Electromagnetic Gauge-Invariance from the Symmetry of Local Spin Axes

1. History and Motivation

The roots of electromagnetic gauge-invariance lie (with Ampère) two-hundred years ago [1], and the modern understanding lies with Fock and Weyl one-hundred years ago [2,3]—it is that a $\mathrm{U}\left(1\right)$ phase-rotation of a charged particle’s field does not affect observables. What is the "best" interpretation? From a review of the history of gauge-invariance [4],

While for Electroweak Theory and QCD gauge invariance is of paramount importance, its physical meaning in QED per se does not seem to be extremely profound.

This paper proposes a simple-yet-profound interpretation built upon over a half-century of research. Geometrical inquiries into the connection between, and nature of, the gauge-invariance of electromagnetism and spin began in the 1900s and have continued to this day [5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21]. Most of this work has been performed using Geometric Algebra (GA): A branch of mathematics that employs Clifford algebras with the purpose of expressing truths about Geometry. Yet even attempts that lie outside of GA are, to a high degree, equivalent to GA [18,19,20,21]. However, while the GA and non-GA approaches have managed to establish connections between electromagnetic gauge-invariance and a geometric spin axis, no approach—with perhaps the sole exception of [21]—directly deals with presumed local rotational invariance.

Hestenes was the first to apply GA and to connect the electromagnetic gauge-invariance to a spin axis [5,6,7,8,9,10,11]. He made this connection within the context of Dirac Theory by considering the rotational plane of the Dirac current’s components $J_1$ and $J_2$. Hestenes discovered that the Dirac current’s "phase plane" was related to electromagnetic gauge-invariance, and that the geometric axis of invariance was related to spin—he never framed electromagnetic gauge-invariance as being the spin axis’ rotational invariance. Actually, in and after [10], Hestenes interpreted this connection to mean the electromagnetic gauge-invariance was a physical orbital rotation called "zitterbewegung." The next group to apply GA and to discover the connection was the Doran-Lasenby-Gull group (DLG) [12,13,14]. They built upon Hestenes’ work, and further discussed the connection of electromagnetic gauge-invariance and a spin axis within the context of Pauli Theory [13]. But, like Hestenes, DLG never framed electromagnetic gauge-invariance as being the spin axis’ rotational invariance. Their framing went only so far as to state the geometry of the connection, almost in passing, without any interpretation. All other groups who then built upon Hestenes’ and DLG’s work may be understood in the same way [15,16,17]. Of note is Baylis’ framing, however. In [15], he interprets the invariance as a physical, but intrinsic and for-now-unobservable, rotation. For the remaining non-GA approaches [18,19,20,21], the interpretational framing is much the same as the GA approaches, if with an even smaller emphasis on Geometry.

This work will use the Algebra of Physical Space (APS), which is the geometric algebra of 3-dimensional Euclidean space. The (nonrelativistic) Pauli-Schrödinger equation (PSE) will be derived from first principles, and local phase-rotations about the fermion’s spin axis will result in the conventional gauge transformations of electromagnetism. Then a manifestly gauge-covariant PSE will be constructed. Before concluding, it will be demonstrated how the conventional, "non-geometric" PSE can be obtained by considering minimal left ideals of the APS. To conclude, the derivations will be summarized and discussed. A heavy importance to interpretation will be given.

The author would like to strongly emphasize that this paper presents no new algebraic techniques! It is already known that the APS is related to the conventional Pauli algebra. The novelty of this work comes from the historical review, the method of derivation, the expansion of its interpretation and philosophy, and the discovery of an apparent tension between traditional Gauge Theory and this geometrical understanding. It is the hope of the author that this paper will fill in a conceptual crack within the foundations of Physics.

All algebraic essentials will be covered in Appendix A. To get the most out of this work, the author recommends lightly reading the full paper, then reading the appendix to garner a better mathematical understanding, and finally rereading the full paper.

