The Aharonov–Casher Effect in Biquaternionic Form

Introduction

A charged particle that encircles a region of magnetic flux acquires a phase that depends on the enclosed flux and not on the details of its path; this is the Aharonov–Bohm (AB) effect. In 1984 Aharonov and Casher pointed out the electric dual: a neutral particle carrying a magnetic dipole moment and moving in a region of electric field acquires a phase

$$ \phi_{AC} = \frac{1}{\hbar c^2}\oint_{\mathcal{C}}\left(\mathbf{E}\times\boldsymbol{\mu}\right)\cdot d\mathbf{r}, $$

which likewise depends on the path and the field but not on the speed at which the path is traversed. This is the Aharonov–Casher (AC) effect. The two effects have the same structure — a geometric phase from a gauge connection — with the electric displacement field and the magnetic dipole moment exchanged for the vector potential and the electric charge.

This article treats the AC effect in the biquaternion algebra $\mathbb{B} = \mathbb{C}\otimes_\mathbb{R}\mathbb{H}$ of the read-list articles. The effect is standard and is reproduced exactly. What the framework contributes is the placement of the AC phase within the same algebraic structure as the geometric phases of spin:

1. The AC phase is a holonomy of the spinor. It is the integral of a spin-space connection whose values are Hermitian elements of $\mathbb{M}_+$ built from the electric field and the spin operator. It is therefore the same kind of object as the Berry and Aharonov–Anandan phases of non-relativistic spin.
2. The coupling is a rest-frame magnetic field. A magnetic dipole moving through an electric field sees, in its rest frame, the effective magnetic field $\mathbf{B}' = \mathbf{B} - \frac{1}{c^2}\mathbf{v}\times\mathbf{E}$; the AC phase is the first-order term in $v/c$ of the Zeeman coupling to that field. The derivation uses only the first-order rest-frame transformation, which is the non-relativistic approximation; the exact treatment belongs to the relativistic series.
3. For a spin-1/2 the connection is an $SU(2)$ connection. With $\boldsymbol{\mu} = \gamma\mathbf{S}$ and $\mathbf{S} = \tfrac{\hbar}{2}i\,\mathbf{e}$, the connection is proportional to the spin operator, and its holonomy is a rotor in $\mathbb{H}_{\mathbb{B}}$ together with a central phase — the same two faces as the geometric phase of non-relativistic spin.
4. The AC effect and the AB effect are electric–magnetic duals. The AB phase is $\frac{q}{\hbar}\oint\mathbf{A}\cdot d\mathbf{r}$ for a charge; the AC phase is $\frac{1}{\hbar c^2}\oint(\mathbf{E}\times\boldsymbol{\mu})\cdot d\mathbf{r}$ for a dipole. The electric dipole analogue is the He–McKellar–Wilkens phase.
5. The phase is measurable and has been measured. Neutron interferometry, atom interferometry and electron interferometry all exhibit the AC phase; orders of magnitude are given below.

The notation is that of the read-list articles: $\tilde{S}_k = \tfrac{\hbar}{2}ie_k$, $\tilde\Pi_\pm(\hat\mu) = \tfrac12(e_0\pm i\hat\mu)$, $\tilde{\rho} = \tfrac12(e_0+i\mathbf{r})$, and the trace formula $\mathrm{Tr}(\tilde{P}\tilde{H}) = 2\,\mathrm{Sc}(\tilde{P}\tilde{H})$.

The companion articles supply the pieces: - Companion article The Path Integral in Biquaternionic Form, for the phase from the action and the Lagrangian route. - Companion article The Berry Phase and Geometric Phases in Biquaternionic Form, for the geometric phase and the solid-angle formula. - Companion article Spin-1/2 Quantum Mechanics in Biquaternionic Form, for the spin operator and the idempotents.

