Solutions of the Dirac equation are of the form
exp(-i / h-bar (E t - p x)) * u,
where u is a 4-component Dirac spinor. We in this blog believe that the exp(...) part describes linear motion of the electron, while the spinor is a compact, "quantized" description of circling motion which may be called zitterbewegung.
Erwin Schrödinger calculated in 1930 that the Dirac electron position in a "typical" wave packet (such a packet always contains both a positive E and a negative E component) seems to move at the speed of light in a circle whose radius is
λ_e / (4 π),
where λ_e is the Compton wavelength of the electron. The circling motion is called zitterbewegung. This movement, if it is classical motion, would explain the electron spin-z angular momentum
L = 1/2 h / (2 π).
David Hestenes: the zitterbewegung interpretation of quantum mechanics
David Hestenes (1990) writes about zitterbewegung. Hestenes believes that the complex phase factor
exp(-i / h-bar (E t - p x))
describes real spatial motion in the zitterbewegung circle. Hestenes calls this fundamental assumption the zitterbewegung interpretation of quantum mechanics.
Is it right to assign such a realistic role for a complex phase? Schrödinger's zitterbewegung has an angular velocity which is double of
E / h-bar.
Which angular velocity describes the "real" zitterbewegung?
Suppose that a relativistic electron has its E doubled because it has lots of kinetic energy. Then the angular velocity is double in the above formula. But if the spin is classical motion, time dilation makes it spin at half the speed in the laboratory frame. We conclude that it is a bad idea to interpret the phase factor as classical circling motion.
The magnetic moment of the electron
The magnetic moment of the electron would be explained if the charge of the Dirac electron moves at the speed of light in a circle whose radius is
λ_e / (2 π),
where λ_e is the Compton wavelength of the electron:
λ_e = h / (m_e c).
That is, the charge does a loop whose radius is double the zitterbewegung radius.
Our March 7, 2021 post tried to explain why the electron gyromagnetic ratio is 2. Let us elaborate our explanation.
The Dirac hamiltonian commutes with the total angular momentum operator
J = L + 1/2 S,
not individually with 1/2 S. This suggests that 1/2 S is the spin classical angular momentum, and that has been confirmed experimentally, too.
Why would the magnetic moment then be determined by S, not 1/2 S? If we couple the Dirac equation with minimal coupling to a magnetic field B, we easily get the result that in the hamiltonian, the magnetic moment is determined by S.
Minimal coupling classically is something which works for linear movement of a charge. It also works in determining the magnetic moment of the Dirac electron. That is evidence for the hypothesis that the Dirac spinor does encode classical circling motion.
We conjectured in the March 7, 2021 blog post that in the path integral, the electron cannot draw a circle whose length is only 1/2 of its Compton wavelength. We claimed that instead, the path integral makes the electron to appear moving in a circle whose length is the Compton wavelength. How would the Dirac equation "know" about these intricacies of the path integral?
The Dirac equation "knows" about the spin 1/2 of the electron through the algebra of the gamma matrices.
Open problem. How does the Dirac equation "know" that the electron path integral has to have the circle length at least equal to the Compton wavelength, and knows to set the electron magnetic moment accordingly?
The anomalous magnetic moment of the electron
We were able to explain qualitatively the Schwinger formula for the anomalous magnetic moment by assuming that the static field of the electron has inertial mass, and that the far field cannot take part in the movement of the electron charge in the circle of the radius
λ_e / (2 π).
That reduces the effective mass of the electron and increases its magnetic moment.
Our result strongly suggests that the electron magnetic moment is a result of real physical circling motion of the electron charge. That is, zitterbewegung is very real physical motion.
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