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Quaternionic Aharonov-Bohm effect

A quaternionic analog of the Aharonov-Bohm effect is developed without the usual anti-hermitian operators in quaternionic quantum mechanics (QQM). A quaternionic phase links the solutions obtained to ordinary complex wave functions, and new theoretical studies and experimental tests are possible for them.


I. INTRODUCTION
Quaternions are non-commutative, generalized complex numbers. An arbitrary quaternion may be written in terms of two complex variables, z and ζ, so that q = z + ζj. ( (1) is the symplectic notation for quaternions and j is a complex unit that does not commute with the complex unit i ; in fact, i j = − j i . There are many sources for the mathematics of quaternions, and we recommend [1]. A generalization of quantum mechanics by introducing quaternions replacing complex numbers has been developed, and an introduction to the subject and an encyclopedic list of references may be found at [2]. Conversely, compared to complex quantum mechanics (CQM), there are few explicit solutions for quaternionic quantum mechanics (QQM). For example, we have several results considering the quaternionic Schrödinger equation [3][4][5][6][7][8], and relativistic solutions obtained from the Dirac equation [9][10][11][12] have also been obtained. Furthermore, extremely few analytic results of QQM have been used for experimental work, and a review of these can be found at [13]. Quite recently, a new study has been carried out [14,15], but observable quaternionic effects have never been detected, despite these efforts. All these previous studies have been conducted considering that QQM should written using anti-hermitian operators, which obey A † = −A . In this article, we propose a simpler QQM, where neither hermiticity nor anti-hermiticity are supposed. At this moment, we cannot say whether this choice will lead to a consistent theory. However, we present some new results that may give us some hope that a consistent quaternionic theory may be built without imposing anti-hermiticity on its operators. If we may develop several physically consistent results and examples, the mathematical formalism will be developed from this set of results that may hopefully be considered in experimental tests. This is, in fact, the historical path that have been followed by quantum mechanics. More recently, theoretical physics have become increasingly mathematical, including QQM. This is not a problem, because the knowledge is never lost and it is a reference to new studies, including this one. However, we must always come back to the physical phenomena in order to ascertain what the mathematical formalism describes.
Thus, in this article, we write a quaternionic Schrödinger equation whose Hamiltonian operator has no defined hermiticity. This Hamiltonian has been obtained by defining a quaternionic momentum operator inspired by the electromagnetic field Hamiltonian. Then we have simply made a transformation that can be schematically written as where p is the momentum, α is the real magnetic vector potential and β is an arbitrary complex vector potential without a previous physical interpretation. We borrowed he notation of (2) from [16], where the complex unity is multiplied on the right, namely (F |i ) = F i = i F . We use this momentum operator to obtain quaternionic wave functions Ψ related to complex wave functions ψ through a unit quaternion, so that where ψ satisfies the time-dependent Schrödinger equation and K is time-independent. We will see that there are nontrivial solutions that obey these conditions, and if we interpret β as having a magnetic character, we will see that a particle Equations (10) are obtained from (8) by the replacement Thus, if H −ε 2 +LL commutes with θ, then φ and χ are either equal or related. Conversely, if they do not commute, then we repeat the procedure employed to obtain (10) from (8) and evaluate the commutativity of the new operators. This analysis enables us to suppose that φ and χ must always be related in order to have a quaternionic solution for the Schrödinger equation. In the next sub-section, we use this conclusion to obtain physically non-trivial quaternionic solutions.

B. CONSTRAINTS TO THE QUATERNIONIC WAVE-FUNCTION
In the previous section, the dependence between the time-independent symplectic components of the wave function (4) has been pointed out. We wonder whether we can use the complex solution of (5), namely the solution that is obtained for the limit within L → 0 in (7). Let us name Ψ the quaternionic solution and ψ the complex solution, such that Let us suppose that the time-independent wave functions are related according to Γ, Θ and Ω are space-dependent real functions and of course |K | = 1. We call (13) the left quaternionic solution; in the appendix we discuss the right quaternionic solution Φ = φK . We notice that, within the complex limit of K , and consequently there is a geometric phase relating the solutions within the complex limit of Ψ. If the proposal (13) is correct, then it generalizes the geometric phase of quantum mechanics. We know that the AB-effect is one of the simplest cases of geometric phases, and then we will try to obtain a quaternionic analogue of the AB-effect. In order to achieve this objective, we need a quaternionic momentum to build a quaternionic Hamiltonian. To achieve this aim, we calculate the gradient of (13) and we get Then we define the momentum operator Π, namely and we observe that the complex unit i must be multiplied on the right of Φ. From (14) and (15) we get the squared momentum, Now, we want to solve the quaternionic Schrödinger equation If V is a real potential, the complex Schrödinger equation is, of course, Using the time-independent equation H φ = E φ and (16) in (17), we get and then the quaternionic and complex wave functions related by (13) have the same energy for every real potential V . Now, we define an arbitrary quaternionic momentum given by with α an arbitrary real vector and β an arbitrary complex vector. We notice that the probability density is preserved in this model. Using (17) and (20), we get the continuity equation The conservation of the probability density ascertained by (21) is an important consistency test. Furthermore, it is an example of non-anti-hermitian quaternionic quantum model. Quaternionic quantum mechanics has been developed using anti-hermitian operators [2], and thus we have a counterexample showing that a consistent non-anti-hermitian model is possible. Substituting Φ in (17), and using (12), (20) and (18), we get (22) generates equations for Θ, Γ, and Ω that permit us to determine the quaternionic wave function and its spectrum. Solving these constraints may be a difficult task, as we see from the derivatives of K , which we write schematically as ∇ ∇ ∇K = pe iΓ + qe iΩ j and where In order to obtain solutions, we use (23) to split (22) into pure complex and pure quaternionic equations. The complex equation reads and the quaternionic term is v + i sin Θ ∇ ∇ ∇ · · · α + 2i α · · ·q + sin ΘQ · · · Q e −iΩ − cos Θ ∇ ∇ ∇ · · ·β + 2β · · · p e iΓ φ + 2 q + i sin Θ α e −iΩ − cos Θβe iΓ · · · ∇ ∇ ∇φ = 0. (26) Equations (25) and (26) are constraints that α, β, Γ, Θ and Ω satisfy for each complex solution φ. The equations are not necessarily identical, and may differ from a complex factor, that can be either a constant or a function. Let us call λ the factor that relates (25) and (26), then the coefficients for φ must satisfy u − i cos Θ ∇ ∇ ∇ · · · α − 2i α · · · p + cos ΘQ · · · Q = −λ cos Θ ∇ ∇ ∇ · · ·β + 2β · · · p (27) v + i sin Θ ∇ ∇ ∇ · · · α + 2i α · · ·q + sin ΘQ · · · Q = 1 λ sin Θ ∇ ∇ ∇ · · · β + 2β · · ·q .

