Vacuum and spacetime signature in the theory of superalgebraic spinors

Formulas for vacuum state vector and operators of the Lorentz transformations and gauge charge transformations of spinors are derived in the superalgebraic representation of spinors. Five operator analogs of five Dirac gamma matrices exist in the superalgebraic approach as well as two additional operator gamma matrices. They are constructed from Grassmann densities and derivatives with respect to them. We show that second copy of gamma operators exist and they are Lorentz invariant. They are constructed from operators of creation and annihilation. We show that the condition for the existence of spinor vacuum imposes restrictions on possible variants of Clifford algebras of gamma operators: only real algebra with one timelike basis Clifford vector corresponding to the zero gamma matrix in the Dirac representation can be realized. In this case, the signature of the four-dimensional spacetime, in which there is a vacuum state, 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).


Introduction
The question of the origin of the dimension and the spacetime signature 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-Millss 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 majoram and pseudo-Maioran 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].
Another 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][9][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 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 fermion 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].
The author develops an approach to the theory of spacetime, allowing to solve 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 (Dirac gamma matrices) are composite.
In [15,16], it was shown 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 their adjoint 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 θ a (p), a = 1, 2, 3, 4, and derivatives ∂ ∂θ a (p) with respect to them were introduced, with CAR-algebra Superalgebraic analogsγ µ (1.2) are constructed for Dirac gamma matrices γ µ from these densities, we call them gamma operators. (1.2) They convert and their linear combinations in the same way that Dirac matrices convert matrix columns and their linear combinations. The theory is automatically secondarily quantized and does not require normalization of operators.
In the proposed theory, in addition to analogs of the Dirac matrices, there are two additional gamma operators and , the rotation operator in whose plane (gauge transformation) is analogous to the charge operator of the second quantization method [18]: In [18], it was shown that transformations of densities and , while maintaining their CAR-algebra of creation and annihilation operators, provide transformations of field operators of the form , whereγ ab = 1 2 (γ aγb −γ bγa ); a, b = 0, 1, 2, 3, 4, 6, 7, and dω ab = −dω ba -real infinitesimal transformation parameters. The multiplier ¼ is added in (1.4) compared to [18] to correspond to the usual transformation formulas for spinors in the case of Lorentz transformations.
Expressions for operatorsγ ab are given in (2.5)-(??) -they will be important later. 3 Vacuum and discrete analogs of Grassmann densities In [17], the author proposed a method for constructing a state vector of a vacuum. Let's analyze it in more detail. We divide the momentum space into infinitely small volumes. We introduce operators At the same time, given (2.2), There is no silent summation over the index that numbers discrete volumes. For example, it does not exist at index j in (3.1)-(3.2). For indexes enclosed in triangular brackets (for example, in (3.4)), there is also no silent summation.
The expression 1 3 p j δ i j in (3.1)-(3.2) is a discrete analogue of the delta function δ(p−p ). In addition, due to the anticommutativity of all b k (p) and b l (p ) as well as allb k (p) and b l (p ) it is obvious that We introduce operators and determine through them the fermionic vacuum operator Ψ V where the product goes over all physically possible values of j. In this case, we will assume that all volumes 3 p j are formed by Lorentz rotations from the volume 3 p j=0 corresponding to p = 0, and the grid of angles ω µν of these rotations is discrete.
Further, it will often be convenient to represent (3.5) in the form is the product of factors in (3.5), independent of p j . Replace in the formulas with participation ofγ a andγ ab continuous operators b k (p) andb k (p) to discrete B k (p j ) andB k (p j ), and the integral´d 3 p ... to the sum j 3 p j ... . In this case, all formulas using continuous operators b k (p) andb k (p) are replaced by completely similar ones using discrete ones, with the replacement of the delta function δ(p − p ) by 1 3 p j δ i j , where p i corresponds to p, and p j corresponds to p . We will use for operatorsγ a = j 3 p jγ a (p j ) andγ ab = j 3 p jγ ab (p j ) after such a replacement the same notation as for the corresponding continuous ones, and we will call suchγ a as discrete gamma operators, andγ ab as discrete gamma operators of rotations.