The Effect

A Magnetic Dipole in an Electric Field

A magnetic dipole $\boldsymbol{\mu}$ at rest in a magnetic field $\mathbf{B}$ has Zeeman energy $-\boldsymbol{\mu}\cdot\mathbf{B}$ and, if the field is constant and the dipole state is an eigenstate, its state carries the exponent phase $-\frac{1}{\hbar}\int E\,dt = +\frac{1}{\hbar}\int\boldsymbol{\mu}\cdot\mathbf{B}\,dt$. For a dipole moving through an electric field, the relevant field is the one in its own rest frame. To first order in $v/c$ the rest-frame magnetic field is

$$ \mathbf{B}' = \mathbf{B} - \frac{1}{c^2}\,\mathbf{v}\times\mathbf{E}, $$

the standard field transformation of a low-velocity Lorentz boost (the electric field in turn acquires a term $\mathbf{E}' = \mathbf{E} + \mathbf{v}\times\mathbf{B}$, which does not couple to a magnetic dipole at this order). The additional coupling from the motional term is therefore

$$ \frac{1}{c^2}\,\boldsymbol{\mu}\cdot\left(\mathbf{v}\times\mathbf{E}\right) = \frac{1}{c^2}\left(\mathbf{E}\times\boldsymbol{\mu}\right)\cdot\mathbf{v}, $$

using the cyclic invariance of the scalar triple product. A term linear in the velocity enters the action as a coupling of the same kind as the minimal coupling $q\mathbf{A}\cdot\mathbf{v}$ of a charge, and it is the action that fixes the phase. Writing the coupling with the sign convention that makes the magnetic-dipole phase the electric–magnetic analogue of the Aharonov–Bohm phase,

$$ L_{AC} = \frac{1}{c^2}\,\boldsymbol{\mu}\cdot\left(\mathbf{v}\times\mathbf{E}\right) = \frac{1}{c^2}\left(\mathbf{E}\times\boldsymbol{\mu}\right)\cdot\mathbf{v}, $$

the phase accumulated along the path is

$$ \Delta\phi = \frac{1}{\hbar}\int L_{AC}\,dt = \frac{1}{\hbar c^2}\int_{\mathcal{C}}\left(\mathbf{E}\times\boldsymbol{\mu}\right)\cdot d\mathbf{r}, $$

using $\mathbf{v}\,dt = d\mathbf{r}$. For a closed path $\mathcal{C}$ this is the AC phase quoted above — the standard result of Aharonov and Casher, with the same sign convention that gives the Aharonov–Bohm phase $+\frac{q}{\hbar}\oint\mathbf{A}\cdot d\mathbf{r}$. It is independent of the duration of the journey and of the speed profile, which is what makes it geometric; reversing the direction of travel reverses the sign of $\phi_{AC}$, as a geometric phase must.

The Phase as a Holonomy

The AC phase is the integral of a connection on parameter space, in the sense of the preceding article. Define the AC connection one-form

$$ \mathcal{A}_{AC} = \frac{1}{\hbar c^2}\left(\mathbf{E}(\mathbf{r})\times\boldsymbol{\mu}\right)\cdot d\mathbf{r}, $$

whose values are generators acting on the spin degrees of freedom. Then $\phi_{AC} = \oint\mathcal{A}_{AC}$, and for a path that is not closed the same integral defines a relative phase between the endpoints, exactly as the Pancharatnam connection does. The connection is neither flat nor exact: away from the charges $\mathbf{E}$ is curl-free, but that is not enough to make the closed integral vanish, and $\mathcal{A}_{AC}$ is not the differential of a single-valued function. Two features are responsible — the curvature of the connection in a general field, and the linking of the path with a charge — and they are separated in the subsection below. The AC effect is the statement that a magnetic dipole is sensitive to the electric field through this connection.

