{r-1} \ldots T_{1} (G_{0} + A)}\), with each \({T_{j}}\) being a truncated unitary matrix.