Pseudo-Hermiticity in Quantum Physics

• Pseudo-Hermiticity is a generalization of Hermiticity defined by an invertible Hermitian metric, ensuring that operator spectra are real or come in complex conjugate pairs.
• It underpins the analysis of PT-symmetric and nonunitary systems in quantum mechanics, facilitating spectral similarity transformations and safeguarding generalized conservation laws.
• Applications span electromagnetic waveguides, quantum field theories, and pseudo-entropy in quantum information, offering practical insights into exceptional points and spectral instabilities.

Pseudo-Hermiticity is a generalization of Hermiticity in operator theory, with a central role in modern quantum physics, non-Hermitian quantum field theory, spectral theory, and the analysis of open systems. An operator $A$ is called pseudo-Hermitian if there exists an invertible Hermitian operator $\eta$ such that $A^\dagger = \eta A \eta^{-1}$. This classically encompasses both Hermitian and many non-Hermitian operators, providing a spectral structure that allows for real eigenvalues or conjugate pairs and forms the spectral and dynamical foundation for regimes with nonunitary, non-Hermitian, or PT-symmetric dynamics in quantum, optical, and field-theoretic contexts.

1. Mathematical Definition and Spectral Structure

A linear operator $A$ is called (strongly) $\eta$-pseudo-Hermitian if there exists an invertible Hermitian operator $\eta$ (the metric or intertwining operator) for which

$A^\dagger = \eta A \eta^{-1}.$

When $\eta$ is indefinite, $A$ is pseudo-Hermitian, and if $\eta$ is positive-definite, $A$ is quasi-Hermitian. This generalizes the standard notion of Hermiticity (the special case $\eta = I$). The definition guarantees that if $A$ is diagonalizable, its spectrum is real or comes in complex conjugate pairs. Even for nondiagonalizable operators, spectra display this symmetric structure, and under certain conditions generalized completeness relations remain valid (Mostafazadeh, 2010, Mostafazadeh et al., 2010, Gavrilik et al., 2015).

The eigenvalue problem for pseudo-Hermitian $A$ parallels Hermitian theory via a similarity transformation:

$h = \eta^{1/2} A \eta^{-1/2}$

which, for quasi-Hermitian $A$, is Hermitian. This underlies the spectral reality of PT-symmetric and more general non-Hermitian, antilinear symmetric systems (Zhang et al., 2018, Zhang et al., 2019).

2. Symmetry, PT-Symmetry, and Pseudo-Hermiticity

Every finite-dimensional PT-symmetric operator is necessarily pseudo-Hermitian, whether or not it is diagonalizable (Zhang et al., 2018, Zhang et al., 2019). PT-symmetry requires $PTH = HP T$, with $P$ a linear and $T$ an anti-linear operator (often complex conjugation). If $H$ is PT-symmetric, there always exists a Hermitian, invertible $\eta$ making $H$ pseudo-Hermitian. The spectral instabilities and PT-symmetry breaking correspond precisely to resonance (collision) between eigenmodes of opposite Krein signature (action), as captured by the generalized Krein product $\langle\psi | \eta | \psi\rangle$. This relation extends to pseudo-Hermitian (also called G-Hamiltonian) systems regardless of diagonalizability (Zhang et al., 2018, Zhang et al., 2019).

PT-symmetry breaking, associated with exceptional points, does not occur when eigenmodes of the same signature coincide—only when positive- and negative-signature eigenmodes resonate. This unifies the physical understanding of PT-symmetry breaking across quantum, optical, and classical systems.

3. Physical Applications: Wave Propagation and Field Theories

a. Electromagnetic and Waveguide Systems

Maxwell's equations in inhomogeneous, dispersive, or anisotropic media can be recast so the generator of dynamics (for example, $$\Omega^2 = \epsilon^{-1} \mathcal{D} \mu^{-1} \mathcal{D}$$) is pseudo-Hermitian (Mostafazadeh, 2010, Mostafazadeh et al., 2010, Chen et al., 2016). The similarity transformation to a Hermitian operator gives a basis for eigenfunction expansion, even in the presence of non-Hermitian permittivity or permeability.

In waveguide mode evolution, Hamiltonians generating transfer matrices are pseudo-Hermitian (with respect to the flux operator) and often also anti-PT symmetric. This produces structured eigenvalue quartets (real, imaginary, or complex conjugate pairs), which govern modal transitions and are directly observable in scattering and transmission spectroscopy (Chen et al., 2016, Loran et al., 2021, Jin, 2022).

b. Quantum Field Theory: Pseudo-Reality and Gauge/Gravity Coupling

Pseudo-Hermiticity is foundational for constructing consistent non-Hermitian quantum field theories. The notion of "pseudo-reality" for quantum fields requires the field satisfy a duality relation, $\phi^d(x) = \phi(x)$, where $\phi^d(x) = \eta \phi^\dagger(x) \eta^{-1}$ (Chernodub et al., 15 Jan 2025). This resolves the Hermiticity puzzle: ensuring correct transformation under Poincaré symmetry and unifying analytic continuation approaches with first-principle constructions. Minimal coupling to gauge fields and gravity can be implemented by requiring the gauge and metric fields themselves be pseudo-real, which preserves local gauge invariance and consistency of the stress-energy tensor (Chernodub et al., 15 Jan 2025).

