Clifford algebra Cl(0,6) approach to beyond the standard model and naturalness problems

The mathematical imaginary number $i$ is ubiquitous in physics theories. In the case of quantum mechanics, the imaginary number makes its appearance in the commutation relation of position operator $\hat{X}$ and momentum operator $\hat{P}$

where $\hbar$ is the Planck constant. Consequently, the quantum wave function is complex-valued.

On the other hand, the imaginary number also shows up in the gauge transformation of a classical field

which is essential in determining the electric charge property of $\psi$.

We customarily treat the imaginary number $i$ in both examples as the same. It may come as a surprise that the imaginary number in the second case is different from the first one. The imaginary number in gauge transformation (2) is actually a unit pseudoscalar in disguise

where $\gamma_{a}$ are no other than the celebrated gamma operators discovered by Paul Dirac in 1928. The Dirac gamma operators satisfy the Clifford algebra $Cl(1,3)$ anticommutation relations

where $\eta_{ab}=diag(1,-1,-1,-1)$.

In view that $I^{2}=-1$, pseudoscalar $I$ can be regarded as a surrogate for imaginary number $i$. As we will learn later in this paper, replacing imaginary number $i$ with pseudoscalar $I$ in gauge transformation (2) such as $\psi\rightarrow\psi e^{I\theta}$ leads to a novel definition of charge conjugation without particle-antiparticle interchange. And for that matter, the original imaginary number $i$ shall be banished from the definition of classical fields such as classical spinor and Higgs fields. Instead, imaginary number $i$ should be reserved for its proper domain which is field quantization.

Historically, the Dirac operators $\gamma_{a}$ are represented as gamma matrices. Due to the dichotomy between fermion states as columns and operators as matrices in the conventional formalism of quantum field theory (QFT), the aforementioned association of pseudoscalar with imaginary number would run into inconsistencies. This identification can only be achieved in an unconventional way by forgoing the traditional matrix and column representation and enlisting the aid of the Clifford algebra approach [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 12, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32], whereby both the algebraic spinor states and Dirac’s gamma operators can be expressed in the same algebraic space. Note that we have already been using the Clifford algebraic non-matrix format, such as in Eq. (4).

where $e_{\mu}^{a}$ and $\omega_{\mu}^{ab}$ are real-valued, $\omega_{\mu}^{ab}=-\omega_{\mu}^{ba}$, and $a,b,\mu=0,1,2,3$. Throughout this paper the summation convention for repeated indices is adopted. The four distinct $\{\gamma_{a}\}$ and six distinct $\{\gamma_{a}\gamma_{b};a<b\}$ are called vectors and bi-vectors of Clifford algebra. We denote vierbein as $\hat{e}_{\mu}$ and spin connection as $\hat{\omega}_{\mu}$ rather than $e_{\mu}$ and $\omega_{\mu}$ to accentuate the fact that they are Clifford-valued.

The Lorentz gauge approach to gravity is also known as Einstein-Cartan gravity. The spin connection gauge field $\hat{\omega}_{\mu}$, associated with the local Lorentz group $SO(1,3)$ (or $Spin(1,3)$ when fermions are involved), plays the role of the gauge fields in Yang-Mills theory. In the gauge theory of gravity, the space-time metric $g_{\mu\nu}$ is defined as a composite field

where $\left\langle\ldots\right\rangle$ stands for the Clifford-scalar part of the enclosed expression. Thus vierbein $\hat{e}_{\mu}$ can be deemed as the “square root” of metric.

Given that the gravity-related fields $\hat{e}_{\mu}$ and $\hat{\omega}_{\mu}$ are vector-valued and bi-vector-valued respectively in the Clifford algebraic space of $Cl(1,3)$, it’s tempting to wonder whether the other interactions in nature such as the electroweak and strong gauge fields can take values in the Clifford algebraic space as well. The answer is a resounding yes, provided that one has to go beyond the confines of the familiar Clifford algebra $Cl(1,3)$.

Learning from the above experience that we arrived at $Cl(1,3)$ via splitting the “imaginary number” $I$ into four $\gamma_{a}$ operators in (3), we may go one step further by decomposing Dirac’s gamma zero operator $\gamma_{0}$ into its underlying components [27]

where the additional trio of gamma operators {$\Gamma_{1}$, $\Gamma_{2}$, $\Gamma_{3}$} satisfy the anticommutation relations

and anticommute with Dirac’s original trio {$\gamma_{1}$, $\gamma_{2}$, $\gamma_{3}$}

Collectively, these six elements

constitute the orthonormal vector basis of the real Clifford algebra $Cl(0,6)$, which is sometimes labeled as $Cl_{0,6}$ or $Cl_{0,6}(R)$ in the literature. Note that while the Clifford algebra has been extended, in our model we assume that the underlying space-time manifold remains 4-dimensional.

