XXI International Meeting of Physical Interpretations of Relativity Theory IOP Publishing Journal of Physics: Conference Series 1557 (2020) 012031 doi:10.1088/1742-6596/1557/1/012031 Spacetime and inner space of spinors in the theory of superalgebraic spinors V V Monakhov Saint Petersburg State University, Saint Petersburg, Russia E-mail: [email protected] Abstract. We constructed gamma operators which are superalgebraic analogs of the Dirac gamma matrices as well as two additional gamma operators which have no analogs in the Dirac theory. We found a new mechanism of the left-right symmetry breaking which is absent in the usual theory of Dirac spinors. We constructed Lorentz invariant gamma operators from operators of creation and annihilation of spinors. These gamma operators are also analogs of the Dirac gamma matrices, however they are not related to Lorentz transformations but generate vector fields as affine spinor connections. We have shown that the theory is equivalent to an extended version of the Pati-Salam theory. 1. Introduction The question of the origin of the spacetime has long attracted the attention of physicists. At the same time, there are different approaches in attempts to substantiate the observed dimension and the spacetime signature. 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. At the same time, multiplets of matter fields for supersymmetric Yang-Mills theories exist only when the dimension of spacetime is less than or equal to 10 [2]. Subsequently, the main attention was paid to the theory of superstrings and supermembranes. Various versions of these theories were combined into an 11-dimensional M-theory [3,4]. In [5], the most general properties of the theories of supersymmetry and supergravity in spaces of various dimensions and signatures were analyzed. Proceeding from the possibility of the existence of Majorana and pseudo-Majorana spinors in such spaces, it was shown that supersymmetry and supergravity of M-theory can exist in 11-dimensional and 10-dimensional spaces with arbitrary signatures, although depending on the signature the theory type will differ. Later, other possibilities were shown for constructing variants of M-theories in spaces of different signatures [6]. Other approaches are Kaluza-Klein theories. For example, in [7] it was shown that in the theories of Kaluza-Klein in some cases it is possible not to postulate, but to determine from the dynamics not only the dimension of the spacetime, but also its signature. In [8-10], an attempt was made to find a signature based on the average value of the quantum fluctuating metric of spacetime. An attempt was made in [11] to explain the dimension and signature of spacetime from the anthropic principle and the possibility of causality, in [12] from the existence of equations of Content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. Published under licence by IOP Publishing Ltd 1 XXI International Meeting of Physical Interpretations of Relativity Theory IOP Publishing Journal of Physics: Conference Series 1557 (2020) 012031 doi:10.1088/1742-6596/1557/1/012031 motion for fermions and bosons coinciding with four-dimensional ones, in [13] from the possibility of existence in spacetime classical electromagnetism. In all the above approaches, the fermionic vacuum operator in the second quantization formalism is not constructed and the restrictions imposed by such a construction are not considered. Therefore, the possibility of the existence of a vacuum and fermions is not discussed. In particular, the vacuum should be a Lorentz scalar and have zero spin, but in the theory of algebraic spinors, which more generally describes spinors than the Dirac matrix theory, Clifford vacuum has the transformational properties of the spinor component, and not the scalar [14]. We develop an approach to the theory of spacetime, solving this problem. It is based on the theory of superalgebraic spinors { an extension of the theory of algebraic spinors, in which the generators of Clifford algebras (analogs of Dirac gamma matrices) are composite. 