Path Integral Representation Research Articles

Local-time representation of path integrals.

We derive a local-time path-integral representation for a generic one-dimensional time-independent system. In particular, we show how to rephrase the matrix elements of the Bloch density matrix as a path integral over x-dependent local-time profiles. The latter quantify the time that the sample paths x(t) in the Feynman path integral spend in the vicinity of an arbitrary point x. Generalization of the local-time representation that includes arbitrary functionals of the local time is also provided. We argue that the results obtained represent a powerful alternative to the traditional Feynman-Kac formula, particularly in the high- and low-temperature regimes. To illustrate this point, we apply our local-time representation to analyze the asymptotic behavior of the Bloch density matrix at low temperatures. Further salient issues, such as connections with the Sturm-Liouville theory and the Rayleigh-Ritz variational principle, are also discussed.

Path integral representation for polymer quantized scalar fields

According to loop quantum gravity, matter fields must be quantized in a background-independent manner. For scalar fields, such a background-independent quantization is called polymer quantization and is inequivalent to the standard Schrödinger quantization. It is therefore important to obtain predictions from the polymer quantized scalar field theory and compare with the standard results. As a step towards this, we develop a path integral representation for the polymer quantized scalar field. We notice several crucial differences from the path integral for the Schrödinger quantized scalar field. One important difference is the appearance of an extra summation at each point in the path integral for the polymer quantized theory. A second crucial difference is the loss of manifest Lorentz symmetry for a polymer quantized theory on Minkowski space.

Boundary conditions as dynamical fields

The possibility of treating boundary conditions in terms of a bilocal dynamical field is formalized in terms of a boundary action. This allows for a simple path-integral perturbation theory approach to physical effects such as radiation from a time-dependent boundary. The nature of the action which governs the dynamics of the bilocal field is investigated for a limited case (which includes the Robin boundary conditions).

Studying the optical second-order interference pattern formation process with classical light in the photon counting regime.

The formation process of the second-order interference pattern is studied experimentally in the photon counting regime by superposing two independent single-mode continuous-wave lasers. Two-photon interference based on the superposition principle in Feynman's path integral theory is employed to interpret the experimental results. The second-order interference pattern of classical light can be formulated when, with high probability, there are only two photons in the interferometer at one time. The studies are helpful in understanding the second-order interference of classical light in the language of photons. The method and conclusions can be generalized to the third- and higher-order interference of light and interference of massive particles.

Third order wave equation in Duffin-Kemmer-Petiau theory: Massive case

Within the framework of the Duffin-Kemmer-Petiau (DKP) formalism a more consistent approach to the derivation of the third order wave equation obtained earlier by M. Nowakowski [Phys.Lett.A {\bf 244} (1998) 329] on the basis of heuristic considerations is suggested. For this purpose an additional algebraic object, the so-called $q$ - commutator ($q$ is a primitive cubic root of unity) and a new set of matrices $\eta_{\mu}$ instead of the original matrices $\beta_{\mu}$ of the DKP algebra are introduced. It is shown that in terms of these $\eta_{\mu}$ matrices we have succeeded in reducing a procedure of the construction of cubic root of the third order wave operator to a few simple algebraic transformations and to a certain operation of the passage to the limit $z \rightarrow q$, where $z$ is some complex deformation parameter entering into the definition of the $\eta$ - matrices. A corresponding generalization of the result obtained to the case of the interaction with an external electromagnetic field introduced through the minimal coupling scheme is carried out and a comparison with M. Nowakowski's result is performed. A detailed analysis of the general structure for a solution of the first order differential equation for the wave function $\psi(x; z)$ is performed and it is shown that the solution is singular in the $z \rightarrow q$ limit. The application to the problem of construction within the DKP approach of the path integral representation in parasuperspace for the propagator of a massive vector particle in a background gauge field is discussed.

A note on classical and quantum unimodular gravity

We discuss unimodular gravity at a classical level, and in terms of its extension into the UV through an appropriate path integral representation. Classically, unimodular gravity is simply a gauge fixed version of General Relativity (GR), and as such it yields identical dynamics and physical predictions. We clarify this and explain why there is no sense in which it can "bring a new perspective" to the cosmological constant problem. The quantum equivalence between unimodular gravity and GR is more of a subtle question, but we present an argument that suggests one can always maintain the equivalence up to arbitrarily high momenta. As a corollary to this, we argue that whenever inequivalence is seen at the quantum level, that just means we have defined two different quantum theories that happen to share a classical limit.

