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Vector potential

The vector potential, denoted as \mathbf{A}, is a vector field in classical electromagnetism defined such that the magnetic field \mathbf{B} is given by \mathbf{B} = \nabla \times \mathbf{A}.^[1] This relation, introduced by James Clerk Maxwell in his 1865 A Dynamical Theory of the Electromagnetic Field and further developed in his 1873 Treatise on Electricity and Magnetism, where it was termed "electromagnetic momentum," provides a mathematical framework for deriving magnetic fields from underlying current distributions without direct action-at-a-distance.^[2] Together with the scalar electric potential φ, the vector potential fully describes electromagnetic fields via \mathbf{E} = -\nabla \phi - \frac{\partial \mathbf{A}}{\partial t}, enabling a unified treatment of electric and magnetic phenomena in both static and dynamic cases.^[3] Unlike the scalar potential, which is unique up to a constant, the vector potential possesses gauge freedom: it can be transformed as \mathbf{A}' = \mathbf{A} + \nabla \lambda (with a corresponding adjustment \phi' = \phi - \frac{\partial \lambda}{\partial t} for the scalar potential) without altering the physical fields \mathbf{E} or \mathbf{B}, since the curl of a gradient vanishes.^[1] This non-uniqueness, known as gauge invariance, is a cornerstone of electromagnetic theory and simplifies calculations, particularly in the Lorenz gauge where \nabla \cdot \mathbf{A} + \frac{1}{c^2} \frac{\partial \phi}{\partial t} = 0 (in SI units, \nabla \cdot \mathbf{A} + \mu_0 \epsilon_0 \frac{\partial \phi}{\partial t} = 0).^[4] In magnetostatics, the vector potential is explicitly given by \mathbf{A}(\mathbf{r}) = \frac{\mu_0}{4\pi} \int \frac{\mathbf{J}(\mathbf{r}')}{|\mathbf{r} - \mathbf{r}'|} dV', where \mathbf{J} is the current density, mirroring the Biot-Savart law but in potential form.^[3] The vector potential plays a pivotal role beyond classical electromagnetism, notably in quantum electrodynamics where it couples to charged particles in the Hamiltonian, influencing wave functions even in regions of zero magnetic field, as exemplified by the Aharonov-Bohm effect.^[3] In relativistic formulations, \mathbf{A} and φ combine into the four-potential A^\mu = (\phi/c, \mathbf{A}), a Lorentz-covariant object that unifies the description of electromagnetic interactions in special relativity and satisfies the inhomogeneous Maxwell equations through the four-current J^\mu.^[5] Applications span antenna design, where \mathbf{A} aids in radiation pattern calculations, to particle physics, underscoring its enduring significance in theoretical and applied physics.

Mathematical Foundations

Definition and basic formulation

In electromagnetism, the vector potential \mathbf{A} is defined as a vector field in three-dimensional Euclidean space such that the magnetic field \mathbf{B} satisfies the relation \mathbf{B} = \nabla \times \mathbf{A}.^[6]^[7] This representation is possible because the magnetic field is divergence-free, \nabla \cdot \mathbf{B} = 0, which follows from the absence of magnetic monopoles in nature; consequently, any solenoidal (divergence-free) vector field can be expressed as the curl of another vector field.^[6]^[7] The divergence-free condition ensures compatibility with the curl operator, as the divergence of any curl vanishes identically: \nabla \cdot (\nabla \times \mathbf{A}) = 0.^[6] The vector potential \mathbf{A} possesses three components, one for each spatial dimension, yet it includes an inherent arbitrariness known as gauge freedom, where one degree of freedom can be chosen freely without affecting the physical magnetic field \mathbf{B}.^[6]^[7] This non-uniqueness arises because \mathbf{A} and \mathbf{A} + \nabla \chi (for any scalar function \chi) yield the same curl.^[6] An important integral interpretation of \mathbf{A} follows from Stokes' theorem: the circulation of \mathbf{A} around any closed loop equals the magnetic flux of \mathbf{B} through a surface bounded by that loop,

\oint_C \mathbf{A} \cdot d\mathbf{l} = \iint_S (\nabla \times \mathbf{A}) \cdot d\mathbf{S} = \iint_S \mathbf{B} \cdot d\mathbf{S},

where C is the loop and S is the spanning surface.^[6] This relation highlights \mathbf{A}'s role in encoding the topology of magnetic fields via path integrals.^[6]

Relation to the magnetic field

The magnetic field \mathbf{B} is related to the vector potential \mathbf{A} by the equation

\mathbf{B} = \nabla \times \mathbf{A}.