2. Spin Axes and Electromagnetism

Let $\widehat{\mathbf{s}}\in {\mathbb{G}}_3^1$ be the unit vector describing the nonrelativistic spin axis of a massive fermion. In a classical application of the geometry, this encodes for the physical orientation of the particle in space; in a conventional-quantum application of the geometry, this encodes for the orientation of the particle on the Bloch-sphere; in a relational-quantum application of the geometry, this encodes the abstract-relational (no interaction-independence) orientation of the particle in space. Regardless of the interpretation, $\widehat{\mathbf{s}}$ is the fermion’s spin axis. The geometric symmetry of the spin axis is

$$\mathrm{Spin}(2,\widehat{\mathbf{s}})=\{U\in \mathrm{Spin}\left(3\right)|U\widehat{\mathbf{s}}{U}^{†}=\widehat{\mathbf{s}}\},$$

(1)

where $U=e^{-i\omega \widehat{\mathbf{s}}}$, for some $\omega \in \mathbb{R}$, is called a phase-rotor. This group is isomorphic to $\mathrm{U}\left(1\right)$. Suppose the spin axis now depends upon some parameter $\tau \in \mathbb{R}$: $\widehat{\mathbf{s}}\mapsto \widehat{\mathbf{s}}(\tau )$. Let ${\widehat{\mathbf{s}}}_0$ now represent the $\tau$-independent initial spin axis. As the spin axis remains a unit vector, the largest group in the APS that satisfies this requirement is

$$\mathrm{U}\left(2\right)=\{e^{iA}\in {\mathbb{G}}_3|A=A^{†}\}.$$

(2)

The simplest $\mathrm{U}\left(2\right)$ transformation that is $\tau$-dependent is linear in its exponential,

$$\Psi (\tau )=e^{-iH\tau },$$

(3)

and it implies that the $\tau$-dependent spin axis is of the form $\widehat{\mathbf{s}}=\Psi {\widehat{\mathbf{s}}}_0{\Psi }^{†}$. Because the spin axis’ $\tau$-dependence is fully parameterized by $\Psi$, it suffices to consider the change with respect to $\tau$ of only the transformation:

$${\displaystyle \frac{d}{d\tau }}\Psi =-iH\Psi .$$

(4)

This is provoking—identifying the parameter $\tau$ with time and the Hermitian object H with the scaled Hamiltonian $H/\hslash$ naturally results in the Pauli-Schrödinger equation (PSE),

$$i\hslash {\displaystyle \frac{d}{d\tau }}\Psi =H\Psi .$$

(5)

The over-curve notation is meant to denote operators in the algebra without confusing them with unit vectors or bivectors. In the case that this equation describes a time-dependent spin axis, this is the Pauli equation. In the case that this equation describes a time-independent spin axis, this is the Schrödinger equation. A quote from [7] is highly relevant:

Consistency [with Pauli theory] requires that the Schrödinger theory be regarded as describing an electron in an eigenstate of spin.

Henceforth, consider the spin axis to be a field over space $\mathbf{x}\in {\mathbb{G}}_3^1$. Then $\Psi$ is likewise a field. Recall the rotational symmetry of the spin axis, expressed in EQ. 1. Explicitly,

$$e^{-i\omega \widehat{\mathbf{s}}}\widehat{\mathbf{s}}{e}^{i\omega \widehat{\mathbf{s}}}\leftrightarrow e^{-i\omega \widehat{\mathbf{s}}}\Psi =\Psi e^{-i\omega {\widehat{\mathbf{s}}}_0},$$

(6)

this rotational symmetry generates local phase-rotations on $\Psi$. Let the Hamiltonian be

$$H(·)=-{\displaystyle \frac{\hslash }{2m}}{\nabla }^2(·)+V(·){\widehat{\mathbf{s}}}_0,$$