Thanks to the recognition of $\gamma_{0}$ as a composite tri-vector, we are able to extend Dirac’s Clifford algebra from $Cl(1,3)$ to $Cl(0,6)$, with $Cl(1,3)$ being a sub-algebra of $Cl(0,6)$. With it, we can define an algebraic spinor as a linear combination of all $2^{6}=64$ basis elements of $Cl(0,6)$. Considering that there are 16 Weyl fermions with $16\times 2=32$ complex components (i.e. $64$ real components) within each of the three fermion families including the right-handed neutrino, an algebraic spinor of the real $Cl(0,6)$ with 64 real degrees of freedom is a perfect match for representing one generation of fermions.

The geometrical wealth of Clifford algebra $Cl(0,6)$ can be exploited to establish a unified theory of gravity and the standard model [27, 29, 30] based on two central tenets: gauge symmetry and quantum condensation. The tenet of gauge symmetry stipulates that both Lorentz and internal symmetries should be treated on an equal footing as gauge symmetries, while the tenet of quantum condensation dictates that the fundamental energy scales of the universe such as the Planck scale are emergent and resulted from multi-spinor quantum condensation which is an ordered quantum phase induced by the dynamical symmetry breaking mechanism.

According to the central tenet of gauge symmetry, the spin connection Lorentz gauge field governing gravity and the Yang-Mills gauge fields governing the standard model interactions are associated with beyond the standard model (BSM) symmetries

where $Spin(1,3)_{L}$ and $Spin(1,3)_{R}$ are the left- and right-handed local Lorentz gauge groups, and the extended left-handed weak gauge group $Spin(1,3)_{WL}$ encompasses the standard model left-handed weak gauge group $SU(2)_{WL}$.

The BSM symmetries (11) are accompanied by an extended vierbein valued in a 16-dimensional multivector subspace of $Cl(0,6)$ as opposed to the traditional vierbein (5) valued in the 4-dimensional vector space of $Cl(1,3)$. The extended vierbein transforms as a vector under both Lorentz and the extended weak $Spin(1,3)_{WL}$ gauge transformations. Therefore, a new kind of gravi-weak interplay is permitted between the extended vierbein and the extended weak gauge field. This gravi-weak interplay is otherwise impossible since the traditional vierbein is invariant under the weak $SU(2)_{WL}$ gauge transformation. Note that the gravi-weak interplay is not in conflict with the Coleman-Mandula theorem [40], since there is no nontrivial mixing between the Lorentz groups ($Spin(1,3)_{L}$ and $Spin(1,3)_{R}$) and the internal gauge groups including the extended weak group $Spin(1,3)_{WL}$.

According to the second central tenet of quantum condensation, we investigate the vierbein field $\hat{e}_{\mu}$ as a composite entity emerging from a Cooper-pair-like fermion-antifermion quantum condensation via the dynamical symmetry breaking mechanism. Hence the standard model fermion fields are the origin of space-time metric. Given that vierbein can be viewed as the “square root” of metric, the standard model fermions can be considered as the “quarter root” of metric, which speaks to the fact that fermions have spin $1/2$ while gravitons have spin $2$. Consequently, quantum gravity is realized indirectly via the quantized standard model spinor fields which underlie the composite space-time metric. Note that due to the chirality of the vierbein-related condensations and the chirality of spin connection fields associated with left- and right-handed local Lorentz gauge groups $Spin(1,3)_{L}$ and $Spin(1,3)_{R}$, there are left- and right-handed gravitational interactions.

We propose that all the regular Lagrangian terms, be it the fermion kinetic term or gravity and cosmological constant terms, are of quantum condensation origin. Therefore, both the Planck scale and the cosmological constant scale are emergent and resulted from quantum condensations. We advocate a multi-scale renormalization procedure for quantum condensation, which is a paradigm shift from the conventional QFT renormalization approach (or the Wilsonian renormalization group approach) characterized by a single renormalization scale. The cosmological constant problem can thus be evaded if we exercise extreme caution in the renormalization procedure that entails multiplications of divergent Feynman integrals.