2. Analogs of Dirac gamma matrices We showed in [15,16] that using Grassmann variables and derivatives with respect to them, one can construct an analog of matrix algebra, including analogs of matrix columns of 4-spinors and matrix rows of conjugate spinors. But at the same time, the spinors and their conjugates exist in the same space { in the same algebra. In [17,18], this approach was developed { Grassmann densities θa(p), a = 1; 2; 3; 4, and @ derivatives @θa(p) with respect to them were introduced, with CAR-algebra @ ; θj(p0) = δjδ(p − p0): (1) @θi(p) i Superalgebraic analogsγ ^µ of Dirac gamma matrices γµ are constructed of these densities, we call them gamma operators: Z @ @ @ @ γ^0 = d3p θ1(p) + θ2(p) + θ3(p) + θ4(p); ∗ ; @θ1(p) @θ2(p) @θ3(p) @θ4(p) Z @ @ @ @ γ^1 = d3p − θ4(p)θ1(p) + − θ3(p)θ2(p); ∗ ; @θ1(p) @θ4(p) @θ2(p) @θ3(p) Z @ @ @ @ γ^2 = i d3p − − θ4(p)θ1(p) + + θ3(p)θ2(p); ∗ ; (2) @θ1(p) @θ4(p) @θ2(p) @θ3(p) Z @ @ @ @ γ^3 = d3p − θ3(p)θ1(p) − + θ4(p)θ2(p); ∗ ; @θ1(p) @θ3(p) @θ2(p) @θ4(p) Z @ @ @ @ γ^4 = iγ^5 = i d3p + θ3(p)θ1(p) + + θ4(p)θ2(p); ∗ : @θ1(p) @θ3(p) @θ2(p) @θ4(p) @ @ 3 4 They convert @θ1(p) ; @θ2(p) ; θ (p); θ (p) and their linear combinations in the same way that Dirac matrices γη convert matrix columns and their linear combinations. The theory is automatically secondarily quantized and does not require normalization of operators. Generalization of Dirac conjugation of a spinor Ψ is Ψ = (M^ Ψ)+; (3) where M^ is an analog of the matrix constructed from gamma operators [18]. In the proposed theory, if we let M^ =γ ^0, and equation (3) corresponds to usual Dirac conjugation, in addition to analogs of the Dirac matrices, there are two additional gamma operatorsγ ^6 andγ ^7 [18]: 6 R 3 h @ @ 2 1 @ @ 4 3 i γ^ = i d p 1 2 + θ (p)θ (p) − 3 4 − θ (p)θ (p); ∗ ; @θ (p) @θ (p) @θ (p) @θ (p) (4) 7 R 3 h @ @ 2 1 @ @ 4 3 i γ^ = d p @θ1(p) @θ2(p) − θ (p)θ (p) + @θ3(p) @θ4(p) − θ (p)θ (p); ∗ : 2 XXI International Meeting of Physical Interpretations of Relativity Theory IOP Publishing Journal of Physics: Conference Series 1557 (2020) 012031 doi:10.1088/1742-6596/1557/1/012031 3. Possible transformations of superalgebraic spinors and breaking of the left-right symmetry a @ We showed in [18] that transformations of densities θ (p) and @θa(p) , while maintaining their CAR-algebra of creation and annihilation operators, provide transformations of field operators Ψ of the form 1 Ψ0 = 1 + iγ^ad! + γ^abd! Ψ; (5) a 4 ab ab 1 a b b a whereγ ^ = 2 (^γ γ^ − γ^ γ^ ); a; b = 0; 1; 2; 3; 4; 6; 7 and d!ab = −d!ba { real infinitesimal transformation parameters. The multiplier 1=4 is added in equation (5) compared to [18] to correspond to the usual transformation formulas for spinors in the case of Lorentz transformations. We can use other variants of conjugation. The most common form of the operator M^ for the spacetime with signature (p; q) which is consistent with Lorentz transformation is [18] ^ 1 2 p 1 2 q M = c+γ^+γ^+ ::: γ^+ + c−γ^−γ^− ::: γ^−; (6) k where c+ and c− are complex constants,γ ^+ are gamma-operators with positive signature and k γ^− are gamma-operators with negative one. If the dimension n = p + q of the spacetime is odd n = 2m + 1 then the center of the Clifford algebra is nontrivial. In this case terms of equation (6) up to the element of this center are proportional one to another, and exists the only possible variant of the generalization of the Dirac conjugation. If the dimension of the spacetime is even n = 2m then ^ 1 2 p M =γ ^+γ^+ ::: γ^+(c+ + c−γ^); (7) where p 2 1 1 2 q γ^ = "γ^+ ::: γ^+γ^+γ^−γ^− ::: γ^−; (8) and we choose " = 1 or " = i so that (^γ)2 = 1. Variants when c+ = 0 or c− = 0 were investigated in [18]. In these cases signature of the spacetime and formula of generalized Dirac conjugation are one-to-one related. Variant with ^ ^ 1 M = c+ corresponds to the Euclidian space, variant with M = c+γ^+ corresponds to the Lorentz signature of the spacetime, and so on. In these cases decomposition (5) performs, however corresponding number of gamma operators (2) and (4) get imaginary unit as multiplier. In the 4-dimensional case we haveγ ^ =γ ^5. In [19] we proved that only one timelike axisγ ^0 is possible, this will be discussed later. That is why we can rewrite condition (7) as 0 0 5 M^ = c+(^γ + λγ^ γ^ ); (9) where λ = c−=c+ = λ1 + iλ2. Part of M^ corresponding to the imaginary part of λ can be diagonalized by the unitary transformation M^ 0 = T MT^ −1; (10) where T = exp(iγ^5'). 0 0 That is why in the case of λ1 = 0 and up to a non-essential phase multiplier we get M^ =γ ^ , and this case is equivalent to the case M^ =γ ^0. Otherwise we have in the common case ^ 0 0 0 0 5 M = c+γ^ (1 + λ1γ^ ); (11) where λ0 is real and c0 = p 1 up to a non-essential phase multiplier.