Search Results (11)

This review devoted to the centenary of Alexander Friedmann’s prediction of the Universe expansion presents the results obtained by him in 1922 and 1924 and an overview of their further developments. Special attention is paid to the role of mathematics, which enabled Friedmann to perform a radical departure from the conventional practice of considering our universe as a static system. The effect of particle creation in the expanding universe is discussed concurrently with the earlier investigated phenomenon of pair creation from a vacuum by an external electric field. The numbers of scalar and spinor particles created at different stages of the Universe’s evolution are presented, and the possible role of the effect of the creation of particles in the formation of relativistic plasma and cold dark matter after the inflationary period is noted. It is stressed that by introducing the concept of the expanding universe, Friedmann made a contribution towards the understanding of the world around us that is compatible with those made by Ptolemy, Copernicus, and Newton in previous epochs. Full article

A new approach to the Dirac equation and the associated hadronic symmetries is proposed. In this approach, we linearize the second Casimir operator of the Lorentz Group, which is defined by the energy–momentum four-vector and the fermion spin, thereby using the spinor-helicity representation instead of the three-vector representation of the particle momentum and spin vector. We then expand the so-obtained standard Dirac equation by employing an inner abstract “hadronic” isospin, initially describing a $SU\left(2\right)$ fermion doublet. Application of the spin-helicity representation of that isospin leads to the occurrence of a quadruplet of inner states, revealing the $SU\left(4\right)$ symmetry via the isospin helicity operator. This further leads to two independent fermion state spaces, specifically, singlet and triplet states, which we interpret as $U\left(1\right)$ symmetry of the leptons and $SU\left(3\right)$ symmetry of the three quarks, respectively. These results indicate the genuinely very different physical nature of the strong $SU\left(4\right)$ symmetry in comparison to the chiral $SU\left(2\right)$ symmetry. While our approach does not require the a priori concept of grand unification, such a notion arises naturally from the formulation with the isospin helicity. We then apply the powerful procedures developed for the electroweak interactions in the SM, in order to break the $SU\left(4\right)$ symmetry by means of the Higgs mechanism involving a scalar Higgs field as an $SU\left(4\right)$ quadruplet. Its finite vacuum creates the masses of the three vector bosons involved, which can change the three quarks into a lepton and vice versa. Finally, we consider a toy model for calculation of the strong coupling constant of a Yukawa potential. Full article

This paper surveys recent progress in our search for an appropriate internal space algebra for the standard model (SM) of particle physics. After a brief review of the existing approaches, we start with the Clifford algebras involving operators of left multiplication by octonions. A central role is played by a distinguished complex structure that implements the splitting of the octonions $\mathbb{O}=\mathbb{C}\oplus {\mathbb{C}}^3$, which reflect the lepton-quark symmetry. Such a complex structure on the 32-dimensional space $\mathcal{S}$ of $C{\ell }_{10}$ Majorana spinors is generated by the $C{\ell }_6(\subset C{\ell }_{10})$ volume form, ${\omega }_6={\gamma }_1\cdots {\gamma }_6$, and is left invariant by the Pati–Salam subgroup of $Spin\left(10\right)$, $G_{\mathrm{PS}}=Spin\left(4\right)\times Spin\left(6\right)/{\mathbb{Z}}_2$. While the $Spin\left(10\right)$ invariant volume form ${\omega }_{10}={\gamma }_1\dots {\gamma }_{10}$ of $C{\ell }_{10}$ is known to split $\mathcal{S}$ on a complex basis into left and right chiral (semi)spinors, $\mathcal{P}=\frac{1}{2}(1-i{\omega }_6)$ is interpreted as the projector on the 16-dimensional particle subspace (which annihilates the antiparticles).The standard model gauge group appears as the subgroup of $G_{\mathrm{PS}}$ that preserves the sterile neutrino (which is identified with the Fock vacuum). The ${\mathbb{Z}}_2$-graded internal space algebra $\mathcal{A}$ is then included in the projected tensor product $\mathcal{A}\subset \mathcal{P}C{\ell }_{10}\mathcal{P}=C{\ell }_4\otimes C{\ell }_6^0$. The Higgs field appears as the scalar term of a superconnection, an element of the odd part $C{\ell }_4^1$ of the first factor. The fact that the projection of $C{\ell }_{10}$ only involves the even part $C{\ell }_6^0$ of the second factor guarantees that the color symmetry remains unbroken. As an application, we express the ratio $\frac{m_H}{m_W}$ of the Higgs to the W boson masses in terms of the cosine of the theoretical Weinberg angle. Full article

