Abstract
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.
Highlights
The question of the origin of the dimension and signature of spacetime has a long history.There are different approaches to substantiate the observed dimension and the spacetime signature.One of the main directions is the theory of supergravity
If we consider a particular primitive idempotent I, we can always define the Grassmann variables in such a way they serve as creation operators, and, the derivatives with respect to them serve as annihilation operators
Borštnik’s interpretation of Grassmann variables and derivatives with respect to them was very different from that proposed in this article and had nothing to do with the theory of algebraic spinors
Summary
The question of the origin of the dimension and signature of spacetime has a long history. One of the main directions is the theory of supergravity It was shown in [1] that the maximum dimension of spacetime, at which supergravity can be built, is equal to 11. Proceeding from the possibility of the existence of Majorana and pseudo-Majorana spinors in such spaces, it was shown that the supersymmetry and the supergravity of M-theory can exist in 11-dimensional and 10-dimensional spaces with arbitrary signatures, depending on the signature, the theory type differs. Other possibilities were shown for constructing variants of M-theories in spaces of different signatures [6]. In [7], it was shown that, in some cases, it is possible to postulate, and to determine from the dynamics the dimension of the spacetime as well as its signature. The possibility of the existence of the vacuum and fermions is not discussed
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