# Relativistic dynamics of point magnetic moment

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## Abstract

The covariant motion of a classical point particle with magnetic moment in the presence of (external) electromagnetic fields is revisited. We are interested in understanding extensions to the Lorentz force involving point particle magnetic moment (Stern–Gerlach force) and how the spin precession dynamics is modified for consistency. We introduce spin as a classical particle property inherent to Poincaré symmetry of space-time. We propose a covariant formulation of the magnetic force based on a ‘magnetic’ 4-potential and show how the point particle magnetic moment relates to the Amperian (current loop) and Gilbertian (magnetic monopole) descriptions. We show that covariant spin precession lacks a unique form and discuss the connection to \(g-2\) anomaly. We consider the variational action principle and find that a consistent extension of the Lorentz force to include magnetic spin force is not straightforward. We look at non-covariant particle dynamics, and present a short introduction to the dynamics of (neutral) particles hit by a laser pulse of arbitrary shape.

## 1 Introduction

- (i)
- (ii)
questions regarding how elementary magnetic dipoles (e.g. neutrons) interact with external fields [3, 4];

- (iii)
particle dynamics in ultra strong magnetic fields created in relativistic heavy ion collisions [5, 6];

- (iv)
magnetars, stellar objects with extreme \(\mathcal{O}(10^{11})\,{\mathrm {T}}\) magnetic fields [7, 8];

- (v)
the exploration of particle dynamics in laser generated strong fields [9];

- (vi)
neutron beam guidance and neutron storage rings [10]; and

- (vii)

- 1.The magnetic moment \(\varvec{\mu }\) has an interaction energy with a magnetic field \( \,{\varvec{\mathcal{B}}}\)The corresponding Stern–Gerlach force \( \,{\varvec{\mathcal{F}}}_{\mathrm {SG}}\) has been written in two formats$$\begin{aligned} E_m=-{\varvec{\mu }} \cdot \,{\varvec{\mathcal{B}}}. \end{aligned}$$(1)The name ‘Amperian’ relates to the loop current generating the force. The ‘Gilbertian’ model invokes a magnetic dipole made of two magnetic monopoles. These two forces written here in the rest frame of a particle are related [3, 4]. We will show that an internal spin based magnetic dipole appears naturally; it does not need to be made of magnetic monopoles or current loops. We find that both force expressions in Eq. (2) are equivalent; this equivalence arises from covariant dynamics we develop and requires additional terms in the particle rest frame complementing those shown in Eq. (2).$$\begin{aligned} \,{\varvec{\mathcal{F}}}_{\mathrm {SG}}\equiv \left\{ \begin{array}{lr} {\varvec{\nabla }} ({\varvec{\mu }}\cdot \,{\varvec{\mathcal{B}}}) , &{}\ \mathrm {Amperian\ Model}, \\ ({\varvec{\mu }}\cdot {\varvec{\nabla }} ) \,{\varvec{\mathcal{B}}}, &{} \mathrm {Gilbertian\ Model}. \end{array} \right. \end{aligned}$$(2)
- 2.The torque \(\,{\varvec{\mathcal{T}}}\) that a magnetic field \(\,{\varvec{\mathcal{B}}}\) exercises on a magnetic dipole \(\varvec{\mu }\) tends to align the dipole with the direction of a magnetic field \(\,{\varvec{\mathcal{B}}}\)The magnetic moment is defined in general in terms of the product of Bohr magneton \(\mu _{\mathrm {B}}\) with the gyromagnetic ratio$$\begin{aligned} \,{\varvec{\mathcal{T}}}\equiv \displaystyle \frac{d\varvec{s}}{dt}={\varvec{\mu }} \times \,{\varvec{\mathcal{B}}}= g \mu _{\mathrm {B}} \;\frac{{\varvec{s}}}{\hbar /2}\times \,{\varvec{\mathcal{B}}}, \quad \mu _{\mathrm {B}}\equiv \displaystyle \frac{e\hbar }{2m}.\quad \end{aligned}$$(3)
*g*, \(|\mu |\equiv g\mu _{\mathrm {B}}\). In Eq. (3) we used \(|{\varvec{s}}|=\hbar /2\) for a spin-1/2 particle; a more general expression will be introduced in Sect. 3.1.1.

