Real-to-Hermitian Taxonomy Overview

• Real-to-Hermitian Taxonomy is a layered classification that organizes structures based on real conditions, Hermitian properties, and intermediate compatibility criteria.
• It unifies diverse domains such as non-Hermitian quantum theory, complex geometry, and tensor analysis by leveraging PT-symmetry, pseudo-Hermiticity, and dynamic metrics.
• The taxonomy refines traditional dichotomies by incorporating interfaces, boundary conditions, and explicit correspondence theorems to bridge ‘real’ and Hermitian frameworks.

Searching arXiv for the cited papers to ground the response. Search query: arXiv (Znojil, 2017) Hermitian non-Hermitian interfaces in quantum theory “Real-to-Hermitian Taxonomy” denotes a family of classification problems in which one organizes mathematical or physical structures by how “real” conditions, Hermitian structures, and intermediate compatibility notions are related. In the literature represented here, this taxonomy does not appear as a single universal theorem. Rather, it appears as a recurrent pattern across several domains: non-Hermitian quantum theory with real spectra, PT symmetry and pseudo-Hermiticity, boundary-condition classifications of differential operators, Hermitian curvature notions on complex manifolds, and higher-order Hermitian tensor forms. The common theme is that manifest Hermiticity is often only one distinguished subclass inside a broader landscape controlled by positivity, antiunitary or metric symmetries, admissible domains, or canonical Hermitian representatives (Znojil, 2017).

1. Scope of the taxonomy

In operator theory and mathematical physics, the basic taxonomic question is whether “Hermitian” and “non-Hermitian but physically admissible” define disjoint classes. Several of the cited works answer negatively. Znojil formulates the problem in terms of two representations of the same unitary dynamics: the textbook Hermitian Schrödinger picture with $\mathfrak h=\mathfrak h^\dagger$ on ${\cal H}^{(textbook)}$, and a non-Hermitian Schrödinger picture with $H$ acting on an auxiliary space ${\cal H}^{(unphysical)}$, related by a non-unitary Dyson map $\Omega$ and metric $\Theta=\Omega^\dagger\Omega$ (Znojil, 2017). The graph-theoretic work of Znojil then separates auxiliary-space pseudo-Hermitization, based on an indefinite pseudometric ${\cal P}$, from physical Hilbert-space Hermitization based on a positive metric $\Theta$ (Znojil, 2011). Finite-dimensional PT-symmetric classifications broaden the picture further by showing that Hermitian matrices can occur as special cases of PT-symmetric or generalized PT-symmetric classes, rather than as a separate alternative (Wang et al., 2010, Deng et al., 2012).

A closely related line of work shifts the organizing principle from Hermiticity to antilinearity. Mannheim argues that the decisive structure behind time-independent inner products and the spectral pattern “real or complex-conjugate paired” is not Hermiticity itself but an antilinear symmetry, realized as PT in nonrelativistic settings and as CPT under the paper’s relativistic assumptions (Mannheim, 2015). This suggests that a real-to-Hermitian taxonomy is often better understood as a hierarchy from raw “real-spectrum” or “real-structured” data, through auxiliary symmetries or metrics, to full Hermitian or self-adjoint realizations.

Outside operator theory, analogous taxonomies appear in complex geometry and tensor analysis. Real bisectional curvature was introduced precisely to refine holomorphic sectional curvature in the non-Kähler Hermitian setting, where the Kähler symmetries are absent and the usual scalar curvature notions no longer organize the geometry adequately (Yang et al., 2016). In tensor algebra, degree-$k$ $t$-Hermitian forms provide a structured Hermitian lift of tubal and Fourier-slice data, with canonical Hermitian tensor representatives and slicewise classical Hermitian forms (Dobes, 24 Feb 2026). A plausible implication is that “Real-to-Hermitian Taxonomy” names a recurring strategy: replace a naive dichotomy by a layered classification in which real, symmetric, antiunitary, indefinite-metric, and Hermitian structures are connected by explicit correspondence theorems.

2. Operator-theoretic classes: Hermitian, pseudo-Hermitian, PT-symmetric, and generalized PT-symmetric

The most developed version of the taxonomy occurs for finite-dimensional or discretized Hamiltonians. A standard Hermitian Hamiltonian satisfies ${\cal H}^{(textbook)}$0. A more general self-adjointness notion uses a positive metric ${\cal H}^{(textbook)}$1 such that

${\cal H}^{(textbook)}$2

which makes ${\cal H}^{(textbook)}$3 self-adjoint in the inner product ${\cal H}^{(textbook)}$4 (Deng et al., 2012). In this sense, Hermitian is a special case with ${\cal H}^{(textbook)}$5, while self-adjointness with respect to a dynamic metric is broader.

