Dirac’s Quantum Theory of Magnetic Monopoles

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    Paul Dirac (1902 Aug 08 – 1984 Oct 20) titled his 1931 paper "Quantised Singularities in the Electromagnetic Field" and in it examined what the still-new wave theory of quantum mechanics had to say about the possibility and some of the consequences of such things existing. He began by recalling his use of the relativistic version of the quantum theory to hypothesize the existence of positive electrons (now called positrons), ran through a brief review of the quantum theory, then got right into his subject.

    It might help us if we recall that in 1931 no one had yet obtained any observational evidence for the existence of the positron. Dirac had hypothesized the existence of a positive electron in 1928 when he found positively charged solutions to his relativistic version of the quantum mechanical wave equation. In 1932 Carl David Anderson (1905 Sep 03 – 1991 Jan 11) found evidence for the actual existence of the positron in a photograph he had made of a Wilson cloud chamber that incorporated a 6-millimeter thick lead plate and a magnetic field. From the curvature of the thin condensation trail that the particle created Anderson calculated the particle’s mass and electric charge and verified that it acted exactly like a positively charged electron.

In his 1931 paper, in the section titled "Non-integrable Phases for Wave Functions", Dirac reminded his readers that we fundamentally describe a quantum system with a wave function,

(Eq’n 1)

in which the amplitude A and the phase γ both denote functions of the spatio-temporal coordinates relevant to the system. "The indeterminacy in ψ," he wrote, "then consists in the possible addition of an arbitrary constant to the phase γ." That arbitrary constant shifts the wave packet that ψ represents around in phase space in accordance with Heisenberg’s indeterminacy principle. "We may assume that γ has no definite value at a particular point, but only a definite difference in values for any two neighboring points," Dirac added. So only dγ between two nearby points has any importance and it may represent an inexact differential; which means, in the equation

(Eq’n 2)

the number N does not necessarily equal zero.

    That non-integrability of the phase (if dγ is inexact) must not produce any ambiguity in the application of the quantum theory to real systems. The Born probability density (ρ=ψ*ψ) does not depend on the phase of ψ, so we have no problem with it. But consider two different states, ψm and ψn, that the system might occupy. The integral

(Eq’n 3)

must yield a real number, because it calculates the probability of the system jumping from one state to the other. That fact necessitates that dγ represent an exact differential (that means, N=0 in Equation 2).

    In order for the integral of Equation 3 to yield only a real number the phase change in ψm*ψn around any closed loop necessarily equals zero. That means, also necessarily, that the phase change around the loop for ψn stands equal and opposed to that for ψm*, which means that the phase change for ψm equals that for ψn (because the phase changes appear in the exponential as purely imaginary numbers). Dirac wrote, "We thus get the general result: The change in phase of a wave function round any closed curve must be the same for all the wave functions." We thus infer that the change in phase is independent of which state of the system we consider (or the system occupies) and, thus, that it comes from the dynamical system in which the quantum system participates. If our quantum system consists of a single particle, then the non-integrable part of the phase must come from its environment.

    To pursue the implications of that fact we write, as Dirac did,

(Eq’n 4)

in which the modulus of ψ1 equals the modulus of ψ and ψ1 has a definite phase at every point. Thus we put the indeterminacy of the phase into the factor exp[iβ]. The phase β does not have a definite value at each point, but it does have a definite difference between closely neighboring points, so it has definite derivatives:

(Eq’ns 5)

Because of the indeterminacy encoded into β those derivatives do not necessarily satisfy the conditions of integrability,

(Eq’ns 6)

If one or more of those equations does not equal zero, we can, nonetheless, integrate the phase. We need only remember that, in that case, the result depends upon the path that the integration follows. By way of Stokes’ theorem we can calculate the change of the phase as

(Eq’n 7)

Note that ψ1 does not enter at all into this calculation of the change in phase.

    If we want to extract a description of our particle’s linear momentum and energy from the wave function of Equation 4, we get

(Eq’ns 8)

That means that if ψ satisfies any wave equation that includes operators p and E, then ψ1 satisfies the corresponding wave equation in which the operators have become p+ k and E- kt. So we see that when ψ satisfies a wave equation for a free particle, then ψ1 satisfies the corresponding equation for a particle immersed in a forcefield: the four-vector [k, kt] thus represents the potentials of the forcefield. Looking at it in reverse, we see that the effect of the forcefield on the particle is what makes the phase non-integrable.

