Path integrals for awkward actions

We show how to construct path integrals in theories of scalar fields when the action is not quadratic in the time derivatives of the fields. In such theories, the hamiltonian is not quadratic in the canonical momenta and may be unknown. The weight functions of the resulting path integrals can assume negative or complex values. We show how to compute them by combining the Monte Carlo method with parallel numerical integration and a lookup table. This “Atlantic City” method lets one study a wide class of theories of scalar fields.

The approximation of multiple integrals whose weight functions assume negative or complex values is a long-standing problem. The Atlantic City method solves it in all the field theories we have examined.