On the Electric Charge Quantization from the Aharonov-Bohm Potential

Transcription

1 On the Electric Charge Quantization from the Aharonov-Bohm Potential F.A. Barone and J.A. Helayël-Neto Centro Brasileiro de Pesquisas Físicas Rua Dr. Xavier Sigaud 150, Urca Rio de Janeiro, RJ, Brasil arxiv:quant-ph/05031v1 8 Mar 005 Abstract The purpose of this paper is to show that we can define different vector potentials in the problems of Aharonov-Bohm type, in such a way that they give the same physical results as the one originally proposed by Aharonov and Bohm, except in the regions of space where the fields become singular. We argue that a modified quantization condition comes out for the electric charge that may open up the way for the understanding of fractional charges. One does not need any longer to rely on the existence of a magnetic monopole to justify electric charge quantization. The main goal of this paper is to show that, even in a quantum-mechanical context, we can take an Aharonov-Bohm-like potential as a gauge field, once we consider some restrictions on these gauge potentials, and, as a consequance, we are led to a quantization condition for the electric charge which does not involve the existence of magnetic monopoles and which is in agreement with the fractional character of quark charges and the existence of anti-particles. Before start discussing electric charge quantization with no appeal to monopoles, let us briefly recall the main aspects of Dirac s quantization, which intrinsically involves the existence of magnetic monopoles. To describe the field strength of a point-like magnetic monopole g, Dirac used the vector potential of a so-called Dirac s string [1, ], which, if considered to be lying along the z < 0 semi-axis, gives the vector potential A I = g 1 cosθ r sin θ ˆφ, (1) where we used spherical coordinates where φ is the azimuthal angle. With this description, Dirac avoided the introduction of a scalar potential of magnetic nature and, consequently, the trouble of establishing an electrodynamics with a pair of four potentials [3]. Even so, expression fbarone@cbpf.br helayel@cbpf.br 1

2 (1) exhibits the problem of not reproducing the field strength of the magnetic monopole in the whole space; in fact [, 4] A I = g rˆr + 4πgΘ( z)δ(x)δ(y)ẑ, () which encompasses the field of the magnetic monopole plus a singular one along the string (in the above expression δ(x) is the Dirac delta function and Θ(ξ) is the step function which, as usually adopted, is non-defined for ξ = 0). In the classical context, this is not actually a problem. Since we are interested in regions of space which do not include the z < 0 semi-axis, we can consider the potential (1) valid only in the domain 0 θ < π. To describe the true magnetic field in the z < 0 semi-axis, we can take the string, for instance, on the z > 0 semi-axis, which gives the potential A II = g 1 + cos θ r sin θ ˆφ, (3) that is not to be considered along z > 0 semi-axis. The potentials (1) and (3) are related by the singular gauge transformation A I A II 1 = g r sin θ ˆφ = (gφ) (4) which is not defined along the z axis. With this approach, the string can be seen as a purely mathematical artifact, valid only in the classical context. Dirac showed that the freedom of describing the magnetic monopole field using string fields, placed in arbitrary regions, still remains in the quantum context only if the so-called Dirac s quantization [1,, 3] qg = hc n (5) is valid, where n is an integer and q stands for the possible electric charges a wave function may have. After Dirac s seminal paper on magnetic monopoles, there appeared several others in the literature [, 3], all of them presenting quantization conditions for the electric charge, but involving magnetic monopoles as well. We aim here at reassessing charge quantization in the absence of monopoles. To do that, in this paper, we reconsider the Aharonov-Bohm experiment on the diffraction of charged particles. We adopt this scenario to make our ideas more clear; however, we could actually propose our discussion without mentioning the Aharonov-Bohm effect. In their well-known papers of 1959 and 1961, Aharonov and Bohm [5, 6] considered an infinitely long solenoid inside which we could have a uniform and constant magnetic field; outside no magnetic field is present. Let us, first, investigate classical aspects of the Aharonov-Bohm set-up. The magnetic field generated by an infinitely long solenoid of radius R, with its axis lying on the z axis, is given, in cylindrical coordinates, by B sol (ρ, φ, z) = { BI = Bẑ, ρ < R B E = 0, ρ > R

