Pseudo-Hermitian Physics

• Pseudo-Hermiticity is a framework where non-Hermitian Hamiltonians are reformulated via a metric operator, ensuring real spectral properties and unitarity.
• It employs biorthogonal eigenbases and canonical transformations to establish classical-quantum correspondence and facilitate novel experimental applications.
• The paradigm underpins advances in quantum field theory, magnonics, and sensing by bridging traditional Hermitian and extended non-Hermitian physics.

Pseudo-Hermitian physics encompasses the study of non-Hermitian operators—particularly Hamiltonians—that, despite lacking Hermiticity in the standard inner product, may still exhibit real spectra and support unitary evolution when equipped with a suitably chosen, generally nontrivial, inner product. Central to this field is the construction and application of a positive-definite or indefinite metric operator that redefines the inner product space, thereby enabling a broader class of quantum, classical, and statistical models with robust physical and mathematical properties.

1. Fundamentals of Pseudo-Hermiticity

A linear operator $H$ on a Hilbert space $\mathcal{H}$ is called pseudo-Hermitian if there exists a bounded, invertible, Hermitian metric operator $\eta$ such that

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

If $\eta$ can be chosen positive-definite, $H$ is termed quasi-Hermitian. In this case, the inner product $$\langle \psi | \phi \rangle_\eta = \langle \psi | \eta | \phi \rangle$$ renders $H$ self-adjoint, ensuring a real spectrum and conservation of probability under time evolution

$U(t) = e^{-iHt},\qquad U^\dagger \eta U = \eta\,.$

Pseudo-Hermiticity generalizes Hermiticity (the special case $\eta=I$) and forms the mathematical basis for PT-symmetric quantum mechanics, where the metric can be the parity-time operator ($\eta=PT$), but encompasses significantly broader phenomena (0810.5643, Berakdar et al., 2 Jan 2026).

A structurally important aspect is that if a physical symmetry group (e.g., the Poincaré group in relativistic QFT) is present and the Hamiltonian is only pseudo-Hermitian, all symmetry generators must likewise be pseudo-Hermitian with respect to the chosen metric operator, maintaining the closure and invariance of the associated algebra (Sablevice et al., 2023).

2. Canonical Models, Biorthogonality, and Classical–Quantum Correspondence

In pseudo-Hermitian systems, eigenfunctions occur in right and left sets, $\{ |R_n\rangle \}$ and $\{ |L_n\rangle \}$, satisfying biorthogonality ($\langle L_m | R_n \rangle = \delta_{mn}$). The metric operator $\eta$ can be constructed via the biorthogonal basis, and observables are diagonalizable in these bases. The Petermann factor and phase rigidity quantify the non-orthogonality of eigenvectors, with divergences near exceptional points (EPs) where eigenvectors coalesce (Berakdar et al., 2 Jan 2026).

For two-level systems driven by complex fields (e.g., generalized Rabi problems or damped spin precession), pseudo-Hermitian Hamiltonians can be mapped via canonical transformations to real-field systems, and quantized such that the metric-induced isometry connects different realizations. The classical–quantum correspondence is preserved, with Bloch equations under the modified metric precisely matching classical pseudoclassical trajectories, provided the pseudo-Hermiticity condition holds (e.g., for certain submanifolds of complex parameters) (Raimundo et al., 2020).

3. Quantum Field Theory and Pseudo-Hermitian Extensions

Pseudo-Hermiticity provides a structural foundation for extending quantum field theories (QFTs) beyond the Hermitian paradigm. By requiring all generators of the proper Poincaré group (momentum $P^\mu$ and Lorentz $J^{\mu\nu}$) to satisfy pseudo-Hermiticity with respect to a common metric $\eta$, one can construct fully consistent relativistic QFTs (Sablevice et al., 2023). This enforces that field operators and their duals transform appropriately under the dual representation, allowing Lorentz- and gauge-invariant Lagrangian densities that are non-Hermitian in the canonical inner product but Hermitian in the $\eta$-inner product.

Recent developments on "pseudo-reality" extend these notions by introducing a field-theoretic condition $\phi^\sharp(x) = \phi(x)$, where the η-dual (or "sharp") is defined as $\phi^\sharp(x) = \eta^{-1} \phi^\dagger(x) \eta$. This principle unifies analytic-continuation approaches (e.g., $g \to ig$ in coupling constants) with first-principles constructions, ensures correct gauge and gravitational couplings, and resolves longstanding issues such as the Hermiticity Puzzle in non-Hermitian QFTs (Chernodub et al., 15 Jan 2025).

Scattering theory in pseudo-Hermitian QFT must distinguish in/out sectors with generally distinct metrics, linked by a metric projector. Probability conservation and pseudo-unitarity are maintained globally, with the S-matrix constructed via η-Hermitian interaction picture and transformed under Lorentz and discrete symmetries implemented with two sector-dependent pseudo-unitary representations. The CPT theorem extends to this setting without modification (Leng et al., 31 Oct 2025).

4. Exceptional Points, Spectral Structure, and Symmetry Classes

Pseudo-Hermitian Hamiltonians can exhibit exceptional points (EPs), parameter values where two or more eigenvalues and their associated eigenvectors coalesce, leading to non-diagonalizable (Jordan block) structures. The algebraic-geometric structure of these EPs is described using the discriminant of the characteristic polynomial, and their presence is associated with rich phenomena such as spontaneous $\mathcal{PT}$-symmetry breaking (when real eigenvalues merge and become complex pairs) (Barnett, 2023).

