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Register a new userUnderstanding the second quantization of fermions in Clifford and in Grassmann space -- New way of second quantization of fermions, Part II
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Norma Susana Mankoc Borstnik
Publication date
2020
fields
Physics
and research's language is
English
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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.
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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.
Based on Lorentz invariance and Born reciprocity invariance, the canonical quantization of Special Relativity (SR) has been shown to provide a unified origin for the existence of Diracs Hamiltonian and a self adjoint time operator that circumvents Paulis objection. As such, this approach restores to Quantum Mechanics (QM) the treatment of space and time on an equivalent footing as that of momentum and energy. Second quantization of the time operator field follows step by step that of the Dirac Hamiltonian field. It introduces the concept of time quanta, in a similar way to the energy quanta in Quantum Field Theory (QFT). An early connection is found allready in Feshbachs unified theory of nuclear reactions. Its possible relevance in current developments such as Feshbach resonances in the fields of cold atom systems, of Bose-Einstein condensates and in the problem of time in Quantum Gravity is noted. .
Quantum point contacts (QPCs) are cornerstones of mesoscopic physics and central building blocks for quantum electronics. Although the Fermi wave-length in high-quality bulk graphene can be tuned up to hundreds of nanometers, the observation of quantum confinement of Dirac electrons in nanostructured graphene systems has proven surprisingly challenging. Here we show ballistic transport and quantized conductance of size-confined Dirac fermions in lithographically-defined graphene constrictions. At high charge carrier densities, the observed conductance agrees excellently with the Landauer theory of ballistic transport without any adjustable parameter. Experimental data and simulations for the evolution of the conductance with magnetic field unambiguously confirm the identification of size quantization in the constriction. Close to the charge neutrality point, bias voltage spectroscopy reveals a renormalized Fermi velocity ($v_F approx 1.5 times 10^6 m/s$) in our graphene constrictions. Moreover, at low carrier density transport measurements allow probing the density of localized states at edges, thus offering a unique handle on edge physics in graphene devices.
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.
The recent theoretical prediction and experimental realization of topological insulators (TI) has generated intense interest in this new state of quantum matter. The surface states of a three-dimensional (3D) TI such as Bi_2Te_3, Bi_2Se_3 and Sb_2Te_3 consist of a single massless Dirac cones. Crossing of the two surface state branches with opposite spins in the materials is fully protected by the time reversal (TR) symmetry at the Dirac points, which cannot be destroyed by any TR invariant perturbation. Recent advances in thin-film growth have permitted this unique two-dimensional electron system (2DES) to be probed by scanning tunneling microscopy (STM) and spectroscopy (STS). The intriguing TR symmetry protected topological states were revealed in STM experiments where the backscattering induced by non-magnetic impurities was forbidden. Here we report the Landau quantization of the topological surface states in Bi_2Se_3 in magnetic field by using STM/STS. The direct observation of the discrete Landau levels (LLs) strongly supports the 2D nature of the topological states and gives direct proof of the nondegenerate structure of LLs in TI. We demonstrate the linear dispersion of the massless Dirac fermions by the square-root dependence of LLs on magnetic field. The formation of LLs implies the high mobility of the 2DES, which has been predicted to lead to topological magneto-electric effect of the TI.
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