(7)

where $m\in {\mathbb{R}}^{+}$ is a fermion’s mass, $\nabla \in {\mathbb{G}}_3^1$ is the vector gradient, and $V\in \mathbb{R}$ is some potential. Considering a phase-rotated element ${\Psi }^{\prime }=\Psi e^{-i\omega {\widehat{\mathbf{s}}}_0}$, and inserting it into the PSE alongside this Hamiltonian,

$$\begin{array}{ccc}\hfill i\hslash {\displaystyle \frac{d}{d\tau }}\left(\Psi e^{-i\omega {\widehat{\mathbf{s}}}_0}\right)& =\hfill & -{\displaystyle \frac{{\hslash }^2}{2m}}{\nabla }^2\left(\Psi e^{-i\omega {\widehat{\mathbf{s}}}_0}\right)+V\Psi e^{-i\omega {\widehat{\mathbf{s}}}_0}{\widehat{\mathbf{s}}}_0\hfill \\ \hfill & \leftrightarrow \hfill & \hfill \\ \hfill i\hslash {\displaystyle \frac{d}{d\tau }}\Psi e^{-i\omega {\widehat{\mathbf{s}}}_0}& +\hfill & {\displaystyle \frac{d\omega }{d\tau }}\Psi e^{-i\omega {\widehat{\mathbf{s}}}_0}{\widehat{\mathbf{s}}}_0\hfill \\ \hfill & =\hfill & -{\displaystyle \frac{{\hslash }^2}{2m}}\left({\nabla }^2\Psi -2i(\nabla \omega ·\nabla )\Psi {\widehat{\mathbf{s}}}_0-{(\nabla \omega )}^2-i{\nabla }^2\omega \Psi {\widehat{\mathbf{s}}}_0\right)e^{-i\omega {\widehat{\mathbf{s}}}_0}\hfill \\ \hfill & +\hfill & V\Psi e^{-i\omega {\widehat{\mathbf{s}}}_0}{\widehat{\mathbf{s}}}_0,\hfill \end{array}$$

(8)

one gets a relatively nasty expression. But again, this is provoking—the form is precisely that which would cancel out an electromagnetic gauge transformation, given $\theta =q\chi /(\hslash c)$ for electromagnetic gauge-phase $\chi \in \mathbb{R}$ and charge q.

To completely demonstrate that the spin axis’ rotational invariance indeed generated an electromagnetic gauge transformation, an explicit construction of a manifestly gauge-covariant PSE—where the gauge group comes from the symmetry of the spin axis—will now be offered. A gauge-covariant derivative (operator) $\mathcal{D}$ must satisfy

$${\mathcal{D}}^{\prime }(\Psi e^{-i{\displaystyle \frac{q}{\hslash c}}\chi {\widehat{\mathbf{s}}}_0})=\left(\mathcal{D}\Psi \right)e^{-i{\displaystyle \frac{q}{\hslash c}}\chi {\widehat{\mathbf{s}}}_0}.$$

(9)

Primes indicate different gauges. The motivation for forming covariant derivatives comes from the known gauge transformations of electromagnetism. The known transformation for the scalar potential $\varphi \in \mathbb{R}$ is

$$\varphi \mapsto {\varphi }^{\prime }=\varphi -{\displaystyle \frac{1}{c}}{\displaystyle \frac{d\chi }{d\tau }}.$$

(10)

With this, the covariant time derivative must be

$$D_{\tau }(·)={\displaystyle \frac{d}{d\tau }}(·)-i{\displaystyle \frac{q}{\hslash }}\varphi (·){\widehat{\mathbf{s}}}_0.$$

(11)

Confirming that the gauge condition of EQ. 9 is satisfied:

$$\begin{array}{cc}\hfill D_{\tau }^{\prime }{\Psi }^{\prime }& =\left({\displaystyle \frac{d}{d\tau }}\Psi -i{\displaystyle \frac{q}{\hslash }}\left({\varphi }^{\prime }+{\displaystyle \frac{d\chi }{d\tau }}\right)\Psi {\widehat{\mathbf{s}}}_0\right)e^{-i{\displaystyle \frac{q}{\hslash c}}\chi {\widehat{\mathbf{s}}}_0}\hfill \\ \hfill & =\left({\displaystyle \frac{d}{d\tau }}\Psi -i{\displaystyle \frac{q}{\hslash }}\varphi \Psi {\widehat{\mathbf{s}}}_0\right)e^{-i{\displaystyle \frac{q}{\hslash c}}\chi {\widehat{\mathbf{s}}}_0}=\left(D_{\tau }\Psi \right)e^{-i{\displaystyle \frac{q}{\hslash c}}\chi {\widehat{\mathbf{s}}}_0}.\hfill \end{array}$$

