Time-Dependent Gauge Transformations

• Time-dependent gauge transformations are defined as changes in gauge parameters that explicitly vary with time, reorganizing canonical variables while preserving observable physics.
• They introduce a compensating derivative term to link gauge-related operators, ensuring dynamic equivalence in systems from electromagnetism to stochastic processes.
• These transformations find applications in diverse fields such as quantum mechanics, lattice gauge theories, and non-Hermitian systems, highlighting their role in unifying different physical descriptions.

Time-dependent gauge transformations are transformations whose parameters or implementing maps depend explicitly on time, or more generally on spacetime position. In the standard electromagnetic form they act on the potentials as

$$\phi'=\phi-\frac{1}{c}\frac{\partial \chi}{\partial t}, \qquad \mathbf A'=\mathbf A+\nabla \chi,$$

while in Hamiltonian, stochastic, and constrained formulations they appear through an inhomogeneous derivative term such as

$$H' = UHU^{-1} - i\dot U U^{-1}, \qquad Q' = \Lambda Q\Lambda^{-1}+\dot{\Lambda}\Lambda^{-1}, \qquad H_{g'}^t = U_{gg'} H_g^t U_{gg'}^\dagger + i\dot U_{gg'}U_{gg'}^\dagger$$

(Reiss, 2013, Caruso et al., 2016, Stokes et al., 2 Apr 2025). In the surveyed literature, the defining issue is therefore not the preservation of the measurable content alone, but the way explicit time dependence reorganizes canonical variables, generators, constraints, and the relation between different gauges.

1. Universal algebraic form

A recurring structure is a similarity-type transformation supplemented by a derivative term that is present only because the gauge map depends on time. In continuous-time Markov theory,

$$Q'=\Lambda Q\Lambda^{-1}+\dot{\Lambda}\Lambda^{-1}, \qquad \mathbf p'=\Lambda \mathbf p.$$

For gauge-linked non-Hermitian systems,

$$h_t' = A_t h_t A_t^{-1} + i\hbar\, (\partial_t A_t) A_t^{-1}.$$

For time-dependent constrained light-matter theories,

$$H_{g'}^t = U_{gg'} H_g^t U_{gg'}^\dagger + i\dot U_{gg'}U_{gg'}^\dagger.$$

These formulas are presented as exact transformation laws, not approximations, and in each case the derivative term is the distinguishing feature of the time-dependent case (Caruso et al., 2016, Luiz et al., 2017, Stokes et al., 2 Apr 2025).

Framework	Transformation law	Role of derivative term
Markov generators	$Q'=\Lambda Q\Lambda^{-1}+\dot{\Lambda}\Lambda^{-1}$	makes time-dependent local equivalence possible
Hermitian/non-Hermitian Hamiltonians	$$h_t' = A_t h_t A_t^{-1} + i\hbar(\partial_t A_t)A_t^{-1}$$	links gauge-related Hamiltonians
Constrained canonical theories	$$H_{g'}^t = U_{gg'} H_g^t U_{gg'}^\dagger + i\dot U_{gg'}U_{gg'}^\dagger$$	restores correct time-dependent equivalence

This algebraic pattern also appears in classical and quantum mechanics when an action is shifted by a total derivative or when an external gauge field is transformed. The canonical variables or the wave-function representation change, but the abstract dynamics is formulated as unchanged after the compensating derivative term is included. The literature therefore treats time-dependent gauge transformations less as passive relabelings than as covariant reparametrizations of evolution equations and generators (Govaerts, 2023).

2. Electromagnetic paradigm, Hamiltonians, and observables

The standard electromagnetic example remains the reference point. The gauge transformation

$$\phi'=\phi-\frac{1}{c}\frac{\partial \chi}{\partial t}, \qquad \mathbf A'=\mathbf A+\nabla \chi$$

preserves the electric and magnetic fields, but for time-dependent $\chi$ the Hamiltonian does not transform by simple unitary conjugation alone. One formulation states that a unitary transformation of an operator should obey $$H' = UHU^{-1} - i\dot U U^{-1}, \qquad Q' = \Lambda Q\Lambda^{-1}+\dot{\Lambda}\Lambda^{-1}, \qquad H_{g'}^t = U_{gg'} H_g^t U_{gg'}^\dagger + i\dot U_{gg'}U_{gg'}^\dagger$$0, whereas the Hamiltonian satisfies

$$H' = UHU^{-1} - i\dot U U^{-1}, \qquad Q' = \Lambda Q\Lambda^{-1}+\dot{\Lambda}\Lambda^{-1}, \qquad H_{g'}^t = U_{gg'} H_g^t U_{gg'}^\dagger + i\dot U_{gg'}U_{gg'}^\dagger$$1

From this it is argued that a gauge transformation is not, in general, a unitary transformation in the usual operator sense (Reiss, 2013).

