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arXiv:1408.1635 · Applied Mathematical Sciences

Functional integral transition elements of a massless oscillator

Giovanni Modanese

Published Jan 1, 2016• 1 citation
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Abstract

The massless harmonic oscillator is a rare example of a system whose Feynman path integral can be explicitly computed and receives its main contributions from regions of the functional space that are far from the classical and semiclassical configurations near the stationary point of the action. The functional average $\langle q_m^2 \rangle$ of the square of the coordinate at a time $t_m$ which is intermediate between the initial and final time gives a measure of the amplitude of quantum fluctuations with respect to the classical path. This average, or "transition element", is divergent in the massless limit, signaling a quantum runaway. We show that the divergence is not due to the continuum limit and formulate the conjecture that the divergent contributions come from regions where the action $S$ is constant and therefore the interference factor $e^{-iS/\hbar}$ does not oscillate. For most systems these regions have zero functional measure and thus give a null contribution to the path integral, but this is not the case for the massless oscillator. We study the simplest functional subspace with constant action, namely the one with $S=0$, which is connected to the classical solutions but extends to infinity, like an hyperplane through the origin; this subspace turns out to be infinite-dimensional. Some possible applications and developments are mentioned.

DOI
10.12988/ams.2016.68237

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Path integration methods are of crucial importance to quantum mechanics and quantum field theory. There are multiple ways the path integral can be constructed, one method uses the link between the Fokker-Planck and the Langevin equations, as is covered in this thesis. This is related to the standard derivation of the path integral in physics, in which the time is discretised. We study the mathematical problems related to the Feynman path integral, in particular the impossibility of a Lebesgue-type measure on the space of paths. It is discussed how oscillatory integrals can be used to have a well-defined `integral' on an infinite dimensional space. This formalism is subsequently applied to both the harmonic and the anharmonic oscillator. We prove an infinite dimensional oscillatory integral exists and obtain a convergent expression for both these cases, under conditions. Examples of these conditions are the initial wavefunctions being Schwartz functions as well as conditions on the endtime, angular frequency and coupling constant. Finally, a possible approach to establish an oscillatory integral for the hydrogen atom is discussed. It is proven that the result is no longer independent of the sequence of projection operators, which is a key step towards a rigorous path integral for this system.
Abstract Functional integrals are basic tools to study first quantum mechanics (QM), and quantum field theory (QFT). The path integral formulation of QM is well suited to the study of systems with an arbitrary number of degrees of freedom. It makes a smooth transition between nonrelativistic QM and QFT possible. The Euclidean functional integral also emphasizes the deep connection between QFT and the statistical physics of systems with short-range interactions near a continuous phase transition. The path integral representation of the matrix elements of the quantum statistical operator e-β H for Hamiltonians of the simple separable form p2/2m +V(q) is derived. To the path integral corresponds a functional measure and expectation values called correlation functions, which are generalized moments, and related to quantum observables, after an analytic continuation in time. The path integral corresponding to the Euclidean action of a harmonic oscillator, to which is added a time-dependent external force, is calculated explicitly. The result is used to generate Gaussian correlation functions and also to reduce the evaluation of path integrals to perturbation theory. The path integral also provides a convenient tool to derive semi-classical approximations.
Even though only the classical action explicitly appears in the Feynman path integral, quantum fluctuations show up in the fluctuating nature of Feynman paths. This prevents a classical interpretation of the latter objects. Still, it makes sense to ask whether the Feynman paths have any relation to the classical dynamical trajectories in configuration space-a possibility naturally suggested by the semiclassical approximation. Unfortunately, no such relation exists (in general). Things are quite different for the generalized Feynman paths, which actually contribute in a novel path integral giving an alternative representation of the quantum mechanical propagator. It is shown that these paths fluctuate about a classical dynamical trajectory (in configuration space) because of a certain white noise. That is to say, the generalized Feynman paths are fluctuating curves that collapse on a classical dynamical trajectory as h to 0. This result throws new light on the interplay between classical and quantum mechanics and hints at a new approach to the quantum theory.
This thesis investigates the quantum harmonic oscillator (QHO) through the path integral formulation. It begins with a review of classical dynamics and the standard operator approach to establish a foundational framework. The core of the work involves the systematic construction and evaluation of the QHO propagator, demonstrating how the transition amplitude naturally separates into a classical contribution and quantum fluctuations. Finally, the discussion extends to systems with infinite degrees of freedom, illustrating how the oscillator structure serves as a bridge to quantum field theory and the emergence of second quantization.
AbstractThis work investigates a variational perspective on the Feynman path integral, aiming to explore connections between stationary action, probability amplitude, integrating over path evolutions, and the semiclassical limit. The variational analysis employed should be viewed as conceptually explorative and heuristic rather than strictly formal. It explores how variations of path evolutions in the path integral at the level of a single infinitesimal slice can yield meaningful insight into the behavior of quantum trajectories. A local variational analysis of the free particle and a particle near a constant potential provides insight into how the stationary-action principle underlies both classical and quantum behavior, through a local relationship between the classical slice-action and a functional over the phase-weighting of quantum path evolutions. In particular, this functional is proportional to the slice-action itself up to additive and multiplicative constants. This proportionality implies that the stationarity of this functional is equivalent to the stationarity of the slice-action, which in turn generates the semiclassical stationary path evolution. The resulting formulation produces results consistent with known short-time propagators for the free particle and a particle near a constant potential, including tunneling behavior.
As an alternative formulation of quantum mechanics, path integral is based on the notion of transition amplitude which gives the wave function of a quantum system at a time t f by acting on the wave function at an earlier time ti.We show that for a general quadratic form for the Lagrangian of the system, transition amplitude has the form f (t f -ti)e i h S class., where S class. is the classical action.We then present an algebraic method to evaluate the function f (t f -ti) without refereing to the path integral calculations.We examine the presented method to the cases of free particle and harmonic oscillator and obtain their propagators.