ABSTRACT
In the standard formulation of quantum mechanics, observables are represented by operators in a Hilbert space and states by vectors in it. Quantum field theory too is formulated in terms of field operators. States in field theory include the vacuum state, one particle states, two particle states and so on. One does not measure operators or state vectors, however; measurements involve eigenvalues of operators or transition probabilities between states. It turns out that one can formulate quantum mechanics and field theory without using operators. Indeed, the probability amplitude for a transition from a position q1 at time t1 to a position q2 at time t2 can be expressed as a path integral ∫
dt L(q(t),q˙(t)), (5.1)
where q(t) runs over all possible functions satisfying the conditions
q(t1) = q1, q(t2) = q2 (5.2)
and the integral over q(t) is an integral over all such paths of the exponential evaluated for each path. The measure of integration is not well defined a priori, but it is possible to give prescriptions that make sense. In classical mechanics, positions at two different times determine a path from the second order differential equation of motion, which yields a definite solution if two conditions are imposed. In contrast, quantum mechanics does not allow us to talk about trajectories. The path integral formulation manifestly refers to the myriad paths that may be taken by a particle in going from one point to another. It goes so far as to assign a probability amplitude to each path, and it is related to the classical action corresponding to that path. However, it is complex and it also involves a superposition, characteristic of quantum mechanics.
