Classical Path Integral Research Articles

An atomistic theory of nucleation: Self-organization via non-equilibrium work and fluctuations

Insights from non-equilibrium statistical mechanics, highlighting the role of work and fluctuations at the microscale, are applied toward the development of a fundamental, rigorous and purely atomistic theory of nucleation. Nanoscale fluctuations in order, density and heat influence the local nucleation rate by orders of magnitude, necessitating their inclusion through a modern approach. Coarse-graining over the underlying Hamiltonian dynamics allows derivation of a microscale expression for the nucleation rate in terms of a classical path integral over far from equilibrium trajectories and their associated work. Second law violating states at the microscale, as found from the dynamics of small critical nucleation clusters, contribute exponentially to the observable macroscale nucleation rate.

All-Mode Quantum-Classical Path Integral Simulation of Bacteriochlorophyll Dimer Exciton-Vibration Dynamics.

We use the quantum-classical path integral (QCPI) methodology to report numerically exact, fully quantum mechanical results for the exciton-vibration dynamics in the bacteriochlorophyll dimer, including all 50 coupled vibrational normal modes of each bacteriochlorophyll explicitly with parameters obtained from spectroscopic Huang-Rhys factors. We present a coordinate transformation that maps the dimer on a spin-Boson Hamiltonian with a single collective bath. We consider two vibrational initial conditions which correspond to a Franck-Condon excitation or to modes initially equilibrated with the excited monomer. Our calculations reveal persistent, underdamped oscillations of the electronic energy between the two pigments at room temperature. Static disorder leads to additional damping, but the population dynamics remains oscillatory. The population curves exhibit atypical, nonsmooth features that arise from the complexity of the bacteriochlorophyll vibrational spectrum and which cannot be captured by simple analytical spectral density functions.

Exploiting classical decoherence in dissipative quantum dynamics: Memory, phonon emission, and the blip sum

The mechanisms that lead to the quenching of the coherence of a quantum system interacting with a dissipative environment are analyzed within the path integral/influence functional framework. Classical and quantum contributions are identified and linked to stimulated and spontaneous phonon emission processes. Three ways of exploiting decoherence in numerical calculations are discussed. The first is based on the decay of memory effects with time separation and leads to well-known iterative quasi-adiabatic propagator path integral (i-QuAPI) schemes. The other two mechanisms exploit the prominent role of classical decoherence. Removal of classical memory is possible through the introduction of auxiliary variables, leading to efficient and accurate quantum–classical path integral (QCPI) methods. A third way is proposed, which involves systematic corrections to the fully incoherent limit. An efficient procedure for summing all paths associated with ‘blips’ (short off-diagonal path pair segments) is described. The ideas are illustrated with numerical calculations of tunneling dynamics.

Quantum-classical path integral. I. Classical memory and weak quantum nonlocality

We consider rigorous path integral descriptions of the dynamics of a quantum system coupled to a polyatomic environment, assuming that the latter is well approximated by classical trajectories. Earlier work has derived semiclassical or purely classical expressions for the influence functional from the environment, which should be sufficiently accurate for many situations, but the evaluation of quantum-(semi)classical path integral (QCPI) expressions has not been practical for large-scale simulation because the interaction with the environment introduces couplings nonlocal in time. In this work, we analyze the nature of the effects on a system from its environment in light of the observation [N. Makri, J. Chem. Phys. 109, 2994 (1998)] that true nonlocality in the path integral is a strictly quantum mechanical phenomenon. If the environment is classical, the path integral becomes local and can be evaluated in a stepwise fashion along classical trajectories of the free solvent. This simple "classical path" limit of QCPI captures fully the decoherence of the system via a classical mechanism. Small corrections to the classical path QCPI approximation may be obtained via an inexpensive random hop QCPI model, which accounts for some "back reaction" effects. Exploiting the finite length of nonlocality, we argue that further inclusion of quantum decoherence is possible via an iterative evaluation of the path integral. Finally, we show that the sum of the quantum amplitude factors with respect to the system paths leads to a smooth integrand as a function of trajectory initial conditions, allowing the use of Monte Carlo methods for the multidimensional phase space integral.

