Pseudo-Hermitian Quantum Mechanics

• Pseudo-Hermitian quantum mechanics is a generalization of conventional quantum theory that employs a redefined inner product using a metric operator to render non-Hermitian Hamiltonians Hermitian.
• It constructs the physical Hilbert space with a positive-definite metric to ensure unitary time evolution and observables with real expectation values.
• The framework extends to incorporate PT symmetry, quasi-Hermiticity, and perturbative techniques for deriving equivalent Hermitian Hamiltonians, impacting diverse areas like condensed matter and quantum field theory.

Pseudo-Hermitian quantum mechanics is a generalization of conventional quantum mechanics that accommodates non-Hermitian Hamiltonians with entirely real spectra through a careful redefinition of the Hilbert space inner product. The central idea is that for a densely defined, diagonalizable operator $H$ acting on a Hilbert space, one may endow the state space with a new inner product via a metric operator $\eta$ such that $H$ becomes Hermitian with respect to this inner product. This ensures unitary time evolution and real-valued expectation values, even when $H \neq H^\dagger$ in the original sense. Pseudo-Hermitian quantum mechanics thereby encompasses and clarifies the structure and physical interpretation of PT-symmetric, quasi-Hermitian, and more general non-Hermitian systems that have drawn significant physical and mathematical interest.

1. Pseudo-inner Products, Metric Operators, and the Distinction from Indefinite-Metric Theories

The foundational object in this framework is a pseudo-inner product, meaning a sesquilinear form $(\psi, \phi)_\eta = (\psi, \eta\phi)$ where $\eta$ is an invertible Hermitian linear operator (the metric operator) and $(\cdot,\cdot)$ is an underlying reference (typically positive-definite) inner product. Conventional Hilbert spaces use positive-definite inner products, enforcing $(\psi, \psi) > 0$ for all nonzero $\psi$. In pseudo-Hermitian theory, positivity of the norm with respect to $\eta$ is not always assumed at first; more generally, $\eta$ may be indefinite, leading to indefinite-metric spaces where $(\psi, \psi)_\eta$ may be negative or zero for nonzero $\psi$.

However, the standard pseudo-Hermitian quantum mechanics program restricts focus to positive-definite choices of $\eta$ when constructing the "physical Hilbert space"—thereby ensuring a unitary time evolution and a probabilistic quantum interpretation. Unlike in indefinite-metric quantum field theories, where the metric is fixed (often indefinite) from the outset (leading to ghosts and negative-probability states), in pseudo-Hermitian settings the metric $\eta$ is treated as a degree of freedom to be determined by the structure of $H$; and ultimately, it is chosen so that $H$ is observable and dynamics are unitary.

Key relations:

• Pseudo-Hermiticity condition:

$H^\dagger = \eta H \eta^{-1}$

• Pseudo-inner product:

$(\psi|\phi)_\eta := (\psi|\eta\phi)$

with $\eta$ Hermitian and invertible (positive-definite for physical models).

• Ordinary Hermiticity is the special case $\eta = 1$.

The notion of the metric operator is thus central: it determines the physical probabilistic content of the theory and allows for many physically equivalent (though kinematically distinct) representations of quantum mechanics (0810.5643).

2. Construction of the Physical Hilbert Space and Observables

Given a non-Hermitian (but diagonalizable) $H$ with real spectrum, one constructs the physical Hilbert space as follows:

• Start with a reference Hilbert space $\mathscr{H}$ and a set of eigenvectors $\{ \psi_n \}$ of $H$.
• Define a linear span $\mathcal{S}$ of the eigenvectors.
• Introduce a positive-definite metric $\eta$, and define a new inner product on $\mathcal{S}$ by $(\phi, \psi)_\eta = (\phi, \eta\psi)$.
• Complete $\mathcal{S}$ with respect to the norm induced by $(\cdot,\cdot)_\eta$ to yield the physical Hilbert space $\mathscr{H}_\mathrm{phys}$.

Observables are required to be self-adjoint with respect to the new inner product; i.e., for any observable $\mathcal{O}$,

$\mathcal{O}^\dagger = \eta \mathcal{O} \eta^{-1}$

with domain chosen such that this relation is meaningful.

The most general method to construct an equivalent Hermitian operator $h$ is via a similarity transformation:

$h = \rho H \rho^{-1},\quad \rho = \sqrt{\eta}$

Here, $h$ acts in a standard Hilbert space with the usual inner product, and $H$ and $h$ are isospectral; the "physical content" of the theory is thus preserved in this Hermitian shadow representation (Mostafazadeh, 2012).

3. Pseudo-Hermitian Hamiltonians, PT-Symmetry, Quasi-Hermiticity, and Antilinear Symmetries

Pseudo-Hermitian quantum mechanics unifies and clarifies the role of PT symmetry (parity–time), quasi-Hermiticity, and other antilinear symmetries. A Hamiltonian $H$ with an antilinear symmetry (e.g., PT symmetry) may possess a real spectrum—even if not Hermitian—provided the symmetry is unbroken.