This definition ensures that \nabla \cdot \mathbf{B} = 0, as required by Maxwell's equations in free space, because the divergence of the curl of any sufficiently smooth vector field vanishes identically:

\nabla \cdot (\nabla \times \mathbf{A}) = 0.

The relation \mathbf{B} = \nabla \times \mathbf{A} thus automatically incorporates the solenoidal nature of the magnetic field without magnetic monopoles.^[8]^[6] The existence of a vector potential \mathbf{A} such that \nabla \times \mathbf{A} = \mathbf{B} for any solenoidal field \mathbf{B} (i.e., \nabla \cdot \mathbf{B} = 0) is guaranteed by Helmholtz's theorem in vector calculus. The theorem decomposes any sufficiently regular vector field into an irrotational part (gradient of a scalar) and a solenoidal part (curl of a vector), with the decomposition unique up to boundary conditions, such as the fields decaying appropriately at infinity. For \mathbf{B}, the zero divergence implies it is purely solenoidal, so \mathbf{B} = \nabla \times \mathbf{A} holds, where \mathbf{A} satisfies suitable regularity conditions. Alternatively, in simply connected domains, the Poincaré lemma provides a local existence proof via homotopy arguments, confirming that solenoidal fields admit a vector potential.^[9]^[10] To construct \mathbf{A} explicitly from a given \mathbf{B}, one approach without gauge fixing involves line integrals along paths from a reference point, but such constructions are path-dependent: the line integral \int \mathbf{A} \cdot d\mathbf{l} from a fixed origin to \mathbf{r} equals the magnetic flux through any surface spanning the path, and different paths yield different values differing by the flux through the closed loop they form, per Stokes' theorem. Gauge fixing resolves this ambiguity. In the Coulomb gauge, where \nabla \cdot \mathbf{A} = 0, a particular solution for localized \mathbf{B} (vanishing at infinity) is given by the Biot-Savart-like volume integral

\mathbf{A}(\mathbf{r}) = \frac{1}{4\pi} \int \frac{\mathbf{B}(\mathbf{r}') \times (\mathbf{r} - \mathbf{r}')}{|\mathbf{r} - \mathbf{r}'|^3} \, dV',

which satisfies \nabla \times \mathbf{A} = \mathbf{B} and the gauge condition under the stated assumptions. This formula arises from the general solution to the vector Poisson equation \nabla^2 \mathbf{A} = -\nabla \times \mathbf{B} in the gauge, analogous to the scalar Green's function solution. A simple example is a uniform magnetic field \mathbf{B} = B_0 \hat{z} throughout space. In the symmetric (Coulomb) gauge, a suitable vector potential is

\mathbf{A} = \frac{B_0}{2} (-y, x, 0),

which yields \nabla \times \mathbf{A} = B_0 \hat{z} upon direct computation: the z-component is \partial A_y / \partial x - \partial A_x / \partial y = B_0 / 2 + B_0 / 2 = B_0, while the other components vanish. This choice is rotationally symmetric around the field direction and commonly used in applications like charged particle motion in uniform fields. The integral formula above reproduces this \mathbf{A} up to a gauge transformation for the uniform case, though boundary conditions at infinity require care for non-localized fields.^[6]

Role in Electromagnetism

In magnetostatics

In magnetostatics, the magnetic field \mathbf{B} produced by steady currents satisfies Ampère's circuital law in differential form, \nabla \times \mathbf{B} = \mu_0 \mathbf{J}, where \mathbf{J} is the current density and \mu_0 is the permeability of free space.^[11] Given that \mathbf{B} = \nabla \times \mathbf{A}, substitution into Ampère's law yields \nabla \times (\nabla \times \mathbf{A}) = \mu_0 \mathbf{J}.^[11] Applying the vector identity \nabla \times (\nabla \times \mathbf{A}) = \nabla (\nabla \cdot \mathbf{A}) - \nabla^2 \mathbf{A} results in the equation

