Time-Dependent Dyson Map

• The time-dependent Dyson map is an invertible, generally non-unitary operator that transforms a time-dependent non-Hermitian Hamiltonian into its Hermitian counterpart via a gauge-like derivative term.
• It constructs a dynamic metric, ρ(t) = η†(t)η(t), ensuring unitary evolution by redefining the physical inner product even when the Hamiltonian changes with time.
• Versatile construction methods—including algebraic Lie-group techniques, Lewis–Riesenfeld invariants, and perturbative expansions—demonstrate its utility in modeling quantum systems and mending broken PT-symmetry.

Searching arXiv for recent and foundational papers on time-dependent Dyson maps to ground the article in the provided literature. A time-dependent Dyson map is an invertible, generally non-unitary operator $\eta(t)$, or equivalently $\Omega(t)$ in some notational conventions, that relates an explicitly time-dependent non-Hermitian Hamiltonian $H(t)$ to a Hermitian counterpart $h(t)$ by a similarity transformation supplemented by a gauge-like derivative term. In its standard form,

$$h(t)=\eta(t)\,H(t)\,\eta^{-1}(t)+i\hbar\,\dot\eta(t)\,\eta^{-1}(t),$$

while the associated instantaneous metric is $\rho(t)=\eta^\dagger(t)\eta(t)$, defining the physical inner product in the non-Hermitian picture (Fring, 2022). The construction is central to time-dependent quasi-Hermitian and $\mathcal{PT}$-symmetric quantum mechanics because it separates the generator of evolution from the observable content of the theory, allows unitary evolution in a dynamical Hilbert space, and provides an algebraic bridge between non-Hermitian models and Hermitian representatives across finite- and infinite-dimensional settings (Fring et al., 2015).

1. Formal definition and basic equations

The time-dependent Dyson map is introduced by relating solutions of two Schrödinger equations, one in a non-Hermitian picture and one in a Hermitian picture. Depending on convention, one writes either $|\phi(t)\rangle=\eta(t)|\Psi(t)\rangle$ or $|\Psi(t)\rangle=\eta(t)|\psi(t)\rangle$; the resulting operator identity is the same time-dependent Dyson equation, written either as

$$h(t)=\eta(t)\,H(t)\,\eta^{-1}(t)+i\hbar\,\dot\eta(t)\,\eta^{-1}(t)$$

or equivalently

$\Omega(t)$0

(Fring et al., 2015). In the Lie-algebraic formulation developed for bosonic realizations of $\Omega(t)$1 and $\Omega(t)$2, the non-Hermitian generator is frequently taken in the unified form

$\Omega(t)$3

with $\Omega(t)$4 and $\Omega(t)$5, where $\Omega(t)$6 for $\Omega(t)$7 and $\Omega(t)$8 for $\Omega(t)$9 (Cius et al., 2022).

Once $H(t)$0 is known, the metric operator is defined by

$H(t)$1

or $H(t)$2 in alternative notation, and expectation values are evaluated in the corresponding time-dependent inner product (Cius et al., 2022). In this framework, observables in the Hermitian picture are mapped into the non-Hermitian picture by similarity transformation, and the evolution operator becomes unitary with respect to the metric rather than the bare Hilbert-space norm (Fring, 2022).

2. Metric dynamics, unitarity, and observability

A central structural distinction in the literature concerns whether the metric is allowed to depend explicitly on time. When $H(t)$3 is time-dependent, the relevant relation is the time-dependent quasi-Hermiticity equation

$H(t)$4

which guarantees conservation of the metric norm $H(t)$5 and therefore unitary evolution in the physical inner product (Fring et al., 2015). In this formulation, however, the non-Hermitian Hamiltonian $H(t)$6 ceases to be an observable; the observable Hamiltonian is instead its Hermitian partner $H(t)$7 (Fring et al., 2015).

A different program imposes observability of $H(t)$8 itself by demanding a time-independent metric. This is achieved through the Schrödinger-like condition

$H(t)$9

under which the Dyson equation reduces to the pure similarity map

$h(t)$0

and the quasi-Hermiticity condition becomes

$h(t)$1

(Luiz et al., 2016). In the gauge-linked formulation, the same condition causes an infinite chain of time-dependent non-Hermitian Hamiltonians to collapse to a single observable Hamiltonian, since the gauge operators become time-independent or commute with the common metric (Luiz et al., 2017).

These two positions are not merely stylistic variants; they encode different admissibility criteria. One strand of the literature emphasizes unitary dynamics with a time-dependent metric and accepts that $h(t)$2 is non-observable (Fring et al., 2015). Another derives time-dependent Dyson maps designed to preserve both unitarity and observability by enforcing metric stability (Luiz et al., 2016). The distinction is foundational for interpreting spectra, observables, and the role of the generator of evolution.