Pseudo-Hermiticity replaces the positive metric by an indefinite Hermitian involution ${\cal H}^{(textbook)}$6, yielding

${\cal H}^{(textbook)}$7

In the graph models of Znojil, the corresponding object ${\cal H}^{(textbook)}$8 yields self-adjointness only in an “ad hoc Krein or, more precisely, Pontryagin space,” and therefore does not yet provide a standard probabilistic interpretation (Znojil, 2011). The decisive upgrade occurs only when one has a positive-definite metric ${\cal H}^{(textbook)}$9 satisfying

$H$0

so that $H$1 is Hermitian in the redefined physical Hilbert space (Znojil, 2011).

PT symmetry gives a different but overlapping classification. In the matrix taxonomy of Gong and Wang, ordinary PT symmetry is defined by

$H$2

with $H$3 a real involution and $H$4 complex conjugation, while generalized PT symmetry keeps only the combined antiunitary involution $H$5 satisfying $H$6 and

$H$7

Their central structural statement is that generalized PT-symmetric matrices are the broadest class among ordinary PT symmetry, $H$8-pseudo-Hermiticity, and generalized PT symmetry, and that all self-adjoint matrices possess generalized PT symmetry (Deng et al., 2012). Zhang, Song, and Ma then sharpen the low-dimensional picture by showing that in their formulation every Hermitian matrix is a special case of a PT-symmetric matrix, and that for $H$9 matrices the most general PT-symmetric Hamiltonian coincides with the most general matrix Hamiltonian having only real eigenvalues (Wang et al., 2010).

The 2024 classification of ensembled ${\cal H}^{(unphysical)}$0 pseudo-Hermitian and PT-symmetric matrices makes the equivalence more explicit in that dimension. For invertible Hermitian ${\cal H}^{(unphysical)}$1, the condition

${\cal H}^{(unphysical)}$2

defines ${\cal H}^{(unphysical)}$3-pseudo-Hermiticity, and the paper proves that all ${\cal H}^{(unphysical)}$4 ${\cal H}^{(unphysical)}$5-pseudo-Hermitian matrices are PT-symmetric; indeed, in ${\cal H}^{(unphysical)}$6 invertible-metric pseudo-Hermiticity and PT symmetry coincide (Abraham et al., 2024). The same work partitions PT-symmetric matrices into four cells ${\cal H}^{(unphysical)}$7, separating Hermitian, anti-Hermitian-like, scalar, and genuinely non-normal PT-symmetric sectors (Abraham et al., 2024). This supports a taxonomy in which Hermitian operators are not an external class but one geometrically distinguished cell inside a broader PT/pseudo-Hermitian landscape.

3. Interfaces, admissibility domains, and the role of metrics and boundary conditions

A major correction to earlier dichotomies is the claim that the Hermitian and hiddenly Hermitian families need not be disjoint. Znojil formulates an interface criterion through a parameter family ${\cal H}^{(unphysical)}$8 for which both the Hermitian and non-Hermitian descriptions are workable on an interval and the metric tends continuously to the identity,

${\cal H}^{(unphysical)}$9

This turns the overlap

$\Omega$0

into a nonempty region rather than an empty set, provided one allows weakly nonlocal interactions instead of insisting on locality in both pictures (Znojil, 2017). In the explicit $\Omega$1 tridiagonal model, the metric can be chosen diagonal and positive, with entries tending smoothly to $\Omega$2 as the non-Hermiticity parameter tends to zero, giving a constructive example of an interface (Znojil, 2017).

The graph-based Hermitization program refines the same issue by distinguishing real spectrum, existence of a pseudometric $\Omega$3, and existence of a positive metric $\Omega$4. For the $\Omega$5 loop graph Hamiltonians $\Omega$6, a sparse pseudometric exists rather generally, but an explicit positive diagonal metric is constructed only for the symmetric subfamily $\Omega$7 (Znojil, 2011). This yields a concrete ladder

$\Omega$8

with nonuniqueness of $\Omega$9 exhibited by the family $\Theta=\Omega^\dagger\Omega$0 (Znojil, 2011). A plausible implication is that a real-to-Hermitian taxonomy in quantum models is often parameter-domain-based rather than absolute: the same Hamiltonian family may move among real-spectrum, pseudo-Hermitian, quasi-Hermitian, and inadmissible sectors as couplings vary.