It’s worth mentioning that Clifford algebra $Cl(0,6)$ is capable of accommodating enveloping groups larger than the BSM groups (11). For example, the spin group $Spin(4,4)$ and the real symplectic group $Sp(8,R)$ are embedded in the $Cl(0,6)$ geometric structure. The Lorentz gauge group $Spin(1,3)$ and the extended weak group $Spin(1,3)_{WL}$ are two commuting subgroups of $Spin(4,4)$. The Pati-Salam’s $SU(4)$ [41], which is a subgroup of the real symplectic group $Sp(8,R)$, is isomorphic to the six-dimensional rotation group $Spin(6)$. We settle for a parsimonious set of subgroups (11) due to chirality considerations and lack of experimental evidence (such as proton decay) supporting any larger unification groups such as $SU(4)$, whereas there are various clues suggestive of the BSM symmetries (11).

According to the conventional wisdom, the usual candidates for gauge transformations are bi-vector-related rotations. Thus the symmetry group of $Cl(0,6)$ seems to be restricted to $Spin(6)$/$SO(6)$ generated by all the 15 bi-vectors of $Cl(0,6)$. Then how can $Cl(0,6)$ accommodate the BSM symmetries discussed above? This is because of two advantages of the Clifford algebra approach that are not enjoyed by the conventional approach. First of all, the permissible group generators of the Clifford algebra approach could involve both bi-vector and non-bi-vector Clifford elements, generalizing the traditional notion of bi-vector rotations. For example, the Lorentz boosts and the left-handed weak-boosts are generated by $Cl(0,6)$ 4-vectors rather than bi-vectors.

Secondly, a Clifford algebraic spinor $\psi$ allows double-sided gauge transformations

where transformations $V$ and $U$ are independent of each other. And by definition, $V$ and $U$ commute with each other. Specifically, the gravi-weak symmetries in (11) (the Lorentz gravity $Spin(1,3)_{L}\times Spin(1,3)_{R}$ and the extended weak symmetries $Spin(1,3)_{WL}\times Spin(1,1)_{WR}\times U(1)_{WR}$) belong to the left-sided gauge transformation $V$, while the color and B-L symmetries in (11) ($SU(3)_{C}\times U(1)_{B-L}$) belong to the right-sided gauge transformation $U$. These double-sided gauge transformations are otherwise impossible in the conventional column fermion formalism. Note that the left-handed and right-handed fermions transform independently under the chiral left-sided gauge transformations $Spin(1,3)_{L}\times Spin(1,3)_{WL}$ and $Spin(1,3)_{R}\times Spin(1,1)_{WR}\times U(1)_{WR}$ respectively, whereas the left-handed and right-handed fermions transform in unison under the right-sided gauge transformations $SU(3)_{C}\times U(1)_{B-L}$.

All in all, the above two features of the Clifford algebra approach are the underlying mechanism which enables us to explore symmetries that go beyond the conventional approach.

In this paper, we present a detailed account of how BSM symmetries (11) are broken by a cascade of symmetry breaking processes, triggered by the nonzero vacuum expectation values (VEV) acquired by the Clifford-valued vierbeins and Higgs fields. The first stage of symmetry breaking starts with the vierbeins acquiring nonzero flat space-time VEVs, or equivalently with the metric $g_{\mu\nu}$ acquiring the Minkowski VEV $\eta_{\mu\nu}$. As a result, the pseudo-weak gauge symmetry (the coset $Spin(1,3)_{WL}/SU(2)_{WL}$ and $Spin(1,1)_{WR}$) , local Lorentz gauge symmetry and diffeomorphism symmetry are lost. The residual gauge symmetries are $SU(2)_{WL}\times U(1)_{WR}\times SU(3)_{C}\times U(1)_{B-L}$, plus a remnant global Lorentz symmetry which is the synchronization of the global portion of Lorentz gauge and diffeomorphism symmetries.

The next step of symmetry breaking is triggered by the Majorana-Higgs field, which is a Higgs-like field in addition to the standard model Higgs field. At this stage, the Majorana-Higgs field assumes a nonzero VEV and breaks the local gauge symmetries down to the standard model symmetries $SU(3)_{C}\times SU(2)_{WL}\times U(1)_{Y}$. Consequently, the neutrino is endowed with a lepton number-conserving Majorana mass which is much heavier than the Dirac mass. A very small effective mass can thus be derived for the neutrino via the seesaw mechanism [42].

