2010/03/19 by Carl M. Bender, Bender, Carl M.
Mathematics · Physics and Astronomy · #FOS: Physical sciences #High Energy Physics - Theory (hep-th) #Mathematical Physics (math-ph) #Quantum Mechanics and Non-Hermitian Physics #Quantum Physics (quant-ph) #hep-th #math-ph #math.MP #quant-ph
paper · pdf · doi:10.48550/arxiv.1003.3881
8 pages, 1 figure
arxiv created 2010/03/19 · openalex publication_date 2010/03/19 · arxiv updated 2010/03/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In 1992 Bender, Boettcher, and Lipatov proposed in two papers a new and unusual nonperturbative calculational tool in quantum field theory. The objective was to expand the Green's functions of the quantum field theory as Taylor series in powers of the space-time dimension D. In particular, the vacuum energy for a massless ϕ2N (N=1,2,3,...) quantum field theory was studied. The first two Taylor coefficients in this dimensional expansion were calculated \it exactly and a set of graphical rules were devised that could be used to calculate approximately the higher coefficients in the series. This approach is mathematically valid and gives accurate results, but it has not been actively pursued and investigated. Subsequently, in 1998 Bender and Boettcher discovered that PT-symmetric quantum-mechanical Hamiltonians of the form H=p2+x2(ix)ε, where ε≥0, have real spectra. These new kinds of complex non-Dirac-Hermitian Hamiltonians define physically acceptable quantum-mechanical theories. This result in quantum mechanics suggests that the corresponding non-Dirac-Hermitian D-dimensional ϕ2(iϕ)εquantum field theories might also have real spectra. To examine this hypothesis, we return to the technique devised in 1992 and in this paper we calculate the first two coefficients in the dimensional expansion of the ground-state energy of this complex non-Dirac-Hermitian quantum field theory. We show that to first order in this dimensional approximation the ground-state energy is indeed real for ε≥0.