In cosmology, pseudo-Hermitian field theories with (e.g.) two complex scalar fields and broken antilinear symmetry exhibit unique dynamical features—such as generation of accelerated expansion even in the absence of vacuum energy, due to competition between instability induced by a complex spectrum and Hubble damping. The result is a sustained, nontrivial cosmological evolution fundamentally tied to the pseudo-Hermitian structure (Copeland et al., 23 Jul 2025).

4. Quantum Information: Pseudo-Entropy and Entanglement Structure

Pseudo-Hermitian transition matrices arise in the formulation of "pseudo-entropy" (notably, pseudo-Rényi entropy), a generalization of entanglement entropy based on non-Hermitian reduced transition matrices (Guo et al., 2022, Guo et al., 2023). The requirement of pseudo-Hermiticity for these matrices ensures that their eigenvalues either are real or appear in complex conjugate pairs; thus, the logarithmic term of the pseudo-entropy, which captures nonlocal correlation structure, is real whenever pseudo-Hermiticity holds.

Explicitly, for transition matrices $T_A^{(O|O')}$, pseudo-Hermiticity with respect to a suitable metric $\eta$ guarantees reality or complex-conjugate pairing in spectra; thus, the logarithmic entropic terms are real. In 2d CFTs, the universal divergence in the second pseudo-Rényi entropy near the lightcone is controlled by the conformal dimension $h$ of the operator, with

$\Delta S^{(2)} \sim 2 h \log |u|$

where $u$ is a suitable kinematic invariant (Guo et al., 2023). For $n\geq3$, the divergence structure reflects theory-dependent details via the OPE.

Modular theory (Tomita–Takesaki) naturally identifies the relevant $\eta$ for transition matrices built from wedge-localized or parity-mirrored operators, yielding a modular operator $\Delta_\Omega^{1/2}$ acting as the metric (Guo et al., 2022). This framework explains observed reality and positivity properties of pseudo-entropies in QFT settings.

5. Conservation Laws and Symmetry Protection

In scattering theory, pseudo-Hermiticity protects generalized conservation laws even in non-Hermitian setups. While standard energy conservation ($|r|^2 + |t|^2 = 1$) fails, pseudo-Hermiticity enforces an energy-difference conservation law:

$\bar r^* r + \bar t^* t = 1$

where $r, t$ (and $\bar r, \bar t$) are reflection and transmission amplitudes for a non-Hermitian system and its Hermitian conjugate. This law is protected provided the system is pseudo-Hermitian, and fails in its absence—for example, in anti-PT-symmetric systems without pseudo-Hermiticity (Xu et al., 2023). In unitary scattering across generic platforms (optics, mesoscopic transport, quantum networks), block-structured pseudo-Hermiticity (with Hermitian subblocks at port connections) ensures that post-scattering amplitudes satisfy $S S^\dagger = I$ (Jin, 2022).

6. Pseudo-Hermitian Random Matrix Models and Krein Spaces

Pseudo-Hermitian random matrix models employ ensembles of matrices Hermitian with respect to an indefinite metric, generalizing Wigner-Dyson classes to PT-symmetric or more general non-Hermitian scenarios (Feinberg et al., 2021). The spectral theory is governed by the signature of the metric: a finite fraction of eigenvalues are real (condensed along the real axis), and the remainder come in complex-conjugate pairs forming symmetric blobs in the complex plane. This framework is natural from the perspective of Krein space theory, where indefinite inner products underpin the structure, and forms the foundation for the statistics of open, disordered, and PT-symmetric quantum systems.

7. Outlook and Broader Implications

Pseudo-Hermiticity, and its recent extension via pseudo-reality, provides a robust algebraic and geometric foundation for the extension of quantum mechanics and field theory into non-Hermitian, antilinear symmetric, and open regimes while preserving core physical features such as spectral reality, unitarity in generalized inner products, and symmetry-protected conservation laws. It unifies and systematizes the paper of exceptional points, dynamical instabilities, nonlocal entanglement properties, and dynamics in both flat and dynamical spacetimes. Ongoing research addresses further generalizations (e.g., to weak pseudo-Hermiticity, quantum statistical mechanics, higher-rank symmetry-protected topological phases, and gravitational coupling), as well as computational and experimental realizations in photonic, condensed matter, and cosmological systems.