Dirac Phase Method in Quantum Systems

• The Dirac Phase Method is a unifying framework that extracts non-integrable phase factors arising from gauge connections, time-dependent dynamics, and singularities.
• It systematically utilizes extended phase space and constrained quantization techniques to derive phase effects in time-dependent Hamiltonians, monopole fields, and Berry phases.
• The method enables precise extraction of topological invariants and quantization conditions in quantum dynamics, with applications in magnetic quantum oscillations and quantum simulations.

The Dirac Phase Method comprises a family of geometric and algebraic constructions in quantum theory, classical mechanics, gauge field theory, and condensed matter physics, united by the identification and extraction of non-integrable phase factors—“Dirac phases”—arising due to gauge connections, time-dependent structure, or singularities such as monopoles and nodal lines. These phases play fundamental roles in quantization, path integrals, representation theory, geometric phases (e.g., Berry phase), and experimental signatures in modern topological materials. The Dirac Phase Method provides a systematic approach to construct, interpret, and utilize these phases, with implementations ranging from the quantization condition of magnetic monopoles to the boundary terms in Feynman propagators for time-dependent Hamiltonians, and the analysis of geometric phases in parameter spaces and quantum oscillation experiments.

1. Dirac Phase in Time-Dependent Hamiltonian Systems

Time-dependent Hamiltonians $H(q,p,t)$ in classical and quantum mechanics generally lack conserved energy and resist conventional canonical transformations due to the explicit time dependence. To address these issues, the Dirac Phase Method adopts an extended phase space, treating $t$ as a variable with conjugate momentum $p_t$, thus forming the enlarged symplectic manifold $(q,t;p,p_t)$. This construction renders time-reparametrizations and time-dependent canonical transformations ordinary within this extended space.

Dirac's formalism for constrained systems is then employed. The primary constraint,

$\phi(q,t;p,p_t) \equiv p_t + H(q,p,t) \approx 0,$

encodes the gauge freedom associated with time reparametrization. Secondary constraints do not emerge. The system’s dynamics are generated by a total Hamiltonian $H_T = \lambda\,\phi$ with arbitrary multiplier $\lambda$ reflecting gauge arbitrariness. Gauge-fixing, e.g., $n(q,t;p,p_t) \equiv t - T \approx 0$, leads to a second-class pair $(\phi,n)$, and the induced Dirac bracket structure projects onto the physical (reduced) phase space. Crucially, the process produces a time-independent Lewis invariant $I(q,p,t)$ as a reparametrization-invariant Dirac observable, e.g., for the time-dependent harmonic oscillator, and maps the dynamics to an autonomous system with a computable phase boundary term.

Upon quantization, the path integral is constructed over the extended space with constraints enforced by Dirac $\delta$-functions. Canonical transformations induced by generating functions reducing $H$ to $I$ yield the Feynman propagator as

$$K(q_f, t_f; q_i, t_i) = [\mathcal{G}(T_f)\mathcal{G}(T_i)]^{-1} \int \mathcal{D}Q\,\mathcal{D}P\, \exp\left\{i\int_{T_i}^{T_f}[P Q' - I(Q,P)]dT\right\} \, e^{-i\int_{Q_i}^{Q_f}A'(Q,T) D(Q,T) dQ}.$$

The boundary term in the action emerges as a quantum phase—identified as the “Dirac phase”—which, in prior literature, had been inserted by hand and is here derived from first principles (Garcia-Chung et al., 2017).

2. Dirac Phase in Gauge Theory and Quantization Conditions

In the context of gauge fields, the Dirac phase emerges in the geometrization of the Dirac monopole and associated quantization conditions. The vector potential for a monopole of strength $g$ is constructed as a line integral along a chosen “Dirac string” $C(\mathbf{n})$:

$$\mathbf{A}_{\mathbf{n}}(\mathbf{r}) = g\int_{C(\mathbf{n})} \frac{(\mathbf{r}-\mathbf{r}')\times d\mathbf{r}'}{|\mathbf{r}-\mathbf{r}'|^3}.$$

Gauge equivalence under deformations of the string $C\to C'$ yields a gauge function $\Lambda(\mathbf{r})$, typically multi-valued and discontinuous across the surface $\sigma$ spanned between strings. Under a “phase gauge” transformation $\psi\to e^{ie\Lambda}\psi$, with $\mathbf{A}$ fixed, global single-valuedness of the wavefunction is enforced. This leads, by geometric boundary considerations, to the quantization of the product $eg$:

$e\,g = \frac{\ell}{2}, \quad \ell\in\mathbb{Z},$

without invoking canonical commutation relations. The method generalizes to a Euclidean four-dimensional setting, where the Dirac string is a world-line defect, and the large gauge transformation corresponds to a winding around this defect (Kim, 27 Apr 2024).