\nabla (\nabla \cdot \mathbf{A}) - \nabla^2 \mathbf{A} = \mu_0 \mathbf{J},

which governs the vector potential \mathbf{A} in the presence of current sources.^[11] The explicit solution for \mathbf{A} in magnetostatics can be obtained by analogy to the scalar potential in electrostatics, assuming the Coulomb gauge \nabla \cdot \mathbf{A} = 0. For a volume current distribution, the vector potential at position \mathbf{r} is given by the integral

\mathbf{A}(\mathbf{r}) = \frac{\mu_0}{4\pi} \int \frac{\mathbf{J}(\mathbf{r}')}{|\mathbf{r} - \mathbf{r}'|} \, dV',

where the integration is over the volume containing the currents.^[12] For a thin wire carrying current I, this reduces to a line integral along the wire path:

\mathbf{A}(\mathbf{r}) = \frac{\mu_0 I}{4\pi} \int \frac{d\mathbf{l}'}{|\mathbf{r} - \mathbf{r}'|}.

^[12] In the Coulomb gauge, the governing equation simplifies further to the Poisson equation \nabla^2 \mathbf{A} = -\mu_0 \mathbf{J}, which is directly solvable for localized current distributions. This gauge choice eliminates the \nabla (\nabla \cdot \mathbf{A}) term, making the computation of \mathbf{A} analogous to that of the electrostatic potential from charge densities. A representative example is the vector potential due to an infinite straight wire along the z-axis carrying uniform current I. In cylindrical coordinates (\rho, \phi, z), the vector potential takes the form \mathbf{A} = -\frac{\mu_0 I}{2\pi} \ln(\rho) \hat{z}, directed parallel to the wire and varying logarithmically with the perpendicular distance \rho.^[13] This expression arises from evaluating the line integral, noting the azimuthal symmetry and infinite extent of the wire.^[13] The vector potential \mathbf{A} has dimensions of magnetic field times length, corresponding to units of tesla-meter (T·m) or weber per meter (Wb/m) in the SI system.^[14] These units reflect its role in linking magnetic flux to spatial variations, as the flux through a surface is \Phi = \oint \mathbf{A} \cdot d\mathbf{l}.^[11]

In electrodynamics

In electrodynamics, the vector potential \mathbf{A} is extended to describe time-varying fields by introducing a scalar potential \phi, forming the electromagnetic four-potential. The electric field \mathbf{E} and magnetic field \mathbf{B} are then expressed in terms of these potentials as \mathbf{E} = -\nabla \phi - \frac{\partial \mathbf{A}}{\partial t} and \mathbf{B} = \nabla \times \mathbf{A}. These relations satisfy two of Maxwell's equations, \nabla \cdot \mathbf{B} = 0 and \nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t}, automatically, while the remaining equations involving sources lead to coupled wave equations for the potentials.^[15] Substituting the potential expressions into Maxwell's equations yields inhomogeneous wave equations. For the vector potential, the equation is \nabla^2 \mathbf{A} - \frac{1}{c^2} \frac{\partial^2 \mathbf{A}}{\partial t^2} - \nabla \left( \nabla \cdot \mathbf{A} + \frac{1}{c^2} \frac{\partial \phi}{\partial t} \right) = -\mu_0 \mathbf{J}, where \mathbf{J} is the current density, c is the speed of light, and \mu_0 is the vacuum permeability. The scalar potential satisfies a similar form: \nabla^2 \phi - \frac{1}{c^2} \frac{\partial^2 \phi}{\partial t^2} - \frac{\partial}{\partial t} \left( \nabla \cdot \mathbf{A} + \frac{1}{c^2} \frac{\partial \phi}{\partial t} \right) = -\frac{\rho}{\epsilon_0}, with \rho the charge density and \epsilon_0 the vacuum permittivity. These equations couple \phi and \mathbf{A} through the gauge-dependent term, reflecting the non-uniqueness of the potentials.^[15] In the Lorentz gauge, where \nabla \cdot \mathbf{A} + \frac{1}{c^2} \frac{\partial \phi}{\partial t} = 0, the equations decouple into independent wave equations: \nabla^2 \mathbf{A} - \frac{1}{c^2} \frac{\partial^2 \mathbf{A}}{\partial t^2} = -\mu_0 \mathbf{J} and \nabla^2 \phi - \frac{1}{c^2} \frac{\partial^2 \phi}{\partial t^2} = -\frac{\rho}{\epsilon_0}. The general solutions are the retarded potentials, which account for the finite propagation speed of electromagnetic signals:

