10 The Aharonov–Bohm effect

Con­sider once more the two-​​slit exper­i­ment with elec­trons. What hap­pens if we place a thin, long sole­noid right behind (or right in front of) the slit plate, between the two slits. If no cur­rent is flowing in the sole­noid, we observe the usual inter­fer­ence pat­tern. But if a cur­rent is flowing, the inter­fer­ence pat­tern is shifted side­ways (right or left, depending on the direc­tion of the current).

This effect could easily have been pre­dicted in the late 1920’s, by which time all the nec­es­sary physics was in place. Yet by many accounts it was pre­dicted three decades later, in 1959, by Yakir Aharonov and David Bohm.[1] (Actu­ally it was first pre­dicted two decades later,[2] but it made a splash only after the pub­li­ca­tion of the paper by Aharonov and Bohm. This was fol­lowed by an actual demon­stra­tion of the effect in less than a year.) Why did it take that long for this effect to be pre­dicted and taken note of?

Take another look at the rec­tangle on the right side of Figure 2.17.1. The flux of the mag­netic field through this rec­tan­gle, we observed, deter­mines the dif­fer­ence between the actions of the paths A→B→C and A→D→C. Think of these paths as leading from the elec­tron gun G through either L or R to a detector D at the screen: G→L→D and G→R→D. As we glean from Equa­tion 2.17.1, the action for each path depends on the vector poten­tial along the path. The vector poten­tial A (as well as the mag­netic field B) asso­ci­ated with a current-​​bearing sole­noid are shown in Figure 3.10.1. If this sole­noid is intro­duced into the setup of the two-​​slit exper­i­ment, the action for the left path (G→L→D) is increased, while the action for the right path (G→R→D) is reduced. Step­ping up the cur­rent in the sole­noid increases the dif­fer­ence between these actions. Since this dif­fer­ence deter­mines the posi­tions of the maxima and minima at the screen, the inter­fer­ence pat­tern is shifted to the left as a result.

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solenoid

Figure 3.10.1 A sole­noid, the cur­rent J flowing through it, and the resulting fields A and B.

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There are sev­eral rea­sons why it took that long to pre­dict this remark­able effect. For one, clas­sical elec­tro­dy­namics can be for­mu­lated exclu­sively in terms of the elec­tric field E and the mag­netic field B. For another, while the four com­po­nents of V and A uniquely deter­mine the six com­po­nents of E and B, they them­selves are not not unique. E and B are invariant under the sub­sti­tu­tions

(3.10.1)xxxV → V−∂tα,xxxAx → Ax+∂xα,xxxAy → Ay+∂yα,xxxAz → Az+∂zα,

where α is a func­tion of the space­time coor­di­nates t,x,y,z.

On the other hand, it was well known that the Schrödinger equa­tion could accom­mo­date elec­tro­mag­netic effects only in terms of the poten­tials V and A, and that the actions asso­ci­ated with loops are also invariant under these sub­sti­tu­tions. (Since these sub­sti­tu­tion change the actions for G→L→D and G→R→D by equal amounts, the dif­fer­ence between the two actions, which equals the action asso­ci­ated with the loop G→L→D→R→G, remains unchanged.) Yet physi­cists were unable to dis­abuse them­selves of the notion that E and B are phys­i­cally real, while the poten­tials “have no phys­ical meaning and are intro­duced solely for the pur­pose of math­e­mat­ical sim­pli­fi­ca­tion of the equa­tions,” as Fritz Rohrlich wrote.[3] The gen­eral idea at the time was that the elec­tro­mag­netic field is a phys­ical entity in its own right, that it is locally acted upon by charges, that it locally acts on charges, and that it medi­ates the action of charges on charges by locally acting on itself. At the heart of this notion was the so-​​called prin­ciple of local action, felic­i­tously artic­u­lated by DeWitt and Graham[4] in an Amer­ican Journal of Physics resource letter:

physi­cists are, at bottom, a naive breed, for­ever trying to come to terms with the “world out there” by methods which, how­ever imag­i­na­tive and refined, involve in essence the same ele­ment of con­tact as a well-​​placed kick.

With the notable excep­tion of Roger Boscovich, a Croa­tian physi­cist and philoso­pher who flour­ished in the 18th Cen­tury, it does not seem to have occurred to anyone that local action is as unin­tel­li­gible as the apparent ability of mate­rial objects to act where they are not, which the prin­ciple of local action has sup­pos­edly done away with. (The impres­sion that local action is intel­li­gible rests on such familiar expe­ri­ences as pulling a rope or pushing a stalled car. As soon as we take a micro­scopic look at what hap­pens between the pushing hands and the car, we dis­cover that this seem­ingly local action involves net inter­atomic or inter­mol­e­c­ular repul­sive forces acting at a distance.)

If it is believed that elec­tro­mag­netic effects on charges are pro­duced via a con­tin­uous sequence of local cause-​​effect rela­tions, then some­thing like the Aharonov–Bohm effect cannot be fore­seen, for the values of E and B along the alter­na­tive elec­tron paths can be made arbi­trarily small by making the sole­noid suf­fi­ciently long. And even if A were a phys­ical entity and as such respon­sible for the Aharonov–Bohm effect, it could not pro­duce it locally since the local values of A are arbitrary.

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1. [↑] Aharonov, Y., and Bohm, D. (1959). Sig­nif­i­cance of elec­tro­mag­netic poten­tials in quantum theory. Phys­ical Review 115, 485–491.

2. [↑] Ehren­berg, W., and Siday, R.E. (1949). The Refrac­tive Index in Elec­tron Optics and the Prin­ci­ples of Dynamics. Pro­ceed­ings of the Phys­ical Society B 62, 8–21.

3. [↑] Rohrlich, F (1965). Clas­sical Charged Par­ti­cles, Addison–Wesley, pp. 65–66.

4. [↑] DeWitt, B.S., and Graham, R.N. (1971). Resource letter IQM-​​1 on the inter­pre­ta­tion of quantum mechanics, Amer­ican Journal of Physics 39, 724–738.