Feynman integral and perturbation theory in quantum tomography

We present a definition for tomographic Feynman path integral as representation for quantum tomograms via Feynman path integral in the phase space. The proposed representation is the potential basis for investigation of Path Integral Monte Carlo numerical methods with quantum tomograms. Tomographic Feynman path integral is a representation of solution of initial problem for evolution equation for tomograms. The perturbation theory for quantum tomograms is constructed.     

Relativistic quantum mechanics of supersymmetric particles

The quantum mechanics of an electron in an external field is developed by Hamiltonian path integral methods. The electron is described classically by an action which is invariant under gauge supersymmetry transformations as well as worldline reparametrizations. The simpler case of a spinless particle is first reviewed and it is pointed out that a strictly canonical approach does not exist. This follows formally from the gauge invariance properties of the action and physically it corresponds to the fact that particles can travel backwards as well as forward in coordinate time. However, appropriate application of path integral techniques yields directly the proper time representation of the Feynman propagator. Next we extend the formalism to systems described by anticommuting variables. This problem presents some difficulty when the dimension of the phase space is odd, because the holomorphic representation does not exist. It is shown, however, that the usual connection between the evolution operator and the path integral still holds provided one indludes in the action the boundary term that makes the classical variational principle well defined. The path integral for the relativistic spinning particle is then evaluated and it is shown to lead directly to a representation for the Feynman propagator in terms of two proper times, one commuting, the other anticommuting, which appear in a symmetric manner. This representation is used to derive scattering amplitudes in an external field. In this step the anticommuting proper time is integrated away and the analysis is carried in terms of one (commuting) proper time only, just as in the spinless case. Finally, some properties of the quantum mechanics of the ghost particles that appear in the path integral for constrained systems are developed in an appendix.     

Unified treatment of the bound states of the Schioberg and the Eckart potentials using Feynman path integral approach

We obtain analytical expressions for the energy eigenvalues of both the Schioberg and Eckart potentials using an approximation of the centrifugal term.In order to determine the e-states solutions,we use the Feynman path integral approach to quantum mechanics.We show that by performing nonlinear space-time transformations in the radial path integral,we can derive a transformation formula that relates the original path integral to the Green function of a new quantum solvable system.The explicit expression of bound state energy is obtained and the associated eigenfunctions are given in terms of hypergeometric functions.We show that the Eckart potential can be derived from the Schioberg potential.The obtained results are compared to those produced by other methods and are found to be consistent.     

Star-product approach to quantum field theory: The free scalar field

The star-quantization of the free scalar field is developed by introducing an integral representation of the normal star-product. A formal connection between the Feynman path integral in the holomorphic representation and the star-exponential is established for the interacting scalar fields.     

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.     

N-body saddle-point approximation

By applying the saddle-point approximation to the N-body Feynman path integral formulation, the classical Hartree-Fock Molecular Orbital (M.O.) equations of quantum chemistry are obtained.     

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     

Phase space representation of quantum dynamics

We discuss a phase space representation of quantum dynamics of systems with many degrees of freedom. This representation is based on a perturbative expansion in quantum fluctuations around one of the classical limits. We explicitly analyze expansions around three such limits: (i) corpuscular or Newtonian limit in the coordinate-momentum representation, (ii) wave or Gross-Pitaevskii limit for interacting bosons in the coherent state representation, and (iii) Bloch limit for the spin systems. We discuss both the semiclassical (truncated Wigner) approximation and further quantum corrections appearing in the form of either stochastic quantum jumps along the classical trajectories or the nonlinear response to such jumps. We also discuss how quantum jumps naturally emerge in the analysis of non-equal time correlation functions. This representation of quantum dynamics is closely related to the phase space methods based on the Wigner-Weyl quantization and to the Keldysh technique. We show how such concepts as the Wigner function, Weyl symbol, Moyal product, Bopp operators, and others automatically emerge from the Feynmann's path integral representation of the evolution in the Heisenberg representation. We illustrate the applicability of this expansion with various examples mostly in the context of cold atom systems including sine-Gordon model, one- and two-dimensional Bose-Hubbard model, Dicke model and others.     

