0912.2635 (Philip D. Mannheim)
Philip D. Mannheim
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.
View original:
http://arxiv.org/abs/0912.2635
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