(12)

The known gauge transformation for the vector potential $\mathbf{A}\in {\mathbb{G}}_3^1$ is

$$\mathbf{A}\mapsto {\mathbf{A}}^{\prime }=\mathbf{A}+\nabla \chi .$$

(13)

Then the covariant gradient must be

$$\mathbf{D}(·)=\nabla (·)+i{\displaystyle \frac{q}{\hslash c}}\mathbf{A}(·){\widehat{\mathbf{s}}}_0.$$

(14)

Once more confirming the gauge condition:

$$\begin{array}{cc}\hfill {\mathbf{D}}^{\prime }{\Psi }^{\prime }& =\left(\nabla \Psi +i{\displaystyle \frac{q}{\hslash c}}\left({\mathbf{A}}^{\prime }-\nabla \chi \right)\Psi {\widehat{\mathbf{s}}}_0\right)e^{-i{\displaystyle \frac{q}{\hslash c}}\chi {\widehat{\mathbf{s}}}_0}\hfill \\ \hfill & =\left(\nabla \Psi +i{\displaystyle \frac{q}{\hslash c}}\mathbf{A}\Psi {\widehat{\mathbf{s}}}_0\right)e^{-i{\displaystyle \frac{q}{\hslash c}}\chi {\widehat{\mathbf{s}}}_0}=\left(\mathbf{D}\Psi \right)e^{-i{\displaystyle \frac{q}{\hslash c}}\chi {\widehat{\mathbf{s}}}_0}.\hfill \end{array}$$

(15)

So, in the presence of the electromagnetic potential $A=c\varphi +\mathbf{A}$,

$$i\hslash D_{\tau }\Psi =-{\displaystyle \frac{{\hslash }^2}{2m}}{\mathbf{D}}^2\Psi ,$$

(16)

is the manifestly gauge-covariant PSE for $H(·)=-{\hslash }^2{\mathbf{D}}^2(·)+q\varphi (·){\widehat{\mathbf{s}}}_0$. The Zeeman term is included automatically, and will be seen in the next subsection. This completes the proof that the local rotational symmetry of spin axes generates electromagnetic gauge symmetry. It is worth emphasizing that within the notation of the APS, squared operators such as ${\mathbf{D}}^2(·)$ do not mean to square the operator first and then apply it to the operand $(·)$—the square represents successive applications of the operator.

2.1. Connection to Conventional Theory

The time-dependent transformation $\Psi \in \mathrm{U}\left(2\right)$ may be in general written as

$$\Psi =e^{i\beta }{R}_s,$$

(17)

where $R_s\in \mathrm{Spin}\left(3\right)$ satisfies $\widehat{\mathbf{s}}=R_s{\widehat{\mathbf{s}}}_0{R}_s^{†}$ and $\beta \in \mathbb{R}$. This purely complex phase $e^{i\beta }$ is interpreted, by most in the field of Geometric Algebra, as creating a mixture of particle and antiparticle states [13,22]. But there is also an argument for its interpretation through electromagnetic dual symmetry [23]. In truth, its meaning is poorly understood. Some within the field seem to be aware of this [19]—the distinction between $\mathrm{U}\left(1\right)$ contributions like $e^{i\beta }$ and $\mathrm{Spin}(2,{\widehat{\mathbf{s}}}_0)$ contributions disappears within minimal left ideals (irreducible representations). Interestingly, however, the kinetic contribution of the Hamiltonian in EQ. 7 can only come from such a $\mathrm{U}\left(1\right)$ term. Again, a full understanding of this term is missing. But, what is known is this new rotor, $R_s$, which is a Pauli spinor. It may be generally decomposed as

$$R_s=cos{\displaystyle \frac{\theta }{2}}{R}_{\uparrow }+sin{\displaystyle \frac{\theta }{2}}{R}_{\downarrow },$$