Classical and Quantum Mechanics via Supermetrics in Time

Koopman-von Neumann in the 30's gave an operatorial formululation of Classical Mechanics. It was shown later on that this formulation could also be written in a path-integral form. We will label this functional approach as CPI (for classical path-integral) to distinguish it from the quantum mechanical one, which we will indicate with QPI. In the CPI two Grassmannian partners of time make their natural appearance and in this manner time becomes something like a three dimensional supermanifold. Next we introduce a metric in this supermanifold and show that a particular choice of the supermetric reproduces the CPI while a different one gives the QPI.

Black Holes: Interfacing the Classical and the Quantum

The central idea of this paper is that forming the black hole horizon is attended with the transition from the classical regime of evolution to the quantum one. We offer and justify the following criterion for discriminating between the classical and the quantum: creations and annihilations of particle-antiparticle pairs are impossible in the classical reality but possible in the quantum reality. In flat spacetime, we can switch from the classical picture of field propagation to the quantum picture by changing the overall sign of the spacetime signature. To describe a self-gravitating object at the final stage of its classical evolution, we propose to use the Foldy–Wouthuysen representation of the Dirac equation in curved spacetimes, and the Gozzi classical path integral. In both approaches, maintaining the dynamics in the classical regime is controlled by supersymmetry.

Koopman‐von Neumann formulation of classical Yang‐Mills theories: I

AbstractIn this paper we present the Koopman‐von Neumann (KvN) formulation of classical non‐Abelian gauge field theories. In particular we shall explore the functional (or classical path integral) counterpart of the KvN method. In the quantum path integral quantization of Yang‐Mills theories concepts like gauge‐fixing and Faddeev‐Popov determinant appear in a quite natural way. We will prove that these same objects are needed also in this classical path integral formulation for Yang‐Mills theories. We shall also explore the classical path integral counterpart of the BFV formalism and build all the associated universal and gauge charges. These last are quite different from the analog quantum ones and we shall show the relation between the two. This paper lays the foundation of this formalism which, due to the many auxiliary fields present, is rather heavy. Applications to specific topics outlined in the paper will appear in later publications.

Projective geometry and special relativity

In this paper we present the Koopman-von Neumann (KvN) formulation of classical non-Abelian gauge field theories. In particular we shall explore the functional (or classical path integral) counterpart of the KvN method. In the quantum path integral quantization of Yang-Mills theories concepts like gauge-fixing and Faddeev-Popov determinant appear in a quite natural way. We will prove that these same objects are needed also in this classical path integral formulation for Yang-Mills theories. We shall also explore the classical path integral counterpart of the BFV formalism and build all the associated universal and gauge charges. These last are quite different from the analog quantum ones and we shall show the relation between the two. This paper lays the foundation of this formalism which, due to the many auxiliary fields present, is rather heavy. Applications to specific topics outlined in the paper will appear in later publications.

Quantum behavior of deterministic systems with information loss: Path integral approach

’t Hooft’s derivation of quantum from classical physics is analyzed by means of the classical path integral of Gozzi et al. It is shown how the key element of this procedure—the loss of information constraint—can be implemented by means of Faddeev–Jackiw’s treatment of constrained systems. It is argued that the emergent quantum systems are identical with systems obtained in Blasone et al. [Phys. Rev. A 71 (2005) 052507] through Dirac–Bergmann’s analysis. We illustrate our approach with two simple examples—free particle and linear harmonic oscillator. Potential Liouville anomalies are shown to be absent.

