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.
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.
This chapter introduces the path integral formulation of quantum mechanics as a rigorous and systematic alternative to canonical quantization. It begins by defining the transition probability amplitude (propagator) and deriving the path integral by discretizing time and summing over all possible intermediate paths between initial and final spacetime points. The chapter shows the final form of the propagator as a functional integral over all paths, weighted by the exponential of the action. This formulation provides an intuitive understanding that paths near the classical trajectory (which minimizes the action) interfere constructively, dominating the overall amplitude.
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.
18.1 Introduction The Feynman path-integral (PI) approach to quantum mechanics (QM) has proven to be an invaluable technical tool with applicability far beyond the non-relativistic QM from which it sprang (Feynman and Hibbs, 1965). The PI approach is based on a fundamental quantum principle, which is that nothing can or should be said with certainty about a system under observation (SUO) when it is not actually being observed. This is in complete contrast to classical mechanics (CM), which asserts that an SUO exists in a well-defined state at all times, regardless of whether it is being observed or not. It is a remarkable fact that what appears to be a metaphysical point actually underpins the principles of quantum physics. The quantum principle that we cannot say precisely in classical terms what an SUO is doing when we are not observing it has a powerful corollary: if we cannot say what the SUO is doing when we are not observing it, then we have to allow for the possibility that it could be doing anything, or at least anything that would be consistent with required conservation laws, behind our backs. Following a lead from Dirac (1933), Feynman developed and popularized a mathematical way of expressing this idea: the path integral. We shall discuss his approach in the case of an SUO consisting of a Newtonian point particle moving in one spatial dimension. Discrete time (DT) comes into the picture right at the start of the formulation.
All the chapters of the book have examined and studied various aspects of path integrals and Hamiltonians, which in turn exemplified different aspects of quantum mathematics. The principles of quantum mathematics, stated in Chapter 2, can be summarized as follows: • The fundamental degrees of freedom F form the bedrock of the quantum system. • A linear vector state space V based on the degrees of freedom F provides an exhaustive description of the quantum system. • Operators O, which includes the Hamiltonian H , represent the physical properties of the degree of freedom and act on the state space V . Observable quantities are the matrix elements of the physical operators. • A spacetime description of quantum indeterminacy is encoded in the Lagrangian L and the Dirac–Feynman formula relates it to the Hamiltonian. • The path integral provides a representation of all the physical properties of a quantum system. In particular, the path integral yields the correlation functions of the degrees of freedom as well as the probability amplitudes for quantum transitions. • The interconnection of the path integral with the underlying Hamiltonian and state space is a specific feature of quantum mathematics that distinguishes path integration from functional integration in general.
Path integrals are a central tool when it comes to describing quantum or thermal fluctuations of particles or fields. Their success dates back to Feynman who showed how to use them within the framework of quantum mechanics. Since then, path integrals have pervaded all areas of physics where fluctuation effects, quantum and/or thermal, are of paramount importance. Their appeal is based on the fact that one converts a problem formulated in terms of operators into one of sampling classical paths with a given weight. Path integrals are the mirror image of our conventional Riemann integrals, with functions replacing the real numbers one usually sums over. However, unlike conventional integrals, path integration suffers a serious drawback: in general, one cannot make non-linear changes of variables without committing an error of some sort. Thus, no path-integral based calculus is possible. Here we identify which are the deep mathematical reasons causing this important caveat, and we come up with cures for systems described by one degree of freedom. Our main result is a construction of path integration free of this longstanding problem.
In 1948, Richard Feynman published the first paper on his new approach to non-relativistic quantum mechanics. Before Feynman's work there were two mathematical formulations of quantum mechanics. Schrödinger's formulation was based on PDE (the Schrödinger equation) and states representation by wave functions, so it was in the framework of analysis and differential equations. The other formulation was Heisenberg's matrix algebra. Initially, they were thought to be competing. The proponents of one claimed that the other was “ wrong. ” Within a couple of years, John von Neumann had proved that they are equivalent.\nAlthough Feynman's theory was not fundamentally new, it nonetheless offered an entirely fresh and different perspective: via a precise formulation of Bohr's correspondence principle, it made quantum mechanics similar to classical mechanics in a precise sense. In addition, Feynman's approach made it possible to explain physical experiments, and, via diagrams, link them directly to computations. What resulted was a very powerful device for computing energies and scattering amplitudes - the famous Feynman's diagrams.\nIn his formulation, Feynman aimed at representing the solution to the non-relativistic Schrödinger equation in the form of an “ average ” over histories or paths of a particle. This solution is commonly known as the Feynman path integral. It plays an important role in the theory but appears as a postulate based on intuition coming from physics rather than a justified mathematical object. This is why Feynman's vision has caught the attention of many mathematicians as well as physicists.\nThe papers of Gelfand, Cameron, and Nelson are among the first, and more substantial, attempts to supply Feynman's theory with a rigorous mathematical foundation. These attempts were followed by many others, but unfortunately all of them were not quite satisfactory. The difficulty comes from a need to define a measure on an infinite-dimensional space of continuous functions that represent all possible paths of a particle. This Feynman's measure has to produce an integral with the properties requested by Feynman. In particular, the expression for the Feynman measure has to involve the non-absolutely integrable Fresnel integrands. The non-absolute integrability of the Fresnel integrands makes the measure fail to be positive and to have the countably additive property. Thus, a well-defined measure in the case of the Feynman path integral does not exist.\nExtensive research has been done on the methods of relating the Feynman path integral to the integral with respect to the Wiener measure. The method of analytic continuation in mass defines the Feynman path integral as a certain limit of the Wiener integrals. Unfortunately, this method can be used as definition for only almost all values of the mass parameter in the Schrödinger equation. For physicists, this is not a satisfactory result and needs to be improved.\nIn this work we examine those questions which originally led to the Feynman path integral. By now we know that Feynman's “ dream ” cannot be realized as a positive and countably additive measure on the path-space. Here, we offer a new way out by modifying Feynman's question, and thereby achieving a solution to the Schrödinger equation via a different kind of averages in the path-space.\nWe give our version of the question that Feynman “ should have asked ” in order to realize the elusive path integral. In our formulation, we get a Feynman path integral as a limit of linear functionals, as opposed to the more familiar inductive limits of positive measures, traditionally used for constructing the Wiener measure, and related Gaussian families. We adapt here an approach pioneered by Patrick Muldowney. In it, Muldowney suggested a Henstock integration technique in order to deal with the non-absolute integrability of the kind of Fresnel integrals which we need in our solution to Feynman's question. By applying Henstock's theory to Fresnel integrals, we construct a complex-valued “ probability distribution functions ” on the path-space. Then we use this “ probability ” distribution function to define the Feynman path integral as an inductive limit. This establishes a mathematically rigorous Feynman limit, and at the same time, preserves Feynman's intuitive idea in resulting functional. In addition, our definition, and our solution, do not place any restrictions on any of the parameters in the Schrödinger equation, and have a potential to offer useful computational experiments, and other theoretical insights.