3. Algebraic parameterizations and constructive methods

A major virtue of the time-dependent Dyson map is that it can often be constructed algebraically. In Lie-algebraic models one typically chooses $h(t)$3 as an exponential of generators. For $h(t)$4 and $h(t)$5 realizations, one uses either the global Hermitian form

$h(t)$6

or its Gauss decomposition

$h(t)$7

with real $h(t)$8 and complex $h(t)$9 related to $$h(t)=\eta(t)\,H(t)\,\eta^{-1}(t)+i\hbar\,\dot\eta(t)\,\eta^{-1}(t),$$0 through explicit hyperbolic parametrizations (Cius et al., 2022). Hermiticity of the transformed Hamiltonian then yields coupled real ODEs for the metric parameters; when $$h(t)=\eta(t)\,H(t)\,\eta^{-1}(t)+i\hbar\,\dot\eta(t)\,\eta^{-1}(t),$$1, the reduced equations involve $$h(t)=\eta(t)\,H(t)\,\eta^{-1}(t)+i\hbar\,\dot\eta(t)\,\eta^{-1}(t),$$2, $$h(t)=\eta(t)\,H(t)\,\eta^{-1}(t)+i\hbar\,\dot\eta(t)\,\eta^{-1}(t),$$3, and $$h(t)=\eta(t)\,H(t)\,\eta^{-1}(t)+i\hbar\,\dot\eta(t)\,\eta^{-1}(t),$$4 driven by the real and imaginary parts of the complex frequency $$h(t)=\eta(t)\,H(t)\,\eta^{-1}(t)+i\hbar\,\dot\eta(t)\,\eta^{-1}(t),$$5 (Cius et al., 2022).

For two-level systems, the map is often factorized in terms of Pauli generators. In the fractional-time setting, a Hermitian ansatz of the form

$$h(t)=\eta(t)\,H(t)\,\eta^{-1}(t)+i\hbar\,\dot\eta(t)\,\eta^{-1}(t),$$6

is used to convert the non-unitary Mittag–Leffler evolution operator of the fractional-time Schrödinger equation into a unitary propagator in the enlarged Hilbert space (Cius et al., 2022).

Lewis–Riesenfeld invariants provide a second major route. If $$h(t)=\eta(t)\,H(t)\,\eta^{-1}(t)+i\hbar\,\dot\eta(t)\,\eta^{-1}(t),$$7 and $$h(t)=\eta(t)\,H(t)\,\eta^{-1}(t)+i\hbar\,\dot\eta(t)\,\eta^{-1}(t),$$8 are invariants of the non-Hermitian and Hermitian systems, respectively, then

$$h(t)=\eta(t)\,H(t)\,\eta^{-1}(t)+i\hbar\,\dot\eta(t)\,\eta^{-1}(t),$$9

and the problem of constructing $\rho(t)=\eta^\dagger(t)\eta(t)$0 is shifted from the Hamiltonians to the invariants (Fring et al., 2021). This is particularly useful because the invariant equation is first order and linear in time, whereas direct solution of the Dyson equation may lead to nonlinear auxiliary equations. The same invariant-based logic underlies the infinite-series construction of Dyson maps and its symmetry analysis (Fring et al., 2021).

A third route uses complex point transformations, mapping an exactly solvable time-independent reference problem to an explicitly time-dependent non-Hermitian target. The induced non-Hermitian invariant is then pseudo-Hermitian, and the Dyson map is obtained as the adjoint action from that invariant to a Hermitian one (Fring et al., 2021). A fourth route is perturbative: one expands

$\rho(t)=\eta^\dagger(t)\eta(t)$1

and solves the order-by-order Dyson equations recursively. This bypasses guesswork in choosing $\rho(t)=\eta^\dagger(t)\eta(t)$2 or $\rho(t)=\eta^\dagger(t)\eta(t)$3 and naturally exposes integration-constant ambiguities that label inequivalent metrics (Fring et al., 2020).

4. Non-uniqueness, gauge linkage, and infinite families

Time-dependent Dyson maps are generically non-unique. In one formulation, starting from $\rho(t)=\eta^\dagger(t)\eta(t)$4 and a Dyson map $\rho(t)=\eta^\dagger(t)\eta(t)$5, one may generate a chain of non-Hermitian Hamiltonians $\rho(t)=\eta^\dagger(t)\eta(t)$6 linked by unitary gauge operators

$\rho(t)=\eta^\dagger(t)\eta(t)$7

with transformation law

$\rho(t)=\eta^\dagger(t)\eta(t)$8

(Luiz et al., 2017). Although the Hamiltonians in the chain are non-observable in general, the matrix elements of observables associated with them are linked to each other, and when the gauges are global they become identical (Luiz et al., 2017).