3 = BΘ(R ρ)ẑ. (6) On the solenoid, ρ = R, the magnetic field is not defined; it is a singularity region. In the context of Classical Electrodynamics, an interesting choice for the vector potential is A(ρ, z, φ) = = (Bρ/)ˆφ, ρ < R (γ/ρ)ˆφ, ρ > R [ Bρ Θ(R ρ) + γ ] Θ(ρ R) ˆφ. (7) ρ where γ is a constant. Notice that, outside the solenoid, ρ > R, the potential (7) has the same structure as the Dirac s gauge field (4). Using simple results on distributions, the vector potential (7) gives the magnetic field B = A = BΘ(R ρ)ẑ + which, in general, is not the field (6) generated by the solenoid. In their well-known papers, Aharonov and Bohm took ( γ R BR ) δ(ρ R)ẑ, (8) γ = γ AB = BR (9) in the potential (7), in such a way to have the correct magnetic field (6), but, in fact, the γ parameter is not restricted to the value (9), not even by the Stoke s Theorem, once the flux of magnetic field in the physical region (all space excluding the solenoid) is correctly given by any value of γ, as pointed out in the work of reference [9]. The fact that the fields (6) and (8) are not the same can be interpreted in two distinct ways: i) Notice that, once we are interested in regions of the space which do not include the solenoid (ρ = R), we can use the field (8), given by the potential (7), with any value of the γ parameter to describe the field strength. If we are interested in regions which include the solenoid, we must restrict ourselves to the Aharonov-Bohm γ parameter (9), once we are considering the field (6) as the correct one. Thus, similarly to the case of Dirac s quantization, we can have more than one potential to describe the electromagnetic field in different regions of the space. In analogy to the Dirac s case, there are overlap regions where both the potentials are defined; in our case, the whole space except the solenoid. In these regions, the potentials must be linked by a gauge transformation. In the interior of the solenoid, the potentials are the same but, in its exterior region, we must have where we used equation (9). A γ A γab = (γ γ AB ) 1 ρ ˆφ = 3 [ ( γ BR ) ] φ, (10)

4 We call attention here for the similarity between the gauge transformations (4) and (10). ii) On the other hand, we can say that, on the solenoid, the field is not defined. It is a singularity region for the field strength, since there is a surface charge density. Therefore, obtaining a divergent contribution in this region is not properly a surprise and the field (8) can actually describe the magnetic field in the whole space, once in the physical regions the field strength is perfectly defined. From this approach, the delta-type contribution in (8) is not seen as a problem, once it exists only in a non-physical region. In the context of Quantum Mechanics, the situation is not so simple: a vector potential of the form (7) can yield measurable physical phenomena, even in regions where there is a vanishing field strength. These phenomena, usually referred to as Aharonov-Bohm effect, have their explanations based on the fact that, by virtue of equation (7), when we consider wave functions whose domain is the exterior region of the solenoid (a non-simply conected domain), these ones become no more single-valued functions; they acquire a phase factor whenever we take a closed path around the solenoid. So, for a wave function ψ outside the solenoid, we have ψ(r, φ + π, z) = exp ( iq A dl ) ψ(r, φ, z) ρ>r = exp ( πiqγ)ψ(r, φ, z), (11) where q is the charge of the considered wave function and the sub-index ρ > R indicates that we are taking a path outside the solenoid. With the Aharonov-Bohm γ-parameter (9), we have for equation (11) ψ(r, φ + π, z) = exp( iqπbr )ψ(r, φ, z). (1) In this way, Aharonov and Bohm showed that the γ-parameter in equation (7) is not completely free in the quantum context, as it was in the Classical Electridynamics. Despite the Aharonov-Bohm result, in order to investigate if there is some freedom for the γ parameter in the quantum context, instead of the standard value (9), let us take γ = BR + κ, (13) where κ is, in principle, any real constant. Notice that equation (13) is equivalent to the gauge transformation (10) with which is similar to the case worked out by Dirac (4). Inserting (13) into equation (7), and using (11), we have A γ A γab = κ ρ ˆφ, (14) ψ(r, φ + π, z) = exp( iqπbr ) exp( πiqκ)ψ(r, φ, z). (15) For the choice (13) to be consistent with the standard γ-value (9), the physical results obtained with both γ s must be the same; this is guaranteed if, and only if, the phase differences presented in equations (1) and (15) are the same. This condition is satisfied by taking qκ = n q,κ, where 4