While the conservation of electrical charge is rooted in gauge invariance symmetry, the magnitude of electrical charge has remained a riddle. The situation is similar for the case of the magnetic moment \({\varvec{\mu }}\): spin properties are rooted in the Poincaré symmetry of space-time, however, the strength of spin interaction with magnetic field, Eqs. (1) and (3), is arbitrary but unique for each type of (classical) particle. Introducing the gyromagnetic ratio *g* we in fact create an additional conserved particle quality. This becomes clearer when we realize that the appearance of ‘*e*’ does not mean that particles we study need to be electrically charged.

First principle considerations of point particle relativistic dynamics experience some difficulties in generating Eqs. (2) and (3), as a rich literature on the subject shows – we will cite only work that is directly relevant to our approach; for further 70+ references see the recent numerical study of spin effects and radiation reaction in a strong electromagnetic field [9].

For what follows it is important to know that the spin precession Eq. (3) is a result of spatial rotational invariance which leads to angular and spin coupling, and thus spin dynamics can be found without a new dynamical principle as has been argued e.g. by Van Dam and Ruijgrok [13] and Schwinger [14]. Similar physics content is seen in the work of Skagerstam and Stern [15, 16], who considered the context of fiber bundle structure focusing on Thomas precession.

Covariant generalization of the spin precession Eq. (3) is often attributed to the 1959 work by Bergmann–Michel–Telegdi [17]. However we are reminded [18, 19, 20] that this result was discovered already 33 years earlier by Thomas [21, 22] at the time when the story of the electron gyromagnetic ratio \(g=2\) was unfolding. Following Jackson [18] we call the corresponding equation TBMT. Frenkel, who published [23, 24] at the same time with L.H. Thomas explored the covariant form of the Stern–Gerlach force, a task we complete in this work.

There have been numerous attempts to improve the understanding of how spin motion back-reacts into the Lorentz force, generating the Stern–Gerlach force. In the 1962 review Nyborg [25] summarized efforts to formulate the covariant theory of electromagnetic forces including particle intrinsic magnetic moment. In 1972 Itzykson and Voros [26] proposed a covariant variational action principle formulation introducing the inertia of spin *I*, seeking a consistent variational principle but they found that no new dynamical insight resulted in this formulation.

Our study relates most to the work of Van Dam and Ruijgrok [13]. This work relies on an action principle and hence there are in the Lorentz force inconsistent terms that violate the constraint that the speed of light is constant, see e.g. their Eq. 3.11 and remarks: ‘ The last two terms are \(\mathcal{O}(e^2)\) and will be omitted in what follows.’ Other authors have proposed mass modifications to compensate for terms, a step which is equally unacceptable. For this reason our approach is intuitive, without insisting on ‘in principle there is an action’. Once we have secured a consistent, unique covariant extension of the Lorentz force, we explore the natural variational principle action. We find it is not consistent and we identify the origin of the variational principle difficulties.

We develop the concept of the classical point particle spin vector in the following Sect. 2. Our discussion relates to Casimir invariants rooted in space-time symmetry transformations. Using Poincaré group generators and Casimir eigenvalues we construct the particle momentum \(p^\mu \) and particle space-like spin pseudo-vector \(s^\mu \). In Sect. 3 we present a consistent picture of the Stern–Gerlach force (Sect. 3.1) and generalize the TBMT precession equation (Sect. 3.2) to be linear in both, the EM field and EM field derivatives. We connect the Amperian form of SG force (3.1.1) with the Gilbertian force (3.1.2). We discuss non-uniqueness of spin dynamics (3.2.3) with consideration of the impact on muon \(g-2\) experiments. We show in Sect. 4 that the natural choice of action for the considered dynamical system does not lead to a consistent set of equations; in this finding we align with all prior studies of Stern–Gerlach extension to the Lorentz force.

In the final part of this work, Sect. 5, we show some of the physical consequences of this theoretical framework. In Sect. 5.1 we present a more detailed discussion of dynamical equations for the case of a particle in motion with a given \({\varvec{\beta }}={\varvec{v}}/c\) and \(\,{\varvec{\mathcal{E}}}\), \(\,{\varvec{\mathcal{B}}}\) in the laboratory. In Sect. 5.2 we study the solution of the dynamical equations for the case of an EM light wave pulse hitting a neutral particle. We have obtained exact solutions of this problem, details will follow under separate cover [27]. The concluding Sect. 6 is a brief summary of our findings.