A more explicit infinite-series mechanism starts from two distinct seed maps, $\rho(t)=\eta^\dagger(t)\eta(t)$9 and $\mathcal{PT}$0, for the same $\mathcal{PT}$1. Defining

$\mathcal{PT}$2

one obtains a non-Abelian gauge relation between the Hermitian partners,

$\mathcal{PT}$3

and, subject to Hermiticity conditions on the adjoint action of $\mathcal{PT}$4 on the Lewis–Riesenfeld invariants, one generates an infinite family

$\mathcal{PT}$5

(Fring et al., 2021). The associated ambiguity in the metric is tied directly to symmetries of the invariants, notably operators such as $\mathcal{PT}$6 and $\mathcal{PT}$7 (Fring et al., 2021).

This non-uniqueness is not merely formal. Different admissible choices of $\mathcal{PT}$8 lead to different metrics and therefore to different inner products and expectation values. The perturbative literature makes this dependence explicit: homogeneous solutions added at any perturbative order leave the Dyson relation intact at that order but change the final metric and the expectation values of observables such as the instantaneous energy operator (Fring et al., 2020). Conversely, imposing observability through a time-independent metric suppresses this proliferation and collapses the gauge-linked chain (Luiz et al., 2017).

5. Representative physical realizations

A striking application occurs in bosonic two-mode realizations of $\mathcal{PT}$9 and $|\phi(t)\rangle=\eta(t)|\Psi(t)\rangle$0, where

$|\phi(t)\rangle=\eta(t)|\Psi(t)\rangle$1

for $|\phi(t)\rangle=\eta(t)|\Psi(t)\rangle$2, and

$|\phi(t)\rangle=\eta(t)|\Psi(t)\rangle$3

for $|\phi(t)\rangle=\eta(t)|\Psi(t)\rangle$4 (Cius et al., 2022). Even when $|\phi(t)\rangle=\eta(t)|\Psi(t)\rangle$5 contains no cross-operators, the time-dependent metric $|\phi(t)\rangle=\eta(t)|\Psi(t)\rangle$6 contains $|\phi(t)\rangle=\eta(t)|\Psi(t)\rangle$7 and $|\phi(t)\rangle=\eta(t)|\Psi(t)\rangle$8, so the Hermitian picture couples the two modes. As a consequence, an initial product state evolves into an entangled state. The effect is quantified by the linear entropy

$|\phi(t)\rangle=\eta(t)|\Psi(t)\rangle$9

which grows with the squeeze parameter $|\Psi(t)\rangle=\eta(t)|\psi(t)\rangle$0, diverging at finite time for $|\Psi(t)\rangle=\eta(t)|\psi(t)\rangle$1 in the $|\Psi(t)\rangle=\eta(t)|\psi(t)\rangle$2 case, while oscillating in the $|\Psi(t)\rangle=\eta(t)|\psi(t)\rangle$3 case with a maximum fixed by the total boson number (Cius et al., 2022).

Time-dependent Dyson maps are also used to rehabilitate broken $|\Psi(t)\rangle=\eta(t)|\psi(t)\rangle$4 regimes. For the two-dimensional oscillator with non-Hermitian $|\Psi(t)\rangle=\eta(t)|\psi(t)\rangle$5 coupling, a four-parameter Dyson map built from a closed operator algebra yields a positive metric

$|\Psi(t)\rangle=\eta(t)|\psi(t)\rangle$6

throughout the unbroken, broken, and exceptional-point regimes, provided the parameters remain real (Fring et al., 2018). The transformed Hamiltonian decouples into two Swanson-type Hamiltonians with time-dependent coefficients, and the construction renders the model consistent even where the time-independent theory had partially complex spectrum (Fring et al., 2018). Closely related two-dimensional explicitly time-dependent models have been solved exactly by comparing direct Dyson-equation methods with Lewis–Riesenfeld invariants; the latter bypass the dissipative Ermakov–Pinney equation and lead to real instantaneous energy expectation values in the broken $|\Psi(t)\rangle=\eta(t)|\psi(t)\rangle$7 regime (Fring et al., 2018).

In finite-dimensional settings, a two-level model

$|\Psi(t)\rangle=\eta(t)|\psi(t)\rangle$8

has been treated with a Hermitian Dyson map expanded in the Pauli basis. The consistency conditions reduce to an auxiliary Ermakov–Pinney equation, and the physically relevant observable becomes the non-Hermitian energy operator

$|\Psi(t)\rangle=\eta(t)|\psi(t)\rangle$9

whose instantaneous eigenvalues remain real even when the original Hamiltonian is in its broken $$h(t)=\eta(t)\,H(t)\,\eta^{-1}(t)+i\hbar\,\dot\eta(t)\,\eta^{-1}(t)$$0 phase (Fring et al., 2017).

