On the Gyromagnetic and Gyrogravito-Magnetic Ratios of the Electron

Abstract

The magnetic dipole moment of the Kerr–Newman metric, defined by mass $M$, electrical charge $Q$ and angular momentum $J$, is $\mu =QJ/M$, corresponding, for all values of $J/M$, to a gyromagnetic ratio $g_1=2$, which is also the value of the intrinsic gyromagnetic ratio of the electron, as first noted by Carter. Here, we argue that this result can be understood in terms of the particle-wave complementarity principle. For $\mu$ can only be defined at asymptotic spatial infinity, where the metric appears to describe a spinning point particle, and therefore setting $M=m$, $Q=e$, we necessarily have a model of the electron. From the Dirac equation  we can construct a covariantly conserved four-current $J_i$ that is the source of the electromagnetic  field generated by the charge $e$. The result $g_1=2$ then follows from the minimal gauge principle $\partial _j\rightarrow \partial _j -\mathrm {i}e A_j$ which is implicit in the formulation of the spinorial wave equation, and which can also be justified from the line action for a spin-1/2 point particle interacting with an external electromagnetic  field, due to Berezin and Marinov. By contrast, analysis of the gyrogravito-magnetic effect, investigated classically by Wald and quantum mechanically by Adler et al., yields the result $\tilde{g}=1$ in all non-relativistic cases, which can be explained from the principle of equivalence. The results are in accord with the correspondence principle.

Appendices

Appendix 1: The Probability-Current Conservation Law for the Schrödinger  Equation

In paper II, we derived the probability-current continuity equation for the Schrödinger equation (II 25), assuming the De Donder [34]–Lanczos [35, 36] gauge condition (84), in the chronometrically invariant  form given by Eq. (II 29),

$$\begin{aligned} ^{*}\partial _t\rho _\mathrm{P} + {^{*}\varvec{\partial }} \centerdot {^{*}{\varvec{j}}_\mathrm{P}} = 0, \end{aligned}$$

(161)

where

$$\begin{aligned} ^{*}\partial _t=\frac{1}{\sqrt{h}}\partial _0 ,\quad \partial _{\alpha } = {^{*}\partial _{\alpha }} +g_{\alpha }\partial _0 \end{aligned}$$

(162)

and the probability density $\rho _\mathrm{P}$ and three-vector current $^{*}{\varvec{j}}_\mathrm{P}$ are defined by Eqs. (II 30), now assuming a total electrical charge $ne$,

$$\begin{aligned} \rho _\mathrm{P} =\psi \psi ^{*},\quad ^{*}{\varvec{j}}_\mathrm{P} =\frac{\mathrm {i}}{2{{\mathcal {M}}}} \left[ \psi \left( ^{*}\varvec{\partial }\psi ^{*}\right) - \left( ^{*}\varvec{\partial }\psi \right) \psi ^{*}\right] -\frac{ne}{{\mathcal {M}}}\psi \psi ^{*} {\varvec{A}}'. \end{aligned}$$

(163)

As in paper II, a pre-star $^{(*\dots )}$ denotes chronometric invariance, discussed by Zel’manov [52] (see also paper I), a post-star $^{(\dots *)}$ denotes complex conjugation, and the three-vector ${\varvec{A}}'$ is defined from the four-vector potential $A_i$ by

$$\begin{aligned} \left( {\varvec{A}}'\right) ^{\alpha } = -A^{\alpha },\quad \left( {\varvec{A}}'\right) _{\alpha } = \gamma _{\alpha \beta } \left( {\varvec{A}}'\right) ^{\beta }. \end{aligned}$$

(164)

Setting $n=2$, we obtain the theory of superconductivity describing a Cooper pair of electrons.

Here, we also assume that the background  space-time  is stationary

$$\begin{aligned} \partial _0 g_{ij} = 0, \end{aligned}$$

(165)

in which case the electromagnetic  Lorenz gauge condition

$$\begin{aligned} \partial _j\left( \sqrt{-g} A^j\right) = 0 \end{aligned}$$

(166)

reduces in three dimensions to

$$\begin{aligned} \text {div} \left( \sqrt{h} {\varvec{A}}'\right) = 0. \end{aligned}$$

(167)

The purpose of this Appendix is to elucidate the meaning of Eq. (161) as the normal four-vector conservation law expressed in coordinate  form,

$$\begin{aligned} \partial _l \left( \sqrt{-g} g^{lm} j_m \right) = 0, \end{aligned}$$

