Oscillators with imaginary coupling: spectral functions in quantum mechanics and quantum field theory

Bruno W. Mintz; Itai Y. Pinheiro; Rui Aquino

REVIEW 2 major objections 6 minor 105 references

Oscillators with imaginary coupling: spectral functions in quantum mechanics and quantum field theory

T0 review · 2 major / 6 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read The paper shows that two scalar fields coupled by an imaginary bilinear term form a consistent unitary quantum theory in the weak-coupling regime, with real spectrum and positive spectral functions for the physical rotated fields.

desk verdict The central spectral functions use double-angle weights where the inverse rotation gives single-angle weights; the paper is internally inconsistent and not acceptable as written. read the letter →

arxiv 2412.14064 v1 pith:HY42Y2ZW submitted 2024-12-18 quant-ph cond-mat.str-elhep-ph

Bruno W. Mintz , Itai Y. Pinheiro , Rui Aquino This is my paper · ORCID

classification quant-phcond-mat.str-elhep-ph MSC 81Q1281T10 PACS 03.65.-w11.10.-z

keywords pseudo-HermitianquantummechanicsPTsymmetryimaginarycouplingspectralfunctionKallen-Lehmannrepresentationmetricoperatornon-Hermitianfieldtheoryexceptionalpoints

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper establishes that a pair of harmonic oscillators, and its field-theory generalization of two scalar fields, coupled through an imaginary bilinear term $i g a \phi$ is not an inconsistent theory but a pseudo-Hermitian one. In the weak-coupling regime the Hamiltonian is $\eta$-pseudo-Hermitian, so its spectrum is real and time evolution is unitary once probabilities are computed with the $\eta$-inner product. The reason the original fields $\phi$ and $a$ seem to produce pathological spectral functions is that they are not observables; the rotated fields $\Phi$ and $A$, built from the metric operator, are the physical degrees of freedom and have positive spectral functions. The paper concludes that positivity violation in a spectral function is evidence that the operators involved are not self-adjoint with respect to the correct inner product, and conjectures that this is a generic signal of a nontrivial metric, and possibly of a $\mathcal{PT}$-broken phase elsewhere in parameter space.

What carries the argument

The central device is the metric operator $\eta = \exp\!\left\{-2\theta\int \frac{d^3p}{(2\pi)^3}\,[\tilde{\phi}(-p)\tilde{\pi}_a(p) - \tilde{a}(-p)\tilde{\pi}_\phi(p)]\right\}$, with $\tanh(2\theta) = 2g/(m_a^2-m_\phi^2)$. This operator enforces the pseudo-Hermiticity relation $H^\dagger = \eta H \eta^{-1}$, defines the physical inner product $\langle\psi|\varphi\rangle_\eta = \langle\psi|\eta\varphi\rangle$, and generates the rotated fields $\Phi = \eta^{-1/2}\phi\,\eta^{1/2} = \phi\cosh\theta + i a\sinh\theta$, with an analogous expression for $A$. The rotation diagonalizes the Hamiltonian into two decoupled free fields, which makes the real spectrum and the Fock space explicit; it also singles out the genuine observables, namely the $\eta$-Hermitian operators, which are exactly those whose spectral functions are non-negative.

What would settle it

A direct check would be to construct the one-particle sector of the field theory and test positivity of $\eta$: compute the matrix elements $\langle p|\eta|q\rangle$ in the Fock basis and look for negative eigenvalues, since any negative eigenvalue in the weak-coupling regime would contradict the paper's central claim. Equivalently, a numerical diagonalization of a discretized version of the Hamiltonian at $g$ slightly below $|m_a^2-m_\phi^2|/2$ should yield only real energies; finding a complex pair below that threshold would refute the pseudo-Hermitian spectral picture.

Watch

Extended reading notes

Core claim

For the Hamiltonian $H = \int d^3x \left[\tfrac{1}{2}\pi_\phi^2 + \tfrac{1}{2}(\nabla\phi)^2 + \tfrac{m_\phi^2}{2}\phi^2 + \tfrac{1}{2}\pi_a^2 + \tfrac{1}{2}(\nabla a)^2 + \tfrac{m_a^2}{2}a^2 + i g a\phi\right]$, the paper exhibits a metric operator $\eta$ (Eq. 71) such that $H^\dagger = \eta H \eta^{-1}$ in the weak-coupling regime $2g < |m_a^2 - m_\phi^2|$. The spectrum is therefore real and identical to that of two free scalar fields with masses $M_\Phi$ and $M_A$ given by Eq. (79), and the $\eta$-inner product restores unitary time evolution. The rotated fields $\Phi$ and $A$ defined by the imaginary rotation in Eqs. (74)-(75) are $\eta$-Hermitian observables, and their propagators have the standard positive spectral functions $\rho(s)=2\pi\delta(s-M^2)$. The original fields $\phi$ and $a$ are Hermitian only at one instant and lose self-adjointness under time evolution; their spectral functions in Eqs. (94) are not everywhere positive. The claimed lesson is that such positivity violation indicates the operator is not an observable of the theory rather than a sign of instability.

Load-bearing premise

The whole construction rests on the assumption that the field-theory metric operator is a genuine, positive operator on the state space; if it is not positive, the $\eta$-inner product is not a valid inner product and the rotated fields cannot be counted as physical observables.