Symmetry-based constructions (e.g., anticommuting matrix pencils) and representation-theoretic approaches allow the explicit construction of broad classes of pseudo-Hermitian operators—including in lattice and block models—whose spectral properties, locality, and PT-breaking regimes can be controlled analytically (Barnett, 2023, Srivastava et al., 2013).

5. Physical Applications and Experimental Realizations

Pseudo-Hermitian frameworks have proliferated across physics:

• Quantum channels: The natural representation of completely positive trace-preserving maps leads to pseudo-Hermitian operators whose spectra encode open-system dynamics, enabling robust Hamiltonian parameter estimation through sequential measurement statistics (Jin et al., 14 Sep 2025).
• Magnonics and spin systems: Coupled macrospins, magnonic crystals, and hybrid devices with engineered gain/loss, spin-orbit torques, or cavity coupling, realize pseudo-Hermitian effective Hamiltonians. Consequences include mode amplification, non-reciprocal propagation, magnon cloaking, non-Hermitian skin effect, enhanced sensitivity near EPs, and PT-assisted Floquet engineering (Berakdar et al., 2 Jan 2026, Connick et al., 26 Feb 2025).
• Wireless power transfer: Multi-port circuits described by symmetric tridiagonal pseudo-Hermitian Hamiltonians display robust, frequency-locked modes and exceptional point–induced abrupt frequency transitions, enabling efficiency and sensing capabilities over a wide parameter range beyond standard PT symmetry (Hao et al., 2022).
• Sensing and metrology: Pseudo-Hermitian sensors (e.g., qubit systems realized via Naimark dilation) exhibit enhanced susceptibility to external fields, with divergent response near coalescence of eigenvectors but without spectral EPs, offering resilience to certain classes of technical noise (Chu et al., 2019).
• Field-theoretic topological defects: In pseudo-Hermitian field theories, vortex defects can locally break underlying antilinear symmetry (e.g., PT symmetry) in their cores, resulting in metastable modes and nontrivial exceptional-point structures, relevant for the decay of cosmological or condensed-matter defect networks (Battye et al., 18 Nov 2025).
• Random matrix theory and statistical mechanics: Pseudo-Hermitian random matrix ensembles (with invariance under pseudo-unitary transformations) differ in level-spacing statistics, exceptional point structure, and application domains (e.g., non-equilibrium statistical mechanics, quantum chaos) from conventional Hermitian classes (Srivastava et al., 2013).

6. Mathematical Techniques and Computational Frameworks

The calculation of metric operators $\eta$ is a central concern: spectral decompositions, perturbative Baker–Campbell–Hausdorff expansions, Lie-algebraic constructions, and differential equations for the kernel $\eta(x,y)$ are prominent techniques (0810.5643, Mehri-Dehnavi et al., 2010, Lan et al., 2024). The spectral collocation method using pseudo-Hermitian Chebyshev differential matrices demonstrates that even non-Hermitian discretizations can converge correctly to physical spectra, provided η-orthogonality is enforced (Lan et al., 2024). Dynamic (time-dependent) generalizations require metric compatibility, leading to generalized vielbein (frame) formalism that transforms the non-Hermitian system to an explicitly Hermitian one, at the expense of time-dependence (Ju et al., 2021).

7. Outlook and Open Problems

Future directions involve extension to non-Abelian gauge theories with pseudo-real gauge fields, pseudo-Hermitian supersymmetry, systematic classification of pseudo-reality maps and higher-spin field representations, and exploration of emergent phenomena in large-scale non-Hermitian many-body systems. Open challenges remain in optimizing construction of metric operators for numerical simulations, establishing complete path-integral formulations, and identifying experimental signatures beyond conventional Hermitian scenarios (Leng et al., 31 Oct 2025, Chernodub et al., 15 Jan 2025, 0810.5643).

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Key References

• (0810.5643) "Pseudo-Hermitian Representation of Quantum Mechanics"
• (Sablevice et al., 2023) "Poincaré symmetries and representations in pseudo-Hermitian quantum field theory"
• (Leng et al., 31 Oct 2025) "Pseudo-Hermitian QFT: relativistic scattering and symmetry structure"
• (Chernodub et al., 15 Jan 2025) "Pseudo-real quantum fields"
• (Berakdar et al., 2 Jan 2026) "Pseudo-Hermitian Magnon Dynamics"
• (Barnett, 2023) "Locality and Exceptional Points in Pseudo-Hermitian Physics"
• (Connick et al., 26 Feb 2025) "Pseudo-Hermitian physics from dynamically coupled macrospins"
• (Chu et al., 2019) "Quantum sensing with a single-qubit pseudo-Hermitian system"
• (Lan et al., 2024) "Pseudo-hermitian Chebyshev differential matrix and non-Hermitian Liouville quantum mechanics"
• (Hao et al., 2022) "Frequency-stable robust wireless power transfer based on high-order pseudo-Hermitian physics"
• (Srivastava et al., 2013) "Pseudo-Hermitian random matrix theory"
• (Battye et al., 18 Nov 2025) "Vortex stability in pseudo-Hermitian theories"
• (Ju et al., 2021) "Flattening the Curve with Einstein's Quantum Elevator: Hermitization of Non-Hermitian Hamiltonians via a Generalized Vielbein Formalism"
• (Raimundo et al., 2020) "Classical-quantum correspondence for two-level pseudo-Hermitian systems"
• (Jin et al., 2011) "Hermitian dynamics in a class of pseudo-Hermitian networks"