arXiv subjects
Publications and source records attributed to N. S. Mankoc Borstnik.
Proceedings for our meeting ``What comes beyond the Standard Models'', which covered a broad series of subjects.
Fifty years ago the {\it 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 an explanation. There are many proposals in the literature for the next step. The {\it spin-charge-family} theory, proposing a simple starting action in $ d\ge (13+1)$-dimensional space with fermions interacting with the gravity only (the vielbeins and the two kinds of the spin connection fields), is offering the explanation for not only all by the {\it standard model} assumed properties of quarks and leptons and antiquarks and antileptons, with the families included, of the vectors gauge fields, of the Higgs's scalar and Yukawa couplings, of the appearance of the {\it dark matter}, of the {\it matter-antimatter asymmetry}, making several predictions, but explains as well the second quantization postulates for fermions and bosons by using the odd and the even Clifford algebra ''basis vectors'' to describe the internal space of fermions and bosons, respectively. Consequently the single fermion and single boson states already anticommute and commute, respectively. I present in this talk a very short overview of the achievements of the {\it spin-charge-family} theory so far, concluding with presenting not yet solved problems, for which the collaborators are very welcome.
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 $γ^a$'s or $\tildeγ^a$'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 $θ^a$'s and $\frac{\partial}{\partial θ_a}$'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 $θ^{a}$'s and $\frac{\partial}{\partial θ_{a}}$'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 Dirac's 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 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 $γ^a$'s. Arranged into irreducible representations of "eigenvectors" of the Cartan subalgebra of the Lorentz algebra $S^{ab}$ $(= \frac{i}{2} γ^a γ^b|_{a \ne b})$ 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 $\tilde{S}^{ab}$ $=\frac{1}{2} \tildeγ^a \tildeγ^b|_{a \ne b}$. Creation operators, carrying the family member and family quantum numbers form the basic vectors. The action of the operators $γ^a$'s, $S^{ab}$, $\tildeγ^a$'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 $γ^a$ 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.
Fermions with the internal degrees of freedom described in Clifford space carry in any dimension a half integer spin. There are two kinds of spins in Clifford space. The spin-charge-family theory,assuming even d=13+1, uses one kind of spins to describe in d=3+1 spins and charges of quarks and leptons and antiquarks and antileptons, while the other kind is used to describe families. The new way of second quantization, suggested by the spin-charge-family theory, is presented. It is shown that the creation and annihilation operators of 1-fermion states, written as products of nilpotents and projectors of an odd Clifford character, fulfill the anticommutation relations as required in the second quantization procedure for fermions: 1-fermion states are in Clifford space already second quantized, the creation operators for any n-fermion second quantized vectors are products of one fermion creation operators, operating on the empty vacuum state. It is demonstrated that also in Grassmann space there exist the creation and annihilation operators of an odd Grassmann character, generating "fermions", which fulfill as well the anticommutation relations for fermions, representing correspondingly the second quantized 1-"fermion" states, in this case with integer spins. Grassmann space offers no families. We discuss the new second quantization procedure of the fields in both spaces. For the Grassmann case we present the action, basic states, solutions of the Weyl equation for free massless "fermions" and discrete symmetry operators. A short overview of the achievements of the spin-charge-family theory is done, and open problems of this theory still waiting to be solved are presented. The Grassmann and the Clifford case are compared in order to better understand open questions in physics of elementary fermion and boson fields and in cosmology.
In Ref. [arXiv:1802.05554v3] one of the authors (N.S.M.B.) studies the second quantization of fermions with integer spin while describing the internal degrees of freedom of fermions in Grassmann space. In this contribution we study the representations in Grassmann space of the groups $SO(5,1)$, $SO(3,1)$, $SU(3) \times U(1)$, and $SO(4)$, which are of particular interest as the subgroups of the group $SO(13,1)$. The second quantized integer spin fermions, appearing in Grassmann space, not observed so far, could be an alternative choice to the half integer spin fermions, appearing in Clifford space. The spin-charge-family theory, using two kinds of Clifford operators --- $γ^a$ and $\tildeγ^a$ --- for the description of spins and charges (first) and family quantum numbers (second), offers the explanation for not only the appearance of families but also for all the properties of quarks and leptons, the gauge fields, scalar fields and others. In both cases the gauge fields in $d \ge(13+1)$ --- the spin connections $ω_{ab α}$ (of the two kinds in Clifford case and of one kind in Grassmann case) and the vielbeins $f^α{}_α$ --- determine in $d=(3+1)$ scalars, those with the space index $α=(5,6,\cdots,d)$, and gauge fields, those with the space index $α=(0,1,2,3)$. While states of the Lorentz group and all its subgroups (in any dimension) are in Clifford space in the fundamental representations of the groups, with the family degrees of freedom included, states in Grassmann space manifest with respect to the Lorentz group adjoint representations, allowing no families.
