Phase-Wave State: Gauge, Geometry & Quantum Aspects

• Phase-wave state is a state where phase is explicitly decomposed alongside amplitude, providing a clear, measurable variable in quantum and wave theories.
• It unifies gauge-fixing approaches in the Schrödinger framework with experimental observables in optics, magnonics, and swarmalator dynamics.
• The framework bridges quantum phase problems, geometric phase tracking, and collective dynamics, offering novel insights into scattering, interference, and signal propagation.

Searching arXiv for papers directly relevant to “phase-wave state” and closely related formulations. “Phase-Wave State” does not denote a single universally fixed construct across contemporary wave and quantum theory. In the cited literature, it refers to several closely related but distinct ideas: a quantum state represented explicitly by an amplitude field and a phase field rather than by a single complex wavefunction; a polarized optical wave whose geometric phase is treated as a continuously evolving property of the wave state itself; a coherently driven spin-wave state whose local precessional phase is directly imaged; a phase-ordered collective state in swarmalator dynamics; and a phase-dependent wavefunction on the circle in Hardy space, used to reformulate the quantum phase problem. Taken together, these usages describe states for which phase is promoted from a hidden parameter or global label to an explicit dynamical, geometric, or measurable degree of freedom (Rau, 30 Jun 2025).

1. Conceptual scope

In the most direct quantum-mechanical sense, a phase-wave state is a state reconstructed from separate amplitude and phase variables. In the radial stationary Schrödinger setting, this appears as

$u(r)=\alpha(r)e^{\pm i\phi(r)}$

or equivalently

$$u(r)=K^{-1/2}(r)\exp\!\left[\pm i\int^r K(r')\,dr'\right],$$

with $\alpha(r)=K^{-1/2}(r)$ and $d\phi/dr=K(r)=\alpha^{-2}(r)$ (Rau, 30 Jun 2025). In this usage, the phase and amplitude are both real quantities and “carry the same information that is contained in the complex wave function.”

A second usage treats phase as a property of the wave state itself rather than only of a closed cycle or of an abstract state-space construction. In optical geometric-phase work, the relevant quantity is the shift of the resultant wave maximum produced by superposition, or, in the polarized-wave formulation, the continuously defined geometric phase $\gamma$ of a Jones state at each point along propagation (Garza-Soto et al., 2022). A related extension states that every polarized wave has a well-defined $\gamma$ at any point in its path, so that the conventional cyclic geometric phase is reinterpreted as $$\Delta\gamma=\gamma_{\text{end}}-\gamma_{\text{start}}$$ (Hagen et al., 2024).

A third usage is operational and experimental. In phase-resolving spin-wave microscopy, a coherently driven magnonic state is characterized by both the local precession amplitude and the local precessional phase $\phi_m(\mathbf r)$ relative to a microwave drive. The wavefronts in the measured images are precisely contours of constant $\phi_m(\mathbf r)$, so the state is defined not only by intensity but by a spatial phase field (Xiong et al., 2024).

In collective dynamics, the term “phase wave” refers to ordered configurations in which position and phase are correlated. In the inertial 1D swarmalator model, the static phase wave is an exact state with one rainbow order parameter maximal and the other vanishing, for example $(r,s)=(1,0)$, while inertia destabilizes this branch and generates the unsteady “thrashing phase wave” (O'Keeffe, 13 Mar 2026).

These usages are not identical, but they share one structural feature: the phase is treated as a constitutive feature of the state rather than as an eliminable accessory.

2. Amplitude–phase quantum states and gauge structure

The most explicit phase-wave-state formalism in the supplied literature is the exact phase–amplitude reformulation of the radial Schrödinger equation

$$u(r)=K^{-1/2}(r)\exp\!\left[\pm i\int^r K(r')\,dr'\right],$$0

with $$u(r)=K^{-1/2}(r)\exp\!\left[\pm i\int^r K(r')\,dr'\right],$$1 (Rau, 30 Jun 2025). The Milne–Young–Wheeler representation writes the exact solution in a JWKB-like form,

$$u(r)=K^{-1/2}(r)\exp\!\left[\pm i\int^r K(r')\,dr'\right],$$2

so that the amplitude is $$u(r)=K^{-1/2}(r)\exp\!\left[\pm i\int^r K(r')\,dr'\right],$$3 and the phase is $$u(r)=K^{-1/2}(r)\exp\!\left[\pm i\int^r K(r')\,dr'\right],$$4.