CHIRAL ANOMALIES VIA CLASSICAL AND QUANTUM FUNCTIONAL METHODS

In the quantum path integral formulation of a field theory model an anomaly arises when the functional measure is not invariant under a symmetry transformation of the Lagrangian. In this paper, generalizing previous work done on the point particle, we show that even at the classical level we can give a path integral formulation for any field theory model. Since classical mechanics cannot be affected by anomalies, the measure of the classical path integral of a field theory must be invariant under the symmetry. The classical path integral measure contains the fields of the quantum one plus some extra auxiliary ones. So, at the classical level, there must be a sort of "cancellation" of the quantum anomaly between the original fields and the auxiliary ones. In this paper we prove in detail how this occurs for the chiral anomaly.

Effective classical theory for real-time SU( N) gauge theories at high temperature

We derive an effective classical theory for real-time SU( N) gauge theories at high temperature. By separating off and integrating out quantum fluctuations we obtain a 3D classical path integral over the initial fields and conjugate momenta. The leading hard mode contribution is incorporated in the equations of motion for the classical fields. This yields the gauge invariant hard thermal loop (HTL) effective equation of motion. No gauge-variant terms are generated as in treatments with an intermediate momentum cut-off. Quantum corrections to classical correlation functions can be calculated perturbatively. The 4D renormalizability of the theory ensures that the 4D counterterms are sufficient to render the theory finite. The HTL contributions of the quantum fluctuations provide the counterterms for the linear divergences in the classical theory.

New formulation of the classical path integral with reparametrization invariance

A classical path integral (CPI) provides a functional integral representation of the kernel which propagates phase space density distributions. In this paper a new formulation of the CPI is developed in which time and energy are promoted to dynamical variables. The reparametrization invariance, inherent in this formalism, is handled by means of the Batalin–Fradkin–Vilkovisky method. The path integral action possesses a set of ISp(2) symmetries connected with reparametrization invariance and an additional set of ISp(2) symmetries connected with the symplectic geometry of the extended phase space. Supersymmetry is also present in the CPI action. This formulation of the CPI allows us to study the dependence on energy of the dynamical evolution of Hamiltonian systems. It naturally incorporates the constraints onto the energy surface.

A path integral for turbulence in incompressible fluids

In this paper a classical path integral is formulated for incompressible fluids that evolve according to the Navier–Stokes equation. The path integral propagates probability distributions deterministically on the space ℊvol of solenoidal velocity fields. We construct a set of ISp(2) charges associated with the geometry of ℊvol and its Poisson structure, and a pair of supersymmetry charges connected with the Hamiltonian. These charges generate exact symmetries of the classical path integral when the viscosity is set equal to zero. When the effect of dissipation is included, the charges associated with the Poisson structure and the Hamiltonian are no longer conserved. Charges that generate Kolmogorov scaling and Galilean transformations are also constructed. The classical path integral is formulated in terms of vorticity as well.

Symmetries of the classical path integral on a generalized phase-space manifold.

In this paper we generalize previous work done on the path-integral approach to classical mechanics and its symmetries. We study in particular the case that the components of the symplectic two-form ${\ensuremath{\omega}}_{\mathrm{ab}}$, expressed in arbitrary coordinates, are allowed to depend on the phase-space coordinates. This lifts the restriction that the path integral and its symmetry generators be expressed only in terms of canonical coordinates. We show, in particular, that an extra term must be added to the anti-Becchi-Rouet-Stora (anti-BRS) charge in order to preserve the ISp(2) symmetry which reflects the geometry of phase space. The cohomology of this new anti-BRS operator is found to be isomorphic to the de Rham cohomology of phase space. The modification of the anti-BRS charge leads to a modification of one of the supersymmetry generators associated with the classical Hamiltonian. Despite this change in the form of the generators, the classical Kubo-Martin-Schwinger conditions can still be derived from this supersymmetry. We also prove that the requirement of supersymmetric invariance of the states results in a new set of equations that, despite their new form, are still satisfied by the Gibbs states on a general phase-space manifold.