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Path Integral, Transition to Qunatum Theory

The transition from classical to quantum in theoretical physics is most elegantly prescribed by path integral. According to classical physics the movement from "here" at time t1 to "there" at time t2 in Figure 12-05 is through the shortest path (the dash line) in evaluating the Action. In quantum theory all paths are possible. Each possible route corresponds to a "history".
Path Integral Each history has associated with it a number, called the amplitude, which defines the probability of that particular path being followed. While the classical path (the dash line) occurs with higher probability, the probability for the other paths vary according to a weighting factor. The probability of going from "here" to "there" is the sum of the probability for all paths. This formulism was originally devised by Richard Feynman for his PhD thesis in early 1940s. Twenty years earlier the transition from classical to quantum had to be formulated with a postulation which can be shown to be equivalent to the method of path integral (see mathematical detail).

Figure 12-05 Path Integral
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