Path Integrals

Some time back I blogged on the topic of path dependence. More recently, I discussed some aspects of the weird world of quantum mechanics, and I gave a short introduction to the action principle of physics. I'd like to tie these ideas more closely together, and say a little about the Lagrangian density. This will lay the groundwork for a future blorticle on particle mass and the Higgs mechanism.

The action principle for a classical particle states that the particle follows that path for which the action, S, is a minimum. The action is defined by

S = integral[a,b] L(x,v) dt

that is, the integral from time a to time b of L(x,v), a quantity known as the Lagrangian function. (I don't have any easy way to typeset formulas at present, so I am forced to improvise.) The Lagrangian is a function of position, x, and velocity, v. For a single classical particle moving in a potential, the Lagrangian is L = T-V, where T is the kinetic energy, T = mv^2/2, and V is the potential. By applying the methods of the calculus of variation, one can derive the familiar Newtonian equations of motion from the action principle. One can also show that the various conservation laws of physics arise from mathematical symmetries of the Lagrangian.

For quantum particles, the action principle takes a slightly different form:

S = integral[a,b] (integral(all space) L(Y, dY/dx) dx) dt

That is, the action of a quantum particle is the integral over all of space and from time a to b of the Lagrangian density L, which depends on the wave function, Y, and its rate of change with position, dY/dx. For an uncharged particle with zero spin (the simplest possible) the Lagrangian density takes the form

L(Y, dY/dx) = (1/2)[(dY/dt)*(dY/dt)-(dY/dx)*(dY/dx)-(mY)^2]

where m is the mass of the particle. When we apply the action principle to this Lagrangian density, we find that the wave equation for the free particle is

d(dY/dx)/dx - d(dY/dt)/dt - m^2 Y = 0

which is called the Klein-Gordon equation. Its solutions are wave functions of the form

Y = A cos(kx - wt - theta)

where A is a normalization constant, k is the wavelength, theta is a phase constant, and w = sqrt(k^2 + m^2). The velocity of the wave is dw/dk = k/sqrt(k^2 + m^2). If the particle is massless, its velocity is 1, the speed of light in the system of units favored by theoretical physicists. If the particle is massive, then the velocity is dw/dk = k/m - (1/2)(k/m)^3 + ... which is always less than the speed of light.

The quantum formula for the action reflects Heisenberg's Uncertainty Principle, which says that we can never be exactly sure of the location of a quantum particle. Since the location is uncertain, there isn't a well-defined path, so we have to integrate the action over all possible paths. The correct wave function for the particle is the wave function for which this integral is stationary.

Of course, paths do not appear explicitly in the quantum action as it is written above. It was the great theoretical physicist and Nobel laureate Richard Feynman who discovered how to express quantum processes in terms of integrals over paths. Feynmann expressed the probability that a particle that is at point a at time s would be found at point b at time t in terms of a propagator function

K(a,b,s,t) = sum(all paths) A exp(iS(a,b,s,t))

That is, each path from (a,s) to (b,t) has a phase, exp(iS(a,b,s,t)), associated with it. Here S is the classical action associated with the path. The propagator is the sum of these phases over all the possible paths between the two locations in space and time.

The path that minimizes the classical action is known as a stationary path, because small deviations from this path yield almost no change to the action. Those paths that deviate only slightly from the stationary path will therefore be nearly in phase and will constructively interfere to yield a large contribution to the propagator. Paths far from the stationary path will not be in phase and will destructively interfere. The net result is that only paths near the classical path of the particle make an important contribution to the propagator. The wave function of a quantum particle is largest along the path that would be followed by a classical particle.

We don't notice quantum effects in the everyday world, because the wave functions of ordinary objects are insignificant except along the classical path. But in the microscopic world, the wave function is spread out around the path enough to produce noticeable quantum behavior.

Feynmann's path integrals are a powerful mathematical tool for calculating quantum behavior. The math is much too formidable to present here. But the path integral is an excellent concept for explaining some of the mysteries of quantum mechanics. My next blorticle in this series will focus on the Lagrangian density and the Higgs mechanism, which may explain why some particles have mass and some do not.