(168)

where the components  $(j_0,j_{\alpha })$ of the four-current $j_m$ are to be determined in terms of the chronometrically invariant  components  $(\rho _\mathrm{p},(j_\mathrm{p})_{\alpha })$. In the gauge (84), Eq. (168) reduces to

$$\begin{aligned} g^{00} \partial _{0} j_0 +\left( g^{\alpha \beta } \partial _{\alpha }j_{\beta } + g^{0\alpha } \partial _{0} j_{\alpha } \right) + g^{0\alpha } \partial _{\alpha }j_{0} = 0. \end{aligned}$$

(169)

Remembering that $g^{00} =h^{-1} - g_{\alpha }g^{\alpha }$, $g^{\alpha \beta } =-\gamma ^{\alpha \beta }$ and $g^{0\alpha }=-g^{\alpha }$, and retaining only terms up to first order in ${\varvec{g}}$, we thus have

$$\begin{aligned} h^{-1} \partial _{0} j_0 -\gamma ^{\alpha \beta *} \partial _{\alpha }j_{\beta } - g^{\alpha } \partial _{\alpha }j_{0} \approx 0. \end{aligned}$$

(170)

Comparing Eqs. (161) and (170), we see that it is possible to identify $\rho _\mathrm{p}$ with $j_0/\sqrt{h}$, assuming Eq. (165), while $^*{\varvec{j}}_\mathrm{p}$ is of the form

$$\begin{aligned} ^*{\varvec{j}}_\mathrm{p} =-{\varvec{j}} -\zeta {\varvec{g}}j_0. \end{aligned}$$

(171)

The second term in Eq. (161) can then be expanded, to first order in ${\varvec{g}}$, as

$$\begin{aligned} \gamma ^{\alpha \beta *} \partial _{\alpha }\left( ^*{\varvec{j}}_\mathrm{p}\right) _{\beta }= & {} -\gamma ^{\alpha \beta *} \partial _{\alpha } j_{\beta } -\zeta \gamma ^{\alpha \beta *} \partial _{\alpha }\left( g_{\beta } j_0\right) \nonumber \\\approx & {} -\gamma ^{\alpha \beta *} \partial _{\alpha } j_{\beta } -\zeta \gamma ^{\alpha \beta } \left( \partial _{\alpha } g_{\beta }\right) j_0 -\zeta g^{\alpha } \partial _{\alpha } j_0. \end{aligned}$$

(172)

To proceed further, we note that the second term on the right-hand side  of Eq. (172) vanishes in the stationary space-time  (165) when the coordinate  condition (84) is applied. For we have

$$\begin{aligned} \gamma ^{\alpha \beta } \partial _{\alpha }g_{\beta }= & {} \partial _{\alpha } g^{\alpha } -\left( \partial _{\alpha } \gamma ^{\alpha \beta } \right) g_{\beta } = -\frac{1}{\sqrt{-g}} \partial _j \left( \sqrt{-g} g^{0j}\right) - \frac{1}{\sqrt{-g}} g^{\alpha } \partial _{\alpha } \left( \sqrt{-g}\right) \nonumber \\&\quad - \left( \partial _{a} \gamma ^{\alpha \beta } \right) g_{\beta } \nonumber \\= & {} -\frac{1}{\sqrt{-g}} g_{\alpha } \partial _{\beta } \left( \sqrt{-g}\gamma ^{\alpha \beta }\right) =0. \end{aligned}$$

(173)

Substitution  of Eqs. (172) and (173) into Eq. (161) yields the equation

$$\begin{aligned} \partial _0\left( \frac{1}{\sqrt{h}} \rho _\mathrm{p} \right) -\gamma ^{\alpha \beta *} \partial _{\alpha } j_{\beta } -\zeta g^{\alpha } \partial _{\alpha } j_0 \approx 0, \end{aligned}$$

(174)

comparison of which with Eq. (170) shows that $\zeta =1$, so that the components  of the four-current $j_i$ are given by

$$\begin{aligned} j_0=\sqrt{h} \rho _\mathrm{p} =\sqrt{h}\psi \psi ^* \end{aligned}$$

(175)

and

$$\begin{aligned} {\varvec{j}}=-^*{\varvec{j}}_\mathrm{p} - {\varvec{g}}j_0 = \frac{\mathrm {i}}{2{{\mathcal {M}}}} \left[ \left( {^*\varvec{\partial }} \psi \right) \psi ^{*} - \psi \left( {^*\varvec{\partial }} \psi ^{*} \right) \right] + \psi \psi ^{*} \left( \frac{ne}{{\mathcal {M}}} {\varvec{A}}'- \tilde{{\varvec{A}}}\right) .\quad \end{aligned}$$

(176)

Note in particular the minus sign in the bracket $\left( \frac{ne}{{\mathcal {M}}}{\varvec{A}}'-\tilde{{\varvec{A}}}\right)$ in the second term on the right-hand side  of Eq. (176), which is due to the combination of classical and quantum effects involved in the calculation, and is in contrast with the expectation of a plus sign, based on the purely classical substitution  rule defined by Eq. (79).