The Spin-1/2 Form

For a spin-1/2 magnetic dipole the moment is proportional to the spin,

$$ \boldsymbol{\mu} = \gamma\,\mathbf{S} = \frac{\hbar\gamma}{2}\,i\,\mathbf{e}, $$

so the AC connection is

$$ \mathcal{A}_{AC} = \frac{\gamma}{2c^2}\left(\mathbf{E}\times i\mathbf{e}\right)\cdot d\mathbf{r}, $$

a one-form with values in the Hermitian subspace $\mathbb{M}_+$ (the spin components $i e_k$ are Hermitian). Its integral is the generator of a rotor; the phase for a given spin state is the expectation

$$ \phi_{AC} = \left\langle \frac{\gamma}{2c^2}\oint\left(\mathbf{E}\times i\mathbf{e}\right)\cdot d\mathbf{r}\right\rangle, $$

evaluated in the state, i.e. through the trace pairing $\mathrm{Tr}(\tilde{\rho}\,\mathcal{A}_{AC})$. For a spin eigenstate along a fixed axis the expectation is $\pm$ the corresponding scalar integral; for a superposition the two components accumulate opposite phases, and the holonomy is a full rotor rather than a central phase.

The Adiabatic Limit and the Solid Angle

If the dipole is transported adiabatically, the spin follows the instantaneous eigenstate of the effective Zeeman Hamiltonian $-\boldsymbol{\mu}\cdot\mathbf{B}'$, and the acquired phase is the Berry phase of the effective-field loop. That phase is

$$ \gamma_{\mathrm{geo}} = -\frac{1}{2}\,\Omega_{\mathrm{sgn}}, $$

one half the oriented solid angle traced by the direction of $\mathbf{B}'$ on the Bloch sphere. The AC integral above and this solid-angle expression agree in the adiabatic limit; the AC form is the more primitive one, since it does not require adiabaticity to be defined, while the solid-angle form exposes the geometry.

Why the Closed Integral Does Not Vanish

The electric field of a static charge distribution is curl-free away from the charges, so a naive application of Stokes' theorem to the AC integral might suggest that $\oint(\mathbf{E}\times\boldsymbol{\mu})\cdot d\mathbf{r}$ vanishes for constant $\boldsymbol{\mu}$. It does not, and three distinct mechanisms are responsible. They should be kept apart, because only the last of them is topological.

1. The moment is an operator. When $\boldsymbol{\mu}$ is a spin operator rather than a fixed vector, its components do not commute with one another, and the integral is path-ordered rather than a number. This is the non-abelian content treated below.

2. The integrand is not curl-free. For a fixed moment the integral is

$$ \oint\left(\mathbf{E}\times\boldsymbol{\mu}\right)\cdot d\mathbf{r} = \boldsymbol{\mu}\cdot\oint d\mathbf{r}\times\mathbf{E}, $$

and although $\nabla\times\mathbf{E} = 0$, the vector $\mathbf{E}\times\boldsymbol{\mu}$ has in general a non-vanishing curl. In a region with no charge and for a constant moment the identity

$$ \nabla\times\left(\mathbf{E}\times\boldsymbol{\mu}\right) = \left(\boldsymbol{\mu}\cdot\nabla\right)\mathbf{E} $$

makes the content of the loop integral explicit: through Stokes' theorem it measures the gradient of the field along the direction of the moment. This contribution is not topological. It depends on the field configuration along the path, and it is present even for a path that can be shrunk to a point without crossing any charge — for instance for a loop lying in the field of a point dipole and enclosing none of it.

1. The path may enclose a charge. If the path cannot be shrunk to a point without crossing a charge — a line or point charge — then no smooth surface spans it, Stokes' theorem does not apply, and the charge contributes a term that depends on the path only through its linking with the charge. This term is topological in the same sense as the AB flux.

In the line-charge geometry below the three mechanisms separate cleanly. The field is radial about the wire and independent of the axial coordinate, so $(\boldsymbol{\mu}\cdot\nabla)\mathbf{E} = 0$ for an axial moment and the second contribution vanishes; the moment may be treated as a fixed vector, so the first vanishes as well; and what remains is the linking term, proportional to the enclosed density $\lambda$. The line-charge AC phase is therefore the exact electric dual of the AB phase of a charge encircling a flux line. For a general field configuration the second contribution is present, and the phase is not a function of the enclosed charge alone.

