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Published by Berlin: Springer From: Antiquariat Bernhardt Kassel, Germany. As mentioned before, any derivation of functional integrals is not attempted here, but just a motivation of their form by analogy. Furthermore, in the case of quantum mechanics the transition amplitude has been considered, whereas now the formula for Greens functions has been written, which is a bit different.
The formulae for functional integrals give rise to some questions. Secondly, these integrals contain oscillating integrands, due to the imaginary exponents; what about their convergence? Moreover, is there a way to evaluate them numerically? In the following it will be discussed, how the introduction of imaginary times helps in answering these questions. Return to quantum mechanics for a moment. Here one can also introduce Greens functions, e. In the following it will be demonstrated that these Greens functions are related to quantum mechanical amplitudes at imaginary times by analytic continuation.
This is the so-called Wick rotation, illustrated in Figure 3. Now we turn to field theory again. They are taken as starting point for non-perturbative investigations of field theories and for constructive studies. Whether it is possible to continue a specific field theory analytically from real to imaginary times and vice versa, depends on certain conditions to be satisfied. For a large class of field theories these conditions have been analyzed and formulated by Osterwalder and Schrader, see Osterwalder and Schrader, , In particular, a Euclidean field theory must satisfy the so-called reflection positivity in order to correspond to a proper field theory in Minkowski space.
This makes Euclidean functional integrals so attractive compared to their Minkowskian counterparts. One might think that in the Euclidean domain everything is unphysical and there is no possibility to get physical results directly from the Euclidean Greens functions. But this is not the case. For example, the spectrum of the theory can be obtained in the following way.
One central question still remains: does the infinite dimensional integration over all classical field configurations, i. In quantum mechanics the path integral representation can be derived as a limit of a discretization in time. It should be noted that the lattice spacing, being a dimensionful quantity, is not a parameter of the discretized theory, which could e. The size of the lattice spacing in physical units is a derived quantity determined by the dynamics. This will be explained in Section "Continuum limit". So a discrete set of variables has to be integrated.
If the lattice is taken to be finite, one just has finite dimensional integrals. Discretization of space-time using lattices has one very important consequence. Due to a non-zero lattice spacing, a cutoff in momentum space arises. In order to begin in a well-defined way one would start with a finite lattice. Of course, one would like to recover physics in a continuous and infinite space-time eventually. The formulation of Euclidean quantum field theory on a lattice bears a useful analogy to statistical mechanics. Functional integrals have the form of partition functions and we can set up the following correspondence:.
This formal analogy allows to use well established methods of statistical mechanics in field theory and vice versa. Even the terminology of both fields is often identical. To mention some examples, in field theory one employs high-temperature expansions and mean field approximations, and in statistical mechanics one applies the renormalization group. An alternative to Euclidean lattice field theory, as described before, is Hamiltonian lattice field theory, introduced by Kogut and Susskind Kogut and Susskind, In this formulation only three-dimensional space is discretized on a lattice, whereas time remains continuous.
Furthermore, time is kept real and is not continued to the Euclidean domain. Hamiltonian lattice field theory allows the application of some analytical methods like strong coupling expansions and perturbation theory. Since it is not suitable for the application of the numerical Monte Carlo method, it doesn't enjoy any more as much attention as in its beginnings, and is not covered in more detail here.
Theories of gauge fields can also be formulated on a space-time lattice.
As details are explained in the Wiki on lattice gauge theories , we shall just indicate the basic elements of lattice gauge theory for gauge group SU N. These group valued variables represent the gauge field. The smallest closed paths on the lattice are the plaquettes , as shown in Figure 6.
We would like to point out that the argument above is just a heuristic motivation. The path integral formulation of quantum mechanics was developed in by Richard Feynman. Some preliminaries were worked out earlier, in the course of his doctoral thesis work with John Archibald Wheeler.
Path Integral Quantum Mechanics
It is a description of quantum theory which generalizes the action principle of classical mechanics. It replaces the classical notion of a single, unique history for a system with a sum, or functional integral , over an infinity of possible histories to compute a quantum amplitude.
This formulation has proved crucial to the subsequent development of theoretical physics, since it provided the basis for the grand synthesis of the 's called the renormalization group which unified quantum field theory with statistical mechanics. It is no surprise, therefore, that path integrals have also been used in the study of Brownian motion and diffusion.
The path integral method is an alternative formulation of quantum mechanics. The canonical approach, pioneered by Schroedinger , Heisenberg and Dirac paid great attention to wave-particle duality and the resulting uncertainty principle by replacing Poisson brackets of classical mechanics by commutators between operators in quantum mechanics.
The Hilbert space of quantum states and the superposition law of quantum amplitudes follows. The path integral starts from the superposition law, and exploits wave-particle duality to build a generating function for quantum amplitudes. Feynman proposed the following postulates :. In order to find the overall probability amplitude for a given process, then, one adds up, or integrates , the amplitude of postulate 3 over the space of all possible histories of the system in between the initial and final states, including histories that are absurd by classical standards.
In calculating the amplitude for a single particle to go from one place to another in a given time, it would be correct to include histories in which the particle describes elaborate curlicues, histories in which the particle shoots off into outer space and flies back again, and so forth. The path integral includes them all.
Not only that, it assigns all of them, no matter how bizarre, amplitudes of equal size ; only the phase , or argument of the complex number , varies. The contributions wildly different from the classical history are suppressed only by the interference of similar histories see below. Feynman showed that his formulation of quantum mechanics is equivalent to the canonical approach to quantum mechanics. Feynman's postulates are somewhat ambiguous in that they do not define what an "event" is or the exact proportionality constant in postulate 3.
The proportionality problem can be solved by simply normalizing the path integral by dividing the amplitude by the square root of the total probability for something to happen resulting in that the total probability given by all the normalized amplitudes will be 1, as we would expect. Generally speaking one can simply define the "events" in an operational sense for any given experiment.
The equal magnitude of all amplitudes in the path integral tends to make it difficult to define it such that it converges and is mathematically tractable. For purposes of actual evaluation of quantities using path-integral methods, it is common to give the action an imaginary part in order to damp the wilder contributions to the integral, then take the limit of a real action at the end of the calculation. In quantum field theory this takes the form of Wick rotation.
There is some difficulty in defining a measure over the space of paths. In particular, the measure is concentrated on " fractallike " distributional paths. Feynman was initially attempting to make sense of a brief remark by Paul Dirac about the quantum equivalent of the action principle in classical mechanics.