A model of nonperturbative vacuum in SU(2) Yang–Mills theory coupled to a nonlinear spinor field is suggested. By analogy with Abelian magnetic monopole dominance in quantum chromodynamics, it is assumed that the dominant contribution to such a vacuum comes from dipole-like field configurations existing in this theory. Using an assumption of the behavior of the number density of dipole-like field configurations whose energy approaches infinity, we derive an approximate expression for the energy density of such nonperturbative vacuum symmetrical under translation that turns out to be finite, unlike the infinite energy density of perturbative vacuum. Using characteristic values of the parameters appearing in the expression for the nonperturbative energy density, it is shown that this density may be of the order of the energy density associated with Einstein’s cosmological constant. The physical interpretation of the spinor field self-coupling constant as a characteristic distance between dipole-like field configurations is suggested. The questions of experimental verification of the nonperturbative vacuum model under consideration and of determining its pressure are briefly discussed. Full article

The theoretical foundations of relativistic electronic structure theory within quantum electrodynamics (QED) and the computational basis of the atomic structure code GRASP are briefly surveyed. A class of four-component basis set is introduced, which we denote the CKG-spinor set, that enforces the charge-conjugation symmetry of the Dirac equation. This formalism has been implemented using the Gaussian function technology that is routinely used in computational quantum chemistry, including in our relativistic molecular structure code, BERTHA. We demonstrate that, unlike the kinetically matched two-component basis sets that are widely employed in relativistic quantum chemistry, the CKG-spinor basis is able to reproduce the well-known eigenvalue spectrum of point-nuclear hydrogenic systems to high accuracy for all atomic symmetry types. Calculations are reported of third- and higher-order vacuum polarization effects in hydrogenic systems using the CKG-spinor set. These results reveal that Gaussian basis set expansions are able to calculate accurately these QED effects without recourse to the apparatus of regularization and in agreement with existing methods. An approach to the evaluation of the electron self-energy is outlined that extends our earlier work using partial-wave expansions in QED. Combined with the treatment of vacuum polarization effects described in this article, these basis set methods suggest the development of a comprehensive ab initio approach to the calculation of radiative and QED effects in future versions of the GRASP code. Full article

We have developed a quantum field theory of spinors based on the algebra of canonical anticommutation relations (CAR algebra) of Grassmann densities in the momentum space. We have proven the existence of two spinor vacua. Operators C and T transform the normal vacuum into an alternative one, which leads to the breaking of the C and T symmetries. The CPT is the real structure operator; it preserves the normal vacuum. We have proven that, in the theory of the Dirac Sea, the formula for the charge conjugation operator must contain an additional generalized Dirac conjugation operator. Full article

Monopole solutions in SU(2) Yang–Mills theory, which interact with massive nonlinear spinor fields, described by the nonlinear Dirac equation, are obtained. These solutions describe a magnetic monopole created by a spherical lump of nonlinear spinor fields. It is shown that the monopole solutions obtained differ in principle from the ‘t Hooft–Polyakov monopole so that (a) it is topologically trivial; (b) the radial magnetic field decreases as $r^{-3}$; (c) the Higgs field is not necessary for its existence. It is demonstrated that the energy spectrum of such a system possesses a global minimum, the appearance of which is due exclusively to the nonlinearity of the Dirac spinor fields. This global minimum can be considered a mass gap, i.e., the energy difference between a vacuum and the next lowest energy state. A similar minimum was found for the energy spectrum of regular solutions to the nonlinear Dirac equation and this minimum is called “the lightest stable particle”. Full article