Superpositions and the Non-Abelian Holonomy

For a spin prepared in an eigenstate of the effective field the AC phase is a single number, the expectation of the connection. For a superposition of the two spin states the two components acquire opposite phases, and the spinor transforms by a genuine $SU(2)$ element. Concretely, a spin-1/2 traversing a region in which the effective field direction rotates is described by a path-ordered exponential

$$ \tilde{U} = \mathcal{P}\exp\!\left(\frac{i}{\hbar c^2}\oint\left(\mathbf{E}\times\boldsymbol{\mu}\right)\cdot d\mathbf{r}\right) = \mathcal{P}\exp\!\left(\frac{i\gamma}{2c^2}\oint \mathbf{E}\times i\mathbf{e}\cdot d\mathbf{r}\right), $$

where the factor $i$ makes the exponent anti-Hermitian: the connection $\mathcal{A}_{AC}$ is $\mathbb{M}_+$-valued, and it is $i\mathcal{A}_{AC}$ that generates a unitary transport, exactly as $-i\tilde{H}/\hbar$ does for the Schrödinger evolution. The holonomy is therefore a genuine unitary, and the cycle-averaged phase for each eigen-component is the corresponding diagonal entry, while the off-diagonal entries rotate the superpositions. On an eigenstate the holonomy reduces to the phase factor $e^{i\phi_{AC}}$, with $\phi_{AC}$ the real eigenvalue of $\oint\mathcal{A}_{AC}$. In the biquaternion language $\tilde{U}$ is a general element of $\mathbb{H}_{\mathbb{B}}$ up to a central phase: a rotor in $\mathrm{SU}(2)$ that acts on the two-component spinor. The abelian AC phase measured with a polarised beam is the trace of $\tilde{U}$ in the eigenbasis; the non-abelian content appears when the spin is prepared in a superposition and recombined.

The AC Connection in Spin-Orbit Form

In a crystal or atomic system with spin-orbit coupling, the electric field seen by a moving electron produces an effective magnetic field of exactly the AC form, and the spin dynamics it drives is the spin-orbit interaction. The Rashba coupling of a two-dimensional electron gas, $\tilde{H}_R = \alpha_R(\mathbf{p}\times\boldsymbol{\sigma})\cdot\hat{z}$, and the Dresselhaus coupling have the same structure: a spin operator contracted with a momentum-dependent effective field. The AC connection is the non-relativistic limit of the spin connection of the Dirac electron, and the solid-angle phase it produces underlies the spin-orbit geometric phases used in spintronics and in the design of spin transistors. The framework does not add to this standard picture; it records that the spin-orbit Hamiltonian and the AC connection are the same $\mathbb{M}_+$ element written in different variables.

The Duality with Aharonov–Bohm

The Two Connections

The AB and AC phases have the same mathematical shape. In the AB effect a charge $q$ moving in a vector potential $\mathbf{A}$ acquires

$$ \phi_{AB} = \frac{q}{\hbar}\oint_{\mathcal{C}}\mathbf{A}\cdot d\mathbf{r} = \frac{q}{\hbar}\Phi_B, $$

the enclosed magnetic flux; the connection is $\frac{q}{\hbar}\mathbf{A}\cdot d\mathbf{r}$, with values in the Lie algebra of $U(1)$. In the AC effect a magnetic dipole $\boldsymbol{\mu}$ moving in an electric field acquires $\phi_{AC}$ as above; the connection is $\frac{1}{\hbar c^2}(\mathbf{E}\times\boldsymbol{\mu})\cdot d\mathbf{r}$, with values in the spin algebra. The dictionary is

Both phases are geometric: neither depends on the speed of traversal, and both are defined modulo $2\pi$. The AC connection is non-abelian for a genuine spin-1/2, since its components do not commute when the effective field direction varies along the path; for a spin prepared in an eigenstate the phase reduces to the abelian expectation.