The same paper illustrates the point with a charged particle in a constant electric field. In the scalar-potential description,

$$H' = UHU^{-1} - i\dot U U^{-1}, \qquad Q' = \Lambda Q\Lambda^{-1}+\dot{\Lambda}\Lambda^{-1}, \qquad H_{g'}^t = U_{gg'} H_g^t U_{gg'}^\dagger + i\dot U_{gg'}U_{gg'}^\dagger$$2

After the time-dependent gauge transformation generated by

$$H' = UHU^{-1} - i\dot U U^{-1}, \qquad Q' = \Lambda Q\Lambda^{-1}+\dot{\Lambda}\Lambda^{-1}, \qquad H_{g'}^t = U_{gg'} H_g^t U_{gg'}^\dagger + i\dot U_{gg'}U_{gg'}^\dagger$$3

the potentials become

$$H' = UHU^{-1} - i\dot U U^{-1}, \qquad Q' = \Lambda Q\Lambda^{-1}+\dot{\Lambda}\Lambda^{-1}, \qquad H_{g'}^t = U_{gg'} H_g^t U_{gg'}^\dagger + i\dot U_{gg'}U_{gg'}^\dagger$$4

and the Hamiltonian becomes

$$H' = UHU^{-1} - i\dot U U^{-1}, \qquad Q' = \Lambda Q\Lambda^{-1}+\dot{\Lambda}\Lambda^{-1}, \qquad H_{g'}^t = U_{gg'} H_g^t U_{gg'}^\dagger + i\dot U_{gg'}U_{gg'}^\dagger$$5

The equation of motion remains $$H' = UHU^{-1} - i\dot U U^{-1}, \qquad Q' = \Lambda Q\Lambda^{-1}+\dot{\Lambda}\Lambda^{-1}, \qquad H_{g'}^t = U_{gg'} H_g^t U_{gg'}^\dagger + i\dot U_{gg'}U_{gg'}^\dagger$$6, and the kinetic energy is unchanged, $$H' = UHU^{-1} - i\dot U U^{-1}, \qquad Q' = \Lambda Q\Lambda^{-1}+\dot{\Lambda}\Lambda^{-1}, \qquad H_{g'}^t = U_{gg'} H_g^t U_{gg'}^\dagger + i\dot U_{gg'}U_{gg'}^\dagger$$7, but the potential-energy interpretation is altered: the transformed description has no scalar potential and yields $$H' = UHU^{-1} - i\dot U U^{-1}, \qquad Q' = \Lambda Q\Lambda^{-1}+\dot{\Lambda}\Lambda^{-1}, \qquad H_{g'}^t = U_{gg'} H_g^t U_{gg'}^\dagger + i\dot U_{gg'}U_{gg'}^\dagger$$8 rather than a manifestly conserved $$H' = UHU^{-1} - i\dot U U^{-1}, \qquad Q' = \Lambda Q\Lambda^{-1}+\dot{\Lambda}\Lambda^{-1}, \qquad H_{g'}^t = U_{gg'} H_g^t U_{gg'}^\dagger + i\dot U_{gg'}U_{gg'}^\dagger$$9 (Reiss, 2013).

A complementary analysis formulates the same issue in terms of canonical versus mechanical observables. The mechanical momentum

$$Q'=\Lambda Q\Lambda^{-1}+\dot{\Lambda}\Lambda^{-1}, \qquad \mathbf p'=\Lambda \mathbf p.$$0

is gauge invariant, whereas the canonical momentum is gauge dependent. For time-dependent gauge functions the Hamiltonian transforms as

$$Q'=\Lambda Q\Lambda^{-1}+\dot{\Lambda}\Lambda^{-1}, \qquad \mathbf p'=\Lambda \mathbf p.$$1

As a result, quantities such as canonical angular momentum can lose their simple commutator conservation law in one gauge even when the corresponding expectation values remain constant. The paper also shows that singular gauge transformations can move physical information from the Hamiltonian into wave-function boundary conditions, producing twisted or multivalued states on a ring (Berche et al., 2016).