The amplitude and phase are constrained by nonlinear equations equivalent to the original linear problem. Replacing the approximate JWKB wavenumber $$u(r)=K^{-1/2}(r)\exp\!\left[\pm i\int^r K(r')\,dr'\right],$$5 by an exact $$u(r)=K^{-1/2}(r)\exp\!\left[\pm i\int^r K(r')\,dr'\right],$$6 yields

$$u(r)=K^{-1/2}(r)\exp\!\left[\pm i\int^r K(r')\,dr'\right],$$7

and, in terms of $$u(r)=K^{-1/2}(r)\exp\!\left[\pm i\int^r K(r')\,dr'\right],$$8,

$$u(r)=K^{-1/2}(r)\exp\!\left[\pm i\int^r K(r')\,dr'\right],$$9

In this formulation, the amplitude satisfies a self-contained nonlinear equation and the phase is obtained afterward by quadrature through $\alpha(r)=K^{-1/2}(r)$0 (Rau, 30 Jun 2025).

The Dashen–Babikov–Calogero method reverses that emphasis. For real solutions,

$\alpha(r)=K^{-1/2}(r)$1

with a companion derivative constraint, one obtains a stand-alone nonlinear phase equation,

$\alpha(r)=K^{-1/2}(r)$2

while the amplitude follows from

$\alpha(r)=K^{-1/2}(r)$3

The contrast is explicit: one exact reformulation produces a self-contained amplitude equation, the other a self-contained phase equation (Rau, 30 Jun 2025).

The paper’s most original move is to interpret this separation as gauge fixing. Eq. (7) is rewritten as

$\alpha(r)=K^{-1/2}(r)$4

and the transformation

$\alpha(r)=K^{-1/2}(r)$5

is presented as the counterpart of the local gauge transformation in quantum electrodynamics. Choosing $\alpha(r)=K^{-1/2}(r)$6 gives the Milne–Young–Wheeler form, while choosing $\alpha(r)=K^{-1/2}(r)$7 returns the original Schrödinger equation. This suggests that the ordinary complex-wavefunction description and the explicit amplitude–phase description are gauge-fixed representations of the same underlying content (Rau, 30 Jun 2025).

The same paper links this formalism to bound states, scattering states, resonances in a multi-channel context, variable-phase scattering theory, and quantum defect theory. In scattering, the asymptotic phase shift is emphasized as directly observable, while the amplitude acquires an adjoint variational role through $\alpha(r)=K^{-1/2}(r)$8 (Rau, 30 Jun 2025).

3. Geometric phase as a wave-state property

A separate but closely allied meaning of phase-wave state arises in geometric-phase theory. One line of work argues that optical geometric phase derives entirely from wave superposition and the resulting shift in the location of the wave maximum. For two scalar waves,

$\alpha(r)=K^{-1/2}(r)$9

their sum is another sinusoid with resultant phase $d\phi/dr=K(r)=\alpha^{-2}(r)$0, and $d\phi/dr=K(r)=\alpha^{-2}(r)$1 is interpreted not merely algebraically but as the spatial shift of the maximum of the sum wave relative to a reference plane (Garza-Soto et al., 2022). In the symmetric convention $d\phi/dr=K(r)=\alpha^{-2}(r)$2,

$d\phi/dr=K(r)=\alpha^{-2}(r)$3

If $d\phi/dr=K(r)=\alpha^{-2}(r)$4, the resultant maximum remains at the midpoint and $d\phi/dr=K(r)=\alpha^{-2}(r)$5; otherwise the maximum shifts toward the stronger component.

For polarized waves with orthogonal components,

$d\phi/dr=K(r)=\alpha^{-2}(r)$6

the phase is defined by the position at which the electric vector reaches maximum displacement along the major axis of the polarization ellipse. The resulting phase formula is

$d\phi/dr=K(r)=\alpha^{-2}(r)$7

or, in the symmetric reference choice $d\phi/dr=K(r)=\alpha^{-2}(r)$8,

$d\phi/dr=K(r)=\alpha^{-2}(r)$9

The claim is not that Jones calculus or Poincaré-sphere methods are incorrect, but that they obscure the physical mechanism by hiding the shift of the wave maximum behind algebraic or geometric constructions (Garza-Soto et al., 2022).