\mathbf{A}(\mathbf{r}, t) = \frac{\mu_0}{4\pi} \int \frac{\mathbf{J}(\mathbf{r}', t - \frac{|\mathbf{r} - \mathbf{r}'|}{c})}{|\mathbf{r} - \mathbf{r}'|} dV',

\phi(\mathbf{r}, t) = \frac{1}{4\pi \epsilon_0} \int \frac{\rho(\mathbf{r}', t - \frac{|\mathbf{r} - \mathbf{r}'|}{c})}{|\mathbf{r} - \mathbf{r}'|} dV'.

These integrals evaluate the sources at the retarded time t - \frac{|\mathbf{r} - \mathbf{r}'|}{c}, ensuring causality.^[16] A representative example is the vector potential for an oscillating electric dipole, modeling radiation from an antenna or atomic transition. For a dipole moment \mathbf{p}(t) = \mathbf{p}_0 \Re[e^{-i\omega t}] at the origin, the far-field vector potential is \mathbf{A}(\mathbf{r}, t) = \frac{\mu_0}{4\pi} \frac{[\ddot{\mathbf{p}}(t_r)]_\perp}{c r}, where t_r = t - r/c is the retarded time and [\cdot]_\perp denotes the transverse component perpendicular to \mathbf{r}. This transverse nature arises because the longitudinal part is absorbed into the scalar potential, highlighting how \mathbf{A} mediates the propagating, radiation part of the magnetic field in time-varying scenarios.^[17] The Lorentz force on a charged particle can also be expressed directly in terms of the potentials. For a particle of charge q and velocity \mathbf{v}, the force is \mathbf{F} = q \left( -\nabla \phi - \frac{\partial \mathbf{A}}{\partial t} + \mathbf{v} \times (\nabla \times \mathbf{A}) \right), which expands to include terms like \mathbf{v} (\mathbf{v} \cdot \nabla) \mathbf{A} upon considering the particle's motion. This formulation underscores the potentials' role in particle dynamics, particularly in accelerator physics where \mathbf{A} influences beam trajectories through its curl.^[18]

Gauge Invariance

Non-uniqueness of the potential

The vector potential \mathbf{A} in electromagnetism is not uniquely determined by the magnetic field \mathbf{B}, as multiple choices of \mathbf{A} can yield the same \mathbf{B} via \mathbf{B} = \nabla \times \mathbf{A}. This ambiguity arises from gauge transformations, which allow for the replacement \mathbf{A}' = \mathbf{A} + \nabla \chi, where \chi is an arbitrary scalar function, without altering \mathbf{B}.^[19]^[20] Similarly, the scalar potential transforms as \phi' = \phi - \frac{\partial \chi}{\partial t} to ensure the electric field \mathbf{E} = -\nabla \phi - \frac{\partial \mathbf{A}}{\partial t} remains invariant.^[19] The invariance of \mathbf{B} under this transformation follows from the vector identity \nabla \times (\nabla \chi) = 0 for any scalar \chi, which implies that adding the gradient of \chi contributes only to the longitudinal (irrotational) component of \mathbf{A}, leaving the transverse (solenoidal) part—which determines \mathbf{B}—unchanged.^[19]^[21] This decomposition of \mathbf{A} into longitudinal and transverse components, via the Helmholtz theorem, underscores that the gauge freedom affects only the non-physical degrees of freedom.^[21] Historically, the non-uniqueness of potentials was highlighted in Heinrich Hertz's 1884 reformulation of Maxwell's equations, where he sought to eliminate potentials altogether in favor of a direct field description to achieve a more consistent action-at-a-distance theory; however, potentials were later reinstated for their mathematical convenience in solving equations.^[22] The implications are profound: the potentials \mathbf{A} and \phi are not directly observable, as physical observables like \mathbf{E} and \mathbf{B} are gauge-invariant, emphasizing that only the fields represent measurable electromagnetic phenomena.^[23] For example, a uniform magnetic field \mathbf{B} = B \hat{z} can be described by the Landau gauge \mathbf{A} = (0, B x, 0) or the symmetric gauge \mathbf{A} = \frac{1}{2} \mathbf{B} \times \mathbf{r} = \left( -\frac{1}{2} B y, \frac{1}{2} B x, 0 \right), both satisfying \nabla \times \mathbf{A} = \mathbf{B} but differing by a gauge transformation.^[11]^[6]