Editorial extensions

If this is right

In the weak-coupling regime the non-Hermitian scalar theory is fully consistent: energies are real and bounded from below, and time evolution is unitary with respect to the $\eta$-inner product.
The physical degrees of freedom are the rotated fields $\Phi$ and $A$; correlation functions built from the original fields will generically violate spectral positivity, so such violation should not by itself be read as a sign of instability.
The boundary $g = |m_a^2-m_\phi^2|/2$ is an exceptional point: below it the spectrum is real, and above it pairs of complex conjugate energies appear in the $\mathcal{PT}$-broken phase, which the paper leaves for future work.
Observables must be $\eta$-Hermitian; an operator that is merely Hermitian at one time will not remain self-adjoint under evolution generated by a non-Hermitian Hamiltonian.
If the closing conjecture is correct, the known positivity violation of the gluon propagator in Yang-Mills theories would be reinterpreted as evidence for a nontrivial Hilbert-space metric and a possible $\mathcal{PT}$-broken phase in the parameter space.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The paper does not prove that the field-theory metric operator is a well-defined positive operator on the full Fock space; if positivity fails, the unitary reformulation would survive only on a restricted physical subspace, so an explicit construction of the Fock-space representation of $\eta$ would be the natural next check.
The proposed interacting extension with a $\lambda(\phi^2+a^2)^2$ term is claimed to have the same metric operator; testing whether the positivity-violating spectral functions persist in that interacting theory would connect the paper's mechanism to nonperturbative settings.
One could try to extract a candidate metric from lattice or functional data of Yang-Mills by asking whether there exists a positive inner product that makes the gluon propagator's spectral function non-negative; a positive answer would support the conjecture, while a no-go result would limit its scope.
In gain-and-loss experiments in optics, mechanics, or ultracold atoms, imaginary couplings are realizable, and the prediction that rotated rather than original coordinates have positive spectral functions could in principle be tested through measured response functions.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Quick take: the stress-test concern is right. I checked the inverse of the rotation (74); the correct weights for Dϕϕ are cosh²θ and -sinh²θ, not cosh2θ and -sinh2θ. The same error appears in the QM correlation function (57) and spectral functions (59) and (91)-(94). This is not a typo: the mixed propagator (93) correctly uses i coshθ sinhθ, so the paper is internally inconsistent. The printed double-angle weights violate the canonical sum rule ∫ ds/(2π) ρ = 1 (they give e^{-2θ}). The qualitative conclusion that the original fields' spectral functions are not everywhere positive survives the correction, but the explicit equations carrying the paper's message are wrong.

What the paper does well: it gives a clear, self-contained review of pseudo-Hermitian mechanics and the two-oscillator model, and it constructs the free-field analogue carefully. It is honest that the QFT model reduces to decoupled free fields, and the conjecture about positivity violation signaling a PT-broken phase is labeled as a conjecture, not a proof. The citations to Refs. [71-75] for the QM model are appropriate.

Soft spots, in order: (1) the algebraic error above is load-bearing; (2) the continuum metric operator (71) is assumed to define a positive inner product on Fock space, but this is not proven, and the mode-integral exponent is not trivial; (3) the QFT extension is the QM model mode by mode, so novelty is thin. The paper's second half is essentially a rewrite of the first half with plane-wave indices.

Who it's for: readers who want a pedagogical walkthrough of pseudo-Hermitian QFT with a worked free-field example. That audience would benefit from a corrected version. As it stands, I would not send it to referees; I'd tell the editor to reject, and explicitly invite a corrected version. The fix is straightforward, and if the authors also address the metric positivity, the paper could be a decent contribution.

Referee Report

2 major / 6 minor

Summary. The paper studies a pair of scalar degrees of freedom coupled by an imaginary bilinear term, both as two quantum harmonic oscillators (Hamiltonian (29)) and as a two-scalar-field quantum field theory (Lagrangian (60)). In the weak-coupling regime, in which the coupling satisfies 2g < |m_a^2 - m_phi^2| (and the analogous inequality in quantum mechanics), the Hamiltonian is eta-pseudo-Hermitian with a formal positive metric eta, and the authors show that the spectrum is real and time evolution is unitary with respect to the eta-inner product. They introduce rotated, eta-Hermitian variables Phi and A (Eqs. (39) and (74)) that diagonalize the Hamiltonian, compute their two-point functions and the corresponding Kallen-Lehmann spectral functions, and show that the latter are non-negative. They then compute the two-point functions of the original variables phi and a and find spectral functions that are not everywhere positive (and in the mixed case complex), which they interpret as the signal that phi and a are not observables of the theory. The paper closes with a clearly labeled conjecture that spectral positivity violation in a theory with real spectrum may signal the existence of a PT-broken phase somewhere in its parameter space. Sections II and III are a review of pseudo-Hermitian quantum mechanics and of spectral functions in that setting.

Significance. The model is elementary and fully solvable, which is also its main strength: it provides a closed-form illustration of how a non-Hermitian Hamiltonian with a positive metric can define a unitary theory, and of how the choice of observables (eta-Hermitian versus merely Dirac-Hermitian) controls the positivity of the Kallen-Lehmann spectral function. This is directly relevant to current discussions of spectral positivity violation in Yang-Mills theory and in condensed-matter systems, and the paper's conjecture gives it a falsifiable edge. The quantitative content is undermined, however, by an algebraic error in the coefficients of the central spectral functions (Eqs. (57)-(59) and (91)-(94)), which as printed violate the canonical sum rule; this error is local and mechanical and, once corrected, the qualitative claims stand. The paper also contains a clear self-assessment of its limits: the strong-coupling/PT-broken regime is explicitly postponed, and the final conjecture is stated as unproven. Overall the work is a useful pedagogical and reference contribution to the pseudo-Hermitian QFT literature, provided the stated corrections are made.

major comments (2)