Continual measurements in the spacetime formation of nonrelativistic quantum mechanics

We study the nature and effects of some continual measurements in nonrelativistic quantum mechanics, a concept introduced by R. P. Feynman in his path integral formulation of quantum mechanics. We prove the existence in various senses of mathematically rigorous objects formally equivalent to the path integrals associated with such measurements, by means of both a limiting definition and more general techniques; and we derive some quantitative results concerning the effects on the wave-function and the numerical value of certain associated probabilities.     

The underlying Brownian motion of nonrelativistic quantum mechanics

Nonrelativistic quantum mechanics can be derived from real Markov diffusion processes by extending the concept of probability measure to the complex domain. This appears as the only natural way of introducing formally classical probabilistic concepts into quantum mechanics. To every quantum state there is a corresponding complex Fokker-Planck equation. The particle drift is conditioned by an auxiliary equation which is obtained through stochastic energy conservation; the logarithmic transform of this equation is the Schrödinger equation. To every quantum mechanical operator there is a stochastic process; the replacement of operators by processes leads to all the well-known results of quantum mechanics, using stochastic calculus instead of formal quantum rules. Comparison is made with the classical stochastic approaches and the Feynman path integral formulation.     

Of Ghosts, Gauge Volumes, and Gauss's Law

The relationship between the canonical operator and the path integral formulation of quantum electrodynamics is analyzed with a particular focus on the implementation of gauge constraints in the two approaches. The removal of gauge volumes in the path integral is shown to match with the presence of zero-norm ghost states associated with gauge transformations in the canonical operator approach. The path integrals for QED in both the Feynman and the temporal gauges are examined and several ways of implementing the gauge constraint integrations are demonstrated. The upshot is to show that both the Feynman and the temporal gauge path integrals are equivalent to the Coulomb gauge path integral, matching the results developed by Kurt Haller using the canonical formalism. In addition, the Faddeev–Popov form for the Feynman gauge and temporal gauge Lagrangian path integrals are derived from the Hamiltonian form of the path integral.     

Fourier path integral Monte Carlo study of a two-dimensional model quantum monolayer

Adsorption isotherms have been constructed for a 2-dimensional ²⁰Ne fluid that represents a quantum monolayer. A quantum distribution function theory is presented and implemented in the computation of the chemical potential as a function of the density of the adsorbed material. The quantum partition function in the canonical ensemble is written in its path integral representation with paths expanded in a Fourier series (Fourier path integral). The multidimensional integrals obtained in this representation are solved using the j-walking Monte Carlo integration technique. The results obtained suggest that as the quantum contributions increase the amount of adsorbed material decreases, compared with classical results. An increment in internal and kinetic energies due to quantum effects is responsible for the reduction in the amount of adsorbed material. As expected, quantum effects are much larger at low temperatures.     

Global Estimates of Errors in Quantum Computation by the Feynman–Vernon Formalism

The operation of a quantum computer is considered as a general quantum operation on a mixed state on many qubits followed by a measurement. The general quantum operation is further represented as a Feynman–Vernon double path integral over the histories of the qubits and of an environment, and afterward tracing out the environment. The qubit histories are taken to be paths on the two-sphere \(S^2\) as in Klauder’s coherent-state path integral of spin, and the environment is assumed to consist of harmonic oscillators initially in thermal equilibrium, and linearly coupled to to qubit operators \(\hat{S}_z\). The environment can then be integrated out to give a Feynman–Vernon influence action coupling the forward and backward histories of the qubits. This representation allows to derive in a simple way estimates that the total error of operation of a quantum computer without error correction scales linearly with the number of qubits and the time of operation. It also allows to discuss Kitaev’s toric code interacting with an environment in the same manner.     

