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Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

Meaning of an individual "Feynman path"

Yakir Aharonov, M. Vardi

Published Apr 15, 1980• 95 citations
Publisher

Abstract

In this article we give an operational meaning to an individual "Feynman path." In other words, we describe a process of dense measurements, made in temporal sequence, which check whether the particle moves along any given trajectory in space-time. We show that in this process the two assumptions of the space-time formulation of quantum mechanics, are realized: (a) The weight that the particle moves along a trajectory that has been checked by this process is the same for all trajectories, and in fact, we show that the particle follows, with probability 1, the trajectory that is being checked. (b) A phase is systematically accumulated, so that, at the end of this process, the state is multiplied by the familiar factor $\mathrm{exp}[(\frac{i}{\ensuremath{\hbar}})\ensuremath{\int}L\mathrm{dt}]$. As an immediate extension of the above formalism, we suggest a setup that measures the relative phase between any two trajectories. Finally, our approach points toward the possibility of extending the Feynman formalism in order to cover more general Hamiltonians.

DOI
10.1103/physrevd.21.2235

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European Journal of Physics•May 1, 1996•3 citations
In this article we give an operational meaning to an individual "Feynman path." In other words, we describe a process of dense measurements, made in temporal sequence, which check whether the particle moves along any given trajectory in space-time. We show that in this process the two assumptions of the space-time formulation of quantum mechanics, are realized: (a) The weight that the particle moves along a trajectory that has been checked by this process is the same for all trajectories, and in fact, we show that the particle follows, with probability 1, the trajectory that is being checked. (b) A phase is systematically accumulated, so that, at the end of this process, the state is multiplied by the familiar factor exp[(i\ensuremathℏ)\ensuremath∫Ldt]\mathrm{exp}[(\frac{i}{\ensuremath{\hbar}})\ensuremath{\int}L\mathrm{dt}]exp[(\ensuremathℏi​)\ensuremath∫Ldt]. As an immediate extension of the above formalism, we suggest a setup that measures the relative phase between any two trajectories. Finally, our approach points toward the possibility of extending the Feynman formalism in order to cover more general Hamiltonians.
A path , in general, is defined by a determinate trajectory in time, from an initial to a final point. The classical trajectory is only one of the possible trajectories, and in quantum mechanics all the possible paths between the initial and final point come into play. Recall that the probability amplitude is a complex number that is assigned to each determinate path. Indeterminate paths are defined as a collection of determinate paths that are experimentally indistinguishable . In the Dirac–Feynman approach, the inherent indeterminacy of the quantum entity is realized by the degree of freedom – in undergoing time evolution – “taking” indeterminate paths [Baaquie (2013e)]. For a quantum degree of freedom evolving from an observed initial state to the observed final state – and with no other observations made – the Feynman path integral is a mathematical construction that computes the probability amplitudes by summing over all the allowed determinate paths of the degree of freedom – discussed in Feynman and Hibbs (1965), Zinn-Justin (1993), Zinn-Justin (2005) and Baaquie (2013e). Probability amplitude and time evolution Recall that the description of a quantum system, at a particular instant,isgiven by its state vector, namely |ψ>. To avoid confusion with the concept of a state vector, the term probability amplitude is used for describing a quantum entity undergoing transitions in time.
Feynman path integrals formalism for non-relativistic quantum mechanics is revisited. A comparison is made with the cases of light progagation (Huygens principle) and Brownian motion. The difficulties for a physical model behind Feynman formalism are pointed out. It is proposed a reformulation where the transition probability from one space-time point to another one is the sum of probabilities of the possible paths. The Born approximation for scattering is derived within the formalism, which suggests an interpretation in terms of particles, without the need of Born assumption that the modulus squared of the wavefunction is a probability density.