[§IV.D, Eqs. (57)-(59); §V.C, Eqs. (91)-(94)] The explicit spectral functions of the original fields are computed with the wrong rotation coefficients. Inverting (74) gives phi = cosh theta Phi - i sinh theta A and a = i sinh theta Phi + cosh theta A; substituting into D_phi phi and using the vanishing of the mixed propagator <Phi A>_eta = 0 yields D_phi phi_eta = cosh^2 theta D_Phi Phi_eta - sinh^2 theta D_A A_eta, not cosh 2 theta D_Phi Phi_eta - sinh 2 theta D_A A_eta as printed in (91), with the same double-angle error in (92) and in the quantum-mechanical results (57)-(59), where (51) implies C_xx_eta = cosh^2 theta C_XX_eta - sinh^2 theta C_YY_eta. The printed weights violate the spectral sum rule: with the normalization of (87)-(94), the integral of rho_phi phi(s)/(2 pi) over s equals cosh 2 theta - sinh 2 theta = e^{-2 theta}, whereas the equal-time commutator [phi, pi_phi] = i delta requires 1; the corrected single-angle weights give cosh^2 theta - sinh^2 theta = 1. The paper itself contains the check needed to expose this error, since Eq. (93) correctly uses the single-angle coefficient i cosh theta sinh theta for the mixed propagator. The qualitative conclusion (rho_phi phi and rho_aa are indefinite while rho_Phi Phi and rho_AA are positive) survives the correction, so the error is fixable, but the central quantitative results are not correct as printed.
[§V.B, Eq. (71)] The continuum metric operator of Eq. (71), eta = exp{-2 theta integral d^3p/(2 pi)^3 [phi_tilde(-p) pi_a(p) - a_tilde(-p) pi_phi(p)]}, is load-bearing for the field-theory claims: the positivity of the eta-inner product, the reality of the spectrum, and the unitarity of time evolution (Eqs. (72)-(76)) all rest on eta being a well-defined positive Hermitian operator on Fock space. The text asserts this without discussion of the operator-ordering subtleties of the exponent (products of fields and momenta at coincident arguments), of the domain of eta, or of a regularization prescription; no proof of self-adjointness or positivity is given for the continuum case, which is not equivalent to the finite-dimensional QM operator (33). The authors should supply at least a formal argument, for example normal-ordering the generator and showing that it is Hermitian with a real spectrum under a point-splitting or lattice regularization, or explicitly relegate the QFT unitarity and positivity claims to formal status.

minor comments (6)

[§IV.A] The name 'Baker-Haussdorff' in Sec. IV.A should be 'Baker-Hausdorff'.
[§V.B, Eqs. (81)-(84)] The notation for the creation and annihilation operators switches from alpha_Phi and alpha_Phi^# in Eqs. (81)-(82) to a_Phi and a_Phi^# in Eqs. (83)-(84) without definition; please make it uniform.
[§IV.A, Eq. (36)] The explicit formulas for omega_x^2 and omega_y^2 implicitly assume a definite sign of Omega_x^2 - Omega_y^2 (the branch of the square root); the convention should be stated.
[§IV.D, Eq. (57)] Eq. (57) is stated without derivation; since the coefficients are the point at issue in the major comment above, a two-line derivation from (51) and the vanishing of the mixed correlation function would be valuable.
[§III, Eqs. (26)-(28)] The formalism proves that eta-Hermiticity of an operator implies a positive spectral function, but it does not establish the converse; the paper's phrasing in Sec. III ('can be evidence') is appropriately cautious, but the summaries in Secs. IV.D and V.C should avoid implying that non-positivity of a spectral function is equivalent to non-eta-Hermiticity of the corresponding operator.
[Throughout] There are several typos and accent problems, including 'eingenvalues' (Sec. I), 'precesely' (Sec. V.C), 'estabilish' (Sec. VI), and the broken accents in 'Kallen-lehmann'; the text should be proofread.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the metric and spectral functions are derived from the stated pseudo-Hermiticity condition and from an independent spectral-function theorem, not from the conclusions they support.

full rationale

The core construction is self-contained. The metric parameter is fixed by the explicit pseudo-Hermiticity condition tanh(2θ) = 2g/(m_A^2 - m_ϕ^2), and the rotated fields Φ and A are defined from the metric via Φ = η^{-1/2} ϕ η^{1/2} = ϕ cosh θ + i a sinh θ. The positivity of ρ_{ΦΦ} and ρ_{AA} follows from the general theorem in Eq. (28), which states that the spectral function of an η-hermitian operator is non-negative; the paper does not assume the desired positivity to choose the metric. The non-positive spectral functions for the original fields ϕ and a are then obtained by substituting the inverse rotation into the two-point functions, which is an independent algebraic step. The concluding conjecture about positivity violation as a sign of PT-breaking is explicitly labeled a conjecture and plays no role in the derivations. The only self-references are to works in progress ([46], [78]), and they are not load-bearing for any result. The skeptic's concern about cosh 2θ versus cosh²θ coefficients in Eqs. (57)-(59) and (91)-(94) is an algebraic-consistency issue rather than circular reasoning; even with the corrected single-angle weights, the central claim that the original fields' spectral functions are not everywhere positive would still hold. Therefore no circularity is present.

Assumptions & free parameters 0 free parameters · 5 assumptions · 1 invented entities

The central construction rests on standard pseudo-Hermitian machinery plus a formal, unproven assumption about the positivity and well-definedness of the continuum field-theory metric. No parameters are fitted to data.

assumptions (5)

standard math A pseudo-Hermitian Hamiltonian with a positive-definite metric operator has real spectrum and unitary time evolution under the eta-inner product.
Invoked throughout Secs. II-IV; standard result from Mostafazadeh and Bender.
standard math The transformation defined by rho = eta^(1/2) maps H to the Hermitian h and preserves the spectrum.
Used in Sec. IV.A to compute real eigenenergies (Eqs. 34-37).
domain assumption The continuum QFT metric operator (71), an exponential of an integral over momentum modes, is a well-defined positive operator on the Fock space in the weak-coupling regime.
No proof of convergence or positivity is given for the 3+1-dimensional field theory; the calculation proceeds formally.
domain assumption The Kallen-Lehmann representation and the spectral sum rule remain valid with the eta-inner product and non-eta-Hermitian fields.
Assumed in Sec. III and used to interpret rho_phi_phi, rho_aa, and rho_a_phi in Sec. V.C.
domain assumption A free-field Fock space exists with a unique vacuum annihilated by the lowering operators, and the eta-inner product is positive on physical states.
Needed for the ground-state correlators (52)-(53) and for the propagator calculations in Sec. V.C.

invented entities (1)

Mode-integral metric operator eta (and rho = eta^(1/2)) in the continuum scalar field theory
purpose: Defines the physical inner product and the eta-Hermitian observables Phi and A; without it H is not self-adjoint and evolution is not unitary.
It is constructed from the oscillator parameters rather than independently measured; its positivity in QFT is assumed, not demonstrated. This is a mathematical tool rather than a new particle or force.