A representation-theoretic treatment sharpens the distinction further. When the action is modified by a total derivative, the classical equations remain unchanged but the canonical momenta shift, and in quantum theory the unitary configuration-space representation of the Heisenberg algebra changes by a local phase. In this formulation, abstract states and abstract observables are gauge invariant, while their wave-function representatives are gauge covariant (Govaerts, 2023).

3. Local spacetime dependence and geometric reinterpretations

Several works reinterpret time-dependent gauge transformations as local changes of spacetime structure rather than merely internal phase rotations. In the gauge theory of the gravitational-electromagnetic field, the synchrony transformation is introduced as

$$Q'=\Lambda Q\Lambda^{-1}+\dot{\Lambda}\Lambda^{-1}, \qquad \mathbf p'=\Lambda \mathbf p.$$2

with commuting synchrony generators $$Q'=\Lambda Q\Lambda^{-1}+\dot{\Lambda}\Lambda^{-1}, \qquad \mathbf p'=\Lambda \mathbf p.$$3. Gauging the synchrony group yields the covariant derivative

$$Q'=\Lambda Q\Lambda^{-1}+\dot{\Lambda}\Lambda^{-1}, \qquad \mathbf p'=\Lambda \mathbf p.$$4

and the local gauge fields transform as

$$Q'=\Lambda Q\Lambda^{-1}+\dot{\Lambda}\Lambda^{-1}, \qquad \mathbf p'=\Lambda \mathbf p.$$5

The paper identifies these synchrony gauge fields with electromagnetism and interprets electromagnetic gauge transformations as local clock-synchronization transformations, hence as changes in simultaneity convention and the one-way speed of light (Bock, 2015).

Twistless torsional Newton–Cartan geometry provides a different geometric realization. In type I TTNC, the Bargmann form transforms as

$$Q'=\Lambda Q\Lambda^{-1}+\dot{\Lambda}\Lambda^{-1}, \qquad \mathbf p'=\Lambda \mathbf p.$$6

and the equation required to set the locally Galilei-invariant potential $$Q'=\Lambda Q\Lambda^{-1}+\dot{\Lambda}\Lambda^{-1}, \qquad \mathbf p'=\Lambda \mathbf p.$$7 to zero is

$$Q'=\Lambda Q\Lambda^{-1}+\dot{\Lambda}\Lambda^{-1}, \qquad \mathbf p'=\Lambda \mathbf p.$$8

In local coordinates this becomes a Hamilton–Jacobi equation. In type II TTNC, the corresponding local gauge fixing is achieved by subleading spatial diffeomorphisms rather than a $$Q'=\Lambda Q\Lambda^{-1}+\dot{\Lambda}\Lambda^{-1}, \qquad \mathbf p'=\Lambda \mathbf p.$$9-type transformation. The result is a local normal form in which the relevant part of the Bargmann form can be removed (Blanckenburg et al., 2024).

A more operator-theoretic geometric construction appears for relativistic fields of quantum harmonic-oscillator states over Minkowski space. The spacetime-dependent unitary

$$h_t' = A_t h_t A_t^{-1} + i\hbar\, (\partial_t A_t) A_t^{-1}.$$0

defines transformed fields $$h_t' = A_t h_t A_t^{-1} + i\hbar\, (\partial_t A_t) A_t^{-1}.$$1, and differentiation yields a self-adjoint gauge potential $$h_t' = A_t h_t A_t^{-1} + i\hbar\, (\partial_t A_t) A_t^{-1}.$$2 through

$$h_t' = A_t h_t A_t^{-1} + i\hbar\, (\partial_t A_t) A_t^{-1}.$$3

The transport operators

$$h_t' = A_t h_t A_t^{-1} + i\hbar\, (\partial_t A_t) A_t^{-1}.$$4

satisfy groupoid composition laws, so gauge transformation is formulated as transport between local frames rather than solely as multiplication by a phase (Naudts, 2020).

