The Standard Model particle content with complete gauge symmetries from the minimal ideals of two Clifford algebras

Building upon previous works, it is shown that two minimal left ideals of the complex Clifford algebra $\mathbb{C}\ell(6)$ and two minimal right ideals of $\mathbb{C}\ell(4)$ transform as one generation of leptons and quarks under the gauge symmetry $SU(3)_C\times U(1)_{EM}$ and $SU(2)_L\times U(1)_Y$ respectively. The $SU(2)_L$ weak symmetries are naturally chiral. Combining the $\mathbb{C}\ell(6)$ and $\mathbb{C}\ell(4)$ ideals, all the gauge symmetries of the Standard Model, together with its lepton and quark content for a single generation are represented, with the dimensions of the minimal ideals dictating the number of distinct physical states. The combined ideals can be written as minimal left ideals of $\mathbb{C}\ell(6)\otimes\mathbb{C}\ell(4)\cong \mathbb{C}\ell(10)$ in a way that preserves individually the $\mathbb{C}\ell(6)$ structure and $\mathbb{C}\ell(4)$ structure of physical states. This resulting model includes many of the attractive features of the Georgi and Glashow $SU(5)$ grand unified theory without introducing proton decay or other unobserved processes. Such processes are naturally excluded because they do not individually preserve the $\mathbb{C}\ell(6)$ and $\mathbb{C}\ell(4)$ minimal ideals.


Introduction
The Standard Model (SM) is currently the best model of particle physics. It provides a hugely predictive mathematical description of the most basic constituents of matter observed in nature, and their interactions via three of the four fundamental forces of nature. However, despite being able to use the theory to make remarkably accurate predictions, the theoretical origin for the mathematical structure of the model remains unexplained. Specifically, why is the internal symmetry group of the SM what it is, when infinitely many other possibilities exist, and why do only some of the representations of the gauge groups correspond to physical states? Furthermore, why does the electroweak force depend on the chirality of physical states? Among others, these remain important open questions.
Grand Unified Theories (GUTs) merge the gauge groups of the SM into a single semi-simple Lie group. However, such GUTs, including the famous Georgi and Glashow's SU (5) model and Georgi's Spin(10) model [1], invariably predict additional gauge bosons, interactions, and proton decay, none of which have thus far been observed.
In 2016, the basis states of the minimal left ideals of the Clifford algebra Cℓ (6) were shown to transform precisely as a single generation of leptons and quarks under the electrocolor group SU (3) C ⊗U (1) EM [2]. The Clifford algebra Cℓ (6) there is generated from the left adjoint actions of the complex octonions C ⊗ O on itself 1 . A Witt decomposition of Cℓ(6) splits the algebra into a basis of nilpotent ladder operators, and the unitary symmetries that preserve this Witt decomposition are SU (3) and U (1). These symmetries act on the minimal left ideals of the algebra, with each basis state of the ideal transforming like a lepton or quark under the unbroken gauge symmetry SU (3) C × U (1) EM .
Similarly, considering the left and right adjoint actions of the complex quaternions C ⊗ H each generate a distinct copy of Cℓ(2), and allows one to describe the spin and chirality of leptons and quarks respectively. Combined, the left and right actions give a representation of the Dirac algebra Cℓ(4) [2,7].
In [7] it is shown that the individual Cℓ(6) and Cℓ(4) results can be combined, and that it is possible to represent one generation of chiral fermions with the SM gauge symmetries in terms of Cℓ(10) minimal ideals. However, this particular representation of a generation of fermions as a minimal left ideal of Cℓ(10) does not preserve the individual Cℓ(6) and Cℓ(4) ideal structures associated with specific leptons and quarks. This is the direct result of how the SU (2) L generators are defined, which do not account for the fact that the two particles in a weak doublet are elements of different Cℓ(6) minimal left ideals. The Cℓ(6) minimal left ideals are invisible to the SU (2) L generators meaning that the two particles making up a weak doublet must have the same Cℓ(6) structure, and differ only in their Cℓ(4) structure. This paper shows that by suitably redefining the SU (2) L generators in such a way that they simultaneously induce transformations in the Cℓ(4) minimal 1 The application of octonions to quark symmetries goes back to the 1970 [3]. More recently division algebras have received some renewed interest. The results of [2] were extended to three generations recently by going beyond the octonions, and considering the next Cayley-dickson algebra of sedenions [4]. Dixon [5] showed that the composition algebra R ⊗ C ⊗ H ⊗ O plays a key role in the architecture of the SM. A gravitational theory based on a R ⊗ C ⊗ H ⊗ O metric has been constructed in [6]. right ideals as well as map between different Cℓ(6) minimal left ideals, it is possible to to embed all the physical states into minimal left ideals of Cℓ(10) in a particularly aesthetic way that preserves both the individual Cℓ(6) minimal left ideal and Cℓ(4) minimal right ideal basis states associated with leptons and quarks. The unitary symmetries that preserve a Witt decomposition of this larger algebra generate SU (5). However, not all of the SU (5) transformations individually preserve the Cℓ(6) minimal left ideals and Cℓ(4) minimal right ideals. Although such transformations would not be excluded if one starts with Cℓ(10) as the fundamental background structure, as one does in the SU (5) and Spin(10) GUTs, they are naturally excluded here. It is these transformations that correspond to unobserved processes, including proton decay. In the model presented here, such unphysical transformations are therefore algebraically excluded.
