10 - The Feynman Integral

Summary

[Mathematics] … there is no study in the world which brings into more harmonious action all the faculties of the mind.

—J. J. Sylvester

Introduction

In the 1920s Norbert Wiener introduced the concept of a measure on the space of continuous functions. As you recall from Chapter 9, this idea arose from his attempts to model the Brownian motion of small particles suspended in a fluid. In the 1940s Richard Feynman (1918–1988) developed an integral on the same space of continuous functions in his efforts to model the quantum mechanics of very small particles such as electrons. To succeed, Feynman's path integral approach to quantum mechanics had to be consistent with Schrödinger's Equation.

Schrödinger's Equation

A frequent correspondent of Albert Einstein, Erwin Schrödinger (1887–1961) made a series of brilliant advances in quantum theory and the general theory of relativity. Our topic here is his breakthrough wave equation, discovered in 1925.

Suppose a particle of mass m is at position x₀ at time t = 0 with a potential V(x₀). The particle may move to position x at time t. The Heisenberg Uncertainty Principle sets accuracy limits on the determination of position x at time t. From physical considerations, then, we assign a probability to each path from x₀ at time t = 0 to x at time t. This probability is P(t, x) = |ψ(t, x)|², where ψ is a complex-valued quantity called the probability amplitude.