Space Inversion of Spinors Revisited: A Possible Explanation of Chiral Behavior in Weak Interactions

We investigate a model in which spinors are considered as being embedded within the Clifford algebra that operates on them. In Minkowski space $M_{1,3}$, we have four independent 4-component spinors, each living in a different minimal left ideal of $Cl(1,3)$. We show that under space inversion, a spinor of one left ideal transforms into a spinor of another left ideal. This brings novel insight to the role of chirality in weak interactions. We demonstrate the latter role by considering an action for a generalized spinor field $\psi^{\alpha i}$ that has not only a spinor index $\alpha$ but also an extra index $i$ running over four ideals. The covariant derivative of $\psi^{\alpha i}$ contains the generalized spin connection, the extra components of which are interpreted as the SU(2) gauge fields of weak interactions and their generalization. We thus arrive at a system that is left-right symmetric due to the presence of a"parallel sector", postulated a long time ago, that contains mirror particles coupled to mirror SU(2) gauge fields.


Introduction
After the seminal paper by Lee and Yang [1], we have become accustomed to the idea that the physical processes associated with weak interactions are not invariant under space inversion. However, in this same paper, Lee and Yang proposed that the introduction of mirror particles restores the invariance of β-decay under space inversion. The idea of mirror particles and the exact parity model has been thereafter pursued and investigated from various points of view in Refs. [2]- [5]. The possibility that mirror particles are responsible for dark matter has been explored in many works [6]. For a review on mirror particles and their phenomenological consequences, see [7].
In this paper, we will show how mirror particles find a natural explanation if we employ the concept of algebraic spinors. It is well known that spinors can be described as elements of the left or right minimal ideals of a Clifford algebra [8,9]. As observed in Ref. [10], such an approach yields a more general and coherent theory than the traditional "column" representation. It incorporates the usual spinor formalism, but in addition, it brings the features that are absent from the usual theory in which spin space is a representation space lying outside of the algebra that operates on it. In a more general theory, a spinor can be considered as being embedded within the Clifford algebra that operates on it. This opens a Pandora's box of possibilities that have been explored in the attempts to find a unified theory of fundamental particles and forces [11]- [13].
We will follow the approach [16,17] in which spinors are constructed in terms of nilpotents formed from the spacetime basis vectors represented as generators of the Clifford algebra Cl (1,3). We find that under space inversion, a left-handed spinor of one ideal transforms into a right-handed spinor of not the same but of a different ideal. We then construct a specific model starting from a generalized Dirac action that is a functional of 16 complex valued fields, components of a generalized algebraic spinor field. Such a 16-component field is written as the 4 × 4 matrix ψ αi , representing four Dirac spinors lying in four different left ideals. A transformation R that is generated by bivectors γ µ ∧ γ ν , µ, ν = 0, 1, 2, 3, can act on ψ αi from the left, in which case R has the role of a Lorentz transformation, or from the right, in which case it behaves as an 'internal' transformation. Our model thus incorporates the internal gauge group SU (2). The covariant derivative occurring in the action contains a generalized spin connection that incorporates not only the ordinary spin connection but also two kinds of SU(2) gauge fields, W µ and W ♮ µ , that are transformed into each other by the space inversion P. The 'internal' components of ψ αi , denoted by the index i that runs over four different left ideals, form two irreducible representations of SU(2) that correspond to ordinary and mirror particles. Thus, we have found a theoretical explanation for the suggestion that there must be a mirror world-a parallel sector-with no common interaction with ordinary particles [2,5]. In this paper, we demonstrate the existence of such a parallel sector for the case of weak interactions only.
Minkowski space M 1,3 ≡ M has been decomposed according to M = N+P into two singular subspaces with null quadratic forms.
We now observe that the product Therefore, f can be interpreted as 'vacuum', andθ a can be interpreted as operators that annihilate f . An object constructed as a superposition, is an element of a minimal left ideal of complexified Cl(1, 3); it is a spinor. The notation of the components in Eq. (6) is adapted to the fact that Ψ is given in terms of a Clifford algebra of a 2-dimensional vector space spanned by θ 1 , θ 2 acting on f . Because θ 1 and θ 2 are null vectors, satisfying θ 2 1 = 0, θ 2 2 = 0, the latter Clifford algebra is in fact the Grassmann algebra N.

Four independent spinors
Thus far, we have considered one possible decomposition of M 1,3 into two singular subspaces, N and P , that are spanned, respectively, over the basis (θ 1 , θ 2 ) and (θ 1 ,θ 2 ). There is another possibility, namely, (θ 1 ,θ 2 ) and (θ 1 , θ 2 ). We can thus consider four different vacua, and can construct four different kinds of spinors, each being in a different minimal left ideal of Cl(1, 3) (more precisely, of its complexified version): An arbitrary element of Cl(1, 3) is the sum of those independent spinors: where the set of elements , forms a spinor basis of Cl(1, 3). We will call the object Φ a generalized spinor. Spinors are thus just particular Clifford numbers. Usually a Clifford number is given in terms of 16 basis elements Γ M ≡ (1, γ µ , γ µν , γ 5 γ µ , γ 5 ), but it can also be given in terms of the 16 basis elements ξÃ ≡ ξ αi , where the first index denotes four spinor components, whereas the second index denotes four independent left minimal ideals of Cl (1,3). In general, ψ αi are complex-valued spacetime fields.