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Cite this review

Pith. "Pith review of Oscillators with imaginary coupling: spectral functions in quantum mechanics and quantum field theory." pith.science (2026). https://pith.science/paper/HY42Y2ZW

@misc{pith2026241214064,
  author       = {Pith},
  title        = {Pith review of: Oscillators with imaginary coupling: spectral functions in quantum mechanics and quantum field theory},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HY42Y2ZW}},
  note         = {Machine review of arXiv:2412.14064}
}

abstract

The axioms of Quantum Mechanics require that the hamiltonian of any closed system is self-adjoint, so that energy levels are real and time evolution preserves probability. On the other hand, non-hermitian hamiltonians with ${\cal{PT}}$-symmetry can have both real spectra and unitary time evolution. In this paper, we study in detail a pair of quantum oscillators coupled by an imaginary bilinear term, both in quantum mechanics and in quantum field theory. We discuss explicitly how such hamiltonians lead to perfectly sound physical theories with real spectra and unitary time evolution, in spite of their non-hermiticity. We also analyze two-point correlation functions and their associated K\"allen-Lehmann representation. In particular, we discuss the intimate relation between positivity violation of the spectral functions and the non-observability of operators in a given correlation function. Finally, we conjecture that positivity violation of some spectral functions of the theory could be a generic sign of the existence of complex pairs of energy eigenvalues (i.e., a ${\cal{PT}}$-broken phase) somewhere in its parameter space.

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Reference graph

Works this paper leans on

105 extracted references · 51 canonical work pages

[1]

El-Ganainy, M

R. El-Ganainy, M. Khajavikhan, D. N. Christodoulides, and S. K. Ozdemir, Communications Physics 2, 37 (2019)

2019
[2]

L. Feng, R. El-Ganainy, and L. Ge, Nature Photonics 11, 752 (2017)

2017
[3]

η−hermitian operator

is to build a convenient connection between the Hilbert spaces. 3 A diﬀerent inner product implies a diﬀerent conjuga- tion rule for operators. With the connection between the reference and the η−inner product given by ( 3), it is con- venient to deﬁne the η−conjugate ˆA# of any operator ˆA so that ⟨ ψ ⏐ ⏐ ⏐ˆAφ ⟩ η = ⟨ ˆA#ψ ⏐ ⏐ ⏐φ ⟩ η , (4) which is equiv...
[4]

Rotter and J

I. Rotter and J. P. Bird, Reports on Progress in Physics 78, 114001 (2015)

work page 2015
[5]

for an η−hermitian hamiltonian ˆH, it follows that (Ei −E∗ j ) ⟨ψi|ψj⟩η = 0. (14) Therefore, if E∗ i ⁄=Ej then ⟨Ei|Ej⟩η = 0, (15) so that orthogonality of states with diﬀerent eigenvalues is guaranteed (in particular, but not exclusively, if such eigenvalues are real). Furthermore, ∀Ei ∈ C we have ‖|Ei⟩‖2 η = ⟨Ei|Ei⟩η = 0, (16) that is, complex energy eig...
[6]

Although (pseudo-)hermiticity implies real eigen- values, the converse is not necessarily true, because the very notion of an adjoint operator (i.e

that supports the reality of the energy levels (but only if the inner-product is positive-deﬁnite [41]). Although (pseudo-)hermiticity implies real eigen- values, the converse is not necessarily true, because the very notion of an adjoint operator (i.e. ˆO† or ˆO#) de- pends on the deﬁnition of a somewhat arbitrary inner product, whereas the spectrum of a...
[7]

known as exceptional points (EPs) [8, 9]. When a non-Hermitian Hamiltonian continuously depends on ex- ∗ bruno.mintz@uerj.br † iypinheiro@yahoo.com.br ‡ ruiaquino@ictp-saifr.org ternal parameters, it could happen that, for certain val- ues of the parameters, two or more eigenvalues coalesce to an EP. However, this is not a usual degeneracy, as observed in...
[8]

Equivalently, ˆOH (t) = ˆU (t)− 1 ˆOS ˆU (t), (20) where ˆU (t) = exp ( −i ˆHt/ℏ ) is the time evolution opera- tor

= ˆOS. Equivalently, ˆOH (t) = ˆU (t)− 1 ˆOS ˆU (t), (20) where ˆU (t) = exp ( −i ˆHt/ℏ ) is the time evolution opera- tor. Note that, for a generic non-hermitian hamiltonian, ˆH † ⁄= ˆH, the evolution operator is not (Dirac-)unitary, ˆU † ⁄= ˆU − 1, so that probability conservation in the theory may be at risk. However, for a pseudo-hermitian hamilto- ni...

Show all 105 references

[9]

rotation

is employed instead of the reference inner product [22, 42, 49]. Observables must be such that not only their eigenval- ues are real, but also that the eigenvectors correspond- ing to diﬀerent eigenvalues are mutually orthogonal. As such, they must be self-adjoint with respect...
[10]

Of course, for a hermitian hamilto- nian, |φn⟩ = |ψn⟩ and ˆη = ˆ1

leads to the rep- resentation of the identity operator as ˆ1 = ∑ n |ψn⟩ ⟨φn| = ∑ n |ψn⟩ ⟨ψn| ˆη, (11) since ⟨φn| = ⟨ψn| ˆη. Of course, for a hermitian hamilto- nian, |φn⟩ = |ψn⟩ and ˆη = ˆ1. Using Eqs. ( 9, 10), it is immediate so see that a possible metric operator can be bui...
[11]

With these new variables, we may write the hamilto- nian ˆHQM as ˆHQM = 1 2m ( ˆP 2 X + ˆP 2 Y ) + m 2 ( ω2 x ˆX 2 +ω2 y ˆY 2 )

is an adequate inner product for the quantum system deﬁned by the hamil- tonian ( 29), the eigenvalues of the operators ( 39) are all real and their eigenvectors are orthogonal with respect to the η−inner product. With these new variables, we may write the hamilto- nian ˆHQM a...
[12]

is positive-deﬁned. Finally, given that [ ˆβi, ˆα† j ] = ˆ1δij [ ˆHQM, ˆβi ] = −ℏωi ˆβi [ ˆHQM, ˆα† i ] = ℏωi ˆα† i, (47) (where no summation over repeated indices is implied) it is straightforward to recover the result (
[13]