In generalized $$h_t' = A_t h_t A_t^{-1} + i\hbar\, (\partial_t A_t) A_t^{-1}.$$5 space-time, gauge transformations are expressed as generalized diffeomorphisms on the full coordinate set $$h_t' = A_t h_t A_t^{-1} + i\hbar\, (\partial_t A_t) A_t^{-1}.$$6, with nonlinear variation

$$h_t' = A_t h_t A_t^{-1} + i\hbar\, (\partial_t A_t) A_t^{-1}.$$7

Here the dependence is on generalized coordinates rather than on physical time alone, but the formalism enlarges the notion of local gauge transformation in a way that subsumes familiar supergravity gauge symmetries (West, 2014).

4. Gauge-parameter flow, BRST structure, and correlation functions

A distinct but closely related line of work treats changes of gauge parameter as canonical flows. In SU($$h_t' = A_t h_t A_t^{-1} + i\hbar\, (\partial_t A_t) A_t^{-1}.$$8) Yang–Mills theory, the gauge parameter $$h_t' = A_t h_t A_t^{-1} + i\hbar\, (\partial_t A_t) A_t^{-1}.$$9 is promoted to a BRST doublet,

$$H_{g'}^t = U_{gg'} H_g^t U_{gg'}^\dagger + i\dot U_{gg'}U_{gg'}^\dagger.$$0

and the effective action satisfies the extended Slavnov–Taylor identity

$$H_{g'}^t = U_{gg'} H_g^t U_{gg'}^\dagger + i\dot U_{gg'}U_{gg'}^\dagger.$$1

Differentiation with respect to $$H_{g'}^t = U_{gg'} H_g^t U_{gg'}^\dagger + i\dot U_{gg'}U_{gg'}^\dagger.$$2 and then setting $$H_{g'}^t = U_{gg'} H_g^t U_{gg'}^\dagger + i\dot U_{gg'}U_{gg'}^\dagger.$$3 gives

$$H_{g'}^t = U_{gg'} H_g^t U_{gg'}^\dagger + i\dot U_{gg'}U_{gg'}^\dagger.$$4

so gauge-parameter dependence is generated by a canonical transformation in BV space. Because the generator $$H_{g'}^t = U_{gg'} H_g^t U_{gg'}^\dagger + i\dot U_{gg'}U_{gg'}^\dagger.$$5 depends on $$H_{g'}^t = U_{gg'} H_g^t U_{gg'}^\dagger + i\dot U_{gg'}U_{gg'}^\dagger.$$6, the solution is a Lie series,

$$H_{g'}^t = U_{gg'} H_g^t U_{gg'}^\dagger + i\dot U_{gg'}U_{gg'}^\dagger.$$7

For the transverse gluon propagator this yields a multiplicative flow,

$$H_{g'}^t = U_{gg'} H_g^t U_{gg'}^\dagger + i\dot U_{gg'}U_{gg'}^\dagger.$$8

(Quadri, 2014).

In QCD linear covariant gauges, Landau–Khalatnikov–Fradkin transformations and Nielsen identities are presented as equivalent consequences of the same generalized Slavnov–Taylor identity. The extension

$$H_{g'}^t = U_{gg'} H_g^t U_{gg'}^\dagger + i\dot U_{gg'}U_{gg'}^\dagger.$$9

allows the derivative with respect to $Q'=\Lambda Q\Lambda^{-1}+\dot{\Lambda}\Lambda^{-1}$0 to be treated as a BRST-exact insertion. On one side, correlators of gauge-invariant composite fields such as $Q'=\Lambda Q\Lambda^{-1}+\dot{\Lambda}\Lambda^{-1}$1 are $Q'=\Lambda Q\Lambda^{-1}+\dot{\Lambda}\Lambda^{-1}$2-independent and generate LKFT relations after Stueckelberg expansion. On the other, differentiation of ordinary correlators yields Nielsen identities. The explicit one-loop transformation of the gluon propagator displays a fixed longitudinal contribution,

$Q'=\Lambda Q\Lambda^{-1}+\dot{\Lambda}\Lambda^{-1}$3

and a nontrivial transverse $Q'=\Lambda Q\Lambda^{-1}+\dot{\Lambda}\Lambda^{-1}$4-dependent term (Meerleer et al., 2019).

This body of work concerns flow in gauge-parameter space rather than physical time. It nonetheless places gauge variation in the same formal family as time-dependent canonical transformations: the generator depends on the evolution parameter, naive exponentiation fails, and the correct solution is non-autonomous.