Extension theory yields an even sharper version of this domain sensitivity. For the real even limit-circle differential expression

$\Theta=\Omega^\dagger\Omega$1

Curgus, Langer, and Najman classify closed extensions into Hilbert-space self-adjoint, $\Theta=\Omega^\dagger\Omega$2-symmetric, and $\Theta=\Omega^\dagger\Omega$3-self-adjoint classes by explicit boundary conditions at $\Theta=\Omega^\dagger\Omega$4 (Azizov et al., 2011). In the limit-circle case, these notions generally differ. For separated $\Theta=\Omega^\dagger\Omega$5-dimensional extensions, $\Theta=\Omega^\dagger\Omega$6 symmetry and $\Theta=\Omega^\dagger\Omega$7-self-adjointness coincide; for mixed $\Theta=\Omega^\dagger\Omega$8-dimensional extensions, all three notions are distinct and are characterized by explicit matrix constraints (Azizov et al., 2011). The same paper also shows that some $\Theta=\Omega^\dagger\Omega$9-symmetric ${\cal P}$0-dimensional and ${\cal P}$1-dimensional extensions have empty resolvent set, ${\cal P}$2, with ${\cal P}$3 (Azizov et al., 2011). This directly counters the misconception that symmetry alone selects a physically acceptable operator.

A more representation-sensitive caution appears in the 2024 study of the inverted oscillator in an ${\cal P}$4 realization. There a formally Hermitian Hamiltonian is assigned purely imaginary discrete eigenvalues using nonlocalized dual eigenstates and a biorthogonal normalization, while a non-Hermitian interaction is related by a non-unitary but Hermitian similarity transformation to a Hermitian Hamiltonian whose effective spectrum crosses from imaginary to real at a critical coupling identified as an exceptional point (Liu et al., 2024). The paper itself thereby underscores that in any taxonomy of “Hermitian versus real spectrum,” the adjoint operation, state space, and admissible boundary conditions must be specified.

4. Geometry: real bisectional curvature and real connections on Hermitian manifolds

In complex geometry, the taxonomy is not about operators with real spectra but about which “real” curvature or connection notions best refine Hermitian structures. Yang and Zheng introduce real bisectional curvature for a Hermitian manifold ${\cal P}$5 with Chern curvature tensor ${\cal P}$6 by

${\cal P}$7

where ${\cal P}$8 is a unitary frame and ${\cal P}$9 are not all zero (Yang et al., 2016). This curvature notion contains holomorphic sectional curvature as a special case, agrees with it in the Kähler or Kähler-like case, and is strictly stronger in general Hermitian geometry (Yang et al., 2016). The strictness is exhibited by local examples where $\Theta$0 but $\Theta$1, and similarly with negative sign (Yang et al., 2016). The resulting taxonomy places real bisectional curvature between holomorphic sectional curvature and stronger full-tensor positivity conditions.

This strengthened notion supports rigidity theorems not available for $\Theta$2 alone. For compact Hermitian manifolds with constant real bisectional curvature $\Theta$3, Yang and Zheng prove that $\Theta$4, and if $\Theta$5 then the metric is balanced, the first three Chern-Ricci tensors vanish, and the Chern curvature satisfies

$\Theta$6

(Yang et al., 2016). Zheng and collaborators then prove in complex dimension three that vanishing real bisectional curvature forces full Chern flatness: any compact Hermitian threefold with $\Theta$7 is Chern flat (Zhou et al., 2021). This gives a dimension-three closure of the zero-curvature side of the taxonomy: the stronger Hermitian “real” scalarized condition collapses the geometry to the flat model.

A second geometric taxonomy relates real and Hermitian connections. Yang develops a precise dictionary between real $\Theta$8- and $\Theta$9-preserving connections on the real tangent bundle, denoted $k$0, and Hermitian connections on $k$1, denoted $k$2, proving a canonical isomorphism

$k$3

(Wang et al., 2019). Under this correspondence, the real Chern connection

$k$4

is the real counterpart of the usual Chern connection, while the Gauduchon family on the Hermitian side is lifted to a family of real $k$5-connections (Wang et al., 2019). Curvature components are preserved under the correspondence, but the natural real Ricci trace of the real Chern connection corresponds to the third and fourth Chern-Ricci tensors rather than directly to the first (Wang et al., 2019). The paper’s real Chern-Einstein condition is equivalent to

$k$6

and if $k$7 on a compact Hermitian manifold then the metric is in fact Kähler-Einstein (Wang et al., 2019). This suggests that geometric real-to-Hermitian taxonomy is governed not only by tensor identities but also by which contraction and which tangent bundle are regarded as primary.