Assume that we have a particle carrying an electric charge of -e moving in an electromagnetic field. For the potentials we thus have

(Eq’ns 9)

From those potentials we get the fields themselves,

(Eq’ns 10)

    Having thus laid out the quantum description of an electrically charged particle in an electromagnetic forcefield, Dirac went into the section he titled Nodal Singularities. And I will note in passing that I have simply copied the equations from Dirac’s paper without converting the cgs form that Dirac used into the MKS form that I usually use in these essays.

    Dirac showed that the condition for an unambiguous physical interpretation of Equation 3 requires that Δβ around a closed loop have the same value for all of the wave functions available to the system. By Equation 7 interpreted through Equations 10 we have

(Eq’n 11)

in which s represents the surface bounded by the closed loop. That equation tells us that the phase difference around the loop stands in direct proportion to the magnetic flux gathered within the loop. But note that we also have as true to mathematics the statement that

(Eq’n 12)

in which n represents any integer. Thus we have, in general,

(Eq’n 13)

We believe that we must have n=0 necessarily. We infer that belief from the fact that ψ, in obedience to the appropriate quantum wave equation, represents a smooth, continuous function of the coordinates, except in certain rare circumstances that we can ignore. Because we have a continuous wave function, we know that Δβ must get smaller as the loop around which we integrate gets smaller. We can make the loop arbitrarily small, so we infer that n=0 because the phase cannot differ by any integer multiple of 2π around a minuscule loop without creating a discontinuity.

We do, however, have a possible exception to that rule. Where ψ=0, if such a place exists, the phase stands undefined and has no meaning. We must recalculate the phase from Equation 4 in the form

(Eq’n 14)

For the differential that contributes to that equation, we have

(Eq’n 15)

in which ψ and ψ1 both equal zero. In this case we integrate the equation by way of the Cauchy integral formula and get Δβ=2πn for some integer n. Dirac also noted that the state ψ=0 traces a line through space and he called that a nodal line. That nodal line acts much like a minuscule version of the solenoid that creates the Aharonov-Bohm effect, which no one knew about in 1931.

    Again, we integrate dβ around a closed loop that bounds a surface. If we define the direction in which we carry out the integration as west-to-east, then we can properly orient the surface bounded by the loop by identifying its north and south sides. Note that the surface can bulge in any way we can imagine. Take two identical loops, their surfaces bulging away from each other, and bring them into perfect coincidence in such a way that their directions of integration oppose each other. In that case Δβ=0 for the whole system and the closed surface thereby created will have the same orientation, either north or south on the outside, everywhere. If, in this system, n does not equal zero, then one or more nodal lines terminate inside the enclosed volume. As Dirac put it, "Since this result applies to any closed surface, it follows that the end points of nodal lines must be the same for all wave functions. Those end points are then points of singularity in the electromagnetic field."

    Equation 13 remains valid, so we can calculate the magnetic flux crossing the closed surface as

(Eq’n 16)

From that relation we can calculate the amount of bare magnetic charge contained within the closed surface,

(Eq’n 17)

The location of the nodal line is unobservable, so we have Dirac’s quantization rule,

(Eq’n 18)

    The nodal line has the character of the axis of a system of cylindrical coordinates. Like a person walking across, say, the South Pole, and suffering a discontinuous change in their longitude, a particle crossing the nodal line suffers no physical effect due to the change. Thus we get no change in the phase of the particle’s wave function, because the nodal line cannot be detected, so we come back to Equation 13 and get Equation 16 again.

    However, we have mooted the deduction of the quantization of electric charge by our deducing that quantization, via the finite-value theorem, from the conservation law pertaining to electric charge. We have already determined that free magnetic charge would permit violations of the conservation law pertaining to angular momentum. Taking those two facts together, along with the fact that no experiment designed to find magnetic monopoles has ever done so, we can infer that magnetic monopoles simply do not exist. Lest we criticize Dirac too strongly, I’ll note that he only claimed that his analysis allowed the monopoles to exist, but did not mandate them: an alternate method of deducing the quantization of electric charge, he noted, would render his analysis moot.

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