(18)

with (polar) angle $\theta ={cos}^{-1}(\widehat{\mathbf{s}}·{\widehat{\mathbf{s}}}_0)$. These two sub-rotors, $R_{\uparrow }$ and $R_{\downarrow }$, are respectively called the identity- and flip-rotors. They are so-named because

$$R_{\uparrow }{\widehat{\mathbf{s}}}_0{R}_{\uparrow }^{†}={\widehat{\mathbf{s}}}_0\mathrm{and}{R}_{\downarrow }{\widehat{\mathbf{s}}}_0{R}_{\downarrow }^{†}=-{\widehat{\mathbf{s}}}_0.$$

(19)

For a unit bivector $\widehat{B}\in {\mathbb{G}}_3^2$ such that $\widehat{B}{\widehat{\mathbf{s}}}_0{\widehat{B}}^{†}=-{\widehat{\mathbf{s}}}_0$, and for phases ${\omega }_{\uparrow /\downarrow }\in \mathbb{R}$, it follows that the general forms of these rotors are

$$R_{\uparrow }=e^{-i{\omega }_{\uparrow }{\widehat{\mathbf{s}}}_0}\mathrm{and}{R}_{\downarrow }=\widehat{B}{e}^{-i{\omega }_{\downarrow }{\widehat{\mathbf{s}}}_0}.$$

(20)

Therefore, the Pauli spinor generating $\widehat{\mathbf{s}}$ is

$$R_s=cos{\displaystyle \frac{\theta }{2}}{e}^{-i{\omega }_{\uparrow }{\widehat{\mathbf{s}}}_0}+sin{\displaystyle \frac{\theta }{2}}\widehat{B}{e}^{-i{\omega }_{\downarrow }{\widehat{\mathbf{s}}}_0}.$$

(21)

Let ${\ell }_{+}=(1+{\widehat{\mathbf{s}}}_0)/2$ be an idempotent projector within the APS, defined in the direction of the reference axis. Via leftwise multiplication of ${\ell }_{+}$ by any element of the APS, a minimal left ideal is formed [17]. This ideal, ${\mathbb{G}}_3{\ell }_{+}$, is isomorphic to the conventional Pauli irreducible representation, ${\mathbb{C}}^2$. Multiplying $\Psi$ by ${\ell }_{+}$ and applying EQ. 21,

$$\begin{array}{cc}\hfill \psi =e^{i\beta }{R}_s{\ell }_{+}& =cos{\displaystyle \frac{\theta }{2}}{e}^{i(\beta -{\omega }_{\uparrow })}{\ell }_{+}+sin{\displaystyle \frac{\theta }{2}}\widehat{B}{e}^{i(\beta -{\omega }_{\downarrow })}{\ell }_{+}\hfill \\ \hfill & \equiv \left[\begin{array}{cc}cos{\displaystyle \frac{\theta }{2}}{e}^{i(\beta -{\omega }_{\uparrow })}& 0\\ sin{\displaystyle \frac{\theta }{2}}{e}^{i(\beta -{\omega }_{\downarrow })}& 0\end{array}\right].\hfill \end{array}$$

(22)

The last line was achieved by choosing ${\widehat{\mathbf{s}}}_0={\sigma }_z$ and $\widehat{B}={\sigma }_{xz}$, then invoking the Pauli matrix representation of EQ. A3. This is clearly equivalent to a Pauli spinor from conventional Pauli Theory! Notice the demonstration of an earlier point: That the contributions from $\mathrm{U}\left(1\right)$ and $\mathrm{Spin}(2,{\widehat{\mathbf{s}}}_0)$ are indistinguishable within the minimal left ideal.