Besides giving a particularly aesthetic embedding of one generations of fermions into minimal ideals of Cℓ(10), our model also provides a clear pathway for extending earlier work [8] which established a curious structural similarity between the basis states of the minimal left ideals of Cℓ(6) and certain braids used in a topological model of leptons and quarks [9]. Incorporating the results of the present paper to extend these earlier results to include the weak interaction will be the focus of a future paper.
2 Electrocolor symmetries for one generations of fermions from Cℓ (6) In [2] it was shown that a Witt decomposition of the Cℓ(6) decomposes the algebra into minimal left ideals whose basis states transform as a single generation of leptons and quarks under the unbroken electrocolor symmetry SU (3) C × U (1) EM . Such a Witt basis of Cℓ(6) can be defined as 2 satisfying the anticommutator algebra of fermionic ladder operators The α i and α † i are nilpotents and each set {α i } and α † i generates a maximal totally isotropic subspace of dimension eight. One can then construct the where ν,d r etc. are suggestively labeled complex coefficients denoting the isospin-up elementary fermions. The conjugate system analogously gives a second linearly independent minimal left ideal of isospin-down elementary fermions S d ≡ Cℓ(6)ω † ω. The representations of the minimal ideals are invariant under the electrocolor symmetry SU (3) C × U (1) EM , whose generators are constructed from the bivectors of the algebra, with each basis state in the ideals transforming as a specific lepton or quark as indicated by their suggestively labeled complex coefficients. That is, the unitary spin transformations that preserve the Witt decomposition are given by In terms of the Witt basis ladder operators, the SU (3) C generators take the form The U (1) EM generator, proportional to the number operator, can be expressed in terms of the Witt basis ladder operators as and gives the electric charge of fermions.
As an illustrative example we consider [Λ 1 , u g ]: 3 Weak symmetries for one generation of fermions from Cℓ (4) 3.1 Towards the weak force So far we have considered two of the full set of eight minimal left ideals of Cℓ(6). One ideal, S u , consists of isospin-up states, and the other, S d consists of isospin-down states. The unitary symmetries SU (3) C and U (1) EM facilitate transitions between states within an ideal. One also notices that multiplying the ideal S u on the right by ω changes the ideal into S d . The same is true for multiplying S d on the right by ω † . Notice that this only works via rightmultiplications, not via left-multiplication. Together, ω and ω † generate a copy of Cℓ (2), which is a subalgebra of Cℓ (6). The SU (2) ω generators are written as follows where it needs to be remembered that this SU (2) ω acts from the right so that [ This SU (2) ω symmetry effects transitions between states of different ideals. In particular, note that the action of this SU (2) ω onto a state changes its electric charge by plus or minus one.
In itself, Cℓ(2) ∼ = SU (2) generated by ω and ω † is not sufficient to describe weak interactions. This is because SU (2) ω does not correspond to a unitary symmetry that preserves the Witt decomposition of Cℓ(6) into minimal ideals. Nonetheless, this Cℓ(2) plays an important role in weak interactions since a weak doublet contains two particles that belong to different Cℓ(6) minimal left ideals and differ in their electric charge by one. Therefore, to describe the SU (2) L weak symmetry, we want to include an additional Clifford algebra, whose Witt decomposition is preserved by SU (2). This algebra is Cℓ(4).