Mathematically, there is a sign ambiguity of $\pm \mathrm {i}^*\partial _t$ in taking the square root of Eq. (86) [Eq. (II.16)] to obtain the Schrödinger equation  (II 25), and if instead of choosing $+ \mathrm {i}^*\partial _t$, corresponding to the positive-energy solution (II 25), we were to choose $- \mathrm {i}^*\partial _t$, corresponding to the negative-energy solution, then we would find that

$$\begin{aligned} j_0=\sqrt{h} \rho _\mathrm{p} =-\sqrt{h}\psi \psi ^* \end{aligned}$$

(177)

and

$$\begin{aligned} {\varvec{j}}=-^*{\varvec{j}}_\mathrm{p} - {\varvec{g}}j_0 = \frac{\mathrm {i}}{2{{\mathcal {M}}}} \left[ \left( {^*\varvec{\partial }} \psi \right) \psi ^{*} - \psi \left( {^*\varvec{\partial }} \psi ^{*} \right) \right] + \psi \psi ^{*} \left( \frac{ne{\varvec{A}}'}{{\mathcal {M}}} + \tilde{o}ver \mathbf{{A}}\right) .\quad \end{aligned}$$

(178)

Appendix 2: The Term $-\mathrm {i}\kappa zF_{lm}\xi ^l\xi ^m$ in the Lagrangian (114)

Here we shall clarify the nature of the term $-\mathrm {i}\kappa zF_{lm}\xi ^l\xi ^m$ in the Lagrangian (114) describing the pseudo-classical electron. The canonical momentum (118) can be written

$$\begin{aligned} p_j=\frac{\partial {L}}{\partial {\dot{x}^j}} =-m^{\prime \prime }u_j -\frac{1}{2}\mathrm {i}\left( \xi _j-u_l\xi ^lu_j\right) \lambda '/z + e A_j, \end{aligned}$$

(179)

contraction of which with $\xi ^j$ yields the equation

$$\begin{aligned} P_j\xi ^j =-m^{\prime \prime }u_j \xi ^j, \end{aligned}$$

(180)

where

$$\begin{aligned} m^{\prime \prime } = m + \mathrm {i}\kappa F_{lm}\xi ^l\xi ^m \end{aligned}$$

(181)

and

$$\begin{aligned} P_j=p_j - eA_j. \end{aligned}$$

(182)

Substituting from Eq. (180) into the constraint Eq. (116), we obtain

$$\begin{aligned} P_j \xi ^j + m^{\prime \prime } \xi _5 = 0. \end{aligned}$$

(183)

Writing the action $S$ in the Hamiltonian  form (125) and assuming the semi-classical approximation (63) for the wave function,

$$\begin{aligned} \psi =\exp {\left( \mathrm {i}{S}\right) }, \end{aligned}$$

(184)

we make the operator replacements (127) and promote the classical constraint (183) into a quantum constraint acting on $\psi$. Simultaneously transforming the Grassmann algebra (113) into the Clifford algebra via Eqs. (131), taking into account Eq. (119) for the quantity $p_j\dot{x}^j$, we obtain the modified Dirac equation, after premultiplication by $\gamma _5$,

$$\begin{aligned} \left[ \mathrm {i}\gamma ^j \left( \partial _j - \mathrm {i}e A_j\right) - m^{\prime \prime \prime } \right] \psi =0, \end{aligned}$$

(185)

where the effective mass is defined by

$$\begin{aligned} m^{\prime \prime \prime } = m -\mathrm {i}\kappa F_{lm} {s}^{lm}. \end{aligned}$$

(186)

Equation (185) is the form of the Dirac equation  resulting from an initial Lagrangian  function that includes the anomalous Pauli term $-\frac{1}{2}\mathrm {i}l_0 F_{lm} {s}^{lm}\bar{\psi }\psi$, where $l_0=-2\kappa$, which reads

$$\begin{aligned} L=\mathrm {i}\bar{\psi } \gamma ^j\left( \partial _j -\mathrm {i}e A_j\right) \psi - m^{\prime \prime \prime } \bar{\psi }\psi . \end{aligned}$$

(187)