To ensure clarity of equivalency, the earlier gauge-covariant derivatives are now to be presented within the minimal ideal ${\mathbb{G}}_3{\ell }_{+}\approx {\mathbb{C}}^2$. Then the gauge-covariant PSE will be placed in its conventional (expanded) representation. The gauge-covariant time derivative is

$$D_{\tau }(·){\ell }_{+}=\left[{\displaystyle \frac{d}{d\tau }}(·)-i{\displaystyle \frac{q}{\hslash }}\varphi (·)\right]{\ell }_{+}.$$

(23)

The gauge-covariant gradient is

$$\mathbf{D}(·){\ell }_{+}=\left[\nabla (·)+i{\displaystyle \frac{q}{\hslash }}\mathbf{A}(·)\right]{\ell }_{+}.$$

(24)

Writing

$$\psi \equiv \left[\begin{array}{cc}cos{\displaystyle \frac{\theta }{2}}{e}^{i(\beta -{\omega }_{\uparrow })}& 0\\ sin{\displaystyle \frac{\theta }{2}}{e}^{i(\beta -{\omega }_{\downarrow })}& 0\end{array}\right]=\left[\begin{array}{cc}{\psi }_{\uparrow }& 0\\ {\psi }_{\downarrow }& 0\end{array}\right],$$

(25)

the conventional representation of EQ. 16, after expanding, is then

$$i\hslash {\displaystyle \frac{d}{d\tau }}\psi +q\varphi \psi =-{\displaystyle \frac{{\hslash }^2}{2m}}{\left(\dot{\nabla }+i{\displaystyle \frac{q}{\hslash }}\mathbf{A}\right)}^2\dot{\psi }+i{\displaystyle \frac{q\hslash }{2mc}}\nabla ·\mathbf{A}\psi -{\displaystyle \frac{q\hslash }{2mc}}\mathbf{B}\psi .$$

(26)

In the expansion, the identity $\nabla \wedge \mathbf{A}=i\mathbf{B}$ is used to obtain the Zeeman term with the magnetic field $\mathbf{B}\in {\mathbb{G}}_3^1$. The overdot notation emphasizes that the gradient acts only on $\psi$. In this case, where the objects in the parentheses are not operators after expansion, the square does mean to first square the parentheses before applying to the operand. This completes the proof of equivalency between the APS result and the conventional theory.

3. Discussion

To recap [4], the modern understanding of electromagnetic gauge-invariance is that "[gauge-invariance’s] physical meaning in QED per se does not seem to be extremely profound." Separate from the mainstream modern understanding, geometrical understanding has been growing for over a half-century. The main contributions were first from Hestenes [5,6,7,8,9,10,11], then from the Doran-Lasenby-Gull group (DLG) [12,13,14], and from several other miscellaneous groups [15,16,17]. Several Geometry-adjacent fields also made contributions [18,19,20,21]. The main realization was that the electromagnetic gauge-invariance was connected to a spin axis. However, the interpretation of this geometric relationship is not agreed upon; Hestenes saw electromagnetic gauge-invariance as related to physical orbital rotation, "zitterbewegung." DLG kept their interpretation agnostic and only stated the bare geometric connection. Baylis interpreted the invariance as related to a physical but intrinsic and unobservable rotation. In short, the general geometrical understanding of the connection between the electromagnetic gauge-invariance and the spin axis is agnostic to its meaning.

This work has presented an argument to enforce meaning onto this geometric connection. The first principle was to assume that a fermion had an orientation—physical for classical considerations, on the Bloch-sphere for conventional-quantum considerations, and abstract-relational (no interaction-independence) for relational-quantum considerations—represented by a vector $\widehat{\mathbf{s}}$ called the spin axis, and from this principle was derived the Pauli-Schrödinger equation (PSE) where the gauge-covariance is directly generated by the spin axis’ rotational symmetry. This manifestly gauge-covariant PSE was also reworked into its expanded form in the conventional theory, proving equivalency between the geometrical and traditional approaches. An interpretation of the connection between electromagnetic gauge-invariance and the spin axis is now forced: The local rotational symmetry of the spin axis generates the electromagnetic gauge group.

