Feynman Path Integral Formulation

56 2 Feynman Path Integral FormulationFig. 2.1 Quantum mechanicalamplitude of transitioningfrom an initial three-geometrydescribed by g at time t initialto a final three-geometrydescribed by g ′ at a latertime t f inal . The full amplitudeis a sum over all interveningmetrics connecting thetwo bounding three-surfaces,weighted by exp(iI/¯h) whereI is a suitably defined gravitationalaction.gg’t initialtfinal(for an illustration see Fig. 2.1), just like the Feynman path integral for a nonrelativisticquantum mechanical particle (Feynman, 1948; 1950; Feynman and Hibbs,1965) expresses quantum-mechanical amplitudes in terms of sums over paths∫A(i → f )= e h ī I path. (2.2)pathsWhat is the precise meaning of the expression in Eq. (2.1)? The remainder of thissection will be devoted to discussing attempts at a proper definition of the gravitationalpath integral of Eq. (2.1). A modern rigorous discussion of path integrals inquantum mechanics and (Euclidean) quantum field theory can be found, for example,in (Albeverio and Hoegh-Krohn, 1976), (Glimm and Jaffe, 1981) and (Zinn-Justin, 2002).2.2 Sum over PathsAlready for a non-relativistic particle the path integral needs to be defined quitecarefully, by discretizing the time coordinate and introducing a short distance cutoff.The standard procedure starts from the quantum-mechanical transition amplitudeA(q i ,t i → q f ,t f )=< q f |e − ī h H(t f −t i ) |q i >, (2.3)and subdivides the time interval into n+1 segments of size ε with t f =(n+1)ε +t i .Using completeness of the coordinate basis |q j > at all intermediate times, one obtainsthe textbook result, here for a non-relativistic particle described by a HamiltonianH(p,q)=p 2 /(2m)+V (q),