The He–McKellar–Wilkens Dual

The further dual is the He–McKellar–Wilkens (HMW) effect: a neutral particle carrying an electric dipole moment $\mathbf{d}$ moving in a magnetic field acquires

$$ \phi_{HMW} = \frac{1}{\hbar}\oint\left(\mathbf{B}\times\mathbf{d}\right)\cdot d\mathbf{r}. $$

The coupling is to the motional electric field $\mathbf{E}_{\mathrm{mot}} = \mathbf{v}\times\mathbf{B}$ seen by the dipole in its own rest frame. The asymmetry with the AC case is the asymmetry of the Lorentz field transformation in SI: the motional magnetic field is $\mathbf{v}\times\mathbf{E}/c^2$, while the motional electric field is $\mathbf{v}\times\mathbf{B}$, so the magnetic-dipole coupling carries the $1/c^2$ and the electric-dipole coupling does not.

The three effects — AB, AC and HMW — form the electric–magnetic dual triangle: charge is dual to magnetic flux, the magnetic dipole to the electric field, and the electric dipole to the magnetic field. In the biquaternion framework all three are integrals of $\mathbb{M}_+$-valued or central connections; the spin-1/2 content is the AC case, and the HMW case couples to the electric dipole, which for a bare spin-1/2 vanishes by parity, as the treatment of the Zeeman and Stark effects records.

The Lagrangian Route

The AC phase can also be obtained from the coupling Lagrangian of a moving dipole, which is the form in which it appears in the path integral. For a point charge in a vector potential the minimal coupling is $L_{AB} = q\,\mathbf{A}\cdot\mathbf{v}$, whose integral over a closed path is the AB phase $\frac{q}{\hbar}\oint\mathbf{A}\cdot d\mathbf{r}$. The magnetic dipole has the analogous coupling

$$ L_{AC} = \frac{1}{c^2}\,\boldsymbol{\mu}\cdot\left(\mathbf{v}\times\mathbf{E}\right) = \frac{1}{c^2}\left(\mathbf{E}\times\boldsymbol{\mu}\right)\cdot\mathbf{v}, $$

whose action integral over a closed path is exactly $\hbar\phi_{AC}$. The electric dipole coupling in a magnetic field is $L_{HMW} = \mathbf{d}\cdot(\mathbf{v}\times\mathbf{B})$, with no $1/c^2$, consistently with the phase above. In each case the coupling is a one-form in velocity contracted with the appropriate moment and field, and the phase is its holonomy. In the biquaternion framework the couplings are elements of $\mathbb{M}_+$ for the dipole cases (they involve the spin or dipole operator) and central for the charge case; the path integral of the companion article The Path Integral in Biquaternionic Form is the natural home for the action formulation, and the AC phase is the holonomy that survives the sum over paths.

Explicit Geometries

Circling a Line Charge

Take a line charge of linear density $\lambda$ along the $\hat{z}$ axis, producing the radial field

$$ \mathbf{E} = \frac{\lambda}{2\pi\epsilon_0 r}\,\hat{r}. $$

Let a magnetic dipole of moment $\boldsymbol{\mu} = \mu_z\hat{z}$ travel once around a circle of radius $R$ in the plane $z = \mathrm{const}$. Along the path $\mathbf{E}$ is radial and $d\mathbf{r} = R\,d\phi\,\hat\phi$ is tangential, so, using $\hat{r}\times\hat{z} = -\hat\phi$,

$$ \left(\mathbf{E}\times\boldsymbol{\mu}\right)\cdot d\mathbf{r} = -E\,\mu_z\,R\,d\phi, $$

and the AC phase is independent of $R$:

$$ \phi_{AC} = -\frac{1}{\hbar c^2}\cdot 2\pi R E\,\mu_z = -\frac{\lambda\,\mu_z}{\epsilon_0\hbar c^2} $$

for the counterclockwise traversal, using $2\pi R E = \lambda/\epsilon_0$. The magnitude depends on the enclosed line charge and on the axial component of the dipole moment, and not on the radius or the speed; the sign is reversed by reversing the traversal. This is the cleanest closed-form instance of the effect, and it is the exact electric dual of the AB phase of a charge encircling a flux line.

