College of Liberal Arts & Sciences

# Mathematical Physics Seminar

**Abstract: **

When quantizing a classical system, one is typically concerned with representing the generalized coordinates $q_{i}$ and conjugate momenta $p_{i}$ as operators $\widehat{q}_{i}$ and $\widehat{p}_{j}$ which act on some Hilbert space, and satisfy the canonical commutation relations, $[\widehat{q}_{i}, \widehat{p}_{j}]=i\delta_{ij}$. For a finite number of coordinates and momenta these representations are almost always unitarily equivalent. However, for theories with an infinite number of $\widehat{q}_{i}$’s and $\widehat{p}_{j}$'s such as those found in quantum field theory, there exists infinitely many inequivalent representations.

At first glance the light-front formulation of quantum field theory does not appear to provide the expected inequivalent representations. By describing the theory in terms of an operator algebra and a vacuum functional, we can see that the inequivalent representations are not eliminated, but rather moved from a property of the vacuum functional, to a property of the algebra itself.

This talk will set the stage by focusing on the case in which we have a finite number of $\widehat{q}_{i}$’s and $\widehat{p}_{j}$'s. We will discuss the Stone-von Neumann Uniquness theorem, which provides criteria which insures that any representation of $\widehat{q}_{i}$’s and $\widehat{p}_{j}$'s is unitarily equivalent to the usual representation found in quantum mechanics. A few surprising examples in which the criteria are not met will also be discussed.