College of Liberal Arts & Sciences

# Math Physics Seminar

**Abstract:**

I discuss Schwinger's representation of quantum systems of a finite number of degrees of freedom to formulate discrete path integral representations of the dynamics. The representation starts with an observable with a finite number of distinct eigenvalues. Using that observable Schwinger constructs complementary pairs of finite dimensional unitary operators that are finite dimensional analogs of the irreducible Weyl algebra. These can be decomposed into products of irreducible sub-algebras that reduce to q-bit gates for the case of $2^N$ degrees of freedom. In the limit of a large number of discrete degrees of freedom these representations can be used to model quantum systems with continuous degrees of freedom. This limit recovers the continuum Weyl algebra. Path integrals can be formulated in the discrete case. They have the advantage that the number of paths for a finite number of time slices is finite. Unitarity is exactly maintained at each time step. In this work the, following Palle and Katya's work, the path integral is interpreted as the expectation value of a potential functional with respect to a complex probability distribution on the space of "paths''. The complex probability exactly factors onto a product of one-step probabilities. An application to scattering from a short-range potential is given. Multi-resolution wavelet bases are used exactly represent a local $\phi^4(x)$ quantum field theory as a theory with an infinite number of discrete local modes. The discrete path integral is illustrated by computing time evolution when this theory is truncated to two coupled modes.