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Register a new userHow does Clifford algebra show the way to the second quantized fermions with unified spins, charges and families, and with vector and scalar gauge fields beyond the {it standard model}
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Added by
Norma Susana Mankoc Borstnik
Publication date
2021
fields
Physics
and research's language is
English
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Fifty years ago the standard model offered an elegant new step towards understanding elementary fermion and boson fields, making several assumptions, suggested by experiments. The assumptions are still waiting for explanations. There are many proposals in the literature for the next step. The spin-charge-family theory of one of us (N.S.M.B.) is offering the explanation for not only all by the standard model assumed properties of quarks and leptons and antiquarks and antileptons, with the families included, of the vectors gauge fields, of the Higgss scalar and Yukawa couplings, but also for the second quantization postulates of Dirac and for cosmological observations, like there are the appearance of the dark matter, of matter-antimatter asymmetry, making several predictions. This theory proposes a simple starting action in d=(13+1)-dimensional space with fermions interacting with the gravity only, what manifests in d=(3+1) as the vector and scalar gauge fields, and uses the odd Clifford algebra to describe the internal space of fermions, what enables that the creation and annihilation operators for fermions fulfill the anticommutation relations for the second quantized fields without Diracs postulates: Fermions single particle states already anticommute. We present in this review article a short overview of the spin-charge-family theory, illustrating shortly on the toy model the breaks of the starting symmetries in d=(13+1)-dimensional space, which are triggered either by scalar fields - the vielbeins with the space index belonging to d>(3+1) - or by the condensate of the two right handed neutrinos, with the family quantum number not belonging to the observed families. We compare properties and predictions of this theory with the properties and predictions of SO(10) unifying theories.
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The internal degrees of freedom of fermions are in the spin-charge-family theory described by the Clifford algebra objects, which are superposition of an odd number of $gamma^a$s. Arranged into irreducible representations of eigenvectors of the Cartan subalgebra of the Lorentz algebra $S^{ab}$ $(= frac{i}{2} gamma^a gamma^b|_{a e b})$ these objects form $2^{frac{d}{2}-1}$ families with $2^{frac{d}{2}-1}$ family members each. Family members of each family offer the description of all the observed quarks and leptons and antiquarks and antileptons, appearing in families. Families are reachable by $tilde{S}^{ab}$ $=frac{1}{2} tilde{gamma}^a tilde{gamma}^b|_{a e b}$. Creation operators, carrying the family member and family quantum numbers form the basic vectors. The action of the operators $gamma^a$s, $S^{ab}$, $tilde{gamma}^a$s and $tilde{S}^{ab}$, applying on the basic vectors, manifests as matrices. In this paper the basic vectors in $d=(3+1)$ Clifford space are discussed, chosen in a way that the matrix representations of $gamma^a$ and of $S^{ab}$ coincide for each family quantum number, determined by $tilde{S}^{ab} $, with the Dirac matrices. The appearance of charges in Clifford space is discussed by embedding $d=(3+1)$ space into $d=(5+1)$-dimensional space.
We present in Part II the description of the internal degrees of freedom of fermions by the superposition of odd products of the Clifford algebra elements, either $gamma^a$s or $tilde{gamma}^a$s, which determine with their oddness the anticommuting properties of the creation and annihilation operators of the second quantized fermion fields in even $d$-dimensional space-time, as we do in Part I of this paper by the Grassmann algebra elements $theta^a$s and $frac{partial}{partial theta_a}$s. We discuss: {bf i.} The properties of the two kinds of the odd Clifford algebras, forming two independent spaces, both expressible with the Grassmann algebra of $theta^{a}$s and $frac{partial}{partial theta_{a}}$s. {bf ii.} The freezing out procedure of one of the two kinds of the odd Clifford objects, enabling that the remaining Clifford objects determine with their oddness in the tensor products of the finite number of the Clifford basis vectors and the infinite number of momentum basis, the creation and annihilation operators carrying the family quantum numbers and fulfilling the anticommutation relations of the second quantized fermions: on the vacuum state, and on the whole Hilbert space defined by the sum of infinite number of Slater determinants of empty and occupied single fermion states. {bf iii.} The relation between the second quantized fermions as postulated by Dirac and the ones following from our Clifford algebra creation and annihilation operators, what offers the explanation for the Dirac postulates.
Both algebras, Clifford and Grassmann, offer basis vectors for describing the internal degrees of freedom of fermions. The oddness of the basis vectors, transferred to the creation operators, which are tensor products of the finite number of basis vectors and the infinite number of momentum basis, and to their Hermitian conjugated partners annihilation operators, offers the second quantization of fermions without postulating the conditions proposed by Dirac, enabling the explanation of the Diracs postulates. But while the Clifford fermions manifest the half integer spins -- in agreement with the observed properties of quarks and leptons and antiquarks and antileptons -- the Grassmann fermions manifest the integer spins. In Part I properties of the creation and annihilation operators of integer spins Grassmann fermions are presented and the proposed equations of motion solved. The anticommutation relations of second quantized integer spin fermions are shown when applying on the vacuum state as well as when applying on the Hilbert space of the infinite number of Slater determinants with all the possibilities of empty and occupied fermion states. In Part II the conditions are discussed under which the Clifford algebras offer the appearance of the second quantized fermions, enabling as well the appearance of families. In both parts, Part I and Part II, the relation between the Dirac way and our way of the second quantization of fermions is presented.
The tremendous phenomenological success of the Standard Model (SM) suggests that its flavor structure and gauge interactions may not be arbitrary but should have a fundamental first-principle explanation. In this work, we explore how the basic distinctive properties of the SM dynamically emerge from a unified New Physics framework tying together both flavour physics and Grand Unified Theory (GUT) concepts. This framework is suggested by the gauge Left-Right-Color-Family Grand Unification under the exceptional $mathrm{E}_8$ symmetry that, via an orbifolding mechanism, yields a supersymmetric chiral GUT containing the SM. Among the most appealing emergent properties of this theory is the Higgs-matter unification with a highly-constrained massless chiral sector featuring two universal Yukawa couplings close to the GUT scale. At the electroweak scale, the minimal SM-like effective field theory limit of this GUT represents a specific flavored three-Higgs doublet model consistent with the observed large hierarchies in the quark mass spectra and mixing already at tree level.
We discuss gauge coupling unification in models with additional 1 to 4 complete vector-like families, and derive simple rules for masses of vector-like fermions required for exact gauge coupling unification. These mass rules and the classification scheme are generalized to an arbitrary extension of the standard model. We focus on scenarios with 3 or more vector-like families in which the values of gauge couplings at the electroweak scale are highly insensitive to the grand unification scale, the unified gauge coupling, and the masses of vector-like fermions. Their observed values can be mostly understood from infrared fixed point behavior. With respect to sensitivity to fundamental parameters, the model with 3 extra vector-like families stands out. It requires vector-like fermions with masses of order 1 TeV - 100 TeV, and thus at least part of the spectrum may be within the reach of the LHC. The constraints on proton lifetime can be easily satisfied in these models since the best motivated grand unification scale is at $sim 10^{16}$ GeV. The Higgs quartic coupling remains positive all the way to the grand unification scale, and thus the electroweak minimum of the Higgs potential is stable.
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