There is an important philosophical implication from this understanding: The electromagnetic gauge-potential and the Pauli spinor/wavefunction exist within the same ontological category. $\Psi$-ontic interpretations, like Everett’s Many-Worlds Interpretation or Bohmian Mechanics, understand the wavefunction to be physically real entities [24]. $\Psi$-onticism implies a similar interpretation of the electromagnetic potential as physically real. $\Psi$-epistemic interpretations, like the Copenhagen interpretation, understand the wavefunction to only represent information or knowledge of a system [25]. $\Psi$-epistemicism likewise implies that the gauge-potential is to be interpreted as only representing information or knowledge of a system. In a relational interpretation—which bears a mixture of onticism and epistemicism—the wavefunction represents an entity with physically real properties that are only defined relative to interacting systems, and in the non-interacting gap are treated as informational or representational [26]. This interpretation would also carry onto the electromagnetic gauge-potential. Whatever the ontology of one thing, the other must share. This philosophical implication is upheld by the fact that both the Pauli spinor and the electromagnetic gauge-potential are gauge-covariant objects who generate gauge-invariant objects; the spin axis (or spin density element) for one, and the electromagnetic field for the other.

The results of this paper immediately imply that Hestenes’ and Baylis’ interpretations, as the invariance of an axis that purely denotes orientation cannot describe any kind of physical rotation—no matter orbital or intrinsic and unobservable—does not follow as a logical result. However, this work does not tell one what the spin axis is besides an orientation. It could be that the orientation comes about from some kind of physical rotation, but the symmetry of the orientation is a geometrical statement that is conceptually separate from physical rotation. So Hestenes and Baylis could be correct about spin and physical rotations, they just cannot claim that the symmetry is proof of them.

There is one seemingly nontrivial objection, however. In traditional Gauge Theory, gauge degrees of freedom (DoF) are internal rather than geometrical [27]. This means that gauge DoF are wholly independent of any geometrical DoF. For example, with some internal gauge group $\mathcal{G}$ and the Lorentz group $\mathrm{Spin}(1,3)$, the total transformation group is $\mathcal{G}\otimes \mathrm{Spin}(1,3)$. The $\mathcal{G}$ DoF are independent of the $\mathrm{Spin}(1,3)$ DoF. But in this work, and in all related geometrical approaches, there is now a connection between the gauge group of electromagnetism and a subgroup of the Lorentz group:

$$\mathrm{U}{\left(1\right)}_{\mathrm{EM}}=\mathrm{Spin}(2,{\widehat{\mathbf{s}}}_0)\subset \mathrm{Spin}(1,3).$$

(27)

Naïvely, this seems to contradict the foundations of Gauge Theory, which in turn could ruin key properties like renormalizability! However, there is no real problem: Leftwise and rightwise multiplication are commuting actions within any associative algebra, so the actions of $\mathrm{Spin}(1,3)$ and $\mathrm{Spin}(2,{\widehat{\mathbf{s}}}_0)$ are independent. In the language of [28], the rightwise action of the gauge group is vertical while the leftwise action of the Lorentz group is horizontal.

There are neither financial nor non-financial competing interests relevant to the author and their work.

Appendix A.1. Spin (3) in G 3

The group $\mathrm{Spin}\left(3\right)\approx \mathrm{SU}\left(2\right)$ is that of the 3-dimensional Euclidean space’s rotations:

$$\mathrm{Spin}\left(3\right)=\{R\in {\mathbb{G}}_3^{+}|R{R}^{†}=R^{†}R=1\}.$$

(A10)

R is called a rotor. Here ${\mathbb{G}}_3^{+}$ is the even subalgebra of ${\mathbb{G}}_3$, which is the subspace of scalars and bivectors (which is closed under multiplication). Also, ${(·)}^{†}$ is the Hermitian conjugate (reversion) which satisfies, for two elements $g,h\in {\mathbb{G}}_3$,

$${\left(gh\right)}^{†}=h^{†}{g}^{†}.$$

(A11)

This group exists due to the basis bivectors of EQ. A8, which form the Lie algebra $\mathfrak{spin}\left(3\right)\approx \mathfrak{su}\left(2\right)$. In general, rotors are exponentials of an arbitrary bivector $\Omega \in {\mathbb{G}}_3^2$:

$$R=e^{-{\displaystyle \frac{1}{2}}\Omega }.$$

(A12)

Bivectors may also be represented as imaginary vectors using the pseudoscalar i: $\Omega =i\mathbf{w}$. Then, using the sandwich product,

$$g\mapsto Rg{R}^{†},$$

(A13)

any multivector can be rotated.