Action of gamma operators on the vacuum
Consider action ofγ 0 on the vacuum (3.6). Sinceγ 0 is a commutator, we havê Taking into account the introduced notation for discrete operators and taking into account the fact that an arbitrary spatial momentum can be obtained from the state with p = 0 (2.1), At the same time B 1 (p j ) means that the result of rotation of a state with p = 0 turns into the state with p = p j .

(4.8)
To understand the meaning of (4.8) we consider the action of the operator of creation of a fermion-antifermion pair 3 The multiplier 3 p is necessary for normalization to the unit probability of finding spinors in the whole space.
Similarly, we find the result of the action ofγ 1 (0) on the multipliers of Ψ V : That means the creation of fermion-antifermion pairs by the operatorγ 1 even at zero momentum, that is, without suppression of this process in the non-relativistic limit.
Thus, the operatorγ 1 in the nonrelativistic limit p → 0 (and, therefore, in general) cannot have eigenvalues on state vectors.
We get the same situation for acting on a vacuum and on state vectors for operatorŝ γ a , a = 1, 2, 3, 4, 6, 7 -they do not annul the vacuum in the nonrelativistic limit and cannot have eigenvalues on state vectors.

Action of gamma operators of rotations on a vacuum
We get the same situation for acting on the vacuum and on state vectors for boostsγ 0a (0) ,a = 1, 2, 3, 4, 6, 7 -they do not annul the vacuum and cannot have eigenvalues on the state vectors.
The invariance of the vacuum during Lorentz rotations exp(γ µν ω µν /4), whereµ, ν = 0, 1, 2, 3, is ensured by the fact that each volume 3 p j passes into another volume 3 p k , and its place is occupied by the third volume 3 p l . Which only leads to a change in the order of the factors Ψ V j in (3.5). These factors commute, so the Lorentz rotations leave the vacuum Ψ V invariant.

