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The Sum Over Histories


The Feynman formulation of Quantum Mechanics builds three central ideas from the de Broglie hypothesis into the computation of quantum amplitudes: the probabilistic aspect of nature, superposition, and the classical limit. This is done by making the following three three postulates:

  1. Events in nature are probabilistic with predictable probabilities P.
  2. The probability P for an event to occur is given by the square of the complex magnitude of a quantum amplitude for the event, Q. The quantum amplitude Q associated with an event is the sum of the amplitudes tex2html_wrap_inline1605 associated with every history leading to the event.
  3. The quantum amplitude associated with a given history tex2html_wrap_inline1605 is the product of the amplitudes tex2html_wrap_inline1609 associated with each fundamental process in the history.

Postulate (1) states the fundamental probabilistic nature of our world, and opens the way for computing these probabilities.

Postulate (2) specifies how probabilities are to be computed. This item builds the concept of superposition, and thus the possibility of quantum interference, directly into the formulation. Specifying that the probability for an event is given as the magnitude-squared of a sum made from complex numbers, allows for negative, positive and intermediate interference effects. This part of the formulation thus builds the description of experiments such as the two-slit experiment directly into the formulation. A history is a sequence of fundamental processes leading to the the event in question. We now have an explicit formulation for calculating the probabilities for events in terms of the tex2html_wrap_inline1605 , quantum amplitudes for individual histories, which the third postulate will now specify.

Postulate (3) specifies the quantum amplitude associated with individual histories in terms of fundamental processes. A fundamental process is any process which cannot be interrupted by another fundamental process. The fundamental processes are thus indivisible ``atomic units'' of history. With this constraint of the choice of fundamental processes, individual histories may always be divided unambiguously into ordered sequences of fundamental events, which is key to making a consistent prescription for computing the amplitudes of individual histories from fundamental processes. The fact that the definition of fundamental processes is not very specific is actually one of the strongest aspects of the Feynman approach. As we will see, we may sometimes discover that we may lump fundamental processes together into larger units which make up new fundamental processes. This procedure is know as renormalization and is one the the great central ideas in managing the infinities in quantum field theory.

The third postulate builds in the classical limit by allowing recovery of the classical physics notion that the probability of an independent sequence of events is the product of the probabilities for each event in the sequence. If we know the sequence of fundamental processes leading to an event, the only contributing history is that sequence of processes. In such a case, we have tex2html_wrap_inline1613 so that then tex2html_wrap_inline1615 , where the tex2html_wrap_inline1617 are just the probabilities for the individual processes in the sequence, and we recover the usual classical probabilistic result.

What remains unspecified by these postulates is the specification of a valid set of fundamental processes and corresponding quantum amplitudes tex2html_wrap_inline1609 for the phenomena we wish to describe. For this information, we must rely upon experimental observations. It is at this point that experimental information is input into the Feynman formulation much like how we inputted experimental information into our formulation when we produced the forms for our operators and the Schrödinger Equations.

A great appeal to the Feynman sum over histories approach is that often we are able to intuit the nature and amplitudes of the fundamental events. A natural way to build the de Broglie hypothesis tex2html_wrap_inline1621 from the Davisson-Germer and G.P. Thomson experiments into the formulation, for instance, would be to ascribe a quantum amplitude of tex2html_wrap_inline1623 for the propagation of a particle with momentum tex2html_wrap_inline1625 across a distance a.

Another common way to infer the fundamental events and associated amplitudes is to determine the amplitudes for fundamental processes from the requirement that the Feynman formulation always give the same results as an already established approach, such as Schrödinger formulation. This latter procedure is referred as construction of Feynman rules, and is also how we determine that the Feynman approach is indeed equivalent to the other formulations of quantum mechanics. We shall follow this procedure in the next section.

next up previous contents
Next: Construction of Feynman Rules Up: Notes on Feynman Diagrams Previous: Introduction

Prof. Tomas Alberto Arias
Thu May 29 15:16:11 EDT 1997