We have developed a quantum field theory of spinors based on the algebra of canonical anticommutation relations (CAR algebra). The proposed approach is based on the use of Grassmann densities in the momentum space and their derivatives with respect to the construction from these densities of both basis Clifford vectors of spacetime and the spinor vacuum. We have shown the existence of two vacua: normal and alternative. We have proven that CPT is the real structure operator in the theory of Krein spaces. C and T operators transform a normal vacuum into an alternative one, which leads to the breaking of these symmetries. Full article

In this work, with the help of the quantum hydrodynamic formalism, the gravitational equation associated with the classical Dirac field is derived. The hydrodynamic representation of the Dirac equation described by the evolution of four mass densities, subject to the theory-defined quantum potential, has been generalized to the curved space-time in the covariant form. Thence, the metric of space-time has been defined by imposing the minimum action principle. The derived gravity shows the spontaneous emergence of the “cosmological” gravity tensor (CGT), a generalization of the classical cosmological constant (CC), as a part of the energy-impulse tensor density (EITD). Even if the classical cosmological constant is set to zero, the CGT is non-zero, allowing a stable quantum vacuum (out of the collapsed branched polymer phase). The theory shows that in the classical macroscopic limit, the general relativity equation is recovered. In the perturbative approach, the CGT leads to a second-order correction to Newtonian gravity that takes contribution from the space where the mass is localized (and the space-time is curvilinear), while it tends to zero as the space-time approaches the flat vacuum, leading, as a means, to an overall cosmological constant that may possibly be compatible with the astronomical observations. The Dirac field gravity shows analogies with the modified Brans–Dicke gravity, where each spinor term brings an effective gravity constant G divided by its field squared. The work shows that in order to obtain the classical minimum action principle and the general relativity limit of the macroscopic classical scale, quantum decoherence is necessary. Full article

A new formalism involving spinors in theories of spacetime and vacuum is presented. It is based on a superalgebraic formulation of the theory of algebraic spinors. New algebraic structures playing role of Dirac matrices are constructed on the basis of Grassmann variables, which we call gamma operators. Various field theory constructions are defined with use of these structures. We derive formulas for the vacuum state vector. Five operator analogs of five Dirac gamma matrices exist in the superalgebraic approach as well as two additional operator analogs of gamma matrices, which are absent in the theory of Dirac spinors. We prove that there is a relationship between gamma operators and the most important physical operators of the second quantization method: number of particles, energy–momentum and electric charge operators. In addition to them, a series of similar operators are constructed from the creation and annihilation operators, which are Lorentz-invariant analogs of Dirac matrices. However, their physical meaning is not yet clear. We prove that the condition for the existence of spinor vacuum imposes restrictions on possible variants of the signature of the four-dimensional spacetime. It can only be (1, $-1$ , $-1$ , $-1$ ), and there are two additional axes corresponding to the inner space of the spinor, with a signature ( $-1$ , $-1$ ). Developed mathematical formalism allows one to obtain the second quantization operators in a natural way. Gauge transformations arise due to existence of internal degrees of freedom of superalgebraic spinors. These degrees of freedom lead to existence of nontrivial affine connections. Proposed approach opens perspectives for constructing a theory in which the properties of spacetime have the same algebraic nature as the momentum, electromagnetic field and other quantum fields. Full article

Space out of a topological defect of the Abrikosov–Nielsen–Olesen (ANO) vortex type is locally flat but non-Euclidean. If a spinor field is quantized in such a space, then a variety of quantum effects are induced in the vacuum. On the basis of the continuum model for long-wavelength electronic excitations originating in the tight-binding approximation for the nearest-neighbor interaction of atoms in the crystal lattice, we consider quantum ground-state effects in Dirac materials with two-dimensional monolayer structures warped into nanocones by a disclination; the nonzero size of the disclination is taken into account, and a boundary condition at the edge of the disclination is chosen to ensure self-adjointness of the Dirac–Weyl Hamiltonian operator. We show that the quantum ground-state effects are independent of the disclination size, and we find circumstances in which they are independent of parameters of the boundary condition. Full article