Doppler, A

J. Doppler, A. A. Mailybaev, J. B¨ ohm, U. Kuhl, A. Girschik, F. Libisch, T. J. Milburn, P. Rabl, N. Moi- seyev, and S. Rotter, Nature 537, 76 (2016)

2016
[14]

(51) It is interesting to note that, although ˆx† = ˆx, this her- miticity condition is not valid in general for t ⁄= 0, i.e., ˆx† H (t) ⁄= ˆxH (t)

= ˆy, ˆpxH (t = 0) = ˆpx, and ˆpyH (t = 0) = ˆpy can be easily be found from ( 39) and ( 50) and is given by ˆxH (t) = cosh θ ˆXH (t) −i sinhθ ˆYH (t) ˆyH (t) = i sinhθ ˆXH (t) + coshθ ˆYH (t) ˆpxH (t) = cosh θ ˆPXH (t) −i sinhθ ˆPY H (t) ˆpyH (t) = i sinhθ ˆPXH (t) + coshθ ˆP...
[15]

original

(or, equivalently, 42), but also the η−hermitian operators make the physics more ap- parent than the (Dirac) hermitian operators used in the original hamiltonian ( 29). In the next subsection, we will show how one can arrive at a similar conclusion from the point of view of co...
[16]

Therefore, it is not surprising that its two-point correlation function violates reﬂection positivity

itself. Therefore, it is not surprising that its two-point correlation function violates reﬂection positivity. In this speciﬁc example, we saw that positivity vio- lation in a correlation function involving only hermitian operators is a consequence of the non-hermiticity of th...
[17]

C. M. Bender, Reports on Progress in Physics 70, 947–1018 (2007)

2007
[18]

is a complex quantity. The hamiltonian is then given by H = ∫ d3xT 00 = ∫ d3x [ 1 2π2 ϕ + 1 2 (∇ϕ)2 + m2 ϕ 2 ϕ2 + 1 2π2 a + 1 2 (∇a)2 + m2 a 2 a2 +igaϕ ] , (63) where the canonically conjugate ﬁeld momenta are πϕ = ∂L ∂(∂0ϕ) = ∂ϕ ∂t , πa = ∂L ∂(∂0a) = ∂a ∂t. (64) The linear mo...
[19]

However, as we will argue next, it is actually quite simple to build a meaningful quantum ﬁeld theory from it

as a starting point. However, as we will argue next, it is actually quite simple to build a meaningful quantum ﬁeld theory from it. B. Canonical quantization Following the standard procedure of canonical quan- tization, let us promote the dynamical variables to (Schr¨ odinger ...
[20]

Pais and G

A. Pais and G. E. Uhlenbeck, Phys. Rev. 79, 145 (1950)

1950
[21]

Property ( 76) ensures that both the hamiltonian and the linear momentum operators are observables of the ﬁeld theory deﬁned by the action ( 60)

and ( 75), the linear momentum operator ( 65) is given by ⃗P = ∫ d3x [ Π Φ (⃗ x)⃗∇Φ(⃗ x) + Π A(⃗ x)⃗∇A(⃗ x) ] (80) which is also η−hermitian, i.e., ⃗P # = ⃗P. Property ( 76) ensures that both the hamiltonian and the linear momentum operators are observables of the ﬁeld theory ...
[22]

(82) Note that the creation operators are given by α#(⃗ p), rather than α†(⃗ p)

and their conju- gate momenta ( 75) in terms of creation and annihilation operators, so that Φ(⃗ x) = ∫ d3p (2π)3 1√ 2Ω Φ (⃗ p) [ αΦ (⃗ p)ei⃗ p·⃗ x+α# Φ (⃗ p)e− i⃗ p·⃗ x ] A(⃗ x) = ∫ d3p (2π)3 1√ 2Ω A(⃗ p) [ αA(⃗ p)ei⃗ p·⃗ x+α# A (⃗ p)e− i⃗ p·⃗ x ] Π Φ (⃗ x) = −i ∫ d3p (2π)3 √...
[23]

is also an imme- diate consequence. The vacuum state is the one annihi- lated by the operators aΦ and aA, and any excited state is obtained from the vacuum by sucessive applications of the creation operators a# Φ and a# A . For example, de- noting the vacuum state by |0⟩, the ...
[24]

C. M. Bender, K. A. Milton, and V. M. Savage, Phys. Rev. D 62, 085001 (2000) , arXiv:hep-th/9907045

2000 arXiv
[25]

do not have an ev- erywhere positive spectral function. This is the case be- cause, even though these operators are hermitian at some given time (say, t = 0), their time evolution (given by a nonhermitian hamiltonian) is such that their hermiticity is lost in general for t ⁄= ...
[26]

Since the hamiltonian is also η−hermitian, propagators of such ﬁelds will have a positive spectral function

will be η−hermitian for every time t. Since the hamiltonian is also η−hermitian, propagators of such ﬁelds will have a positive spectral function. In the following, we calculate such correlation functions, in order to show our claims explicitly. First, let us calculate the pro...
[27]

W.-Y. Ai, J. Alexandre, and S. Sarkar, Phys. Rev. D 107, 025007 (2023) , arXiv:2211.06273 [hep-th]

2023 arXiv
[28]

W.-Y. Ai, C. M. Bender, and S. Sarkar, Phys. Rev. D 106, 125016 (2022) , arXiv:2209.07897 [hep-th]

2022 arXiv
[29]

ac- tive transformation

are eQM (nx,n y) = EQM (nx,n y) nx,n y ∈ N, = ℏωx ( nx + 1 2 ) + ℏωy ( ny + 1 2 ) , (37) where eQM denotes the eigenvalues of ˆhQM and EQM denotes the eigenvalues of ˆHQM . The energy eigenvectors |ψn⟩ of ˆHQM can be calculated from those of ˆhQM , |χ⟩, as [41] |ψn⟩ = ˆρ− 1 |χ...
[30]

Alexandre, J

J. Alexandre, J. Ellis, P. Millington, and D. Seynaeve, Phys. Rev. D 98, 045001 (2018) , arXiv:1805.06380 [hep-th]

2018 arXiv
[31]