3. Dirac Phase in Quantum Geometric Phases and Berry Connection

The method extends to geometric phase analysis in parameter space, linking the Berry phase to a Dirac phase factor via identification of nodal lines and monopole singularities. For a generic two-state Hamiltonian parametrized by $(\theta,\varphi)$, eigenvectors possess nodal lines—Dirac strings—ending at level degeneracies (monopole points). The phase accumulated around a closed loop is

$\Delta\gamma = \frac{1}{2} \Omega(C) + 2\pi m,$

where $\Omega(C)$ is the solid angle subtended and $m$ the winding number around the string. The Berry connection $\mathbf{A}$ and curvature $\mathbf{F}$ arise as the phase-gradient (vector potential) and monopole field of the Dirac construction. The Dirac phase, thus, provides a universal framework for geometric phase phenomena and their monopole interpretation (Zhao, 3 Sep 2024).

System	Dirac Phase Manifestation	Physical Consequence
Time-dependent $H$	Action boundary term via canonical map	Quantizes the Feynman propagator
Monopole gauge field	Multi-valued phase from line integral	Dirac quantization condition
Adiabatic parameter space	Non-integrable phase around nodal line	Berry geometric phase, topology

4. Applications in Quantum Dynamics and Simulation

The Dirac Phase Method admits concrete analytic and computational implementations in diverse settings:

• Quantum Simulation in Curved Spacetimes: In 1+1D, any solution of the massless Dirac equation in a curved, conformally flat metric $ds^2 = \Omega^2(x)(dt^2-dx^2)$ can be generated from a flat-space solution by the phase transformation $\psi(x) = \Omega^{-1/2}(x)\phi(x)$. The curved-spin connection is thus encoded in the Dirac phase, enabling efficient quantum simulation in platforms implementing the free Dirac equation by postprocessing with the local phase (Sabín, 2016).
• Moving-Wall (Dilating) Quantum Systems: For a single particle in an isotropically moving potential boundary, the emergent gauge potential is absorbed into the wavefunction by a scale- and phase-transformation,

$$\Psi(\mathbf{r}, t) = a(t)^{-3/2} e^{i\varphi_D(\mathbf{r},t)} \psi(\mathbf{r}/a(t), t),$$

with Dirac phase $$\varphi_D(\mathbf{r},t) = -\frac{m}{2\hbar} \frac{\dot{a}}{a} r^2$$. This phase is entirely determined by the instantaneous relative expansion rate, $H=\dot{a}/a$, and governs shortcuts to adiabaticity and effective classical-quantum correspondence in expanding geometries, including cosmological models (Mehrafarin et al., 2016).

5. Dirac Phase in Magnetic Quantum Oscillations

In three-dimensional Dirac semimetals, the Dirac phase framework is crucial for extracting topological information from magnetic quantum oscillation (MQO) experiments. Beyond the single-orbit Lifshitz-Kosevich theory, the inclusion of spin-degenerate orbits yields oscillation amplitudes with phase factors,

$$\Delta\sigma_{xx}/\sigma_{xx}^{\mathrm{no}} \propto |\cos\lambda_i| \cos\left(2\pi p \frac{F_i}{B} - \pi + \Theta_i + \delta_i \right),$$

with total phase

$\lambda_i = \phi_{B,i} + \phi_{R,i} + \phi_{Z,i},$

where $\phi_{B,i}$ is Berry, $\phi_{R,i}$ orbital, and $\phi_{Z,i}$ Zeeman phase. An abrupt $\pi$ phase shift in the MQO signal corresponds to a node in $|\cos\lambda_i|$, occurring when the cyclotron mass satisfies $m_c = m_e/2$ for $g_s=2$. This direct “Dirac-phase” method enables unambiguous extraction of the spin-gyromagnetic ratio and topological indices, even in the presence of band-mass anisotropy or nontrivial Fermi pockets (Lee et al., 2020).

6. Synthesis and Theoretical Implications

The Dirac Phase Method interlinks gauge structure, geometric phases, quantization, and dynamical evolution across physics subfields. In all implementations, the method:

• Promotes auxiliary variables (e.g., time, gauge parameters) to dynamical degrees of freedom, encoding invariance or redundancy as constraints.
• Resolves physical observables and reduced phase space via systematic gauge-fixing and computation of Dirac brackets.
• Identifies topologically nontrivial “phases” (boundary terms, geometric phases, non-integrable factors) as consequences of holonomy or monodromy associated with gauge or parameter-space singularities.
• Provides direct analytic recipes for extracting topological invariants, quantization conditions, or observable phase shifts in experiment.

The Dirac phase framework thus equips researchers with a rigorous, geometric, and unifying perspective on phase structures in quantum and classical systems, directly connecting abstract constraints and gauge freedoms to experimentally accessible phenomena and fundamental quantization laws.