At the last stage of symmetry breaking, the standard model electroweak Higgs field acquires nonzero VEV and breaks the standard model symmetries down to $SU(3)_{C}\times U(1)_{EM}$, where $U(1)_{EM}$ is the electromagnetic gauge symmetry. In the context of spontaneous breaking of Peccei-Quinn-like symmetries, the fermion Dirac mass hierarchies can be explained by a three-Higgs-doublet model with the help from reorganizing the lepton-quark configurations of the three generations of standard model fermions. For instance, the first generation electron and neutrino are paired up with the third generation top and bottom quarks (rather than the first generation up and down quarks) to form one unique fermion generation in our model, which is very different from the traditional fermion family assignments.

One point we want to highlight is that all the ingredients of our model, such as fermions, gauge fields, vierbeins, and Higgs fields, share the same Clifford algebraic space of $Cl(0,6)$. For instance, the electromagnetic gauge field $\hat{A}_{\mu}$ is pseudoscalar-valued

where $A_{\mu}$ is real-valued, and we include charge $q$ (such as $q=-1$ for electron) in the definition of $\hat{A}_{\mu}$. The pseudoscalar $I$ is the 6-vector of $Cl(0,6)$ which is the same $Cl(1,3)$ pseudoscalar $I$ in Eq. (3) after expanding the tri-vector $\gamma_{0}$ into its constituents

The electromagnetic gauge field $\hat{A}_{\mu}$ as shown above and the gravity-related fields $\hat{\omega}_{\mu}$/$\hat{e}_{\mu}$ in Eq. (5) take values in various Clifford algebraic subspaces with the same six gamma operators $\{\Gamma_{1},\Gamma_{2},\Gamma_{3},\gamma_{1},\gamma_{2},\gamma_{3}\}$ as the unifying building blocks. The Clifford algebraic gauge field of gravity and Yang-Mills fields are connected with each other through their interactions with the common fermion fields, vierbein fields and Higgs fields, all valued in Clifford algebraic space.

Therefore, the model outlined in this paper is indeed a cohesive unified theory of the standard model and gravity. Note that the Clifford algebra approach differs from the conventional grand unified theories (GUTs) [43, 44, 45, 46, 47] and gravi-GUTs [48, 49, 50, 51, 52, 53, 54, 55, 56] which demand that the gauge coupling constants should be unified. In the Clifford algebra approach, the allowable symmetries are the ones that preserve the invariance properties of spinor bilinears. Hence the symmetries of the model are in a sense derived rather than postulated. The permitted symmetries usually involve a direct product of different groups, suggesting that the individual gauge coupling constants are not necessarily related to each other. To a certain extent, the Clifford algebra approach as advocated here predicts the unpredictability of Weinberg angle $\theta_{W}$. As such, the fusion delineated in this paper is more of a unification via spinors and less of a unification via symmetry groups.

This paper is structured as follows: In Section 2, we introduce the algebraic spinors of $Cl(0,6)$ and explore beyond the standard model gauge symmetries. In Section 3, we investigate spontaneous symmetry breaking due to the non-degenerate vacuum expectation values of various bosonic fields, and study the fermion mass hierarchies and the lepton number-conserving Majorana mass. In Section 4, we contemplate naturalness problems through the lens of quantum condensations and examine the extended gauge symmetries which enable gravi-weak interaction. In the last section we draw our conclusions. Throughout this paper, we adopt the natural units $c=\hbar=1$.

In this section, we introduce algebraic spinors of the real Clifford algebra $Cl(0,6)$ and explore beyond the standard model gauge symmetries. We proceed with expressing the gauge fields and curvatures as multivectors of $Cl(0,6)$. We round out the section with a detailed formulation of the Lagrangian of the algebraic spinors, as well as the Lagrangian of the Yang-Mills and gravitational interactions in terms of Clifford algebra.

Note that the complexified Clifford algebra $Cl(6,C)$ [57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71] and its isomorphic equivalences such as $Cl(7,0)$ [16] and the complex $8\times 8$ matrix $M(8,C)$ algebra [72], either as standalone algebra or in association with octonions or sedenions, have been investigated in connection with the color $SU(3)_{c}$ symmetry and the other standard model symmetries. Nevertheless, in our approach [27, 29, 30] we choose to stick with the real $Cl(0,6)$ which is based on two reasons: The first reason is that an algebraic spinor of the real $Cl(0,6)$ with 64 real basis elements is a perfect match with one generation of 16 Weyl fermions (including the right-handed neutrino) in the standard model with 32 complex degrees of freedom (i.e. $64$ real degrees of freedom).