The spin-charge-family theory predicts the existence of the fourth family to the observed three. The $4 \times 4$ mass matrices --- determined by the nonzero vacuum expectation values of the two triplet scalars, the gauge fields of the two groups of $\widetilde{SU}(2)$ determining family quantum numbers, and by the contributions of the dynamical fields of the two scalar triplets and the three scalar singlets with the family members quantum numbers ($τ^α=(Q, Q',Y')$) --- manifest the symmetry $\widetilde{SU}(2) \times \widetilde{SU}(2) \times U(1)$. All scalars carry the weak and the hyper charge of the standard model higgs field ($\pm \frac{1}{2},\mp \frac{1}{2}$, respectively). It is demonstrated, using the massless spinor basis, that the symmetry of the $4\times4$ mass matrices remains $SU(2) \times SU(2) \times U(1)$ in all loop corrections, and it is discussed under which conditions this symmetry is kept under all corrections, that is with the corrections induced by the repetition of the nonzero vacuum expectation values included.
We investigate bosonization/fermionization for free massless fermions being equivalent to free massless bosons with the purpose of checking and correcting the old rule by Aratyn and one of us (H.B.F.N.) for the number of boson species relative to the number of fermion species which is required to have bosonization possible. An important application of such a counting of degrees of freedom relation would be to invoke restrictions on the number of families that could be possible under the assumption, that all the fermions in nature are the result of fermionizing a system of boson species. Since a theory of fundamental fermions can be accused for not being properly local because of having anticommutativity at space like distances rather than commutation as is more physically reasonable to require, it is in fact called for to have all fermions arising from fermionization of bosons. To make a realistic scenario with the fermions all coming from fermionizing some bosons we should still have at least some not fermionized bosons and we are driven towards that being a gravitational field, that is not fermionized. Essentially we reach the spin-charge-families theory by one of us (N.S.M.B.) with the detail that the number of fermion components and therefore of families get determined from what possibilities for fermionization will finally turn out to exist. The spin-charge-family theory has long been plagued by predicting 4 families rather than the phenomenologically more favoured 3. Unfortunately we do not yet understand well enough the unphysical negative norm square components in the system of bosons that can fermionize in higher dimensions because we have no working high dimensional case of fermionization. But suspecting they involve gauge fields with complicated unphysical state systems the corrections from such states could putatively improve the family number prediction.
The contribution contains the preface to the Proceedings to the 19th Workshop "What Comes Beyond the Standard Models", Bled, July 11 - 19, 2016, published in Bled workshops in physics, Vol.17, No. 2, DMFA-Zaloznistvo, Ljubljana, Dec. 2016, links to (most of) the published contributions and section (by M.Yu. Khlopov) on VIA at Bled 2016.
It is shown that in the spin-charge-family theory, as well as in all the Kaluza-Klein like theories, vielbeins and spin connections manifest in $d=(3+1)$ space equivalent vector gauge fields, when space with $d\ge5$ manifests large enough symmetry. The authors demonstrate this equivalence in spaces with the symmetry of the metric tensor in the space out of $d=(3+1)$ - $g^{στ} = η^{στ} \,f^{2}$ - for any scalar function $f$ of the coordinates $x^σ$, where $x^σ$ denotes coordinates of space out of $d=(3+1)$. Also the connection between vielbeins and scalar gauge fields in $d=(3+1)$ (offering the explanation for the Higgs's scalar) is discussed.
The contribution contains the preface to the Proceedings to the 18th Workshop "What Comes Beyond the Standard Models", Bled, July 11 - 19, 2015, published in Bled workshops in physics, Vol.16, No. 2, DMFA-Zaloznistvo, Ljubljana, Dec. 2015, links to (most of) the published contributions and section (by M.Yu. Khlopov) on VIA at Bled 2015.