A later extension generalizes this claim from closed cycles to continuously evolving wave states. With Jones vector

$\gamma$0

the geometric phase is defined locally by

$\gamma$1

This assigns $\gamma$2 to any polarized wave, not only to a completed loop. Optical elements are sliced into thin pieces,

$\gamma$3

so that $\gamma$4 can be tracked continuously through propagation (Hagen et al., 2024).

Within that framework, the conventional closed-loop geometric phase becomes a change in a continuously defined phase-state quantity,

$\gamma$5

The same work argues that this continuous tracking gives a natural explanation for why Poincaré-sphere constructions use geodesic closures: in the eigenbasis of a homogeneous retarder, geometric phase does not accumulate along the physical path, and the nontrivial contribution is concentrated at basis changes between adjacent optical elements (Hagen et al., 2024).

A related but more kinematic treatment distinguishes the wave-state phase, the Pancharatnam phase, and the full geometric phase for a monochromatic polarized light wave represented by a Hilbert vector,

$\gamma$6

Using the gauge-invariant decomposition

$\gamma$7

the paper shows that a polarizer may induce a nonzero geometric phase even when the initial and final states are in phase according to the Pancharatnam criterion. The point is that the full geometric phase depends on the actual path in ray space, including nonunitary projection-like processes, whereas ordinary interferometry measures only the endpoint overlap phase (Lages et al., 2013).

4. Phase-defined experimental wave states

In contemporary magnonics, a phase-wave state is experimentally realized as a continuously driven spin-wave configuration with both amplitude and local phase resolved in space. The phase-resolving microscope of Ref. (Xiong et al., 2024) uses a 1550-nm infrared strobe synchronized to microwave excitation. The lock-in outputs satisfy

$\gamma$8

and, in imaging with an adjustable microwave detection phase $\gamma$9,

$\gamma$0

Here $\gamma$1 is the magnetic phase of interest, referenced to the coherent drive, and $\gamma$2 is the amplitude of the out-of-plane dynamic magnetization (Xiong et al., 2024).

What is reconstructed experimentally is therefore a local state of the form suggested in the paper as

$\gamma$3

or equivalently a complex field $\gamma$4, though the latter notation is described as equivalent rather than explicitly used. The phase-sensitive images directly map spin-wave wavefronts in the continuous-wave stationary regime, without an optical reference path (Xiong et al., 2024).

The measured propagation phase is modeled as

$\gamma$5

where the first term is instrumental and the second is the magnetic propagation delay from source to probe spot. For backward volume spin waves, the dispersion is written as

$\gamma$6

with $\gamma$7. This ties the phase field directly to dispersion and group velocity (Xiong et al., 2024).

The same work shows that phase is not a decorative observable. In patterned Py/YIG systems, the $\gamma$8-quadrature distinguishes in-phase from anti-phase precession even when amplitude maps are similar; detection-phase tuning moves the observed pattern from in phase to anti-phase; and in a $\gamma$9-wide gap near a Py edge, two hotspots act as coupled spin-wave sources whose relative phase determines destructive versus constructive interference in the gap region (Xiong et al., 2024). A plausible implication is that the “state” in such systems is experimentally phase-organized rather than intensity-defined.

A broader phase-centered interpretation of wavefunction structure is advanced in the theory of phase-sensitive nonadiabatic dressed states. There, coherence is defined as a phase property of the wave function itself, and a multilevel ground dressed state is written as

$$\Delta\gamma=\gamma_{\text{end}}-\gamma_{\text{start}}$$0

The crucial result is that the phases of different virtual components differ only by constants,

$$\Delta\gamma=\gamma_{\text{end}}-\gamma_{\text{start}}$$1

whereas the phase difference between a real component and a virtual component contains a rapidly varying optical term,

$$\Delta\gamma=\gamma_{\text{end}}-\gamma_{\text{start}}$$2

The author interprets these as, respectively, stationary high coherence between different virtual components and rapidly changing phase correlation between real and virtual components (Koprinkov, 12 Jul 2026).