First-principles equation of state and electronic properties of warm dense oxygen.

We perform all-electron path integral Monte Carlo (PIMC) and density functional theory molecular dynamics (DFT-MD) calculations to explore warm dense matter states of oxygen. Our simulations cover a wide density-temperature range of 1-100 g cm(-3) and 10(4)-10(9) K. By combining results from PIMC and DFT-MD, we are able to compute pressures and internal energies from first-principles at all temperatures and provide a coherent equation of state. We compare our first-principles calculations with analytic equations of state, which tend to agree for temperatures above 8 × 10(6) K. Pair-correlation functions and the electronic density of states reveal an evolving plasma structure and ionization process that is driven by temperature and density. As we increase the density at constant temperature, we find that the ionization fraction of the 1s state decreases while the other electronic states move towards the continuum. Finally, the computed shock Hugoniot curves show an increase in compression as the first and second shells are ionized.

Evaluation of photoexcitation and photoionization probabilities by the trajectory method

A new trajectory-based method of transition probability evaluation in quantum system was developed. It is based on a path integral representation of probability and uses Weyl symbols for initial and final states. The method belongs to the efficient initial value representation (IVR) schemes. The pre-exponential factor specific to the semi-classical method is equal to one, and does not need be separately calculated. This eliminates problems with caustics and Maslov indices of trajectories. The method is equally efficient for evaluation of the transition probabilities into separate states and groups of states, including an entire ionization continuum, for example. The capabilities of the method are demonstrated by evaluation of photo-excitation and photo-ionization probabilities in the hydrogen atom exposed to an ultrashort photo-pulse, and total photo-ionization probability in the helium atom.

Continuum tensor network field states, path integral representations and spatial symmetries

A natural way to generalize tensor network variational classes to quantum field systems is via a continuous tensor contraction. This approach is first illustrated for the class of quantum field states known as continuous matrix-product states (cMPS). As a simple example of the path-integral representation we show that the state of a dynamically evolving quantum field admits a natural representation as a cMPS. A completeness argument is also provided that shows that all states in Fock space admit a cMPS representation when the number of variational parameters tends to infinity. Beyond this, we obtain a well-behaved field limit of projected entangled-pair states (PEPS) in two dimensions that provide an abstract class of quantum field states with natural symmetries. We demonstrate how symmetries of the physical field state are encoded within the dynamics of an auxiliary field system of one dimension less. In particular, the imposition of Euclidean symmetries on the physical system requires that the auxiliary system involved in the class’ definition must be Lorentz-invariant. The physical field states automatically inherit entropy area laws from the PEPS class, and are fully described by the dissipative dynamics of a lower dimensional virtual field system. Our results lie at the intersection many-body physics, quantum field theory and quantum information theory, and facilitate future exchanges of ideas and insights between these disciplines.

Two-photon interference with non-identical photons

Two-photon interference with non-identical photons is studied based on the superposition principle in Feynman's path integral theory. The second-order temporal interference pattern is observed by superposing laser and pseudothermal light beams with different spectra. The reason why there is two-photon interference for photons of different spectra is that non-identical photons can be indistinguishable for the detection system when Heisenberg's uncertainty principle is taken into account. These studies are helpful to understand the second-order interference of light in the language of photons.

The first- and second-order temporal interference between thermal and laser light.

The first- and second-order temporal interference between two independent thermal and laser light beams is discussed by employing the superposition principle in Feynman's path integral theory. It is concluded that the first-order temporal interference pattern can not be observed by superposing two independent thermal and laser light beams, while the second-order temporal interference pattern can be observed in the same condition. These predictions are experimentally verified by employing pseudothermal light to simulate thermal light. The relationship between the indistinguishability of alternatives and photons is analyzed. The conclusions are helpful to understand the interference of different kinds of light and the difference between the coherence properties of thermal and laser light.

