When the constraints of a system form a functional group the path integral quantum amplitude can exactly be obtained. It can be shown that the constraints deduced by symmetry just just form the functional group. Most of the interesting systems in physics are such symmetry systems.
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.
Quantum field theories are quantum mechanical systems with an infinite number of degrees of freedom. We start this book with a review of quantum mechanics in a form that is particularly suited for the field theory generalization. This form utilizes the Heisenberg picture and the functional integral representation of probability amplitudes — the transformation functions of Dirac. Some time will be spent developing and discussing the functional or path integral representation of quantum mechanics since this will enable us to learn a new formalism in a simple and familiar physical context — we won't have to learn too many new things all at once. The functional formulation will be illustrated by applying it at some length to the harmonic oscillator which is not only the simplest (and exactly soluble) non-trivial quantum system but which is also the dynamics of a single mode of a free field theory. The purpose of this review of quantum mechanics is to introduce in the simplest form much of the notation and methods used later in the book. Throughout this chapter we will explicitly work only with systems that have a single degree of freedom (one coordinate variable). The extension to systems with several degrees of freedom is straightforward.
Closed systems in Newtonian mechanics obey the principle of Galilean relativity. However, the usual Lagrangian for Newtonian mechanics, formed from the difference of kinetic and potential energies, is not invariant under the full group of Galilean transformations. In quantum mechanics, Galilean boosts require a non-trivial transformation rule for the wave function and a concomitant “projective representation” of the Galilean symmetry group. Using Feynman's path integral formalism, this latter result can be shown to be equivalent to the non-invariance of the Lagrangian. Thus, using path integral methods, the representation of certain symmetry groups in quantum mechanics can be simply understood in terms of the transformation properties of the classical Lagrangian and conversely. The main results reported here should be accessible to students and teachers of physics—particularly classical mechanics, quantum mechanics, and mathematical physics—at the advanced undergraduate and beginning graduate levels, providing a useful exposition for those wanting to explore topics such as the path integral formalism for quantum mechanics, relativity principles, Lagrangian mechanics, and representations of symmetries in classical and quantum mechanics.