5. Collective and hydrodynamic phase-wave states

Outside single-particle quantum mechanics, “phase wave” can describe a collective ordered state. In the inertial 1D swarmalator model, the static phase wave is a configuration in which positions and internal phases are perfectly correlated so that one rainbow order parameter is maximal and the other vanishes. For the $$\Delta\gamma=\gamma_{\text{end}}-\gamma_{\text{start}}$$3-wave,

$$\Delta\gamma=\gamma_{\text{end}}-\gamma_{\text{start}}$$4

while the symmetry-related $$\Delta\gamma=\gamma_{\text{end}}-\gamma_{\text{start}}$$5-wave has $$\Delta\gamma=\gamma_{\text{end}}-\gamma_{\text{start}}$$6 (O'Keeffe, 13 Mar 2026). This is a literal phase-wave branch in the model’s order-parameter space.

With inertia, this static state loses stability through a Hopf mechanism. Linearization leads to the characteristic equation

$$\Delta\gamma=\gamma_{\text{end}}-\gamma_{\text{start}}$$7

and the Hopf threshold

$$\Delta\gamma=\gamma_{\text{end}}-\gamma_{\text{start}}$$8

The resulting oscillatory state is the “thrashing phase wave,” characterized by sustained, multiharmonic oscillations of the rainbow order parameters and occurring in clockwise and counterclockwise symmetric pairs (O'Keeffe, 13 Mar 2026). This is a genuinely different sense of phase-wave state: not an amplitude–phase decomposition of one field, but a collective macroscopic organization of many coupled agents.

Water-wave theory provides another geometric formulation. A narrowband linear wave group is represented by a complex envelope $$\Delta\gamma=\gamma_{\text{end}}-\gamma_{\text{start}}$$9 with $\phi_m(\mathbf r)$0 phase symmetry, and the state space is interpreted as a principal fiber bundle. The total phase decomposes into dynamical and geometric parts,

$\phi_m(\mathbf r)$1

with

$\phi_m(\mathbf r)$2

For the Gaussian envelope family, the crest speed becomes

$\phi_m(\mathbf r)$3

where $\phi_m(\mathbf r)$4 is constant and the slowdown is produced by the geometric term $\phi_m(\mathbf r)$5 (Fedele, 2014). The paper interprets the observed crest slowdown in ocean wave groups as a geometric phase effect induced by shape change of the group. This suggests a phase-wave-state picture in which the physically relevant state is the envelope modulo $\phi_m(\mathbf r)$6 symmetry, and observable kinematics depend on both dynamical and geometric phase.

6. Phase-state representations and mathematical formulations

A more formal usage of phase-wave state appears in the quantum phase problem. One paper starts from the canonical phase state

$\phi_m(\mathbf r)$7

which is an eigenstate of the exponential phase operator

$\phi_m(\mathbf r)$8

Because its number-space amplitudes have constant modulus and a linear phase progression in $\phi_m(\mathbf r)$9, the state is explicitly wave-like in photon-number space. The same work recasts it as a nonlinear coherent state of $\phi_m(\mathbf r)$0, with generators

$\phi_m(\mathbf r)$1

and shows that the phase state can be written in $\phi_m(\mathbf r)$2 coherent-state form (Soto-Eguibar et al., 2014).

A different construction uses phase sampling in the Fock–Bargmann representation. For coherent-state sampling points on a circle,

$\phi_m(\mathbf r)$3

the wavefunction $\phi_m(\mathbf r)$4 is treated as a phase-number wave function $\phi_m(\mathbf r)$5. The number coefficients are recovered from its angular dependence by

$\phi_m(\mathbf r)$6

For truncated Hilbert spaces the reconstruction from discrete phase samples is exact, while for the full space the coherent-state subsystem is “almost complete,” yielding a pseudo-frame and partial reconstruction formulas (Calixto et al., 2011). This is not a phase-eigenstate formalism, but it is a mathematically precise phase-sampled wave-function formalism.

