Temporal discretization is developed as a means of studying the nonperturbative stability of nonlocal actions such as those of string theory and third-quantized gravity. An algorithm is derived for constructing the classical canonical Hamiltonian which generates the natural interpolation of continuous time evolution for Lagrangian higher-difference theories. Using this quantity we prove the analog of Ostrogradski's result for continuum higher derivatives: namely, that every nondegenerate, deterministic higher-difference theory is unstable. This result is nonperturbative and survives quantization. An explicit construction is given for the energy of a general quadratic Lagrangian with arbitrarily high differences.
- Received 9 December 1988
- Published in the issue dated 15 July 1989
© 1989 The American Physical Society