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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 φ4 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].