Orders of Magnitude

With $\hbar c^2 = 1.0546\times10^{-34}\times(2.998\times10^8)^2\,\mathrm{J\,m} \approx 9.48\times10^{-18}\,\mathrm{J\,m}$:

• Neutron. $\mu_n \approx 1.913\,\mu_N \approx 9.66\times10^{-27}\,\mathrm{J/T}$. For a path of length $L = 0.1\,\mathrm{m}$ with $\mathbf{E}\perp d\mathbf{r}$ and $E = 10^{7}\,\mathrm{V/m}$, the phase is $\phi_{AC} = \mu_n E L/(\hbar c^2) \approx 1.02\times10^{-3}\,\mathrm{rad}$, a milliradian-scale phase readily measured by interferometry.
• Electron. $\mu_e \approx 9.28\times10^{-24}\,\mathrm{J/T}$, giving $\phi_{AC} \approx 0.98\,\mathrm{rad}$ for the same field and path — the effect is roughly a thousand times larger for the electron because of the mass ratio in the Bohr magneton.
• Line charge. For $\lambda = 10^{-8}\,\mathrm{C/m}$ and the neutron moment, $\phi_{AC} = \lambda\mu_n/(\epsilon_0\hbar c^2) \approx 1.15\times10^{-6}\,\mathrm{rad}$; the smallness reflects the small enclosed charge.

These are order-of-magnitude estimates; the point is that the phase is linear in the field, the moment and the path length, so it can be tuned into the detectable range.

Experiments

The AC phase has been observed in several systems. The neutron experiment of Cimmino and collaborators used a charged wire in a crystal neutron interferometer and measured the phase shift of the spin-polarised neutron; the atom-interferometry experiments of Sangster and collaborators and, later, of Yanagimachi and collaborators used neutral atoms with an applied electric field; and the electron experiment of König and collaborators observed the phase directly in a semiconductor ring with spin–orbit coupling. The HMW effect has been the subject of proposals and of cold-atom and molecular experiments. All of these are measurements of the standard result; the framework reproduces them and adds no shift.

The Measured Observable

Spin-Dependent Interference

The AC phase is not an overall phase of the beam; it is a spin-dependent phase, and it is measured by interfering the two spin components. In a two-path interferometer an arm that reverses the spin relative to the other converts the AC phase into a relative phase between the two paths, and the fringes shift by $\Delta\phi/2\pi$ periods. Because reversing the spin reverses the sign of $\boldsymbol\mu$ and hence of $\phi_{AC}$, the spin-flip phase difference is twice the single-spin phase,

$$ \Delta\phi_{\uparrow\downarrow} = 2\phi_{AC} = \frac{2}{\hbar c^2}\oint\left(\mathbf{E}\times\boldsymbol{\mu}\right)\cdot d\mathbf{r}. $$

This doubling is what makes the effect accessible: for the neutron example above, the single-spin phase $\approx 1.0\times10^{-3}\,\mathrm{rad}$ corresponds to a fringe shift of $\approx 1.6\times10^{-4}$ of a period, and the spin-flip difference is twice that, $\approx 3.2\times10^{-4}$ of a period. Neutron interferometers resolve phase shifts at this level by counting the full interference pattern, and the experiments use the spin-flip (or the reversal of the field) to isolate the AC term from the ordinary dynamical phases.

