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:

*Events in nature are probabilistic with predictable probabilities**P*.*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 associated with every history leading to the event.*The quantum amplitude associated with a given history is the product of the amplitudes 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 , 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
so that then , where the
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 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
from the Davisson-Germer and G.P. Thomson experiments
into the formulation, for instance, would be to ascribe a quantum
amplitude of for the propagation of a particle with
momentum 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.

Thu May 29 15:16:11 EDT 1997