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Abstract: While Hermiticity of a time-independent Hamiltonian leads to unitary timeevolution, in and of itself, the requirement of Hermiticity is only sufficientfor unitary time evolution. In this paper we provide conditions that are bothnecessary and sufficient. We show that $PT$ symmetry of a time-independentHamiltonian, or equivalently, reality of the secular equation that determinesits eigenvalues, is both necessary and sufficient for unitary time evolution.For any $PT$-symmetric Hamiltonian $H$ there always exists an operator $V$ thatrelates $H$ to its Hermitian adjoint according to $VHV^{-1}=H^{\dagger}$. Whenthe 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 energyeigenvalues of a real secular equation being either real or appearing incomplex conjugate pairs, we thus establish the unitarity of time evolution inboth cases. We also establish the unitarity of time evolution for Hamiltonianswhose energy spectra are not complete. We show that when the energy eigenvaluesof a Hamiltonian are real and complete the operator $V$ is a positive Hermitianoperator, which has an associated square root operator that can be used tobring the Hamiltonian to a Hermitian form. We show that systems with$PT$-symmetric Hamiltonians obey causality. We note that Hermitian theories areordinarily associated with a path integral quantization prescription in whichthe path integral measure is real, while in contrast non-Hermitian but$PT$-symmetric theories are ordinarily associated with path integrals in whichthe measure needs to be complex, but in which the Euclidean time continuationof the path integral is nonetheless real. We show that through $PT$ symmetrythe fourth-order derivative Pais-Uhlenbeck theory can be stabilized againsttransitions to states negative frequency.

Autor: Philip D. Mannheim


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