The second-order interference of two independent single-mode He–Ne lasers

The second-order spatial and temporal interference patterns with two independent single-mode continuous-wave He–Ne lasers are observed when these two lasers are incident to two adjacent input ports of a 1:1 non-polarizing beam splitter, respectively. Two-photon interference based on the superposition principle in Feynman's path integral theory is employed to interpret the experimental results. The conditions to observe the second-order interference pattern with two independent single-mode continuous-wave lasers are discussed. It is concluded that frequency stability is important to observe the second-order interference pattern with two independent light beams.

Path-integral methods for analyzing the effects of fluctuations in stochastic hybrid neural networks.

We consider applications of path-integral methods to the analysis of a stochastic hybrid model representing a network of synaptically coupled spiking neuronal populations. The state of each local population is described in terms of two stochastic variables, a continuous synaptic variable and a discrete activity variable. The synaptic variables evolve according to piecewise-deterministic dynamics describing, at the population level, synapses driven by spiking activity. The dynamical equations for the synaptic currents are only valid between jumps in spiking activity, and the latter are described by a jump Markov process whose transition rates depend on the synaptic variables. We assume a separation of time scales between fast spiking dynamics with time constant tau_{a} and slower synaptic dynamics with time constant τ. This naturally introduces a small positive parameter epsilon=tau _{a}/tau, which can be used to develop various asymptotic expansions of the corresponding path-integral representation of the stochastic dynamics. First, we derive a variational principle for maximum-likelihood paths of escape from a metastable state (large deviations in the small noise limit epsilonrightarrow0). We then show how the path integral provides an efficient method for obtaining a diffusion approximation of the hybrid system for small ϵ. The resulting Langevin equation can be used to analyze the effects of fluctuations within the basin of attraction of a metastable state, that is, ignoring the effects of large deviations. We illustrate this by using the Langevin approximation to analyze the effects of intrinsic noise on pattern formation in a spatially structured hybrid network. In particular, we show how noise enlarges the parameter regime over which patterns occur, in an analogous fashion to PDEs. Finally, we carry out a 1/epsilon-loop expansion of the path integral, and use this to derive corrections to voltage-based mean-field equations, analogous to the modified activity-based equations generated from a neural master equation.

Quantum critical scenario in a three-dimensional XY model with an easy-plane single-ion anisotropy via a renormalization group analysis

The low-temperature critical properties of the three-dimensional spin-1 XY model with an easy-plane single-ion anisotropy in zero magnetic field are explored on the ground of a path-integral representation and of the momentum-shell renormalization group where the single-ion anisotropy parameter drives a quantum phase transition. We address this problem first deriving systematically a suitable quantum Ginzburg–Landau–Wilson functional whose structure suggests, in a transparent way, that the microscopic spin model is equivalent to a continuous O(2)-vector one with transverse-Ising-like intrinsic dynamics (dynamic critical exponent z=1). Then, the relevant physics close to the quantum critical point is extracted from the appropriate one-loop Wilsonian flow equations. In particular we calculate the zero-temperature critical value of the anisotropy parameter for different cubic lattices. We compare our findings with those obtained in the literature by means of alternative methods and of experiments on praseodymium. The critical line ending in the quantum critical point is also calculated and the related shift exponent is determined. Finally, by solving in different asymptotic regimes the self-consistent equation for transverse susceptibility, a finite-temperature quantum critical scenario emerges which is quite similar to the conventional ones.

First-principles simulations and shock Hugoniot calculations of warm dense neon

All-electron path integral Monte Carlo (PIMC) and density functional theory molecular dynamics (DFT-MD) simulations provide a consistent, first-principles investigation of warm dense neon plasmas in the density-temperature range of 1--15 g ${\mathrm{cm}}^{\ensuremath{-}3}$ and ${10}^{4}--{10}^{9}$ K. At high temperatures, DFT-MD becomes intractable because of too many partially occupied bands, while at lower temperatures, PIMC is intractable because of the free-particle approximation of fermion nodes. In combination, PIMC and DFT-MD pressures and internal energies provide a coherent equation of state with a region of overlap in which the two methods cross-validate each other. Pair-correlation functions at various temperatures and densities provide details of the plasma structure and the temperature-driven ionization process. The electronic density of states of neon shows that a gap persists for the highest density-temperature conditions studied here with DFT-MD. Finally, the computed shock Hugoniot curves show an increase in compression as the first and second shells are ionized.

