We establish a necessary and sufficient condition for averages over complex valued weight functions on R^N to be represented as statistical averages over real, non-negative probability weights on C^N. Using this result, we show that many path-integrals for time-ordered expectation values of bosonic degrees of freedom in real-valued time can be expressed as statistical averages over ensembles of paths with complex-valued coordinates, and then speculate on possible consequences of this result for the relation between quantum and classical mechanics.
Following the renewed interest in the topic [1], we revisit the problem of assigning probabilities to classes of Feynman paths passing through specified space-time regions. We show that by assigning of probabilities to interfering alternatives, one already makes an assumption that the interference has been destroyed through interaction with an environment, or a meter. Including the effects of the meter allows to construct a consistent theory, free of logical 'pitfalls' identified in Ref. [1]. Wherever a meter cannot be constructed, or cannot be set to perform the desired decoherence, formally …
We present a path analysis of the condition under which the outcomes of previous observation affect the results of the measurements yet to be made. It is shown that this effect, also known as "signalling in time", occurs whenever the earlier measurements are set to destroy interference between two or more virtual paths. We also demonstrate that Feynman's negative "probabilities" provide for a more reliable witness of "signalling in time", than the Leggett-Garg inequalities, while both methods are frequently subject to failure
It is proved that in non-relativistic quantum mechanics (without spin) the transition probability may be described in terms of particle paths, every path having a (positive) probability. This leads to a stochastic hidden variables theory providing an intuitive picture of particle motion. The change of velocity at every time has a probability that depends on the potential over a large region, at a difference with the local action of classical dynamics. Thus the hidden variables theory is non-local like Bohm's, but not deterministic.
We consider a finite-dimensional quantum system, making a transition between known initial and final states. The outcomes of several accurate measurements, which {\it could be} made in the interim, define virtual paths, each endowed with a probability amplitude. If the measurements are {\it actually made}, the paths, which may now be called "real", acquire also the probabilities, related to the frequencies, with which a path is seen to be travelled in a series of identical trials. Different sets of measurements, made on the same system, can produce different, or incompatible, statistical ensem…
We identify a nontrivial multistate Landau-Zener model for which transition probabilities between any pair of diabatic states can be determined analytically and exactly. In the semiclassical picture, this model features the possibility of interference of different trajectories that connect the same initial and final states. Hence, transition probabilities are generally not described by the incoherent successive application of the Landau-Zener formula. We discuss reasons for integrability of this system and provide numerical tests of the suggested expression for the transition probability matri…
Path integrals represent a powerful route to quantization: they calculate probabilities by summing over classical configurations of variables such as fields, assigning each configuration a phase equal to the action of that configuration. This paper defines a universal path integral, which sums over all computable structures. This path integral contains as sub-integrals all possible computable path integrals, including those of field theory, the standard model of elementary particles, discrete models of quantum gravity, string theory, etc. The universal path integral possesses a well-defined me…
In physics, one is often misled in thinking that the mathematical model of a system is part of or is that system itself. Think of expressions commonly used in physics like "point" particle, motion "on the line", "smooth" observables, wave function, and even "going to infinity", without forgetting perplexing phrases like "classical world" versus "quantum world".... On the other hand, when a mathematical model becomes really inoperative with regard to correct predictions, one is forced to replace it with a new one. It is precisely what happened with the emergence of quantum physics. Classical mo…
A non-Grassmanian path integral representation is given for the solution of the Klein-Gordon and the Dirac equations. The trajectories of the path integral are rendered differentiable by the relativistic corrections. The nonrelativistic limit is briefly discussed from the point of view of the renormalization group.
Although the Hamiltonian formalism is so far favored for quantum computation of lattice gauge theory, the path integral formalism would never be useless. The advantages of the path integral formalism are the knowledge and experience accumulated by classical lattice simulation and manifest Lorentz invariance. We discuss quantum computation of lattice gauge theory in the path integral formalism. We utilize a quantum sampling algorithm to generate gauge configurations, and demonstrate a benchmark test of $Z_2$ lattice gauge theory on a four-dimensional hypercube.
We introduce two new integral transforms of the quantum mechanical transition kernel that represent physical information about the path integral. These transforms can be interpreted as probability distributions on particle trajectories measuring respectively the relative contribution to the path integral from paths crossing a given spatial point (the hit function) and the likelihood of values of the line integral of the potential along a path in the ensemble (the path averaged potential).
Feynman's path integral is generalized to quantum mechanics on p-adic space and time. Such p-adic path integral is analytically evaluated for quadratic Lagrangians. Obtained result has the same form as that one in ordinary quantum mechanics.
In this paper we consider a phase space path integral for general time-dependent quantum operations, not necessarily unitary. We obtain the path integral for a completely positive quantum operation satisfied Lindblad equation (quantum Markovian master equation). We consider the path integral for quantum operation with a simple infinitesimal generator.
In perturbative calculations of quantum mechanical path integrals in curvilinear coordinates, Feynman diagrams involve multiple temporal integrals over products of distributions, which are mathematically undefined. We derive simple rules for their evaluation from the requirement of coordinate independence of path integrals.
A method for calculating the relativistic path integral solution via sum over perturbation series is given. As an application the exact path integral solution of the relativistic Aharonov-Bohm-Coulomb system is obtained by the method. Different from the earlier treatment based on the space-time transformation and infinite multiple-valued trasformation of Kustaanheimo-Stiefel in order to perform path integral, the method developed in this contribution involves only the explicit form of a simple Green's function and an explicit path integral is avoided.
A method of path integral construction without gauge fixing in the holomorphic representation is proposed for finite-dimensional gauge models. This path integral determines a manifestly gauge-invariant kernel of the evolution operator.