The most explicit modern “phase-wave state” language appears in a Hardy-space reformulation of phase. For

$\phi_m(\mathbf r)$7

the phase-dependent wavefunction is defined by the one-sided Fourier series

$\phi_m(\mathbf r)$8

and is required to belong to the Hardy space $\phi_m(\mathbf r)$9 rather than to all of $(r,s)=(1,0)$0 (Korneev et al., 9 Mar 2026). The analyticity requirement automatically enforces positivity of the number spectrum because only nonnegative Fourier modes are present. For the harmonic Hamiltonian $(r,s)=(1,0)$1, the phase-wave state satisfies

$(r,s)=(1,0)$2

so harmonic evolution becomes rigid translation on the circle. With weak anharmonicity,

$(r,s)=(1,0)$3

and the resulting dispersive evolution on $(r,s)=(1,0)$4 is used to explain Talbot revivals, fractional revivals, and fractal interference in multimode waveguides (Korneev et al., 9 Mar 2026).

The same paper argues that a self-adjoint phase operator requires extending the physical Hardy-space representation to full $(r,s)=(1,0)$5, thereby admitting negative Fourier modes and, in the authors’ interpretation, negative-energy states. They describe this extension through a “Dirac sea of phase” analogy (Korneev et al., 9 Mar 2026). Whether one accepts that interpretation or not, the paper makes the state-level claim precise: phase can be represented by a wavefunction on the circle, but only within a restricted function space if one insists on positivity of the spectrum.

7. Terminological boundaries and related usages

The phrase should not be conflated with all superficially similar “phase” terminology. In ultrathin $(r,s)=(1,0)$6-TaS$(r,s)=(1,0)$7, for example, the observed non-equilibrium state in the NC–C hysteresis window is described as “a phase separated state characterized by a complex microstructure consisting of intertwined domains of the ground state of the system and the nearly degenerate metastable state.” The authors explicitly reject the interpretation that it is simply an NC state with altered domain-wall periodicity; it is a mixed-domain metastable texture, not a coherent phase-wave state in the strict sense (Walker et al., 2021).

Likewise, in lattice studies of $(r,s)=(1,0)$8 and $(r,s)=(1,0)$9 scattering, “$$u(r)=K^{-1/2}(r)\exp\!\left[\pm i\int^r K(r')\,dr'\right],$$00-wave” and “$$u(r)=K^{-1/2}(r)\exp\!\left[\pm i\int^r K(r')\,dr'\right],$$01-wave” refer to partial-wave channels in the Lüscher formalism. Those are important “wave” sectors, but they are not phase-wave states in the amplitude–phase, geometric-phase, or Hardy-space sense. The central result there is that the physically relevant near-threshold channel for $$u(r)=K^{-1/2}(r)\exp\!\left[\pm i\int^r K(r')\,dr'\right],$$02 is the $$u(r)=K^{-1/2}(r)\exp\!\left[\pm i\int^r K(r')\,dr'\right],$$03-wave $$u(r)=K^{-1/2}(r)\exp\!\left[\pm i\int^r K(r')\,dr'\right],$$04 state, while the $$u(r)=K^{-1/2}(r)\exp\!\left[\pm i\int^r K(r')\,dr'\right],$$05-wave is smooth and non-resonant (Nagatsuka et al., 2024). This is a different use of “wave.”

A final caution concerns open-path versus endpoint measurements. In polarization theory, the literature distinguishes the full geometric phase from the Pancharatnam phase accessible by ordinary interferometric comparison (Lages et al., 2013). In waveguide-QED tomography of two edge qubits in an open waveguide, the relative phase $$u(r)=K^{-1/2}(r)\exp\!\left[\pm i\int^r K(r')\,dr'\right],$$06 becomes observable through ancilla modulation because the dynamics makes population differences depend on both $$u(r)=K^{-1/2}(r)\exp\!\left[\pm i\int^r K(r')\,dr'\right],$$07 and $$u(r)=K^{-1/2}(r)\exp\!\left[\pm i\int^r K(r')\,dr'\right],$$08 (Greenberg et al., 2022). These results are strongly phase-sensitive, but they address phase readout of a state rather than proposing a general ontology of phase-wave states.

Taken together, the cited work supports a precise but plural conclusion: a phase-wave state is best understood as any state-theoretic description in which phase is retained as an explicit structural variable—whether as a local field paired with amplitude, a continuously evolving geometric attribute of a polarized wave, a directly imageable spatial phase field in magnonics, a collective order parameter in active matter, or a Hardy-space wavefunction on the circle. The specific mathematics and observables differ by discipline, but the recurring theme is the same: phase is not merely appended to the state; it is part of what the state is.