Following the construction of minimal left ideals of Cℓ(6), we here construct the minimal right ideals of Cℓ(4). First define the nilpotents Ω = β 2 β 1 and Ω † = β † 1 β † 2 , from which one constructs the idempotents ΩΩ † and Ω † Ω. Two minimal right ideals are then given by ΩΩ † Cℓ (4) and Ω † ΩCℓ(4), which are four complex-dimensions. Explicitly the ideals are spanned by the states The SU (2) and U (1) unitary symmetries that preserve the ladder operator basis of Cℓ(4) are generated by and Under the action of the SU (2) generators the states Ω † Ω and Ω † Ωβ † 1 β † 2 transform as singlets whereas the states Ω † Ωβ † 1 and Ω † Ωβ † 2 transform into each other as a doublet. The singlet states may then be identified with right handed fermions, and the doublet states with left-handed fermions. Similarly, the states ΩΩ † and ΩΩ † β 1 β 2 are identified with left-handed anti-fermions, whereas ΩΩ † β 1 and ΩΩ † β 2 with right-handed anti-fermions.
Subsequently, we can write two Cℓ(4) minimal right ideals, one for leptons and one for anti-leptons, as where in both cases the coefficients are complex numbers indicating the particle like which the basis states transforms.
As an example, consider [T 1 , ν L ] : Similarly, for the quarks and anti-quarks, the Cℓ(4) minimal right ideals can be written as where the superscripts refers to the color of the quarks. Presently, as Cℓ(4) minimal right ideals, the quark ideals and lepton ideals are identical. We will be able to distinguish between leptons and quarks once we include the Cℓ(6) minimal left ideals.

Combining Cℓ(6) electrocolor and Cℓ(4) weak states
We now combine the earlier results based on Cℓ(6) electrocolor states with the Cℓ(4) weak states of the preceding section. We assume for simplicity here that the two algebras commute so that α i β j = β j α i . Everything that follows still works, with some minor modifications, when α i and β j anticommute. The neutrino ν is represented by the Cℓ(6) minimal left ideal basis state ωω † . Via the Cℓ(4) right ideals of the previous section, we can now include chirality. We then have Similarly, the neutrino's weak doublet partner, the electron e − in its left-and right-handed form can now be written as Notice that the neutrino and electron live in different Cℓ(6) minimal left ideals, but in the same Cℓ(4) minimal right ideal. The red up quark u r with electrocolor symmetry was previously identified with the Cℓ(6) state α † 3 α † 2 ωω † . Via the Cℓ(4) right ideals of the previous section, we can now include chirality. We then have Subsequently, the red down quark d r becomes Again, the red down quark and red up quark belong to different Cℓ(6) ideals but the same Cℓ(4) ideal. The antiparticles live in the conjugate Cℓ(4) minimal right ideal. Hence, for example In summary, the eight weak-doublets are identified as All of the other physical states are weak singlets Now that we can write down chiral fermions in terms of Cℓ (6) and Cℓ(4) minimal ideals, we must next find appropriate SU (2) generators so that the states transform correctly via the weak symmetry SU (2) L . Consider the weak doublet consisting of a left handed neutrino and left handed electron. In terms of the ideals we have: To transform the neutrino into the electron requires not only that β † 1 is transformed into β † 2 via T i in equation (11), but also that the Cℓ(6) ideal, and electric charge are changed. The latter two transformations are mediated by the SU (2) ω generators (8). What is required in the present case then is a combination of the generators (11) and (8) 3 . After some deliberation, the suitable SU (2) L generators can be chosen as As a first example, consider the action of T ′ 1 on the left-handed electron ν L Furthermore, So that − 1 2 T ′ 3 returns the correct weak isospin of physical states. As a second example consider an anti-quark weak doublet. Then, 6 Including spin degrees of freedom In the same way that ω and ω † via right multiplication transform between the two different Cℓ(6) minimal left ideals S u and S d (with ω and ω † generating a Cℓ(2) ∼ = SU (2) subalgebra, so left multiplication by Ω and Ω † facilitates transformations between the two Cℓ(4) minimal right ideals L andL. Subsequently a copy of SU (2) Ω can be generated by Multiplying, for example, a right-handed anti-down quark on the left by Ω † gives Thus far, α † i ωω † Ω † Ωβ † 1 has not yet been identified with a physical state. Considering the action of t 3 ond (3) and α † i ωω † Ω † Ωβ † 1 we find that If we then defineŜ z = 1 2 t 3 we can interpret α † i ωω † ΩΩ † β 2 as a spin-up anti-down quark and α † i ωω † Ω † Ωβ † 1 as a spin-down anti-down quark The same can be done for the other physical states.