5. Nonlinear gauge functionals, constrained theories, and restricted gauge freedom

Time dependence can invalidate naive equivalence between gauges when the gauge transformation itself depends on dynamical fields or when the physical control mechanism enters through time-dependent constraints. In nonlinear gauge-coupled quantum fluids, the basic $Q'=\Lambda Q\Lambda^{-1}+\dot{\Lambda}\Lambda^{-1}$5 transformation is

$Q'=\Lambda Q\Lambda^{-1}+\dot{\Lambda}\Lambda^{-1}$6

If $Q'=\Lambda Q\Lambda^{-1}+\dot{\Lambda}\Lambda^{-1}$7 is an external gauge function, the canonical hydrodynamic equations are form-invariant. If $Q'=\Lambda Q\Lambda^{-1}+\dot{\Lambda}\Lambda^{-1}$8 is a nonlinear gauge functional of the density, the phase equation acquires extra terms, so the transformed equations are no longer identical. For the one-dimensional superfluid with $Q'=\Lambda Q\Lambda^{-1}+\dot{\Lambda}\Lambda^{-1}$9, an attempted gauge removal of the nonlinear vector potential simply reappears as the gauge-pressure contribution in the Hamiltonian density (Buggy et al., 2020).

Time-dependent light-matter theories with holonomic constraints sharpen the problem. A general framework based on explicitly time-dependent constrained Lagrangians shows that different gauges are canonically equivalent only if the time dependence is introduced at the Lagrangian level from the outset. If one starts with a time-independent Hamiltonian and then inserts time dependence by hand, different gauges generally produce non-equivalent canonical theories. The paper defines the irrotational gauge by the condition

$$h_t' = A_t h_t A_t^{-1} + i\hbar(\partial_t A_t)A_t^{-1}$$0

so that naive Hamiltonian-level modulation is correct only in that gauge. In this formulation the Coulomb gauge is not generally irrotational for time-dependent light-matter interactions, and the paper explicitly states that this contradicts the conclusions in Phys. Rev. A 107, 013722 (2023) and Phys. Rev. Research 3, 023079 (2021), while reaffirming the prior treatment reported in Phys. Rev. Research 3, 013116 (2021) (Stokes et al., 2 Apr 2025).

A further restriction arises when the gauge coupling itself depends on spacetime. In Abelian theories the allowed gauge parameter is no longer an arbitrary function of spacetime but must satisfy $$h_t' = A_t h_t A_t^{-1} + i\hbar(\partial_t A_t)A_t^{-1}$$1, and in non-Abelian theory the allowed group element is restricted to $$h_t' = A_t h_t A_t^{-1} + i\hbar(\partial_t A_t)A_t^{-1}$$2. The corresponding consistency conditions are written as

$$h_t' = A_t h_t A_t^{-1} + i\hbar(\partial_t A_t)A_t^{-1}$$3

Gauge invariance therefore survives, but only in a restricted form compatible with the coupling profile (Mohammedi, 2011).

Taken together, these results distinguish two different claims that are often conflated. The first is covariance under a formally defined gauge transformation. The second is equivalence of the resulting canonical theories after explicit time dependence, field dependence, or background dependence is introduced. Several of the surveyed works accept the first while denying the second.

6. Extensions, constructive uses, and nonstandard domains

Time-dependent gauge transformations also function as constructive tools outside standard gauge-field dynamics. For continuous-time Markov processes on denumerable state spaces, the local transformation

$$h_t' = A_t h_t A_t^{-1} + i\hbar(\partial_t A_t)A_t^{-1}$$4

defines an equivalence relation on generators, and for any pair of differentiable matrices $$h_t' = A_t h_t A_t^{-1} + i\hbar(\partial_t A_t)A_t^{-1}$$5 and $$h_t' = A_t h_t A_t^{-1} + i\hbar(\partial_t A_t)A_t^{-1}$$6 of the same dimension there exists a non-singular $$h_t' = A_t h_t A_t^{-1} + i\hbar(\partial_t A_t)A_t^{-1}$$7 satisfying

$$h_t' = A_t h_t A_t^{-1} + i\hbar(\partial_t A_t)A_t^{-1}$$8

When state spaces have different cardinality, the smaller process is first dilated by redundant absorbing states; the same strategy also treats time-dependent state spaces by embedding them into a fixed ambient space (Caruso et al., 2016).

