Making Sense of Non-Hermitian Hamiltonians

TL;DR

This work asks whether Dirac Hermiticity is essential for physical quantum theories or if PT symmetry suffices. By developing a complete PT-symmetric framework, it constructs a consistent Hilbert space via the C operator and a CP T inner product, enabling real spectra and unitary evolution for non-Hermitian Hamiltonians. It demonstrates this across quantum-mechanical models (e.g., i x^3, -x^4) and quantum-field-theoretic contexts (Lee model, iφ^3, -gφ^4), including a detailed treatment of the C operator and mappings to equivalent Hermitian theories. The results reveal a rich class of physically viable theories with potential implications for the Higgs sector, complex crystals, and beyond, suggesting a vast landscape of complex-domain physics that remains consistent with fundamental principles.

Abstract

The Hamiltonian H specifies the energy levels and time evolution of a quantum theory. A standard axiom of quantum mechanics requires that H be Hermitian because Hermiticity guarantees that the energy spectrum is real and that time evolution is unitary (probability-preserving). This paper describes an alternative formulation of quantum mechanics in which the mathematical axiom of Hermiticity (transpose + complex conjugate) is replaced by the physically transparent condition of space-time reflection (PT) symmetry. If H has an unbroken PT symmetry, then the spectrum is real. Examples of PT-symmetric non-Hermitian quantum-mechanical Hamiltonians are H=p^2+ix^3 and H=p^2-x^4. Amazingly, the energy levels of these Hamiltonians are all real and positive! In general, if H has an unbroken PT symmetry, then it has another symmetry represented by a linear operator C. Using C, one can construct a time-independent inner product with a positive-definite norm. Thus, PT-symmetric Hamiltonians describe a new class of complex quantum theories having positive probabilities and unitary time evolution. The Lee Model is an example of a PT-symmetric Hamiltonian. The renormalized Lee-model Hamiltonian has a negative-norm "ghost" state because renormalization causes the Hamiltonian to become non-Hermitian. For the past 50 years there have been many attempts to find a physical interpretation for the ghost, but all such attempts failed. Our interpretation of the ghost is simply that the non-Hermitian Lee Model Hamiltonian is PT-symmetric. The C operator for the Lee Model is calculated exactly and in closed form and the ghost is shown to be a physical state having a positive norm. The ideas of PT symmetry are illustrated by using many quantum-mechanical and quantum-field-theoretic models.

Figures (26)

• Figure 1: Energy levels of the Hamiltonian $H=\hat{p}^2+\hat{x}^2(i\hat{x})^\epsilon$ as a function of the real parameter $\epsilon$. There are three regions: When $\epsilon\geq0$, the spectrum is real and positive and the energy levels rise with increasing $\epsilon$. The lower bound of this region, $\epsilon=0$, corresponds to the harmonic oscillator, whose energy levels are $E_n=2n+1$. When $-1<\epsilon<0$, there are a finite number of real positive eigenvalues and an infinite number of complex conjugate pairs of eigenvalues. As $\epsilon$ decreases from $0$ to $-1$, the number of real eigenvalues decreases; when $\epsilon\leq-0.57793$, the only real eigenvalue is the ground-state energy. As $\epsilon$ approaches $-1^+$, the ground-state energy diverges. For $\epsilon \leq-1$ there are no real eigenvalues. When $\epsilon\geq0$, the $\mathcal{P}\mathcal{T}$ symmetry is unbroken, but when $\epsilon<0$ the $\mathcal{P}\mathcal{T}$ symmetry is broken.
• Figure 2: Stokes wedges in the complex-$x$ plane containing the contour on which the eigenvalue problem for the differential equation (\ref{['e24']}) for $\epsilon= 2.2$ is posed. In these wedges $\psi(x)$ vanishes exponentially as $|x|\to \infty$. The eigenfunction $\psi(x)$ vanishes most rapidly at the centers of the wedges.
• Figure 3: Energy levels of the Hamiltonian $H=p^2+|x|^P$ as a function of the real parameter $P$. This figure is similar to Fig. \ref{['f1']}, but the eigenvalues do not pinch off and go into the complex plane because the Hamiltonian is Hermitian. (The spectrum becomes dense at $P=0$.)
• Figure 4: Stokes wedges in the lower-half complex-$x$ plane for the Schrödinger equation (\ref{['e39']}) arising from the Hamiltonian $H$ in (\ref{['e37']}). The eigenfunctions of $H$ decay exponentially as $|x|\to\infty$ inside these wedges. Also shown is the contour in (\ref{['e41']}).
• Figure 5: Potential of the Hermitian Hamiltonian (\ref{['e55']}) plotted as a function of the real variable $x$ for the case $\epsilon=0$ and $g=0.046$. The energy levels are indicated by horizontal lines. Because $\epsilon=0$, there is no anomaly and the double-well potential is symmetric. Therefore, the mass gap is very small and thus there are no bound states.