Common gauge choices

In electromagnetism, the freedom in choosing the vector potential \mathbf{A} and scalar potential \phi is constrained by selecting specific gauges that simplify the equations of motion for particular problems.^[24] The Coulomb gauge is defined by the condition \nabla \cdot \mathbf{A} = 0, which ensures that \mathbf{A} is transverse to the direction of propagation.^[25] In this gauge, the vector potential satisfies the wave equation \nabla^2 \mathbf{A} - \frac{1}{c^2} \frac{\partial^2 \mathbf{A}}{\partial t^2} = -\mu_0 \mathbf{J}_\perp, where \mathbf{J}_\perp is the transverse component of the current density and \mu_0 is the vacuum permeability.^[25] This choice is particularly useful in magnetostatics and non-relativistic quantum mechanics, as it decouples the scalar potential, allowing it to be solved instantaneously via Poisson's equation, while keeping the vector potential responsible for magnetic effects.^[25]^[24] The Lorenz gauge imposes the condition \nabla \cdot \mathbf{A} + \frac{1}{c^2} \frac{\partial \phi}{\partial t} = 0, where c is the speed of light.^[26] This leads to decoupled wave equations for both potentials, such as \nabla^2 \mathbf{A} - \frac{1}{c^2} \frac{\partial^2 \mathbf{A}}{\partial t^2} = -\mu_0 \mathbf{J} (the d'Alembertian operator acting on \mathbf{A}), enabling solutions in terms of retarded potentials that propagate at the speed of light.^[26] Its relativistic invariance, expressed covariantly as \partial_\mu A^\mu = 0 in four-vector notation, makes it ideal for relativistic electrodynamics and wave propagation problems.^[26]^[24] The temporal gauge, also known as the Weyl gauge, sets \phi = 0, eliminating the scalar potential entirely.^[27] This gauge simplifies the Hamiltonian formulation in quantum electrodynamics by reducing the degrees of freedom and facilitating canonical quantization, though it sacrifices manifest Lorentz covariance.^[27] Compared to other choices, the Coulomb gauge supports instantaneous action for the scalar potential, making it suitable for non-relativistic approximations, while the Lorenz gauge excels in capturing wave propagation and relativistic effects through its covariant structure. The temporal gauge prioritizes simplicity in quantum Hamiltonians over relativistic symmetry, often used when focusing on spatial dynamics.^[27]