A Hamiltonian is called quasi-Hermitian if it is pseudo-Hermitian with respect to a positive-definite $\eta$. PT symmetry is a prominent example: many PT-symmetric Hamiltonians are pseudo-Hermitian, with $\mathcal{PT}$ playing the role of an indefinite metric (especially in finite-dimensional settings).

The theory carefully distinguishes between indefinite-metric settings (where "ghost" states arise, as in early treatments of PT-symmetric quantum mechanics) and pseudo-Hermitian frameworks, where a positive-definite $\eta$ is ultimately chosen to restore physicality.

Charge-conjugation-like operators $C$ and the $\mathcal{CPT}$-inner product are also constructed within this framework, further extending the bridge to PT-symmetric quantum mechanics. The theory generalizes to cover pseudo-Hermitian canonical quantization, the treatment of antilinear symmetries, and other extensions (0810.5643).

4. Methods of Constructing Metric Operators and Equivalent Hermitian Hamiltonians

A range of analytic and perturbative methods for constructing the metric operator $\eta$ exist. For many practical systems, particularly in scattering and many-body applications, perturbative approaches are employed.

• Assume $H = H^{(0)} + H^{(1)}$ with $H^{(1)}$ small; decompose $H^{(1)}$ into Hermitian and anti-Hermitian parts.
• Represent $\eta$ as $\exp(-Q)$ with $Q$ Hermitian; require pseudo-Hermiticity $H^\dagger = \eta H \eta^{-1}$ order-by-order in perturbation theory.
• Solve at first order:

$[H^{(0)}, Q^{(1)}] = -2H^{(1)}_\text{a.h.}$

The anti-Hermitian part is thus "compensated" by a suitable $Q$, rendering $H$ Hermitian in the new metric.

After $\eta$ is determined, the equivalent Hermitian Hamiltonian is constructed by a similarity transformation:

$h = \rho H \rho^{-1}$

with explicit expressions for $h$ that contain nonlocal terms (arising from non-Hermiticity in the original $H$). Notably, the nonlocal effects are not necessarily tied to PT symmetry but rather to the magnitude of the anti-Hermitian part (Mehri-Dehnavi et al., 2010).

These tools allow systematic construction of pseudo-Hermitian quantum systems in a broad range of settings, including exactly solvable models, integrable many-particle systems (e.g., Calogero models, XXZ spin chains), and quantum field theory analogs (Ghosh, 2010).

5. Physical Equivalence, Duality of Descriptions, and Classical Correspondence

A central theme is the duality between descriptions via nonlocal Hermitian and local non-Hermitian Hamiltonians. A given physical system with a non-Hermitian $H$ (originally local in the absence of a metric) can always be mapped to a nonlocal Hermitian $h$, with all measurable predictions coinciding under the appropriate mapping of state spaces and observables. This duality is often nontrivial, with observables in one representation appearing nonlocal in the other. For instance, a non-Hermitian, local scattering problem admits an equivalent Hermitian nonlocal description where energy expectation values and other observables match (Mehri-Dehnavi et al., 2010).

The theory extends quantum-to-classical correspondences, path-integral formulations, and geometric frameworks. Classical systems corresponding to pseudo-Hermitian quantum models (in appropriate semiclassical limits or in Heisenberg picture formulations) are also accessible, with care to recognize that the classical observables must be constructed in terms of the "physical" (i.e., metric-modified) inner product (0810.5643).

6. Applications Across Physics and the Structure of the State Space

Pseudo-Hermitian quantum mechanics has concrete applications in condensed matter physics (non-Hermitian tight-binding models, topological systems), nuclear and atomic physics (resonance phenomena, complex scattering), quantum field theory (non-Hermitian extensions, PT-invariant field theories), cosmology (time-dependent metrics in quantum cosmological models), electromagnetic wave propagation (complex refractive index), open quantum systems (gain/loss, Lindblad-like structures), magnetohydrodynamics, and biophysics. Key physical results include the explanation of real spectra in models with balanced gain and loss, description of exceptional points, and phenomena such as the quantum Brachistochrone effect.

A notable structural result is that the geometry of quantum state space is not fixed: distinct choices of metric lead to kinematically distinct, but physically equivalent, Hilbert spaces. This has ramifications for quantum speed limits (Brachistochrone problem), the structure of the space of observables, and possible extensions to time-dependent Hilbert spaces and geometric quantum theory.

In summary, pseudo-Hermitian quantum mechanics provides a mathematically robust and physically transparent extension of conventional quantum theory. It encompasses non-Hermitian Hamiltonians with real spectra, clarifies the role of PT and other antilinear symmetries, resolves long-standing ambiguities in indefinite-metric treatments, and offers powerful tools for both theoretical exploration and modeling complex quantum systems (0810.5643).