Reversal Tests

The geometric character of the phase is established by the reversal tests. Reversing the direction of travel reverses $\phi_{AC}$; reversing the sign of the enclosed charge reverses it; reversing the spin reverses it; and changing the speed of the particle does not change it. The first three reversals and the speed-independence together distinguish the AC phase from a dynamical phase, which would scale with the traversal time. The neutron experiments used exactly these reversals.

What the Algebra Adds and What It Does Not

Standard physics, transcribed. The AC phase, its derivation from the rest-frame field, the duality with AB, the line-charge closed form, and the experimental values are all standard.

What the algebra organises.

• A common holonomy language. The AC phase is the integral of an $\mathbb{M}_+$-valued connection, exactly as the Berry and Aharonov–Anandan phases of the preceding article are integrals of a spinor connection. The three are the same kind of object, distinguished by which field couples to the spin.
• The spin-1/2 $SU(2)$ structure. With $\boldsymbol{\mu} = \gamma\mathbf{S} = \tfrac{\hbar\gamma}{2}i\mathbf{e}$, the AC connection is proportional to the spin operator, so its holonomy is a rotor in $\mathbb{H}_{\mathbb{B}}$ and its abelian part is the central phase. The abelian/non-abelian distinction is the eigenstate/superposition distinction, as in the geometric-phase article.
• The dual triangle is visible. AB, AC and HMW all have the form of a coupling constant times a field integral; the algebra collects them as integrals of connections built from the field and the appropriate moment operator.
• Parity selects the coupling. A bare spin-1/2 has a magnetic moment but no electric dipole; hence it exhibits AC and not HMW. The selection is the parity argument of the Zeeman–Stark article.

What the algebra does not supply. The gyromagnetic ratio $\gamma$, the field configuration and the path are external data. The algebra supplies the holonomy structure and takes the coupling constants as given.

Open Questions

1. Non-abelian AC phases. For a spin-1/2 in a spatially varying electric field the AC connection is $SU(2)$-valued, and a path that returns the spin direction to itself can nonetheless return the spinor with a non-abelian holonomy. Is there a natural experimental configuration in which the non-abelian character is manifest? The question is the electric analogue of the non-abelian AB phase in degenerate systems.

2. The exact relativistic treatment. The rest-frame field used here is first order in $v/c$. The exact AC phase for a Dirac particle, including the Thomas precession and the spinor structure, belongs to the relativistic series; whether the first-order formula acquires corrections that are algebraically natural is open.

3. Topological robustness. The AB phase is robust because the enclosed flux is quantised in a superconducting ring; the AC phase has no analogous quantisation of the enclosed charge in ordinary matter. Is there a physical configuration in which the AC phase is quantised, and does the algebra single it out?

4. Bound-state and scattering analogues. The AC effect has analogues in scattering (the "AC scattering" of dipoles) and in bound systems. Does the framework's scattering machinery, which belongs to the spin-0 and general sectors, extend to the dipole case?

5. Empirical contact. As everywhere in the series, the reformulation reproduces the standard predictions; whether it implies a measurable deviation is open.

Summary

A neutral particle carrying a magnetic dipole moment $\boldsymbol{\mu}$ and moving along a path $\mathcal{C}$ in an electric field acquires the Aharonov–Casher phase

$$ \phi_{AC} = \frac{1}{\hbar c^2}\oint_{\mathcal{C}}\left(\mathbf{E}\times\boldsymbol{\mu}\right)\cdot d\mathbf{r}, $$

which follows from the rest-frame field $\mathbf{B}' = \mathbf{B} - \frac{1}{c^2}\mathbf{v}\times\mathbf{E}$ and the Zeeman coupling $-\boldsymbol{\mu}\cdot\mathbf{B}'$. The phase is geometric: it depends on the path and the field but not on the speed, and it reverses sign when the traversal is reversed.