Physical Interpretations and Applications

In classical field theory

In classical field theory, the vector potential plays a central role in the relativistic formulation of electromagnetism, where it is combined with the scalar potential to form the four-potential A^\mu = (\phi/c, \mathbf{A}). This four-vector transforms covariantly under Lorentz transformations, ensuring that the electromagnetic fields remain invariant. The electric and magnetic fields are derived from the field strength tensor F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu, with components such as E_i = F_{0i} and B_i = \epsilon_{ijk} F_{jk}, where \epsilon_{ijk} is the Levi-Civita symbol. This antisymmetric tensor encapsulates Maxwell's equations in a compact, relativistic form.^[28]^[29] The Lagrangian formulation of classical electromagnetism further highlights the vector potential's importance, with the Lagrangian density given by \mathcal{L} = -\frac{1}{4\mu_0} F_{\mu\nu} F^{\mu\nu} - A_\mu J^\mu, where J^\mu is the four-current density. Varying the action S = \int \mathcal{L} \, d^4x with respect to A^\mu yields the Euler-Lagrange equations \partial_\mu F^{\mu\nu} = \mu_0 J^\nu, which reproduce Maxwell's equations. This approach treats the four-potential components as independent fields, facilitating the derivation of field equations and interactions in a unified manner.^[30]^[31] Beyond electromagnetism, the vector potential finds analogies in other classical fields. In incompressible fluid dynamics, the velocity field \mathbf{v} can be expressed as the curl of a vector potential \mathbf{A}, such that \mathbf{v} = \nabla \times \mathbf{A}, mirroring the relation \mathbf{B} = \nabla \times \mathbf{A}; this decomposition ensures the divergence-free condition \nabla \cdot \mathbf{v} = 0 and relates to vorticity \boldsymbol{\omega} = \nabla \times \mathbf{v}. Similar structural parallels appear in acoustics, where a velocity potential (often scalar for irrotational flows) analogs the scalar electric potential, though vector forms arise in more complex wave propagations.^[32]^[33] In general relativity, the vector potential is incorporated into the curved spacetime formalism, primarily through weak-field approximations where the metric is nearly Minkowski. The electromagnetic field equations become \nabla_\mu F^{\mu\nu} = \mu_0 J^\nu, with the covariant derivative accounting for curvature, but solutions for A^\mu are limited to perturbative regimes around flat space, as full nonlinear coupling with gravity complicates exact potentials. Gauge invariance persists, allowing transformations that preserve the action while adapting to the geometry. Gauge invariance in the action principle ensures that the electromagnetic action remains unchanged under transformations A^\mu \to A^\mu + \partial^\mu \chi, where \chi is a scalar field satisfying the wave equation \square \chi = 0 in the Lorentz gauge. This redundancy in the potentials does not affect physical observables, as the field strength F_{\mu\nu} is invariant, underscoring the principle's role in maintaining relativistic consistency.^[34]^[35]

In quantum mechanics

In non-relativistic quantum mechanics, the vector potential \mathbf{A} is incorporated into the dynamics of a charged particle through the principle of minimal substitution, where the canonical momentum operator \mathbf{p} = -i\hbar \nabla is replaced by the mechanical momentum \mathbf{p} - q \mathbf{A}, with q denoting the particle's charge. This substitution modifies the Hamiltonian for a particle in an electromagnetic field to H = \frac{1}{2m} (\mathbf{p} - q \mathbf{A})^2 + q \phi, where \phi is the scalar potential and m is the particle mass. The corresponding time-dependent Schrödinger equation becomes i \hbar \frac{\partial \psi}{\partial t} = H \psi, governing the evolution of the wave function \psi. Expanding the kinetic term in the Hamiltonian reveals how the magnetic field \mathbf{B} = \nabla \times \mathbf{A} influences the system: \frac{1}{2m} (\mathbf{p} - q \mathbf{A})^2 = \frac{\mathbf{p}^2}{2m} - \frac{q}{2m} (\mathbf{A} \cdot \mathbf{p} + \mathbf{p} \cdot \mathbf{A}) + \frac{q^2 A^2}{2m}. The cross term yields a paramagnetic contribution \frac{q}{2m} \mathbf{B} \cdot \mathbf{L} involving the orbital angular momentum \mathbf{L} = \mathbf{r} \times \mathbf{p}, while the A^2 term introduces a diamagnetic effect; notably, \mathbf{A} itself affects the phase of \psi beyond the explicit \mathbf{B} dependence. Gauge invariance in quantum mechanics ensures that physical observables remain unchanged under the transformation \mathbf{A}' = \mathbf{A} + \nabla \chi and \phi' = \phi - \frac{\partial \chi}{\partial t}, where \chi is an arbitrary scalar function. The wave function transforms as \psi' = e^{i q \chi / \hbar} \psi, introducing a position-dependent phase that preserves the probability density |\psi|^2 and expectation values of gauge-invariant operators. This phase factor, first recognized in the context of charged particle motion, underscores the non-local influence of \mathbf{A} on quantum interference. A key application arises in the quantum mechanics of an electron in a uniform magnetic field \mathbf{B}, leading to quantized energy levels known as Landau levels. Using the symmetric gauge \mathbf{A} = \frac{[1](/page/1)}{2} \mathbf{B} \times \mathbf{r}, the Hamiltonian yields discrete eigenvalues E_n = \hbar \omega_c (n + \frac{[1](/page/1)}{2}), where \omega_c = |q| B / m is the cyclotron frequency and n = 0, [1](/page/1), 2, \dots; each level is highly degenerate, reflecting the extended nature of the wave functions. The vector potential also connects to geometric phases in quantum mechanics, where the Berry phase acquired during adiabatic transport of a quantum state around a closed path in parameter space equals the line integral of the Berry connection, analogous to the classical line integral of \mathbf{A}. This linkage highlights \mathbf{A}'s role in encoding topological features of quantum evolution.