Many-body open quantum systems,

R. Fazio, J. Keeling, L. Mazza, and M. Schir` o, “Many-body open quantum systems,” (2024), arXiv:2409.10300 [quant-ph]

2024 arXiv
[32]

Alexandre, J

J. Alexandre, J. Ellis, P. Millington, and D. Seynaeve, Phys. Rev. D 99, 075024 (2019) , arXiv:1808.00944 [hep-th]

2019 arXiv
[33]

C. T. e. Fabio Bagarello, Roberto Passante, Non-Hermitian Hamiltonians in Quantum Physics: Selected C ontributions from the 15th International Conference on Non -Hermitian 1st ed., Springer Proceedings in Physics 184 (Springer International Publishing, 2016)

2016
[34]

Michishita and R

Y. Michishita and R. Peters, Phys. Rev. Lett. 124, 196401 (2020)

2020
[35]

passive transfor- mation

results. In this subsection, we will adopt a sort of “passive transfor- mation” point of view, in which we deﬁne new variables and tackle the same problem directly, i.e., in terms of the original hamiltonian ( 29). Let us deﬁne the new variables ˆX = ˆρ− 1 ˆxˆρ = ˆx coshθ +iˆy...
[36]

M. V. Berry and D. H. J. O’Dell, Journal of Physics A: Mathematical and General 31, 2093 (1998)

1998
[37]

for the energy spectrum. As for the usual 2-d harmonic oscillator, the ground state |ψ0⟩ is deﬁned as ˆβX |ψ0⟩ = 0 and ˆβY |ψ0⟩ = 0, (48) whereas the stationary excited states can be obtained from the ground state by repeated application of the rais- ing operators, i.e., |NX,N...
[38]

W. D. Heiss and A. L. Sannino, Journal of Physics A: Mathematical and General 23, 1167 (1990)

1990
[39]

Kato, Perturbation theory for linear operators, Clas- sics in Mathematics (Springer-Verlag, Berlin Heidelberg, 1995)

T. Kato, Perturbation theory for linear operators, Clas- sics in Mathematics (Springer-Verlag, Berlin Heidelberg, 1995)

1995
[40]

G¨ unther, I

U. G¨ unther, I. Rotter, and B. F. Samsonov, Journal of Physics A: Mathematical and Theoretical 40, 8815 (2007)

2007
[41]

S.-B. Lee, J. Yang, S. Moon, S.-Y. Lee, J.- B. Shim, S. W. Kim, J.-H. Lee, and K. An, Phys. Rev. Lett. 103, 134101 (2009)

2009
[42]

is a sum of two decoupled parts, it is easy to see that ˆXH(t) = cos( ωxt) ˆX + sin(ωxt) ˆPX mωx ˆYH (t) = cos( ωyt) ˆY + sin(ωyt) ˆPY mωy ˆPXH (t) = cos( ωxt) ˆPX −mωx sin(ωxt) ˆX ˆPY H (t) = cos( ωyt) ˆPY −mωy sin(ωyt) ˆY, (50) where the Schr¨ odinger picture operators ˆX, ˆ...
[43]

H. Xu, D. Mason, L. Jiang, and J. G. E. Harris, Nature 537, 80 (2016)

2016
[44]

Zhang, X

W. Zhang, X. Ouyang, X. Huang, X. Wang, H. Zhang, Y. Yu, X. Chang, Y. Liu, D.-L. Deng, and L.-M. Duan, Phys. Rev. Lett. 127, 090501 (2021)

2021
[45]

Kawabata, T

K. Kawabata, T. Bessho, and M. Sato, Phys. Rev. Lett. 123, 066405 (2019)

2019
[46]

C. M. Bender and S. Boettcher, Physical Review Letters 80, 5243–5246 (1998)

1998
[47]

C. M. Bender and D. W. Hook, Rev. Mod. Phys. 96, 045002 (2024)

2024
[48]

T. D. Lee, Physical Review 95, 1329 (1954)

1954
[49]

C. M. Bender, S. F. Brandt, J.-H. Chen, and Q. Wang, Physical Review D 71 (2005), 10.1103/physrevd.71.025014

2005 doi
[50]

C. M. Bender and P. D. Mannheim, Phys. Rev. Lett. 100, 110402 (2008) , arXiv:0706.0207 [hep-th]

2008 arXiv
[51]

C. M. Bender, N. Hassanpour, S. P. Klevan- sky, and S. Sarkar, Phys. Rev. D 98, 125003 (2018) , arXiv:1810.12479 [hep-th]

2018 arXiv
[52]

M. E. Peskin and D. V. Schroeder, An Introduction to quantum ﬁeld theory (Addison- Wesley, Reading, USA, 1995)

1995
[53]

Romatschke, Int

P. Romatschke, Int. J. Mod. Phys. A 38, 2350157 (2023) , arXiv:2211.15683 [hep-th] . 15

2023 arXiv
[54]

Lawrence, R

S. Lawrence, R. Weller, C. Peterson, and P. Romatschke, Phys. Rev. D 108, 085013 (2023) , arXiv:2303.01470 [hep-th]

2023 arXiv
[55]

Alkofer and L

R. Alkofer and L. von Smekal, Phys. Rept. 353, 281 (2001) , arXiv:hep-ph/0007355

2001 arXiv
[56]

J. M. Cornwall, Modern Physics Letters A 28, 1330035 (2013) , arXiv:1310.7897 [hep-ph]

2013 arXiv
[57]

Alexandre, P

J. Alexandre, P. Millington, and D. Sey- naeve, Phys. Rev. D 96, 065027 (2017) , arXiv:1707.01057 [hep-th]

2017 arXiv
[58]

Iritani, H

T. Iritani, H. Suganuma, and H. Iida, Physical Review D 80 (2009), 10.1103/physrevd.80.114505

2009 doi
[59]

Raval and B

H. Raval and B. P. Man- dal, Nucl. Phys. B, 114699 (2019) , arXiv:1805.02510 [hep-th]

2019 arXiv
[60]