Causality, symmetries and quantum mechanics

It is argued that there is no evidence for causality as a metaphysical relation in quantum phenomena. The assumptions that there are no causal laws, but only probabilities for physical processes constrained by symmetries, leads naturally to quantum mechanics. In particular, an argument is made for why there are probability amplitudes that are complex numbers. This argument generalizes the Feynman path integral formulation of quantum mechanics to include all possible terms in the action that are allowed by the symmetries, but only the lowest order terms are observable at the presently accessible energy scales, which is consistent with observation. The notion of relational reality is introduced in order to give physical meaning to probabilities. This appears to give rise to a new interpretation of quantum mechanics.     

The Feynman path integral and its optical applications

There is a fruitful analogy between mechanics and optics. To describe the transition from quantum mechanics to classical mechanics, Feynman introduced the concept of an “integral over all paths”. The Feynman integral is used here to describe the transition from wave optics to geometrical optics. We suggest simple mathematical tools that allow use the Feynman integral and its approximation to calculate the radiation transport through an optically inhomogeneous layer and through an aperture in an infinite opaque screen.     

Fractals and quantum mechanics

A new application of a fractal concept to quantum physics has been developed. The fractional path integrals over the paths of the Levy flights are defined. It is shown that if fractality of the Brownian trajectories leads to standard quantum mechanics, then the fractality of the Levy paths leads to fractional quantum mechanics. The fractional quantum mechanics has been developed via the new fractional path integrals approach. A fractional generalization of the Schrodinger equation has been discovered. The new relationship between the energy and the momentum of the nonrelativistic fractional quantum-mechanical particle has been established, and the Levy wave packet has been introduced into quantum mechanics. The equation for the fractional plane wave function has been found. We have derived a free particle quantum-mechanical kernel using Fox's H-function. A fractional generalization of the Heisenberg uncertainty relation has been found. As physical applications of the fractional quantum mechanics we have studied a free particle in a square infinite potential well, the fractional "Bohr atom" and have developed a new fractional approach to the QCD problem of quarkonium. We also discuss the relationships between fractional and the well-known Feynman path integral approaches to quantum mechanics. (c) 2000 American Institute of Physics.     

Electron self-trapping at quantum and classical critical points

Using Feynman path integral technique estimations of the ground state energy have been found for a conduction electron interacting with order parameter fluctuations near quantum critical points. In some cases only singular perturbation theory in the coupling constant emerges for the electron ground state energy. It is shown that an autolocalized state (quantum fluctuon) can be formed and its characteristics have been calculated depending on critical exponents for both weak and strong coupling regimes. The concept of fluctuon is considered also for the classical critical point (at finite temperatures) and the difference between quantum and classical cases has been investigated. It is shown that, whereas the quantum fluctuon energy is connected with a true boundary of the energy spectrum, for classical fluctuon it is just a saddle-point solution for the chemical potential in the exponential density of states fluctuation tail.     

Numerical calculations in elementary quantum mechanics using Feynman path integrals

We show that it is possible to do numerical calculations in elementary quantum mechanics using Feynman path integrals. Our method involves discretizing both time and space, and summing paths through matrix multiplication. We give numerical fesults for various one-dimensional potentials. The calculations of energy levels and wavefunctions take approximately 100 times longer than with standard methods, but there are other problems for which such an approach should be more efficient.     

Dephasing of quantum tunnelling in molecular nanomagnets

Dephasing mechanism of quantum tunnelling in molecular magnets has been studied by means of the spin-coherentstate path integral in a mean field approximation. It is found that the fluctuating uncompensated transverse field from the dipolar-dipolar interaction between molecular magnets contributes a random phase to the quantum interference phase. The resulting transition rate is determined by the average tunnel splitting over the random phase. Such a dephasing process leads to the suppression of quenching due to the quantum phase interference, and to the steps due to odd resonances in hysteresis loop survived, which is in good agreement with experimental observations in molecular nanomagnets Fes and Mn12.