5 n q,κ is an integer number (positive, negative or zero) that depends on the charge q of the wave function and on the parameter κ. With this condition, we assure that the physics produced by the potential (7), with γ given by (13), is the same as the one produced by the potential (7) with the Aharonov-Bohm γ-parameter (9), which gives the magnetic field (6). In order to ensure the independence between the charge q and κ, the latter should be considered merely as a gauge parameter. We can write, without loss of generality, qκ = n q n κ, (16) where n q and n κ are integers that depend respectively on the charge q and the parameter κ. Using in equation (16) the charge of the electron e, for example, we are taken to κ = n en κ e = Nn κ e, (17) where we defined N = n e which is a non-zero integer constant. Replacing equation (17) into (16), we have q = n q N e, (18) i.e., any value of the electric charge q is an integer multiple of a given fraction of the electron charge e. It is interesting to notice that the result (18) is consistent with the existence of anti-particles and with the fractional value of the quark charges. From equation (17), we can see that the possible values of the κ parameter, that, we stress, was completely free in the classical level, becomes quantized in the quantum context. The result (18) could be proposed by means of a slightly different point of view. Equation (13) is equivalent to a gauge transformation of the form A = A + κ ρ ˆφ (19) in a region of the space where the z-axis is excluded. Once we are interested in regions that do not contain the z-axis (the Aharonov-Bohm experiment is an example), in the classical context, it is not really a problem. But, in the context of Quantum Mechanics, except in some singularity regions, the potential A in expression (19) gives the same physics as the one yielded by A only if the condition qκ = n q,κ is satisfied. With the previous arguments, we are led to equation (18). Some years ago, the gauge transformation (19) was used to argue on the non-existence of the Aharonov-Bohm effect [7], and some times, these arguments were refused with remarks based on the Stokes Theorem [8]. As already stressed in the work of reference [9], we would like to say that a transformation of the form (19) is not in disagreement with Stokes Theorem. So, the final outcome of our discussion is that the reassessment of the problem of a magnetig Aharonov- Bohm-like vector potential taken as a gauge field, leads us to the interesting possibility of getting charge quantization, without the usual argument based on the existence of a magnetic monopole. These results could be extended to the case of non-abelian fields, where a quantization condition for the color charges must be obtained too [10]. 5

6 The authors would like to thank M. V. Cougo-Pinto, C. Farina, F.C. Santos, H. Boschi-Filho and F.E. Barone for useful discussions. Professor R. Jackiw is kindly acknowledged for relevant comments, and Professor Nathan Lepora for the help with relevant references. F.A.B. thanks FAPERJ for the invaluable financial help. References [1] P.A.M. Dirac, Proc. Roy. Soc. A 133, 60 (1931); Phys. Rev. 74, 817 (1948). [] P. Goddard and di Olive, Rep. Prog. Phys. 41, 1361 (1978); [3] Juan A. Mignaco, Braz. J. of Physics 31, 35 (001). [4] B. Felsager, Geometry Particles and Fields, edited by C. Claussen, Odense University Press, Gylling (1987). [5] Y. Aharonov and D. Bohm, Phys. Rev. 115, 485 (1959); 13 (4), 1511, (1961). [6] M. Peshkin and A. Tonomura, The Aharonov -Bohm Effect, Lectures Notes in Physics 340, Springer-Verlag, Germany (1989). [7] P. Bocchieri and A. Loinger, Nuovo Cim (1978); P. Bocchieri, A. Loinger and G. Siragusa, Nuovo Cim (1979). [8] D. Bohm and B.J. Hiley, Nouvo Cim (1979); A. Zeilinger, Lett. Nuovo Cim (1979); H.J. Rothe, Nuovo Cim (1981). [9] F.A. Barone and J.A. Helayël-Neto, Proceedings of the Fourth International Winter Conference on Mathematical Methods in Physics, JHEP, Proceedings of Science, WC004/038 [quant-ph/ ]. [10] work in progress 6