Lorentz-invariant gamma operators
It is easy to construct Lorentz-invariant analogsΓ a andΓ ab of superalgebraic representationŝ γ a of Dirac matrices and rotation generatorsγ ab . To do this, it is enough in formulas (1.2)-(1.3), (2.5) replace all operators ∂ ∂θ k (p) by b k (p), and operators θ k (p) byb k (p). For example, and so on.
In the discrete version of the theory, in the operatorsΓ a andΓ ab , as before, continuous operators b k (p) andb k (p) are replaced by discrete B k (p j ) andB k (p j ), and integrals´d 3 p ... by sums j 3 p j ....
The operatorsΓ a andΓ ab are constructed by summing (integrating in the continuous case) over spatial momentums the results of all possible Lorentz rotations of the operators γ a (0) andγ ab (0). As a result of such rotations, ∂ ∂θ k (0) goes to b k (p), and θ k (0) tob k (p) as in the field operators, as inγ a (0) andγ ab (0).
In contrast toγ a andγ ab , in the Lorentz transformations the operatorsΓ a andΓ ab do not change either, since, like for the vacuum, the sum element for some momentum goes into the sum element for another momentum, and the sum element for the third momentum takes its place. As a result, these operators are Lorentz-invariant (and therefore also Lorentzcovariant). For the same reason, if for some values of a and b the operatorγ a (0) orγ ab (0) annuls the vacuum, thenΓ a orΓ ab annuls the vacuum, and ifγ a (0) orγ ab (0) not annuls the vacuum, thenΓ a orΓ ab under the action on the vacuum do not give zero. And for the same reason, ifγ a (0) orγ ab (0) has eigenvalue for the state with p = 0, thenΓ a orΓ ab has corresponding eigenvalue for states with any momentums. That is why operatorsΓ a have the same signature asγ a (0) and, hence, the same signature asγ a .
Therefore, in quantum relativistic field theory, the eigenvalues of the operatorsΓ a and Γ ab are meaningful on the state vectors, and the operatorsγ a andγ ab cannot have eigenvalues at all, since they do not annul the vacuum. Operatorsγ a andγ ab have eigenvalues only in the non-relativistic limit p → 0.
We introduce the superalgebraic analogues [17] of the operators of the number of par-ticlesN 1 ,N 2 and antiparticlesN 3 ,N 4 and the charge operatorQ in the method of second quantization:N Then the physical meaning ofΓ 0 andΓ 67 is obvious, since (6.1) and (6.3) can be rewritten in the form: That is, −Γ 0 is the operator of the total number of spinors and antispinors, andΓ 67 is related to the charge operatorQ by the ratioΓ 67 = iQ. Similarly,Γ jk = iτ l , where j, k, l is cyclic permutation of 1, 2, 3. Moreover,τ l are Lorentz-invariant spin operators, which are analogs of the Pauli matrices. OperatorsŴ k = −imΓ k4 are components of Lorentz invariant analog of spin components of the Pauli-Lyubansky vector. However, physical meaning of operatorsΓ jk ,Ŵ k andΓ µa , where µ = 0, 1, 2, 3, a = 4, 6, 7, is incomprehensible.
It is useful to note that the matrix formalism does not provide the possibility of zero eigenvalues of gamma matrices, in contrast to the proposed theory. Expansion (1.4) generates the expansion of field operators in momenta and leads to the implementation of the Dirac equation [18]. The question arises of what kind of Clifford bases such decomposition is possible.
Multiplyingγ 0 by an imaginary unit will lead to the appearance in the expansion in momenta [18] of exponentially increasing terms, that is, to the impossibility of the existence of normalized solutions. Therefore, Clifford vectorsγ 0 andΓ 0 are time-like and have signature +1 for spacetime where spinors can exist as physical particles.
Multiplication of any of the operatorsγ m (and, consequently,Γ m ) by the imaginary unit due to the presence of the vacuum (3.5) will lead to asymmetry between Clifford vectorsΓ 0 and iΓ m , sinceΓ 0 Ψ V = 0 and iΓ m Ψ V = 0, andΓ 0 can have eigenvalues on the state vectors but iΓ m cannot. The space of Clifford vectors with the same signature must be isotropic, however in this case we obtain a preferred direction. Therefore, other than Γ 0 Clifford vectors could not have the same signature asΓ 0 . Consequently, the condition for the existence of the vacuum imposes restrictions on the possible variants of Clifford algebras: neither complex algebra nor algebras in which at least one of the base vectorŝ Γ m (and henceγ m ) is timelike is suitable. Therefore, all Clifford vectorsΓ m are spacelike (and henceγ m ) -they have a signature of -1, and there is only one basic timelike Clifford vectorΓ 0 (and henceγ 0 ).
Only 16 of the 28 operatorsΓ a andΓ ab in (1.4) annul the vacuum and therefore can have eigenvalues on the state vectors. Therefore, if we require the existence of a decomposition in momenta, that is, the existence of spinors as physical particles, out of seven gamma matricesΓ a (and henceγ a ), one must have a positive signature, and the other six must have a negative signature.
Thus, in the superalgebraic theory of spinors, the signature of a four-dimensional spacetime can only be (1, -1, -1, -1), and there are two additional axesγ 6 andγ 7 with a signature (-1, -1) corresponding to the inner space of the spinor. The reason why they and the axiŝ γ 4 are not additional spatial axes is not yet clear.

Conclusion
The proposed theory has a number of interesting consequences.
-The theory is free from divergences, leading to the need for the normalization of operators [17].
-It leads to an unambiguous signature of spacetime, which coincides with the observable.
-Part of the decomposition terms (1.4) corresponds to the usual field theories available in the framework of the general theory of relativity [19], as well as to theories of bundles [20]. An operatorΓ 67 = iQ and gauge transformation exp(iQω 67 ) automatically arises, whereQ is the charge operator in the second quantization formalism,QΨ = Ψ for the spinor Ψ, andQΨ = −Ψ for its antiparticleΨ.
-The proposed approach to constructing a discrete vacuum is fundamentally different from theories in which the discreteness of spacetime is considered, leading to the loss of Lorentz covariance [21]. The proposed theory is Lorentz covariant and combines the features of discrete and continuous theories.