One purpose of this proceedings-contribution is to show that at least for free massless particles it is possible to construct an explicit boson theory which is exactly equivalent in terms of momenta and energy to a fermion theory. The fermions come as $2^{d/2-1}$ families and the to this whole system of fermions corresponding bosons come as a whole series of the Kalb-Ramond fields, one set of components for each number of indexes on the tensor fields. Since Kalb-Ramond fields naturally (only) couple to the extended objects or branes, we suspect that inclusion of interaction into such for a bosonization prepared system - except for the lowest dimensions - without including branes or something like that is not likely to be possible. The need for the families is easily seen just by using the theorem long ago put forward by Aratyn and one of us (H.B.F.N.), which says that to have the statistical mechanics of the fermion system and the boson system to match one needs to have the number of the field components in the ratio $\frac{2^{d-1}-1}{2^{d-1}}= \frac{\# bosons}{\# fermions}$, enforcing that the number of fermion components must be a multiple of $2^{d-1}$, where $d$ is the space-time dimension. This "explanation" of the number of dimension is potentially useful for the explanation for the number of dimension put forward by one of us (S.N.M.B.) since long in the spin-charge-family theory, and leads like the latter to typically (a multiple of) $4$ families. And this is the second purpose for our work on the fermionization in an arbitrary number of dimensions - namely to learn how "natural" is the inclusion of the families in the way the spin-charge-family theory does.
This is a discussion on degrees of freedom of massless fermion and boson fields, if they are free or weakly interacting. We generalize the gauge fields of $S^{ab}$ - $ω_{abc}$ - and of $\tilde{S}^{ab}$ - $ \tildeω_{abc}$ - of the spin-charge-family to the gauge fields of all possible products of $γ^a$'s and of all possible products of $\tildeγ^a$'s, the first taking care in the {\it spin-charge-family} theory of the spins and charges quantum numbers ($τ^{Ai}=\sum_{a,b} c^{Ai}{}_{ab} \,S^{ab}$) of fermions, the second ($\tildeτ^{Ai}= \sum_{a,b} \tilde{c}^{Ai}{}_{ab}\, \tilde{S}^{ab}$) taking care of the families quantum numbers.
The spin-charge-family theory predicts before the electroweak break four - rather than the observed three - massless families of quarks and leptons. The 4 x 4 mass matrices of all the family members demonstrate in this theory the same symmetry, which is determined by the scalar fields: the two SU(2) triplets (the gauge fields of the family groups) and the three singlets, the gauge fields of the three charges (Q, Q' and Y') distinguishing among family members. All the scalars have, with respect to the weak and the hyper charge, the quantum numbers of the {\it standard model} scalar Higgs: $\pm \frac{1}{2}$ and $ \mp \frac{1}{2}$, respectively. Respecting by the spin-charge-family theory proposed symmetry of mass matrices and assuming (due to not yet accurate enough experimental data) that the mass matrices are hermitian and real, we fit the six free parameters of each family member mass matrix to the experimental data of twice three measured masses of quarks and to the measured quark mixing matrix elements, within the experimental accuracy. Since any 3 x 3 submatrix of the 4 x 4 unitary matrix determines the whole 4 x 4 matrix uniquely, we are able to predict the properties of the fourth family members provided that the experimental data are enough accurate, which is not yet the case. We, however, found out that the new experimental data for quarks fit better to the required symmetry of mass matrices than the old data and we predict towards which value will more accurately measured matrix elements move. The present accuracy of the experimental data for leptons does not enable us to make sensible predictions.
The contribution contains the preface to the Proceedings to the 17th Workshop "What Comes Beyond the Standard Models", Bled, July 20 - 28, 2014, published in Bled workshops in physics, Vol.15, No. 2, DMFA-Zaloznistvo, Ljubljana, Dec. 2014, links to (most of) the published contributions and section (by M.Yu. Khlopov) on VIA at Bled 2014.
We studied properties of spinors in a toy model in $d=(5+1)$, when ${\cal M}^{(5+1)}$ breaks to an infinite disc with a zweibein which makes a disc curved on an almost $S^2$ and with a spin connection field which allows on such a sphere only one massless spinor state, as a step towards realistic Kaluza-Klein theories in non compact spaces. Previously we allowed on $S^2$ two kinds of the spin connection fields, those which are gauge fields of spins in and those which are the gauge fields of the family quantum numbers, both as required for this toy model by the spin-charge-family theory. This time we study, by taking into account families of spinors interacting with several spin connection fields, properties of massless and massive solutions of equations of motion, with the discrete symmetries ($\mathbb{C}_{ \cal N}$, ${\cal P}_{\cal N}$, ${\cal T}_{ \cal N}$) included. We also allow nonzero vacuum expectation values of the spin connection fields and study the masses.
The technique for representing spinors and the definition of the discrete symmetries is used to illustrate on a toy model properties of massless and massive spinors states, in the first and the second quantized picture. Since in this toy model the number of the starting massless representations is well defined as well as the origin of masses and charges in $d=(3+1)$ space, this contribution might help to clarify the problem about Dirac, Weyl and Majorana kinds of representations in physically more interesting cases.