References

[1]
Magnetic Vector Potential - HyperPhysics Concepts
The vector potential is defined to be consistent with Ampere's Law and can be expressed in terms of either current i or current density j (the sources of ...
[2]
Maxwell and the Vector Potential
Here we investigate the introduction of the vector potential in the work of James Clerk Maxwell, and we shall see that the above view is far from accurate in ...
[3]
The Feynman Lectures on Physics Vol. II Ch. 15: The Vector Potential
A current loop—or magnetic dipole—not only produces magnetic fields, but will also experience forces when placed in the magnetic field of other currents.
[4]
Scalar and Vector Potentials - Richard Fitzpatrick
The previous prescription for expressing electric and magnetic fields in terms of the scalar and vector potentials does not uniquely define the potentials. ...
[5]
Potential 4-Vector - Richard Fitzpatrick
Thus, the field equations that govern classical electromagnetism can all be summed up in a single 4-vector equation.
[6]
14 The Magnetic Field in Various Situations - Feynman Lectures
If, for example, the B field is the axial field inside a solenoid, then the vector potential circulates in the same sense as do the currents of the solenoid.
[7]
https://phys.libretexts.org/Bookshelves/Electricity_and_Magnetism/Electromagnetics_II_(Ellingson)/09%3A_Radiation/9.02%3A_Magnetic_Vector_Potential
[8]
5.4: The Vector Potential - Engineering LibreTexts
Oct 3, 2023 · (5.4.1) ∇ ⋅ B = 0 ⇒ B = ∇ × A. where A is called the vector potential, as the divergence of the curl of any vector is always zero. Often ...
[9]
Helmholtz's theorem - Richard Fitzpatrick
There is only one possible steady electric and magnetic field which can be generated by a given set of stationary charges and steady currents.
[10]
Helmholtz' Theorem - Galileo and Einstein
Now we're ready for Helmholtz' theorem: Any reasonably well behaved vector field (and they all are in physics) can be writes as a sum of two fields, one a ...
[11]
[PDF] Vector Potential for the Magnetic Field - UT Physics
field B(r) determines the vector potential A(r) only up to a gradient of an arbitrary scalar ... integral, the remaining integral. ZZ sphere r′. |r − r ...
[12]
[PDF] Vector Potential for the Magnetic Field - UT Physics
The vector potential A(x, y, z) is related to the magnetic field B(x, y, z) by B(x, y, z) = ∇ × A(x, y, z), and is used to calculate the magnetic field.
[13]
Lecture Notes Chapter 1
The Vector Potential. The magnetic field generated by a static current distribution is uniquely defined by the so-called Maxwell equations for magnetostatics:.
[14]
The magnetic vector potential - Richard Fitzpatrick
We wish to find a vector potential ${\bf A}$ whose curl is equal to the above magnetic field, and whose divergence is zero. It is not difficult to see that ...
[15]
[PDF] SI Units of Kinematic and Electromagnetic Quantities
Units Symbol SI Base Units. Length и. Meters, m m m. Mass m. Kilograms, kg ... Magnetic Vector Potential. A = “p/Q”. Tesla-m = Wb/m = N/A. T-m = N/A kg-m/A-s2.
[16]
[PDF] Classical Electromagnetism - Richard Fitzpatrick
... potential. Furthermore, we can automatically satisfy Equa- tion (1.3) by writing. E = −∇φ −. ∂A. ∂t. ,. (1.12) where φ(r, t) is termed the scalar potential.
[17]
[PDF] Chapter 5: Electromagnetic Forces - MIT OpenCourseWare
The Lorentz force equation (1.2.1) fully characterizes electromagnetic forces on stationary and moving charges. Despite the simplicity of this equation, ...
[18]
15.10: Gauge Transformations - Physics LibreTexts
Mar 5, 2022 · Electric and magnetic fields can be written in terms of scalar and vector potentials. However, there are many different potentials which can ...
[19]
Gauge transformations - Richard Fitzpatrick
Gauge transformations. Electric and magnetic fields can be written in terms of scalar and vector potentials, as follows: $\displaystyle { ...
[20]