We present a new method for the consistent construction of time-continuous coherent-state path integrals using the theory of half-form quantization. Through the inversion of the quantization procedure we construct a de-quantization map taking first order operators to their corresponding path integrals. We generalize our results using functional techniques, allowing for the consistent path integral study of more general operators, including higher orders and interactions.
Path integral solutions with kinetic coupling potentials $\propto p_1p_2$ are evaluated. As examples I give a Morse oscillator, i.e., a model in molecular physics, and the double pendulum in the harmonic approximation. The former is solved by some well-known path integral techniques, whereas the latter by an affine transformation.
Both Bohmian mechanics, a version of quantum mechanics with trajectories, and Feynman's path integral formalism have something to do with particle paths in space and time. The question thus arises how the two ideas relate to each other. In short, the answer is, path integrals provide a re-formulation of Schroedinger's equation, which is half of the defining equations of Bohmian mechanics. I try to give a clear and concise description of the various aspects of the situation.
We present an explicit path integral evaluation of the free Hamiltonian propagator on the (D-1)-dimensional pseudosphere, in the horicyclic coordinates, using the integral equation method. This method consists in deriving an integral equation for the propagator that turns out to be of Abel's type.
Efforts to give an improved mathematical meaning to Feynman's path integral formulation of quantum mechanics started soon after its introduction and continue to this day. In the present paper, one common thread of development is followed over many years, with contributions made by various authors. The present version of this line of development involves a continuous-time regularization for a general phase space path integral and provides, in the author's opinion at least, perhaps the optimal formulation of the path integral.
Path integral solutions are obtained for the the PT-/non-PT-Symmetric and non-Hermitian Morse Potential. Energy eigenvalues and the corresponding wave functions are obtained.
All the geometric phases, adiabatic and non-adiabatic, are formulated in a unified manner in the second quantized path integral formulation. The exact hidden local symmetry inherent in the Schrödinger equation defines the holonomy. All the geometric phases are shown to be topologically trivial. The geometric phases are briefly compared to the chiral anomaly which is naturally formulated in the path integral.
We analyze the property of locality with respect to the framework for quantum mechanics based on the path integral formalism. As is well known, this framework makes the same experimental predictions as does the one based on a separable Hilbert space and the Schrödinger equation.
We show how the time-continuous coherent state path integral breaks down for both the single-site Bose-Hubbard model and the spin path integral. Specifically, when the Hamiltonian is quadratic in a generator of the algebra used to construct coherent states, the path integral fails to produce correct results following from an operator approach. As suggested by previous authors, we note that the problems do not arise in the time-discretized version of the path integral.
The derivation of path integrals is reconsidered. It is shown that the expression for the discretized action is not unique, and the path integration domain can be deformed so that at least Gaussian path integrals become probabillistic. This leads to a practical algorithm of sign-problem-free Monte Carlo sampling from the Gaussian path integrals. Moreover, the dynamical influence of Gaussian quantum system (the bath) on any other quantum system can be exactly represented as interaction with classical non-Markovian noise. We discuss the relation of these findings to the Bell's theorem and the Fe…
Path integral Monte Carlo with Green's function analysis allows the sampling of quantum mechanical properties of molecules at finite temperature. While a high-precision computation of the energy of the Born-Oppenheimer surface from path integral Monte Carlo is quite costly, we can extract many properties without explicitly calculating the electronic energies. We demonstrate how physically relevant quantities, such as bond-length, vibrational spectra, and polarizabilities of molecules may be sampled directly from the path integral simulation using Matsubura (temperature) Green's functions (imag…
Starting from the canonical formalism of relativistic (timeless) quantum mechanics, the formulation of timeless path integral is rigorously derived. The transition amplitude is reformulated as the sum, or functional integral, over all possible paths in the constraint surface specified by the (relativistic) Hamiltonian constraint, and each path contributes with a phase identical to the classical action divided by $\hbar$. The timeless path integral manifests the timeless feature as it is completely independent of the parametrization for paths. For the special case that the Hamiltonian constrain…
Through a very careful analysis of Dirac's 1932 paper on the Lagrangian in Quantum Mechanics as well as the second and third editions of his classic book {\it The Principles of Quantum Mechanics}, I show that Dirac's contributions to the birth of the path-integral approach to quantum mechanics is not restricted to just his seminal demonstration of how Lagrangians appear naturally in quantum mechanics, but that Dirac should be credited for creating a path-integral which I call {\it Dirac path-integral} which is far more general than Feynman's while possessing all its desirable features. On top …
This is an introductory review of the connection between homotopy theory and path integrals, mainly focus on works done by Schulman [23] that he compared path integral on SO(3) and its universal covering space SU(2), DeWitt and Laidlaw [15] that they proved the theorem to the case of path integrals on the multiply-connected topological spaces. Also, we discuss the application of the theorem in Aharonov-Bohm effect given by [20,24]. An informal introduction to homotopy theory is provided for readers who are not familiar with the theory.
Representation of the elastic scattering amplitude in the form of the path integral is obtained using the stationary Schroedinger equation. A few methods of evaluation of path integrals for large coupling constants are formulated. The methods are based on the uncertainty correlation. The scattering lengths and cross sections are calculated for the right angled and singular potentials and the Yukawa potential. The comparison with exact results is made.
A fully regulated definition of Feynman's path integral is presented here. The proposed re-formulation of the path integral coincides with the familiar formulation whenever the path integral is well-defined. In particular, it is consistent with respect to lattice formulations and Wick rotations, i.e., it can be used in Euclidean and Minkowskian space-time. The path integral regularization is introduced through the generalized Kontsevich-Vishik trace, that is, the extension of the classical trace to Fourier Integral Operators. Physically, we are replacing the time-evolution semi-group by a holo…
