Ivancevic_Applied-Diff-Geom

994 Applied Differential Geometry: A Modern Introductionproducts and sums, 5 < B|A >=n∑< B|c i >< c i |A > . (6.1)i=16.1.8 Feynman’s Sum–over–HistoriesNow, iterating the Dirac’s expansion principle (6.1) over a complete set ofall possible states of the system, leads to the simplest form of the Feynmanpath integral, or, sum–over–histories. Imagine that the initial and finalstates, A and B, are points on the vertical lines x = 0 and x = n + 1,respectively, in the x − y plane, and that (c(k) i(k) , k) is a given point onthe line x = k for 0 < i(k) < m (see Figure 6.2). Suppose that thesum of projectors for each intermediate state is complete 6 Applying thecompleteness iteratively, we get the following expression for the transitionamplitude:< B|A >= ∑ ∑ ... ∑ < B|c(1) i(1) >< c(1) i(1) |c(2) i(2) > ... < c(n) i(n) |A >,where the sum is taken over all i(k) ranging between 1 and m, and k rangingbetween 1 and n. Each term in this sum can be construed as a combinatorialroute from A to B in the 2D space of the x − y plane. Thus the transitionamplitude for the system going from some initial state A to some final stateB is seen as a summation of contributions from all the routes connectingA to B.Feynman used this description to produce his celebrated path integralexpression for a transition amplitude (see, e.g., [Grosche and Steiner(1998)]). His path integral takes the form∫T ransition Amplitude =< B|A >= Σ, D[x] e iS[x] , (6.2)5 In Dirac’s language, the completeness of intermediate states becomes the statementthat a certain sum of projectors is equal to the identity. Namely, suppose that P i |c i >= 1 for each i. Then< b|a >=< b||a >=< b| X i|c i >< c i ||a >= X i< b|c i >< c i |a > .6 We assume that following sum is equal to one, for each k from 1 to n − 1:|c(k) 1 >< c(k) 1 | + ... + |c(k) m >< c(k) m| = 1.