A NEW CLASS OF AMPLITUDE FUNCTIONS FOR THE STATIONARY PHASE METHOD ON AN ABSTRACT WIENER SPACE

A stochastic oscillatory integral is a probabilistic counterpart to a Feynman path integral, and the stationary phase method for stochastic oscillatory integrals relates to the semi-classical limits in the Feynman path integral theory. A complete stationary phase method for stochastic oscillatory integrals has not been achieved yet. In this paper, a new class of amplitude functions of stochastic oscillatory integrals is introduced, and it is shown that the stationary phase method works for the class.

Higgs mode near superfluid-to-Mott-insulatortransition studied by the quantum Monte Carlo method

In additional to the phonon (massless Goldstone mode) in Galilean invariant superfluid, there is another type of mode known as the Higgs amplitude mode in superfluid with emergent Lorentz invariance. In two dimensions, due to the strong decay into phonons, whether this Higgs mode is a detectable excitation with sharp linear response has been controversial for decades. Recent progress gives a positive answer to this question. Here, we review a series of numerical studies of the linear response of a two-dimensional Lorentz invariant superfluid near the superfluid-Mott insulator quantum critical point (SF-MI QCP). Particularly, we introduce a numerical procedure to unbiasedly calculate the linear response properties of strongly correlated systems. The numerical procedure contains two crucial steps, i.e., one is to use a highly efficient quantum Monte Carlo method, the worm algorithm in the imaginary-time path-integral representation, to calculate the imaginary time correlation functions for the system in equilibrium; and then, the other is, based on the imaginary time correlation functions, to use the numerical analytical continuation method for obtaining the real-time (real-frequency) linear response function. Applying this numerical procedure to the two-dimensional Bose Hubbard model near SF-MI QCP, it is found that despite strong damping, the Higgs boson survives as a prominent resonance peak in the kinetic energy response function. Further investigations also suggest a similar but less prominent resonance peak near SF-MI QCP on the MI side, and even on the normal liquid side. Since SF-MI quantum criticality can be realized by ultracold aotms in optical lattice, the Higgs resonance peak can be directly observed in experiment. In addition, we point out that the same Higgs resonance peak exists in all quantum critical systems with the same universality, namely (2 + 1)-dimensional relativistic U(1) criticality, as SF-MI QCP.

FQHE on curved backgrounds, free fields and large N

We study the free energy of the Laughlin state on curved backgrounds, starting from the free field representation. A simple argument, based on the computation of the gravitational effective action from the transformation properties of Green functions under the change of the metric, allows to compute the first three terms of the expansion in large magnetic field. The leading and subleading contributions are given by the Aubin-Yau and Mabuchi functionals respectively, whereas the Liouville action appears at next-to-next-to-leading order. We also derive a path integral representation for the remainder terms. They correspond to a large mass expansion for a related interacting scalar field theory and are thus given by local polynomials in curvature invariants.

Fokker–Planck equation and path integral representation of the fractional Ornstein–Uhlenbeck process with two indices

This paper considers the Fokker–Planck equation and path integral formulation of the fractional Ornstein–Uhlenbeck process parametrized by two indices. The effective Fokker–Planck equation of this process is derived from the associated fractional Langevin equation. The path integral representation of the process is constructed, and the basic quantities are evaluated.

Blip decomposition of the path integral: exponential acceleration of real-time calculations on quantum dissipative systems.

The real-time path integral representation of the reduced density matrix for a discrete system in contact with a dissipative medium is rewritten in terms of the number of blips, i.e., elementary time intervals over which the forward and backward paths are not identical. For a given set of blips, it is shown that the path sum with respect to the coordinates of all remaining time points is isomorphic to that for the wavefunction of a system subject to an external driving term and thus can be summed by an inexpensive iterative procedure. This exact decomposition reduces the number of terms by a factor that increases exponentially with propagation time. Further, under conditions (moderately high temperature and/or dissipation strength) that lead primarily to incoherent dynamics, the "fully incoherent limit" zero-blip term of the series provides a reasonable approximation to the dynamics, and the blip series converges rapidly to the exact result. Retention of only the blips required for satisfactory convergence leads to speedup of full-memory path integral calculations by many orders of magnitude.