7 Cℓ(10) and SU (5) GUT Combining the Cℓ(6) minimal left ideals and Cℓ(4) minimal right ideals, it is possible to rewrite physical states as basis states of the minimal left ideals of Cℓ(10), as is done in [7]. Given our different SU (2) L generators however means that in the present case the physical states will be represented differently. For example, consider the left handed electron (ignoring spin which is absent in [7]). In [7] this state is represented as e − L = β † 2 ω † ωΩΩ † , where the notation has been adopted to be consistent with the present paper. One sees, due to the lack of α 1 α 2 α 3 that the original Cℓ(6) representation of the electron is not preserved in this construction. Consequently, Q no longer gives the electric charge in this Cℓ(10) scheme. On the other hand, our scheme identifies the left-handed electron as e − L = α 1 α 2 α 3 ω † ωΩ † Ωβ † 2 , which may be rewritten as a Cℓ(10) element as e − L = α 1 α 2 α 3 β † 2 ω † ωΩΩ † . This representation preserved both the electrocolor structure α 1 α 2 α 3 from Cℓ(6) and the weak structure β † 2 from Cℓ(4). Subsequently Q still gives the correct electric charge of the electron.
The unitary symmetry that preserves a Witt decomposition of Cℓ(10) algebra is SU (5), the basis of the Georgi and Glashow GUT. This GUT predicts additional gauge bosons and associated unobserved physical processes, most famously proton decay. However, in the present case, physical states belong simultaneously to a minimal left ideal of Cℓ(6) and a minimal right ideal of Cℓ (4). Although physical states can be rewritten as basis elements of minimal left ideals of Cℓ(10) they do not span these minimal left ideals. Similarly, the SU (3) × U (1) symmetry that preserves the Cℓ(6) Witt decomposition and SU (2) × U (1) symmetry that preserves the Cℓ(4) Witt decomposition also preserve a Witt decomposition of Cℓ (10). The converse is however not true as not all SU (5) unitary symmetries that preserve the Witt decomposition of Cℓ(10) individually preserve the Witt decompositions of Cℓ (6) and Cℓ(4). It is those unitary symmetries of SU (5) that correspond to unobserved physical processes, and these are therefore algebraically excluded in our construction.

Discussion
Two minimal left ideals of Cℓ (6), and two minimal right ideals of Cℓ(4) were shown to transform as a generation of fermions under the unbroken electrocolor group SU (3) C ×U (1) EM , and the weak group SU (2) L respectively. Combining the Cℓ (6) and Cℓ(4) ideal basis states, the leptons and quarks for a single generation transforming correctly under the SM gauge group can be represented, including the chirality and spin of the particles. These combined Cℓ(6) and Cℓ(4) ideal basis states can be embedded into a minimal left ideal of Cℓ(10) in such a way that preserves individually the Cℓ(6) structure and Cℓ(4) structure of physical states. This is an improvement over an earlier model [7] where the individual Cℓ(6) structure and Cℓ(4) structure of physical states is not preserved. This improvement is achieved through redefining the SU (2) L to include factors of ω and ω † in appropriate places to ensure that a transformation within a Cℓ(4) minimal right ideal simultaneously maps between different Cℓ(6) minimal left ideals.
It is important to notice the conceptual difference between the SU (2) generated from (8) and the SU (2) generated from (11). In the former case, ω and ω † span a Cℓ(2) subalgebra of Cℓ (6). This algebra is isomorphic to SU (2) and the generators mediate transitions between the Cℓ(6) ideals S u and S d . This SU (2) symmetry is not a gauge symmetry as it does not transform between states within the same ideal, but rather between states of different ideals. Consequently there are no gauge bosons associated with this SU (2) spin symmetry. In the latter case, the SU (2) generators are constructed from the bivectors of Cℓ(4). This copy of SU (2) contains the symmetries that preserves the Witt decomposition of Cℓ(4).
In our model, unitary symmetries generated from combinations of Cℓ(6) and Cℓ(4) ladder operators, such as α i β † j + β j α † , are naturally excluded because they do not individually preserve the Witt decompositions of Cℓ(6) and Cℓ(4) ideals 4 . It is precisely these generators that are responsible for the unobserved particle processes, including proton decay, in the Georgi-Glashow SU (5) GUT. The present model therefore naturally excludes all of these unobserved processes.
The motivation for this work came from the results of [8] which showed that there is a one-to-one correspondence between the basis states of the minimal left ideals of Cℓ(6) and the braided states in the topological model proposed in [9] 5 . To be able to extend this curious structural similarity to include the weak force, the underlying electrocolor structure generated from Cℓ(6) should remain intact. This is not the case in the Cℓ(10) representation in [7], but is the case for the construction considered here. The extension of [8] to include the weak force will be the focus of an upcoming paper.