### 1.1 Notation

*d*– the limitations of the alphabet force us to adopt the letter

*d*otherwise used to describe the electric dipole to be the elementary magnetic dipole charge. The magnetic dipole charge of a particle we call

*d*converts the spin vector \({\varvec{s}}\) to magnetic dipole vector \({\varvec{\mu }}\),

*c*is needed in SI units since in the EM-tensor \(F^{\mu \nu }\) has as elements \(\,{\varvec{\mathcal{E}}}/c\) and \(\,{\varvec{\mathcal{B}}}\). It seems natural to introduce also \(s^\mu d=\mu ^\mu \), but this object can be confusing therefore we will stick to the product \(s^\mu d\), however we always replace \({\varvec{s}} d\rightarrow {\varvec{\mu }}/c\). Note that we place

*d*to the right of pertinent quantities to avoid confusion such as

*dx*.

We cannot avoid the appearance in the same equation of both magnetic moment \({\varvec{\mu }}\) and vacuum permeability \(\mu _0\).

## 2 Spin vector

A classical intrinsic covariant spin has not been clearly defined or even identified in prior work. In some work addressing covariant dynamics of particles with intrinsic spin and magnetic moment, particle spin is by implication solely a quantum phenomenon. Therefore we describe the precise origin of classical spin conceptually and introduce it in explicit terms in the following.

*m*(with a value specific for any particle type). The second Casimir operator \(C_2\) is obtained from the square of the Pauli–Lubański pseudo-4-vector

As long as forces are small in the sense discussed in Ref. [28] we can act as if the rules of relativity apply to both inertial and (weakly) accelerated frames of reference. This allows us to explore the action of forces on particles in their rest frame where Eq. (10) defines the state of a particle. By writing the force laws in covariant fashion we can solve for the dynamical evolution of \(p^\mu (\tau ),\;s^\mu (\tau )\) as classical numbered variables.

## 3 Covariant dynamics

### 3.1 Generalized Lorentz force

#### 3.1.1 Magnetic dipole potential and Amperian force

We have gone to great lengths in Sect. 2 to argue for the existence of particle intrinsic spin. For all massive particles this implies the existence of a particle intrinsic magnetic dipole moment, without need for magnetic monopoles to exist or current loops. Spin naturally arises in the context of symmetries of Minkowski space-time, it is not a quantum property.

*e*, and an elementary magnetic dipole charge

*d*. The covariant dynamics beyond the Lorentz force needs to incorporate the Stern–Gerlach force. Thus the extension has to contain the elementary magnetic moment of a particle contributing to this force. To achieve a suitable generalization we introduce the magnetic potentialWe use the dual pseudo-tensor since \(s_\mu \) is a pseudo-vector; the product in Eq. (14) results in a polar 4-vector \(B_\mu \). We note that the magnetic dipole potential \(B_\mu \) by construction in terms of the antisymmetric field pseudo-vector \(F^\star _{\mu \nu }\) satisfies

*G*-tensor we note the appearance in the force of the derivative of EM fields which is required if we are to see the Amperian model variant of the Stern–Gerlach force Eq. (2) as a part of generalized Lorentz force.

*G*-tensor Eq. (17). Thus the total 4-force a particle of charge

*e*and magnetic dipole charge

*d*experiences isIn the particle rest frame we have

#### 3.1.2 Gilbertian model Stern–Gerlach force

*eF*by \({\widetilde{F}}\) in a somewhat simpler way compared to the original \(H^{\mu \nu }\) Eq. (18) modification.

*e*and the elementary magnetic moment ‘charge’

*d*Eq. (4) as independent qualities of a point particle. As noted in the introduction it is common to set \( |{\varvec{\mu }}|\equiv g\mu _{\mathrm {B}}\), see above Eq. (3). Hence we can have both, charged particles without magnetic moment, or neutral particles with magnetic moment, aside from particles that have both charge and magnetic moment. For particles with both charge and magnetic moment we can write, using Gilbertian format of forcewhere \(a=(g-2)/2\) is the gyromagnetic ratio anomaly. The Compton wavelength Open image in new window defines the scale at which the spatial field inhomogeneity is relevant; note that inhomogeneities of the field are boosted in size for a particle in motion, a situation which will become more explicit in Sect. 5.1.3.