5. Tensor and multilinear taxonomies

Tensor theory supplies an explicitly algebraic version of the real-to-Hermitian distinction. Wang and collaborators define Hermitian tensors in

$k$8

by the condition

$k$9

and introduce Hermitian rank-1 tensors of the form

$t$0

(Nie et al., 2019). Over $t$1, every Hermitian tensor admits a Hermitian rank decomposition. Over $t$2, this fails in general. The exact characterization is that a real Hermitian tensor is $t$3-Hermitian decomposable if and only if its entries satisfy the pair-swapping symmetry

$t$4

whenever $t$5 for every mode $t$6 (Nie et al., 2019). The decomposable real tensors therefore form a proper subspace of the full real Hermitian tensor space whenever $t$7 and all mode sizes exceed $t$8 (Nie et al., 2019). This is one of the sharpest instances of a real-to-Hermitian taxonomy: complex Hermitian decomposition is universal, whereas real Hermitian decomposition requires an additional modewise symmetry.

A different but related taxonomy is built in third-order tensor $t$9-algebra. Yu, Ling, and coauthors define degree-${\cal H}^{(textbook)}$00 ${\cal H}^{(textbook)}$01-Hermitian forms as tubal analogues of classical Hermitian forms, represented by order-${\cal H}^{(textbook)}$02 ${\cal H}^{(textbook)}$03-Hermitian partially symmetric tensors (Dobes, 24 Feb 2026). Their key structural theorem is a Fourier-slice decomposition: every degree-${\cal H}^{(textbook)}$04 ${\cal H}^{(textbook)}$05-Hermitian form decomposes under the discrete Fourier transform into a ${\cal H}^{(textbook)}$06-tuple of ordinary degree-${\cal H}^{(textbook)}$07 Hermitian forms, and conversely arbitrary collections of classical degree-${\cal H}^{(textbook)}$08 Hermitian forms lift uniquely to a single ${\cal H}^{(textbook)}$09-Hermitian form (Dobes, 24 Feb 2026). In set-theoretic form,

${\cal H}^{(textbook)}$10

Positivity is slicewise: ${\cal H}^{(textbook)}$11-Hermitian positive definiteness is characterized exactly by positivity of the tensor eigenvalues of every Fourier slice, while positivity of matrix eigenvalues after cubically balanced unfolding is only a stronger sufficient condition for a commutant subclass (Dobes, 24 Feb 2026). A plausible implication is that the tensor-algebra version of the taxonomy separates canonical Hermitian representatives, Fourier-slice Hermitian forms, and stronger matrix-flattening positivity notions into a strict hierarchy.

6. Dynamical realizations and broader significance

The taxonomy also appears in dynamical systems where a strictly Hermitian many-body Hamiltonian generates an effectively non-Hermitian dynamical matrix. In a quadratic bosonic chain with four sublattices per cell, the many-body Hamiltonian is Hermitian, but the Heisenberg evolution is governed by the bosonic dynamical matrix ${\cal H}^{(textbook)}$12, which is generically non-Hermitian once pairing is present (Estake et al., 20 Aug 2025). The paper shows that the effective non-Hermitian topology depends on whether the hopping and pairing amplitudes are real or purely imaginary. In the real-parameter regime, the dynamical matrix is unitarily equivalent to four decoupled copies of a sublattice-symmetric non-Hermitian SSH model nSSH2, supporting trivial, nontrivial, and Möbius phases, the last characterized by fractional winding number ${\cal H}^{(textbook)}$13 and having no Hermitian counterpart (Estake et al., 20 Aug 2025). In the purely imaginary-parameter regime, the same Hermitian bosonic architecture instead maps to a different effective non-Hermitian model nSSH1, where the Möbius phase disappears and the system exhibits sublattice-dependent chiral amplification under open boundary conditions (Estake et al., 20 Aug 2025). Here the real-to-Hermitian taxonomy is neither a dichotomy nor a similarity transform, but a selection rule by which microscopic reality conditions on a Hermitian parent choose among different effective non-Hermitian topological classes.

Taken together, these literatures support a broad but technically precise conclusion. “Real-to-Hermitian Taxonomy” does not denote a single canonical classification scheme. It denotes a recurring structural program in which one replaces coarse oppositions—real versus complex, Hermitian versus non-Hermitian, Kähler versus non-Kähler, real tensor versus Hermitian tensor—by explicit intermediate strata. In quantum theory these strata are controlled by positive metrics, pseudometrics, antiunitary symmetries, interfaces, and exceptional-point domains (Znojil, 2017, Mannheim, 2015). In geometry they are controlled by refined real curvature notions and correspondence theorems between real and Hermitian connections (Yang et al., 2016, Wang et al., 2019). In tensor theory they are controlled by canonical Hermitian representatives, Fourier-slice decompositions, and strict real-decomposability conditions (Nie et al., 2019, Dobes, 24 Feb 2026). The cumulative lesson is that Hermitian structure is often not an endpoint opposed to “real” structure, but a distinguished node inside a richer classification architecture.