# Poster presented by John Lawson

**Poster Session**
**Topic:** Theoretical Developments

### New Path Integral for the Microcanonical Density Matrix

**Abstract:** A new path integral representation for the density matrix of the quantum microcanonical ensemble is presented. In contrast to the Feynman path integral which corresponds to matrix elements of the evolution operator $e^{-i\hat{H}t/\hbar}$, the path integral presented in this work is a representation of $\delta(E-\hat{H})$. An immediate benefit of this approach is that the trace $\omega(E)=Tr\delta(E-\hat{H}) = \sum_n \delta(E-E_n)$ corresponds to the density of states and give the spectrum directly. A path integral formalism leads to an exact integral representation for $\langle x^{\prime} | \delta(E-\hat{H}) | x \rangle$. Both phase space and configuration space forms are presented. For simple systems, such as the free particle and harmonic oscillator, exact solutions are possible. For more complicated systems, expansion schemes or numerical evaluations are required. Both perturbative expansions and numerical evaluations are performed for the quantum anharmonic oscillator. Applications to the statistical mechanics of quantum fields will be discussed.

Massimo Campostrini,
`campo@mailbox.difi.unipi.it`