### 3.2 Spin motion

#### 3.2.1 Conventional TBMT

In Eq. (33) \({\widetilde{a}}\) is an arbitrary constant considering that the additional term multiplied with \(u^\mu \) vanishes. On the other hand we can read off the magnetic moment entering Eq. (3): the last term is higher order in \(1/c^2\). Hence in the rest frame of the particle we see that \(2(1+{\widetilde{a}})=g\) i.e. Eq. (33) reproduces Eq. (3) with the magnetic moment coefficient when \({\widetilde{a}} = a\). Therefore, as introduced, \({\widetilde{a}}=a\) is the \(g\ne 2\) anomaly. However, in Eq. (33) we could for example use \({\widetilde{a}} = (g^2-4)/8= a+a^2/2\), which classical limit of quantum dynamics in certain specific conditions implies [12]. In this case \({\widetilde{a}}\rightarrow a\) up to higher order corrections. This means that measurement of \({\widetilde{a}}\) as performed in experiments [1, 2] depends on derivation of the relation of \({\widetilde{a}}\) with *a* obtained from quantum theory. These remarks apply even before we study gradient in field corrections.

#### 3.2.2 Gradient corrections to TBMT

#### 3.2.3 Non-uniqueness of gradient corrections to TBMT

*F*and

*G*replacing the usual EM-tensor \(F^{\mu \nu }\)in the Schwinger solution, Eq. (33). In other words, we explore the dynamics according to

*F*and

*G*tensors could be included in Schwinger solution independently with different constants. Intuition demands that \({\widetilde{a}}={\widetilde{b}}\). However, aside from algebraic simplicity we do not find any compelling argument for this assumption.

*G*tensor Eq. (17) to obtain

As Eq. (35) shows the physical difference between factors \({\widetilde{a}}\) and \({\widetilde{b}}\) is related to the nature of the interaction: the ‘magnetic’ tensor *G* is related to \({\widetilde{b}}\) only. Thus for a neutral particle \(e\rightarrow 0\) we see in Eq. (38) that the torque depends only on \({\widetilde{b}}\). Conversely, when the effect of magnetic potential is negligible Eq. (38) becomes the textbook spin dynamics that depends on \({\widetilde{a}}\) alone.

## 4 Search for variational principle action

At the beginning of earlier discussions of a covariant extension to the Lorentz force describing the Stern–Gerlach force was always a well invented covariant action. However, the Lorentz force itself is not a consistent complement of the Maxwell equations. The existence of radiation means that an accelerated particle experiences radiation friction. The radiation-reaction force has not been incorporated into a variational principle [28, 32]. Thus we should not expect that the Stern–Gerlach force must originate in a simple action.

We seek a path \(x^\mu (\tau )\) in space-time that a particle will take considering an action that is a functional of the 4-velocity \(u^\mu (\tau )=dx^\mu /d\tau \) and spin \(s^\mu (\tau )\). Variational principle requires an action *I*(*u*, *x*; *s*). When *I* respects space-time symmetries, the magnitudes of particle mass and spin are preserved in the presence of electromagnetic (EM) fields. We also need to assure that \(u^2=c^2\) which constrains the form of force and thus *I* that is allowed. Moreover, we want to preserve gauge invariance of the resultant dynamics.

If we replace in our thoughts \({ds_\nu }/{d\tau }\) in Eq. (47) by the TBMT equation Eq. (33) or as would be more appropriate by its extended version Eq. (35), we see that the force \(L_\mathrm {S2}^\mu \) would be quadratic in the fields containing also field derivatives. However, by assumption we modified the action limiting the new term in Eq. (44) to be linear in the fields and derivatives. Finding non linear terms we learn that this assumption was not justified. However, if we add the quadratic in fields term to the action we find following the chain of arguments just presented that a cubic term is also required and so on; with derivatives of fields appearing at each iteration.

We have searched for some time for a form that avoids this circular conundrum, but akin to previous authors we did not find one. Clearly a ‘more’ first principle approach would be needed to create a consistent variational principle based equation system. On the other hand we have presented a formulation of spin dynamics which does not require a variational principle in the study of particle dynamics: as is we have obtained a dynamical equation system empirically. Our failing in the search for an underlying action is not critical. A precedent situation comes to mind here: the radiation emitted by accelerated charges introduces a ‘radiation friction’ which must be studied [28, 32] without an available action, which is also based on empirical knowledge about the energy loss arising for accelerated charges.