III. THE WAVE-FUNCTIONS
Let us first consider the ∇ ∇ ∇Θ = 0 case, where (33) simplifies to In order not to constrain the complex wave function, we impose conditions to get two cases. Now we will examine the first.

A. SOLUTIONS FOR
In this case, the quaternionic solution comprises where L is an arbitrary constant unitary quaternion. This is the simplest quaternionic solution that is related to a complex solution through a geometric phase. It is also the simplest counterpart of the Aharonov-Bohm effect that may be obtained in quaternionic quantum mechanics. Its interpretation is similar to the complex case, but in this case the phase is quaternionic and consequently non-commutative, and this is its physical novelty that can be researched experimentally. It reduces immediately to the complex case for L = 1, and it is thus the simplest quaternionic generalization of the AB-effect.
B. SOLUTIONS FOR ∇ ∇ ∇Θ = 0 AND |λ| 2 = tan 2 Θ In this case, (34) is satisfied for The condition implies that the quaternionic solution comprises This solution reduces to the previous one if ∇ ∇ ∇(Γ − Ω) = 0, and then we see that it is another example of non-commutative quaternionic AB-effect. Furthermore, the fields generated by the vector potentials are interesting because ∇ ∇ ∇× × ×β = 0. Accordingly, Then, as in the previous case, the geometric phase is non-commutative. Furthermore, we have two non-trivial fields encoded in ∇ ∇ ∇ × × × β. Hence there is a big difference in relation to the usual complex AB-effect. In the quaternionic case, there are real fields in the region where the wave function is defined, so that the particle may interact with the field in some manner. This interaction may explain change in the interference pattern observed in the AB-effect. Additionally, some kind of noncommutativity must be observed, and this is the crucial point that must be stressed when researching quaternionic effects. Now, we consider the effect of this choice on the constraints (27-29) and (33). Constraints (29) are trivially satisfied. Conversely, the complex constraints (27-28) generate four real conditions. One of these constraints implies that |∇ ∇ ∇Θ| = 0, which is a contradiction, and then there are no solutions in this case.

IV. ONE EXAMPLE
We compare our results with the usual Aharonov-Bohm effect, where a a long solenoid of radius R generates a magnetic field B . Defining r as a cylindrical radial distance, we get a zero magnetic field for r > R and a constant magnetic field with intensity B for r < R. Outside the solenoid, although B = 0, the magnetic vector potential is finite and given by where Φ = πR 2 B is the magnetic flux andφ is the azimuthal angle direction. Let us consider the case described in section III B, where ∇ ∇ ∇Θ = 0 and |λ| 2 = tan 2 Θ, and the decomposition ofφ onto Cartesian coordinateŝ Remembering that α = (q/ħ)A, and using (37) The fact that (43) is complex is not necessarily a problem. The magnetic field also comes from an imaginary component of (20). Conversely, the physical nature of the field generated by the potential β is not clear. If we suppose that (43) has a magnetic character, then an electrically charged particle that describes a path on the azimuthal directionφ will suffer a force over the radial direction given by the Lorenz expression. This may explain the interference pattern of the AB-effect. Conversely, the non-commutativity of the wave function must also be ascertained in order to verify the quaternionic character of the phenomenon. Then, non-commutativity and a quantitative correspondence between the interference pattern and the radial quaternionic force are the phenomenona that must be experimentally researched in order to verify the quaternionic AB-effect.

V. CONCLUSION
In this article we obtained a quaternionic solution for the AB-effect. This solution was not obtained according to the canonical anti-hermitian quaternionic quantum mechanics, although, a physically meaningful solution was obtained. This fact opens a discussion about the mathematical consistence of non-anti-hermitian QQM, and there is a necessity for results. Expected values, commutators, the Heisenberg formalism and the Ehrenfest theorem are several examples of the consistency tests that must be executer in order to understand whether a non-hermitian QQM makes sense.
If we suppose that a non-hermitian QQM is consistent, we have further interesting problems related to the physical solution described in this paper. The AB-effect is an example of geometric phase, and a quaternionic case opens the possibility that a quaternionic geometric phase must exist. The geometric phase of anti-hermitian QQM is already described [2,17], but a non-anti-Hermitian counterpart has not been developed.
Finally, we have the possible experiments that may be conducted in order to verify the predictions of the article. The possibility of describing the AB-effect through a non-commutative quaternionic wave function seems to be an interesting theme for experimentalists, and it is the only way to determine whether QQM is physically meaningful. Thus, we expect this article to attract interest in QQM because it raises interesting questions regarding conceptual issues, mathematical consistency and experimental tests.