The corresponding Euler-Lagrange equations are (✷ +m2 a)a = −igϕ (✷ +m2 ϕ)ϕ = −iga, (61) where ✷ =∂2/∂t2 − ∇2 is the D’Alembertian operator

is evidently complex. The corresponding Euler-Lagrange equations are (✷ +m2 a)a = −igϕ (✷ +m2 ϕ)ϕ = −iga, (61) where ✷ =∂2/∂t2 − ∇2 is the D’Alembertian operator. Note that, even if the initial and/or boundary condi- tions are such that the ﬁelds a and ϕ are real, the imag- in...
[61]

Beygi, S

A. Beygi, S. P. Klevansky, and C. M. Bender, Physical Review A 99 (2019), 10.1103/physreva.99.062117

2019 doi
[62]

Alexandre, J

J. Alexandre, J. Ellis, and P. Millington, Physical Review D 102 (2020), 10.1103/physrevd.102.125030

2020 doi
[63]

Alexandre, J

J. Alexandre, J. Ellis, and P. Millington, Physical Review D 106 (2022), 10.1103/physrevd.106.065003

2022 doi
[64]

Felski, A

A. Felski, A. Beygi, and S. P. Kle- vansky, Phys. Rev. D 107, 016015 (2023) , arXiv:2210.15503 [hep-ph]

2023
[65]

Unitarity through pt symmetry in quantum quadratic gravit y,

J. Kuntz, “Unitarity through pt symmetry in quantum quadratic gravit y,” (2024), arXiv:2410.08278 [hep-th]

2024 arXiv
[66]

Mostafazadeh, Journal of Mathematical Physics 43, 205–214 (2002)

A. Mostafazadeh, Journal of Mathematical Physics 43, 205–214 (2002)

2002
[67]

von Neumann, Mathematical foundations of quantum mechanics (1955)

J. von Neumann, Mathematical foundations of quantum mechanics (1955)

1955
[68]

P. A. M. Dirac, The principles of quantum mechanics (1947)

1947
[69]

Mostafazadeh, International Journal of Geometric Methods in Modern Physi cs 07, 1191–1306 (2010)

A. Mostafazadeh, International Journal of Geometric Methods in Modern Physi cs 07, 1191–1306 (2010)

2010
[70]

C. M. Bender, D. C. Brody, and H. F. Jones, Physical Review Letters 89 (2002), 10.1103/physrevlett.89.270401

2002 doi
[71]

D. C. Brody, Journal of Physics A: Mathematical and Theoretical 47, 035305 (2013)

2013
[72]

Jia, R.-Y

H. Jia, R.-Y. Zhang, J. Hu, Y. Xiao, S. Zhang, Y. Zhu, and C. T. Chan, Communications Physics 6, 293 (2023)

2023
[73]

J. Ryu, J. Han, and C. e. a. Yi, Commun Phys 7, 109 (2024) , arXiv:2306.06967 [quant-ph]

2024 arXiv
[74]

(79) Note that the form ( 77) of the hamiltonian is mani- festly self-adjoint with respect to the η−inner product, i.e., ˆH # = ˆH

as M 2 Φ = cosh2θm 2 ϕ + sinh2θm 2 a cosh 2θ , M 2 A = cosh2θm 2 a + sinh2θm 2 ϕ cosh 2θ . (79) Note that the form ( 77) of the hamiltonian is mani- festly self-adjoint with respect to the η−inner product, i.e., ˆH # = ˆH. Besides, transformations ( 74) and ( 75) also decompos...
[75]

B. W. Mintz, I. Y. Pinheiro, and R. Aquino, (Work in progress)
[76]

Pauli, Rev

W. Pauli, Rev. Mod. Phys. 15, 175 (1943)

1943
[77]

P. A. M. Dirac, Proceedings of the Royal Society London A 180, 1 (1942)

1942
[78]

P. D. Mannheim, Phil. Trans. Roy. Soc. Lond. A 371, 20120060 (2013) , arXiv:0912.2635 [hep-th]

2013 arXiv
[79]

Kallen, Helv

G. Kallen, Helv. Phys. Acta 25, 417 (1952)

1952
[80]

Lehmann, Il Nuovo Cimento 11, 342 (1954)

H. Lehmann, Il Nuovo Cimento 11, 342 (1954)

1954
[81]

Oehme, International Journal of Modern Physics A 10, 1995 (1995) arXiv:hep-th/9412040 [hep-th]

R. Oehme, International Journal of Modern Physics A 10, 1995 (1995) arXiv:hep-th/9412040 [hep-th]

1995 arXiv
[82]

Nishijima, Czechoslovak Journal of Physics 46, 1 (1996)

K. Nishijima, Czechoslovak Journal of Physics 46, 1 (1996)

1996
[85]

In a her- mitian ﬁeld theory, one has, of course, ˆη = ˆ1, so that the metric does not appear in the calculation of the corre- lation functions

depends explicitly on the choice of the metric operator. In a her- mitian ﬁeld theory, one has, of course, ˆη = ˆ1, so that the metric does not appear in the calculation of the corre- lation functions. Note also that, in a path-integral for- mulation, the two-point function ca...
[86]

Cucchieri, T

A. Cucchieri, T. Mendes, and A. R. Taurines, Physical Review D 71 (2005), 10.1103/physrevd.71.051902

2005 doi
[87]

and ( 88), and their spectral functions are given, respectively, by ρΦΦ (s) = 2πδ(s −M 2 Φ ) (89) and ρAA(s) = 2πδ(s −M 2 A) (90) Note that both spectral functions ( 89) and ( 90) are non- negative, as expected from the general discussion in Sec. ( III). Now, let us consider t...
[88]

Dudal, O

D. Dudal, O. Oliveira, and P. J. Silva, Physical Review D 89 (2014), 10.1103/physrevd.89.014010

2014 doi
[89]

Alkofer, W

R. Alkofer, W. Detmold, C. S. Fischer, and P. Maris, Physical Review D 70 (2004), 10.1103/physrevd.70.014014

2004 doi
[90]

C. S. Fischer and M. Q. Hu- ber, Phys. Rev. D 102, 094005 (2020) , arXiv:2007.11505 [hep-ph]

2020 arXiv
[92]

A. K. Cyrol, L. Fister, M. Mitter, J. M. Pawlowski, and N. Strodthoﬀ, Physical Review D 94 (2016), 10.1103/physrevd.94.054005

2016 doi
[93]

A. K. Cyrol, J. M. Pawlowski, A. Rothkopf, and N. Wink, SciPost Phys. 5, 065 (2018)

2018
[94]

A. C. Aguilar, F. De Soto, M. N. Ferreira, J. Papavas- siliou, J. Rodr ´ ıguez-Quintero, and S. Zafeiropoulos, The European Physical Journal C 80 (2020), 10.1140/epjc/s10052-020

2020 doi
[95]

Of course, inﬁnitely many other interacting pseudo-hermitian theories are possible, with diﬀerent metric operators

can be an interesting toy model for future studies of interacting theories. Of course, inﬁnitely many other interacting pseudo-hermitian theories are possible, with diﬀerent metric operators. Evidently, an important regime of the theory that was not explored in this work is th...

