Recent work has shown that many non-Hermitian but PT-symmetric theories
have a real positive spectrum, and it has even been postulated that
a non-Hermitian but PT-symmetric `−g φ`^{4}
theory could describe the Higgs Boson. These findings suggest that
only studying Hermitian Hamiltonians may be too restrictive. To discover
if these new theories apply to any physical systems, field-theoretic
calculations in four dimensions must be performed. A rigorous
probabilistic formulation of these PT-symmetric theories has not been
developed, and as a result, Monte Carlo methods do not apply. I will
present arguments and show data that give hints for how to construct
a probabilistic formulation for these theories. I will also explain why
a method using the complex Langevin equation should work as a numerical
procedure, and then show results from successful applications
of this method to zero and one dimensional problems where exact results
are known. This numerical method provides further evidence that a real
probability distribution does indeed underly these PT-symmetric theories,
and this method could potentially be used to calculate predictions
of physical quantities in higher dimensions.

C. W. Bernard and V. M. Savage, *“Numerical simulations
of PT-symmetric quantum field theories,”*
Phys. Rev. D
**64**, 085010 (2001)
[hep-lat/0106009].

George T. Fleming ( gfleming@mps.ohio-state.edu ), last updated 15 May 2002.