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arXiv:1211.7168 · International Journal of Modern Physics A

ACTION WITH ACCELERATION I: EUCLIDEAN HAMILTONIAN AND PATH INTEGRAL

Belal Ehsan Baaquie

Published Oct 28, 2013• 8 citations
arXivHTMLPDF

Abstract

An action having an acceleration term in addition to the usual velocity term is analyzed. The quantum mechanical system is directly defined for Euclidean time using the path integral. The Euclidean Hamiltonian is shown to yield the acceleration Lagrangian and the path integral with the correct boundary conditions. Due to the acceleration term, the state space depends on both position and velocity — and hence the Euclidean Hamiltonian depends on two degrees of freedom. The Hamiltonian for the acceleration system is non-Hermitian and can be mapped to a Hermitian Hamiltonian using a similarity transformation; the matrix elements of the similarity transformation are explicitly evaluated.

DOI
10.1142/s0217751x13501376

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An action having an acceleration term in addition to the usual velocity term is analyzed. The quantum mechanical system is directly defined for Euclidean time using the path integral. The Euclidean Hamiltonian is shown to yield the acceleration Lagrangian and the path integral with the correct boundary conditions. Due to the acceleration term, the state space depends on both position and velocity — and hence the Euclidean Hamiltonian depends on two degrees of freedom. The Hamiltonian for the acceleration system is non-Hermitian and can be mapped to a Hermitian Hamiltonian using a similarity transformation; the matrix elements of the similarity transformation are explicitly evaluated.
Introduction to part five The most widely used actions in quantum mechanics have a kinetic term that is the velocity squared of the degree of freedom and a potential term that depends on the degree of freedom. The kinetic term fixes the equal time commutation equation of the degree of freedom, as shown in Section 6.7. Furthermore, the velocity term in the Lagrangian entails that all the possible indeterminate paths obey two boundary conditions, which in turn yields a state space that depends on the degree of freedom only. In Section 11.13, this property of the indeterminate paths was used for deriving the state function of the harmonic oscillator. The action with acceleration has a kinetic term that is given by the acceleration squared of the degree of freedom, in addition to the usual velocity and potential terms. It is an example of higher derivative Lagrangians, discussed in Simon (1990). The higher derivative quantum systems have many remarkable properties not present for the usual cases studied so far. The action with acceleration arises in many diverse fields and has been widely studied; it describes the behaviour of “stiff” polymers, of cell walls, of the formation of microemulsions, the properties of chromoelectric flux lines in quantum chromodynamics, as well as the Big Bang singularity in cosmology.
Path integrals are by and large defined directly in terms of the configuration space representation of the quantum entity's degrees of freedom and employ the Lagrangian description of the quantum entity; most of the path integrals in this book follow this approach. The Hamiltonian provides another independent approach for defining path integrals and is discussed in this chapter. Two important path integrals, which are directly based on the Hamiltonian, are the following: (a) one that is defined on the degree of freedom's phase space , defined as the tensor product of the degree of freedom space and its canonical conjugate momentum space; and (b) path integrals using the coherent state basis instead of the coordinate basis. Path integrals defined on phase space, or for coherent states, are both based on the Hamiltonian. To put in the foreground the role of the Hamiltonian in quantum mechanics, the canonical equations connecting the Lagrangian to the Hamiltonian are discussed. A brief review of Hamiltonian mechanics, also called the canonical equations, is given in Section 5.1 and the connection of symmetries with conservation laws is discussed in Section 5.2. The Hamiltonian is derived from the Lagrangian in Section 5.3, for both Minkowski and Euclidean time. Phase space path integrals are defined in Section 5.4. Canonical quantization based on the Poisson brackets is discussed in Section 5.5, and Dirac brackets required for quantizing constrained systems are derived in Section 5.7. Coherent states and their path integrals are discussed in Sections 5.10 to 5.14.
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.
There is no conservation of probability in the Euclidean formulation of the acceleration Lagrangian. A narrow definition of a Hamiltonian, and the one that holds in quantum mechanics but is not suitable for our case, is that the Hamiltonian defines a probability conserving unitary evolution. A broader definition of the Hamiltonian is adopted that is consistent with both statistical mechanics and quantum mechanics. Namely, the Hamiltonian – which is equivalent to the transfer matrix in statistical mechanics – is the generator of infinitesimal translations in time for both the Minkowski and Euclidean cases. The Hamiltonian derived for the acceleration Lagrangian in Chapter 13 is studied in this chapter using the concepts of the state space and operators. The Hamiltonian for unequal and real frequencies ω 1 ,ω 2 is pseudo-Hermitian and is explicitly mapped to a Hermitian Hamiltonian using a nontrivial differential operator. An explicit expression is obtained for the matrix element of the differential operator. The mapping fails for a critical value of the coupling constants. The Hamiltonian and state space are shown to have real and complex branches that are separated by the critical point. At the critical point, the Hamiltonian is inequivalent to any Hermitian operator, and is shown in Chapter 15 as being equal to an infinite-dimensional block diagonal matrix, with each block being a Jordan matrix.
In this thesis we are investigating a different formalism of non-relativistic quantum mechanics called the path integral formalism. It is a generalization of the classical least action principle. The introduction to this subject begins with the construction of the path integral in terms of the idea of probability amplitudes whose absolute square gives the probability of finding a system in a particular state. Then we show that if the Lagrangian is a quadratic form one needs only to calculate the classical action besides from a time-dependent normalization constant to find the explicit expression of the path integral. We look in to the subject of two kinds of slit-experiments: The square slit, the single- and the double-Gaussian slit. Also, the propagator for constrained paths is calculated and applied to the Aharonov-Bohm effect, which shows that the vector potential defined in classical electrodynamics have a physical meaning in quantum mechanics. It is also shown that the path integral formulation is equivalent to the Schrödinger description of quantum mechanics, by deriving the Schrödinger equation from the path integral. Further applications of the path integral are discussed.