[PDF] The Helmholtz Decomposition and the Coulomb Gauge
Apr 20, 2023 · The Helmholtz decomposition (1)-(2) is an artificial split of the vector field E into two parts, which parts have interesting mathematical ...
[21]
The Long Road to Maxwell's Equations - IEEE Spectrum
Dec 1, 2014 · The key was eliminating Maxwell's strange magnetic vector potential. ... Heinrich Hertz's interest in Maxwell's work is piqued by a Prussian ...
[22]
The physics of gauge transformations - AIP Publishing
Aug 1, 2005 · We review the traditional discussions of classical electromagnetic potentials and gauge transformations in most textbooks on electromagnetism ...<|control11|><|separator|>
[23]
[PDF] From Lorenz to Coulomb and other explicit gauge transformations
Apr 17, 2002 · The Lorenz and Coulomb gauges are limiting cases, v = c and v = , respectively. The gauge function and the potentials are determined, as are the ...
[24]
[PDF] A short proof that the Coulomb-gauge potentials yield the retarded ...
Mar 13, 2011 · After the Lorenz gauge, the most popular gauge in electrodynamics is the Coulomb gauge in which the scalar potential ΦC displays the properties ...
[25]
[PDF] The “Lorenz gauge” is named in honour of Ludwig Valentin ... - arXiv
The importance of the Lorenz gauge comes from its relativistic invariance, from a sim- plification of many calculations in it, etc. A partial analysis of the ...
[26]
[PDF] Mathematical structure of the temporal gauge in quantum ... - arXiv
Let Ω be a state on the Weyl gauge algebra W invariant under the free time evolution, satisfying the energy spectral condition and its restriction ω to Wobs be ...
[27]
Electrodynamics in Relativistic Notation - Feynman Lectures - Caltech
The potentials at P can be computed in either frame. We will consider one example of the usefulness of the idea of the four-potential. What are the vector ...
[28]
[PDF] 5. Electromagnetism and Relativity - DAMTP
This can be thought of as the Galilean boost of electric and magnetic fields. We recognise E+v⇥B as the combination that appears in the Lorentz force law. We'll ...
[29]
[PDF] lagrangian formulation of the electromagnetic field - UChicago Math
Jul 16, 2012 · expressed in terms of a 4-vector called the 4-Potential A = (φ c. , −. −→. A)=(A0,A1,A2,A3). Page 9. LAGRANGIAN FORMULATION OF THE ...<|control11|><|separator|>
[30]
[PDF] Physics 221B Spring 1997 Notes 32 Lagrangian and Hamiltonian ...
The electromagnetic Lagrangian density will consist of a free-field part and a part involving the interaction with matter. The free-field part must be a ...
[31]
The Analogy between Electromagnetics and Hydrodynamics - arXiv
Jul 26, 2022 · Maxwell developed an analogy, where the magnetic field and the vector potential in electromagnetics are compared to the vorticity and velocity in hydrodynamics ...
[32]
[PDF] Analogous formulation of electrodynamics and two-dimensional fluid ...
The vorticity is analogous to electric charge density, and point vortices are the analogs of point charges. The dynamics are equivalent to an action principle ...
[33]
[PDF] Coupling of Electromagnetism and Gravitation in the Weak Field ...
Using weak field approximation, general relativity is expressed like electromagnetism. Every electromagnetic field couples to gravitoelectric and ...
[34]
[PDF] Gravitation in the Weak-Field Limit
In special relativity, electromagnetism is described by a one-form field Aµ(x) in flat spacetime. Similarly, in the weak-field limit gravitation is ...
[35]
Gauge invariance - Scholarpedia
Dec 3, 2008 · Gauge invariance refers to the property that a whole class of scalar and vector potentials, related by so-called gauge transformations, describe the same ...The classical variational... · Classical electromagnetism...
[36]
[PDF] Historical roots of gauge invariance - arXiv
ABSTRACT. Gauge invariance is the basis of the modern theory of electroweak and strong interactions (the so called Standard Model). The roots of gauge ...