## 5 Experimental consequences

### 5.1 Non covariant form of dynamical equations

#### 5.1.1 Laboratory frame

One easily checks that Eqs. (48) and (49) also satisfy Eq. (11): \(u_\mu s^\mu =0\). A classic result of TBMT reported in textbooks is that the longitudinal polarization \(\hat{{\varvec{\beta }}}\cdot {\varvec{s}}\) for \(g\simeq 2\) and \(\beta \rightarrow 1\) is a constant of motion. This shows that for a relativistic particle the magnitude of both time-like and space-like components of the spin 4-vector Eq. (49) can be arbitrarily large, even if the magnitude of the 4-vector is bounded \(s_\mu s^\mu =-{\varvec{s}}^{\,2}\). This behavior parallels the behavior of 4-velocity \(u^\mu u_\mu =c^2 \).

*F*, i.e. with laboratory given \(\,{\varvec{\mathcal{E}}},\;\,{\varvec{\mathcal{B}}}\) EM-fields

*L*with \(Lu|_\mathrm {rest}=u_{\mathrm {L}}\) when used on the left hand side in Eq. (50) produces proper time differentiation of the transformation operator, see also [33]. Such transformation into a co-rotating frame of reference originates the Thomas precession term in particle rest frame for the torque equation. This term is naturally present in the covariant formulation when we work in the laboratory reference frame.

#### 5.1.2 Magnetic potential in the laboratory frame

#### 5.1.3 Field to particle energy transfer

*E*and remembering that \(c^2{\varvec{p}} d{\varvec{p}}=E dE\) we obtain

### 5.2 Neutral particle hit by a light pulse

#### 5.2.1 Properties of equations

The dynamical equations developed here have a considerably more complex form compared to the Lorentz force and TBMT spin precession in constant fields [33]. We need field gradients in the Stern–Gerlach force, and in the related correction in the TBMT equations. Since the new physics appears only in the presence of a particle magnetic moment, we simplify by considering neutral particles. We now show that the external field described by a light wave (pulse) lends itself to an analytical solution effort. This context could be of practical relevance in the study of laser interaction with magnetic atoms, molecules, the neutron and maybe neutrinos.

#### 5.2.2 Invariant acceleration and spin precession

We see in Eq. (77) that the magnitude of the 4-force created by a light pulse and acting on an ultrarelativistic particle is dependent on the square of the product of the 2nd derivative of the pulse function with respect to \(\xi \), \(f^{\prime \prime }(\xi )\), with the Doppler shifted frequency Eq. (72). The value Eq. (77) is negative since acceleration is a space-like vector.

Upon solution of Eq. (79) \(k\cdot s(\tau )\) is known. Given Eq. (71) we also know the dependence of Eq. (67) on proper time \(\tau \). Hence Eq. (64) can be solved for \(u^\mu \) and Eq. (65) can be solved for \(s^\mu \) resulting in an analytical solution of the dynamics of a neutral magnetic dipole moment in the field of a light pulse of arbitrary shape. The full description of the dynamics exceeds in length this presentation and will follow [27].

## 6 Conclusions

- 1.
introduced in Eq. (10) the covariant classical 4-spin vector \(s^\mu \) in a way expected in the context of Poincaré symmetry of space-time;

- 2.
presented a unique linear in fields form of the covariant magnetic moment potential, Eq. (14), which leads to a natural generalization of the Lorentz force;

- 3.
shown that the resultant Amperian, Eq. (19), and Gilbertian, Eq. (26), forms of the magnetic moment force are equivalent;

- 4.
extended the TBMT torque dynamics, Eq. (35), making these consistent with the modifications of the Lorentz force;

- 5.
demonstrated the need to connect the magnetic moment magnitude entering the Stern–Gerlach force with the one seen in the context of torque dynamics, Sect. 3.2.3;

- 6.
shown that variational principle based dynamics has systemic failings when both position and spin are addressed within present day conceptual framework, see Sect. 4;

- 7.
reduced the covariant dynamical equations to laboratory frame of reference uncovering important features governing the coupled dynamics, see Sect. 5.1;

- 8.
obtained work done by variations of magnetic field in space-time on a particle, Eq. (61);

- 9.
shown salient features of solutions of neutral particles with non-zero magnetic moment hit by a laser pulse, see Sect. 5.2.

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