2

The Duistermaat-Heckman theorem, although too complex for me to completely grasp, states that under some conditions, the partition function for a special class of Hamiltonians is semiclassically exact. The harmonic oscillator is exact and should be an element of this class - the problem is that I don't completely understand the theorem, therefore can't generalize it.

I would like to know if, for systems of the form

$$ H(q,p) = \left( \frac{q^2+p^2}{2} \right)^\gamma \, , \quad \gamma \in \mathbb{N} \, ,$$

the partition function is semiclassically exact, and if this exactness implies that the semiclassical propagator is also exact.

Qmechanic
  • 220,844
QuantumBrick
  • 4,183

1 Answers1

1

There are at least 2 types of partition functions:

  1. A finite-dimensional integral $$Z~=~\int \!dq~dp ~\exp\left\{-\frac{i}{\hbar}H(q,p)\right\}$$ with some $U(1)$ circle action, which OP didn't specify. Here a (Wick-rotated/oscillatory) Duistermaat-Heckman theorem applies.

  2. A loop-space functional integral $$Z~=~\int_{q(0)=q(T)}\!{\cal D}q~{\cal D}p ~\exp\left\{\frac{i}{\hbar}\int_0^T\! dt(p\dot{q}-H(q,p))\right\},$$ where the Hamiltonian $H(q,p)$ has no explicit $t$-dependence. Niemi-Tirkkonen equivariant localization to constant loops works generically away from caustics, cf. Refs. 1-3.

References:

  1. R.J. Szabo, arXiv:hep-th/9608068; section 4.6.

  2. A.J. Niemi & O. Tirkkonen, arXiv:hep-th/9206033; eq. (28).

  3. A.J. Niemi & O. Tirkkonen, arXiv:hep-th/9301059; eq. (3.23).

Qmechanic
  • 220,844