In time-dependent non-Hermitian quantum mechanics, a time-dependent Dyson map

$$h_t' = A_t h_t A_t^{-1} + i\hbar(\partial_t A_t)A_t^{-1}$$9

permits the construction of an infinite chain of gauge-linked non-observable Hamiltonians. If the interlinking gauge operators are global,

$$H_{g'}^t = U_{gg'} H_g^t U_{gg'}^\dagger + i\dot U_{gg'}U_{gg'}^\dagger$$0

the observable matrix elements coincide across the chain. If the Dyson maps are instead chosen to enforce observability through

$$H_{g'}^t = U_{gg'} H_g^t U_{gg'}^\dagger + i\dot U_{gg'}U_{gg'}^\dagger$$1

the chain collapses to a single observable Hamiltonian (Luiz et al., 2017).

In tight-binding transport and pulse propagation, a local unitary transformation

$$H_{g'}^t = U_{gg'} H_g^t U_{gg'}^\dagger + i\dot U_{gg'}U_{gg'}^\dagger$$2

removes a time-dependent onsite potential from site $$H_{g'}^t = U_{gg'} H_g^t U_{gg'}^\dagger + i\dot U_{gg'}U_{gg'}^\dagger$$3. The transformed Hamiltonian acquires phase-dressed hoppings, with outgoing terms from site $$H_{g'}^t = U_{gg'} H_g^t U_{gg'}^\dagger + i\dot U_{gg'}U_{gg'}^\dagger$$4 multiplied by $$H_{g'}^t = U_{gg'} H_g^t U_{gg'}^\dagger + i\dot U_{gg'}U_{gg'}^\dagger$$5 and incoming terms by $$H_{g'}^t = U_{gg'} H_g^t U_{gg'}^\dagger + i\dot U_{gg'}U_{gg'}^\dagger$$6. Applied across an entire lead, the phases cancel on internal bonds and survive only at the interface, converting an extended bulk drive into a localized time-dependent boundary condition (Abbout, 11 Mar 2026).

Quantum simulation of lattice gauge theories uses gauge transformations operationally rather than representationally. One proposal combines random local gauge transformations with a Zeno-effect projection circuit built from an ancilla qubit. For a successful projection the average squared suppression factor of unphysical components is

$$H_{g'}^t = U_{gg'} H_g^t U_{gg'}^\dagger + i\dot U_{gg'}U_{gg'}^\dagger$$7

and in a pure 1D SU(2) toy model the measured average suppression factor over 5,050 projection events is

$$H_{g'}^t = U_{gg'} H_g^t U_{gg'}^\dagger + i\dot U_{gg'}U_{gg'}^\dagger$$8

The same paper proposes applying random local gauge transformations during time evolution to slow gauge drift before projection is required (Ball, 2024).

A different Hamiltonian use occurs for conserved charges with explicit time dependence. For a particle in a constant force field, the charge

$$H_{g'}^t = U_{gg'} H_g^t U_{gg'}^\dagger + i\dot U_{gg'}U_{gg'}^\dagger$$9

is conserved even though

$$\phi'=\phi-\frac{1}{c}\frac{\partial \chi}{\partial t}, \qquad \mathbf A'=\mathbf A+\nabla \chi$$0

because the explicit time derivative cancels the Poisson bracket contribution. Gauging this symmetry introduces the first-class constraint $$\phi'=\phi-\frac{1}{c}\frac{\partial \chi}{\partial t}, \qquad \mathbf A'=\mathbf A+\nabla \chi$$1 through

$$\phi'=\phi-\frac{1}{c}\frac{\partial \chi}{\partial t}, \qquad \mathbf A'=\mathbf A+\nabla \chi$$2

thereby allowing arbitrary time-dependent spatial translations (Govaerts, 2019).

A further modification of Yang–Mills theory promotes finite gauge-transformation parameters themselves to dynamical variables and imposes a condition of classicality on them. In this formulation the Hamiltonian separates transverse physical degrees of freedom from longitudinal pure-gauge motion in group space (Gorobey et al., 2020).

Across these diverse settings, time-dependent gauge transformations are not treated as a narrow electromagnetic technicality. They appear as a general mechanism for relating descriptions with derivative-corrected generators, for transferring explicit time dependence between different sectors of a theory, for distinguishing representation change from physical equivalence, and for constructing workable descriptions in stochastic, geometric, lattice, transport, and non-Hermitian problems.