2023
[96]

Dudal, S

D. Dudal, S. P. Sorella, N. Van- dersickel, and H. Verschelde, Physical Review D 77 (2008), 10.1103/physrevd.77.071501

2008 doi
[97]

Vandersickel and D

N. Vandersickel and D. Zwanziger, Phys. Rept. 520, 175 (2012) , arXiv:1202.1491 [hep-th]

2012 arXiv
[98]

Aquino and D

R. Aquino and D. G. Barci, Phys. Rev. B 102, 201110 (2020)

2020
[99]

Aquino and D

R. Aquino and D. G. Barci, Phys. Rev. B 104, 195150 (2021)

2021
[100]

Aquino, N

R. Aquino, N. O. Silvano, and D. G. Barci, Phys. Rev. B 110, 085147 (2024)

2024
[101]

Nanayakkara, Physics Letters A 304, 67 (2002)

A. Nanayakkara, Physics Letters A 304, 67 (2002)

2002
[102]

Fring and T

A. Fring and T. Frith, Journal of Physics A: Mathematical and Theoretical 51, 265301 (2018)

2018
[103]

Beygi, S

A. Beygi, S. P. Klevansky, and C. M. Bender, Physical Review A 91 (2015), 10.1103/physreva.91.062101

2015 doi
[104]

C. M. Bender, A. Felski, N. Hassanpour, S. P. Klevansky, and A. Beygi, Physica Scripta 92, 015201 (2016)

2016
[105]

Felski and S

A. Felski and S. P. Klevan- sky, Phys. Rev. A 98, 012127 (2018) , arXiv:1808.05108 [math-ph]

2018 arXiv
[106]

J. J. Sakurai and J. Napolitano, Mecˆ anica quˆ antica moderna, segunda edi¸ c˜ ao ed. (Book- man, S˜ ao Paulo, 2013)

2013
[107]

Mostafazadeh, Physical Review D 76 (2007), 10.1103/physrevd.76.067701

A. Mostafazadeh, Physical Review D 76 (2007), 10.1103/physrevd.76.067701

2007 doi
[108]

B. W. Mintz and I. Y. Pinheiro, (Work in progress)

Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.

[1] [1]

El-Ganainy, M

R. El-Ganainy, M. Khajavikhan, D. N. Christodoulides, and S. K. Ozdemir, Communications Physics 2, 37 (2019)

2019

[2] [2]

L. Feng, R. El-Ganainy, and L. Ge, Nature Photonics 11, 752 (2017)

2017

[3] [3]

η−hermitian operator

is to build a convenient connection between the Hilbert spaces. 3 A diﬀerent inner product implies a diﬀerent conjuga- tion rule for operators. With the connection between the reference and the η−inner product given by ( 3), it is con- venient to deﬁne the η−conjugate ˆA# of any operator ˆA so that ⟨ ψ ⏐ ⏐ ⏐ˆAφ ⟩ η = ⟨ ˆA#ψ ⏐ ⏐ ⏐φ ⟩ η , (4) which is equiv...

[4] [4]

Rotter and J

I. Rotter and J. P. Bird, Reports on Progress in Physics 78, 114001 (2015)

work page 2015

[5] [5]

for an η−hermitian hamiltonian ˆH, it follows that (Ei −E∗ j ) ⟨ψi|ψj⟩η = 0. (14) Therefore, if E∗ i ⁄=Ej then ⟨Ei|Ej⟩η = 0, (15) so that orthogonality of states with diﬀerent eigenvalues is guaranteed (in particular, but not exclusively, if such eigenvalues are real). Furthermore, ∀Ei ∈ C we have ‖|Ei⟩‖2 η = ⟨Ei|Ei⟩η = 0, (16) that is, complex energy eig...

[6] [6]

Although (pseudo-)hermiticity implies real eigen- values, the converse is not necessarily true, because the very notion of an adjoint operator (i.e

that supports the reality of the energy levels (but only if the inner-product is positive-deﬁnite [41]). Although (pseudo-)hermiticity implies real eigen- values, the converse is not necessarily true, because the very notion of an adjoint operator (i.e. ˆO† or ˆO#) de- pends on the deﬁnition of a somewhat arbitrary inner product, whereas the spectrum of a...

[7] [7]

known as exceptional points (EPs) [8, 9]. When a non-Hermitian Hamiltonian continuously depends on ex- ∗ bruno.mintz@uerj.br † iypinheiro@yahoo.com.br ‡ ruiaquino@ictp-saifr.org ternal parameters, it could happen that, for certain val- ues of the parameters, two or more eigenvalues coalesce to an EP. However, this is not a usual degeneracy, as observed in...

[8] [8]

Equivalently, ˆOH (t) = ˆU (t)− 1 ˆOS ˆU (t), (20) where ˆU (t) = exp ( −i ˆHt/ℏ ) is the time evolution opera- tor

= ˆOS. Equivalently, ˆOH (t) = ˆU (t)− 1 ˆOS ˆU (t), (20) where ˆU (t) = exp ( −i ˆHt/ℏ ) is the time evolution opera- tor. Note that, for a generic non-hermitian hamiltonian, ˆH † ⁄= ˆH, the evolution operator is not (Dirac-)unitary, ˆU † ⁄= ˆU − 1, so that probability conservation in the theory may be at risk. However, for a pseudo-hermitian hamilto- ni...