Understanding the second quantization of fermions in Clifford and in Grassmann space -- New way of second quantization of fermions, Part II

Abstract

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.

45 pages, \published in Proceedings to 23rd Bled Workshop "What Comes beyond the Standard Models", 2020, Zaloznistvo DMFA, Ljubljana, December 2020

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