The second reason for choosing the real $Cl(0,6)$ is that we regard the spinor and gauge fields as classical fields prior to field quantization. As explained in the introduction section, we consign the imaginary number $i$ to the realm of quantum theory. Therefore, the imaginary number $i$ should play no role in the definition of the classical spinor fields, gauge fields and their symmetry transformations, as demonstrated by the replacement of imaginary number $i$ with the pseudoscalar $I$ in the electromagnetic $U(1)$ gauge transformation (2). In our model, we rigorously enforce the rule of avoiding the quantum $i$ in the definition of classical spinors, gauge fields, and the corresponding gauge-covariant derivatives. Note that we can nonetheless maintain a “pseudo-complex representation” with the pseudoscalar $I$ acting as the surrogate imaginary number. As a side note, in our earlier papers [27, 29, 30] we have used notations for imaginary number (denoted as $\hat{i}$) and pseudoscalar (denoted as $i$) which are different from notations in the current paper.

Pioneered by Hestenes [1], one original objective of the Clifford algebra approach to physics is to abandon the imaginary number altogether and replace it with certain element of Clifford algebra. The endeavor of using an effective operator to proxy the imaginary number has been fairly successful in a wide variety of physics domains [11, 15, 73, 20]. However, it has been realized [30, 31] that the initiative championed by Hestenes has its limitations: When it comes to field quantization and quantum loop integral calculations, the imaginary number is indispensable and can not be replaced by a Clifford element. For example, a self-energy loop diagram would yield an imaginary-valued quantum correction due to proper contour integral on the complex plane of Feynman propagators. Therefore, in Section 4 on field quantization, we will formally introduce the imaginary number $i$ in the Clifford functional integral formalism [31] when we study quantum phenomena. In the same Section 4, we will try to demystify the unexpected appearance of the imaginary number $i$ in the classical fermion Lagrangian as the quantum fingerprint left on the classical world.

It is worth noting that the real 6-D Clifford algebra has 7 different signatures: Clifford algebras $Cl(0,6)$, $Cl(3,3)$ and $Cl(4,2)$ are isomorphic to the real $8\times 8$ matrix $M(8,R)$ algebra. On the other hand, Clifford algebras $Cl(6,0)$, $Cl(5,1)$, $Cl(2,4)$ and $Cl(1,5)$ are isomorphic to the quaternion $4\times 4$ matrix $M(4,H)$ algebra. In the literature [74], the merits of various signatures of Clifford algebras have been discussed in connection with the standard model symmetries.

Given the isomorphisms noted above, the $Cl(0,6)$-based model can be faithfully translated to the $Cl(3,3)$, $Cl(4,2)$ or $M(8,R)$ format, which would be basically the equivalent model. For example, in terms of the $Cl(0,6)$ orthonormal vector basis $\{\Gamma_{1},\Gamma_{2},\Gamma_{3},\gamma_{1},\gamma_{2},\gamma_{3}\}$, the orthonormal vector bases of $Cl(3,3)$ and $Cl(4,2)$ are $\{\Gamma_{0}\Gamma_{1},\Gamma_{0}\Gamma_{2},\Gamma_{0}\Gamma_{3},\gamma_{1},\gamma_{2},\gamma_{3}\}$ and $\{\gamma_{1}\gamma_{0},\gamma_{2}\gamma_{0},\gamma_{3}\gamma_{0},\Gamma_{0},\Gamma_{1},\Gamma_{2}\}$ respectively, where $\gamma_{0}=\Gamma_{1}\Gamma_{2}\Gamma_{3}$ and $\Gamma_{0}=\gamma_{1}\gamma_{2}\gamma_{3}$. As such, the time-like $\gamma_{0}$ can be defined as the 6-vector (pseudoscalar) of $Cl(3,3)$, whereas $\gamma_{0}$ can be defined as the 4-vector (product of the first 4 vectors) of $Cl(4,2)$. The mapping between $Cl(0,6)$ and $M(8,R)$ can also be worked out, which will not be explored here in this paper since the physics essence of our model has already been elegantly captured by the purely Clifford algebraic formalism.