The framework also extends beyond ordinary Schrödinger dynamics. In the fractional-time scenario, the evolution operator

$$h(t)=\eta(t)\,H(t)\,\eta^{-1}(t)+i\hbar\,\dot\eta(t)\,\eta^{-1}(t)$$1

is non-unitary in the standard norm, but becomes unitary with respect to a dynamical metric

$$h(t)=\eta(t)\,H(t)\,\eta^{-1}(t)+i\hbar\,\dot\eta(t)\,\eta^{-1}(t)$$2

constructed from a time-dependent Dyson map (Cius et al., 2022). In non-Hermitian spin-boson models, a Dyson map containing a squeezing transformation maps the Schütte–Da Providência Hamiltonian to a Hermitian partner with real instantaneous energy spectrum in an admissible bounded regime; the squeezing term generates a dilatation contribution

$$h(t)=\eta(t)\,H(t)\,\eta^{-1}(t)+i\hbar\,\dot\eta(t)\,\eta^{-1}(t)$$3

so the Hermitian partner acquires a moving-boundary interpretation, and boundary motion opens transitions between sectors differing by two bosonic quanta (Fring et al., 19 May 2026).

6. Conceptual implications, misconceptions, and recurring issues

A persistent misconception is that explicit time dependence of the metric is incompatible with unitary evolution. This claim was explicitly challenged by the demonstration that the time-dependent Dyson equation and the time-dependent quasi-Hermiticity relation can be solved consistently for a time-dependent Dyson map and time-dependent metric operator, at the price of rendering the non-Hermitian Hamiltonian non-observable (Fring et al., 2015). A different misconception is that absence of explicit cross-terms in $$h(t)=\eta(t)\,H(t)\,\eta^{-1}(t)+i\hbar\,\dot\eta(t)\,\eta^{-1}(t)$$4 implies absence of mode entanglement; the two-mode bosonic analysis shows that the metric itself can couple the modes “behind the scenes,” and the entanglement is then generated by the non-locality of the dynamical Hilbert-space structure rather than by a direct interaction term (Cius et al., 2022).

A second recurring issue is the status of the Hamiltonian versus the energy operator. In explicitly time-dependent non-Hermitian systems, the generator of evolution need not coincide with the observable energy. This separation is especially sharp in the “mending” of broken $$h(t)=\eta(t)\,H(t)\,\eta^{-1}(t)+i\hbar\,\dot\eta(t)\,\eta^{-1}(t)$$5 regimes, where the non-Hermitian Hamiltonian may remain non-observable while the corresponding energy operator possesses real instantaneous spectrum and an unbroken antilinear symmetry (Fring et al., 2017).

A third issue is metric ambiguity. The review literature emphasizes that even for a fixed $$h(t)=\eta(t)\,H(t)\,\eta^{-1}(t)+i\hbar\,\dot\eta(t)\,\eta^{-1}(t)$$6 the solution $$h(t)=\eta(t)\,H(t)\,\eta^{-1}(t)+i\hbar\,\dot\eta(t)\,\eta^{-1}(t)$$7 is not unique, and that this ambiguity may lead to infinite series of metric operators (Fring, 2022). The same review also places the Dyson-map program alongside time-dependent Darboux transformations, Lewis–Riesenfeld invariants, point transformations, and approximation methods, and reports that explicitly time-dependent systems can render parameter regions physical that were unphysical in the time-independent theory (Fring, 2022). In the applications discussed there, time dependence not only mends broken $$h(t)=\eta(t)\,H(t)\,\eta^{-1}(t)+i\hbar\,\dot\eta(t)\,\eta^{-1}(t)$$8 phases but also leads to a prolongation of the otherwise rapidly decaying von Neumann entropy, with the sudden death of the entropy stopped at a finite value (Fring, 2022).

Taken together, these developments define the time-dependent Dyson map as more than a technical similarity transform. It is the organizing object that determines the physical metric, controls whether $$h(t)=\eta(t)\,H(t)\,\eta^{-1}(t)+i\hbar\,\dot\eta(t)\,\eta^{-1}(t)$$9 is an observable or merely a generator, exposes hidden gauge structure and invariant symmetries, and, in concrete models, can induce entanglement, restore unitary evolution, regularize broken $\Omega(t)$00 sectors, and recast non-Hermitian dynamics into Hermitian systems with moving boundaries, squeezing, or other explicitly time-dependent effective interactions (Fring et al., 19 May 2026).