Quantum action principle Path integral formulation

dirac (1933), p. 69

another way of saying since hamiltonian naturally function of p , q, exponentiating quantity , changing basis p q @ each step allows matrix element of h expressed simple function along each path. function quantum analog of classical action. observation due paul dirac.

dirac further noted 1 square time-evolution operator in s representation:

e

i
ε
s


,


{\displaystyle e^{i\varepsilon s},}

and gives time-evolution operator between time t , time t + 2ε. while in h representation quantity being summed on intermediate states obscure matrix element, in s representation reinterpreted quantity associated path. in limit 1 takes large power of operator, 1 reconstructs full quantum evolution between 2 states, 1 fixed value of q(0) , later 1 fixed value of q(t). result sum on paths phase, quantum action. crucially, dirac identified in article deep quantum-mechanical reason principle of least action controlling classical limit (see quotation box).