Physics – High Energy Physics – High Energy Physics - Theory
Scientific paper
2009-12-14
Physics
High Energy Physics
High Energy Physics - Theory
latex, 24 pages. Extended and updated version of first version. Prepared for a theme issue in Philosophical Transactions A of
Scientific paper
While Hermiticity of a time-independent Hamiltonian leads to unitary time evolution, in and of itself, the requirement of Hermiticity is only sufficient for unitary time evolution. In this paper we provide conditions that are both necessary and sufficient. Specifically, we show that $PT$ symmetry of a time-independent Hamiltonian, or equivalently, reality of the secular equation that determines its eigenvalues, is both necessary and sufficient for unitary time evolution. For any $PT$-symmetric Hamiltonian $H$ there always exists an operator $V$ that relates $H$ to its Hermitian adjoint according to $VHV^{-1}=H^{\dagger}$. When the energy spectrum of $H$ is complete, Hilbert space norms $<\psi_1|V|\psi_2>$ constructed with this $V$ are always preserved in time. With the energy eigenvalues of a real secular equation being either real or appearing in complex conjugate pairs, we thus establish the unitarity of time evolution in both cases. We establish the unitarity of time evolution for Hamiltonians whose energy spectra are not complete. We show that when the energy eigenvalues of a Hamiltonian are real and complete the operator $V$ is a positive Hermitian operator, which has an associated square root operator that can be used to bring the Hamiltonian to a Hermitian form. We show that systems with $PT$-symmetric Hamiltonians obey causality. We note that Hermitian theories are associated with a path integral quantization prescription in which the path integral measure is real, while in contrast non-Hermitian but $PT$-symmetric theories are associated with path integrals in which the measure needs to be complex, but in which the Euclidean time continuation of the path integral is real. We show that $PT$ symmetry generalizes to higher-derivative field theories the Pauli-Weisskopf prescription for stabilizing against transitions to states with negative frequency.
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