In the biquaternion algebra the phase is the integral of the $\mathbb{M}_+$-valued connection

$$ \mathcal{A}_{AC} = \frac{1}{\hbar c^2}\left(\mathbf{E}\times\boldsymbol{\mu}\right)\cdot d\mathbf{r} = \frac{\gamma}{2c^2}\left(\mathbf{E}\times i\mathbf{e}\right)\cdot d\mathbf{r} $$

for a spin-1/2 of magnetic moment $\boldsymbol{\mu} = \gamma\mathbf{S}$, with the phase given by the trace pairing in the state. In the adiabatic limit it reduces to the solid-angle phase $-\tfrac12\Omega_{\mathrm{sgn}}$ of non-relativistic spin.

The AC effect is the electric dual of the Aharonov–Bohm effect and the magnetic-field dual of the He–McKellar–Wilkens effect: charge is dual to flux, the magnetic dipole to the electric field, and the electric dipole to the magnetic field. For a dipole encircling a line charge of density $\lambda$, the phase is $\phi_{AC} = -\lambda\mu_z/(\epsilon_0\hbar c^2)$ for counterclockwise traversal, independent of radius and speed. For a neutron over $0.1\,\mathrm{m}$ in a field of $10^{7}\,\mathrm{V/m}$ the magnitude of the phase is $\approx 1.0\times10^{-3}\,\mathrm{rad}$; for an electron it is $\approx 0.98\,\mathrm{rad}$. The phase has been measured by neutron, atom and electron interferometry, and the framework reproduces the standard result without modification.

Summary of Notation

Further Reading

• Y. Aharonov and A. Casher, "Topological Quantum Effects for Neutral Particles," Physical Review Letters 53 (1984) 319–321, for the original prediction of the effect.
• Y. Aharonov and D. Bohm, "Significance of Electromagnetic Potentials in the Quantum Theory," Physical Review 115 (1959) 485–491, for the magnetic dual.
• X.-G. He and B. H. J. McKellar, "Topological Phase Due to Electric Dipole Moment and Magnetic Monopole Interaction," Physical Review A 47 (1993) 3424–3425, and A. Wilkens, "Quantum Phase of a Moving Dipole," Physical Review Letters 72 (1994) 5–8, for the He–McKellar–Wilkens effect.
• A. Cimmino, G. I. Opat, A. G. Klein, H. Kaiser, S. A. Werner, M. Arif, and R. Clothier, "Observation of the Topological Aharonov–Casher Phase Shift by Neutron Interferometry," Physical Review Letters 63 (1989) 380–383, for the neutron measurement.
• K. Sangster, E. A. Hinds, S. M. Barnett, and E. Riis, "Measurement of the Aharonov–Casher Phase in an Atomic System," Physical Review Letters 71 (1993) 3641–3644, for the atom-interferometry measurement.
• S. Yanagimachi, M. Kajiro, M. Machiya, and A. Morinaga, "Direct Measurement of the Aharonov–Casher Phase and Tensor Stark Polarizability Using a Calcium Atomic Polarization Interferometer," Physical Review A 65 (2002) 042104, for a second atom-interferometry measurement.
• M. König, A. Tschetschetkin, E. M. Hankiewicz, J. Sinova, V. Hock, V. Daumer, M. Schäfer, C. R. Becker, H. Buhmann, and L. W. Molenkamp, "Direct Observation of the Aharonov–Casher Phase," Physical Review Letters 96 (2006) 076804, for the electron measurement.
• C. R. Hagen, "Exact Equivalence of Spin-1/2 Aharonov–Bohm and Aharonov–Casher Effects," Physical Review Letters 64 (1990) 2347–2349, for the relationship between the two effects.
• J. Anandan, "Electromagnetic Effects in the Quantum Interference of Dipoles," Physics Letters A 138 (1989) 347–352, for the geometric-phase interpretation of the AC effect.
• A. Shapere and F. Wilczek, Geometric Phases in Physics (World Scientific, 1989), for the general theory of geometric phases and holonomies.