The conventional approach to quantum mechanics begins with the classical Hamiltonian and changes observables to noncommuting operators. Canonically conjugate observables have to be noncommuting operators in order to satisfy the Heisenberg uncertainty principle. The dynamics of a quantum mechanical system is given by the time-dependent Schrödinger equation. In the early 1940s Feynman realized that it would be possible to construct a quantum model using the classical Lagrangian approach (a method closely related to the action integral). Dirac had earlier pointed out that eit times the Lagrangian was analogous to a transformation function for the wave function in which the wave function at one instant could be related to the wave function at a next instant in a time interval of t by multiplying with such an exponential function. Feynman found the exponential phase factor to be the action S which is the time integral of the Lagrangian. Further, he noticed that it was necessary to perform integrals over all space variables at every instant of time to find the transition amplitudes. He developed a way for describing and evaluating the integration using the idea of integral over all paths. In Feynman’s approach, the particles are described using amplitudes calculated along the paths they may or may not follow. These amplitudes behave like waves, their phase changes as the quantum system moves along the path and the amplitudes for all paths superpose and interfere. This interference between all paths is called as sum-over-histories and the resulting amplitude is connected in the same way that the amplitude of a wave function is linked to probability as magnitude-squared.
The Feynman path integral formalism for non-relativistic quantum mechanics is revisited. A comparison is made with cases of light propagation (Huygens' principle) and Brownian motion. The difficulties for a physical model applying Feynman's formalism are pointed out. A reformulation is proposed, where the transition probability of a particle from one space-time point to another one is the sum of probabilities of the possible paths. As an application, Born approximation for scattering is derived within the formalism, which suggests an interpretation involving the stochastic motion of a particle rather than the square of a wavelike amplitude.
There has been a recent revival of interest in the notion of a 'trajectory' of a quantum particle. In this paper, we detail the relationship between Dirac's ideas, Feynman paths and the Bohm approach. The key to the relationship is the weak value of the momentum which Feynman calls a transition probability amplitude. With this identification we are able to conclude that a Bohm 'trajectory' is the average of an ensemble of actual individual stochastic Feynman paths. This implies that they can be interpreted as the mean momentum flow of a set of individual quantum processes and not the path of an individual particle. This enables us to give a clearer account of the experimental two-slit results of Kocsis et al.
The phase corresponding to a certain path in the Feynman path integral is derived with the help of the de Broglie hypothesis. Zusammenfassung. Die einem bestimmten Pfad entsprechende Phase im Feynman-Pfadintegral wird mit Hilfe der de Broglie-Hypothese abgeleitet. In the beginning quantum mechanics was built up by analogy with the formulation of classical mechanics in terms of the Hamilton function. In 1933, Dirac [2] announced the idea of formulating quantum mechanics on the basis of the Lagrange function. He started from the analogy between the treatment of classical motion as a canonical transformation and the transformational properties of operators with respect to a time translation. This idea of Dirac was developed further by Feynman [3], who regarded a quantum-mechanical motion from one space-time point to another as a superposition of motions along all possible paths connecting the two points. Each path contributes to the total probability amplitude a term proportional to where is the classical action corresponding to this path and is Planck's constant. Feynman [3] adopted the form of the phase in the expression (1) as a postulate. Another way, however, of deriving the phase is to use the de Broglie hypothesis [1], as shown below. According to this hypothesis, it is possible to associate any particle with a certain frequency proportional to its energy (Hamiltonian function), i.e., and a wave vector proportional to the particle's momentum as Now consider, in the sense of Feynman's formulation, a particle moving along a certain path from point 1 to point 2. During the motion, let and be the radius vectors of the particle at the times t, t+ dt respectively. According to the de Broglie hypothesis, we can associate a plane wave with the particle in the neighbourhood of these points. The phase change of this wave along the particle's path during the motion between the points and consists of space and time phase changes: The term in the parentheses is precisely the value of the Lagrange function at the point . The phase change of the wavefunction corresponding to the whole path can be obtained by integration of the term expressing the infinitesimal phase change: In this way, on the basis of the de Broglie hypothesis, we have reached full agreement with Feynman's postulate: the phase of the probability amplitude corresponding to a certain path is equal to , where is the classical action corresponding to this path. References [1] de Broglie L 1923 Comptes rendus 177 548 - 550 [2] Dirac P A M 1933 Phys. Z. der Sowjetunion 3 64 [3] Feynman R 1948 Rev. Mod. Phys. 2 367