Behavior of spinors under discrete Lorentz transformations
We will now explore how the left-handed and right-handed spinors (8), (9) transform under space inversion P, under time reversal T, and under PT.
Using Eqs. (8), (9) and the above transformations of the Witt basis, we find We see that the left-handed spinor Ψ L does not transform into the right-handed Ψ R of Eq. (9). More precisely, it does not transform into the right-handed spinor of the same ideal of Cl (1,3). Instead, it transforms into the spinor Ψ ♮ R , which, according to Eq. (18), belongs to the third ideal. The transformed spinor Ψ ♮ R is right-handed because it satisfies where we have used Eqs. (1), (2) and {γ 5 , γ µ } = 0, from which it follows that iγ 5 θ 1θ2 = θ 1θ2 . Analogously, the right-handed spinor Ψ R transforms into the spinor Ψ ♮ L , which, according to Eq. (18), also belongs to the third ideal and satisfies Therefore, Ψ ♮ L is a left-handed spinor. By including a new index j that runs over ideals, we have Under the space inversion, Ψ j L → not Ψ j R , and Ψ j R → not Ψ j L , but Ψ j L → Ψ j ′ R , and Ψ j R → Ψ j ′ L , where j = 1, 2, 3, 4 and j ′ = 3, 4, 1, 2. Thus, a left (right)-handed spinor of an ideal j transforms into a right (left)-handed spinor of a different ideal j ′ .
Then, we have Ψ L → Ψ L and Ψ R → −Ψ R . In general, under a Lorentz transformation, including a discrete (i.e., improper) one, an arbitrary element of Cl(1, 3), Eq. (20), transforms according to where Here,Ã = αi andB = βj. BecauseÃ denotes the double index αi, we can arrange the basis elements ξÃ ≡ ξ αi into a matrix as and components ψÃ ≡ ψ αi into a matrix as For instance, under space inversion (21), the Witt basis elements transform according to (21), and the basis elements (34) transform according to (2.13) into new basis elements, whereas the components transform according to (33) as Because we consider the active transformations, then, according to (31), either Eq. (32) or Eq. (33) holds, but both cannot hold at once. A generic Clifford number Φ of (20), a sum of the spinors of four different left ideals, thus transforms into a different object Φ ′ that is expanded, according to Eq. (31), either in terms of a new basis (36) and old components or in terms of the old basis and new components (37).

An action for the generalized spinor field
A possible action for a spacetime-dependent field ψÃ is 1 Here [14,15], are matrix elements of the 16-dimensional, reducible, representation of the generators γ µ . The operation A S takes the scalar part of a generic Clifford number A and multiplies the result by the dimension n = 4 of the minimal left ideal (i.e., the dimension of a spinor of Cl(1, 3)). The symbol ' ‡' denotes reversion, i.e., the operation that reverses the order of all 1-vectors in an expression. Indices are lowered and raised by the metric ZÃB = ξ ‡Ã ξB S = z ij z αβ and its inverse
The interactive term Lagrangian in the action (38) can be written explicitly as If acting within the same left ideal, the field G µ βi ρk for i = k behaves as the ordinary spin connection in curved spacetime. Otherwise, it behaves as a gauge field due to the "internal" local gauge transformations [14,15] that transform one ideal into the other. The group of the latter transformations contain SU(2) as a subgroup.
Let us write Eq. (42) in a more compact form, where all quantities now contain the implicit spinor indices α,β. In the chiral representation, where σ r are the Pauli 2 × 2 matrices. If we write the gauge fields in the following block matrix form, then This interactive Lagrangian has an interesting structure that involves spinor degrees of freedom and "internal" degrees of freedom associated with different left ideals (denoted by indices i, k) and various gauge fields. Left-handed or right-handed spinors of one left ideal, denoted by k, are coupled (via various gauge fields) to the corresponding spinors of another left ideal, denoted by i (where, in particular, it can be i = k). If we split the indices according to i =ī, i and k =k, k, whereī,k = 1, 2 and i, k = 3, 4, then the Lagrangian (47) can be written in the form of The bared indices denote the 1st and 2nd ideals, whereas the underlined indices denote the 3rd and 4th ideals. One can show that a rotation from the right, Φ → Φ ′ = ΦR γr γs , in general transforms the 1st into the 2nd and the 3rd into the 4th ideal. The left-handed field χk L ,k = 1, 2 forms the fundamental representation of SU (2), and it is coupled via the SU(2) gauge field a µīk to χ † iL and via c µīk to χ † iR . The analogous holds for the fields χ k L , χk R , and χ k R . We have seen in Eqs. (36),(37) that space inversion transforms the 1st ideal into the 3rd one and the 2nd ideal into the 4th one. Besides that, space inversion exchanges the upper half of the 4-component spinor with the lower half, i.e., it exchanges χ k L → χ k ′ R and χ k L ← χ k ′ R . Instead of the inversion (21), it is convenient to apply the reflection γ 0 → γ 0 , γ 1 → γ 1 , γ 2 → γ 2 , γ 3 → −γ 3 , because then no changes of sign take place in the transformations of ξ αi , ψ αi , and G αi βj . Using (32), one can show that So, we find that The interaction Lagrangian (47) (which can be written in the form of (48)) is thus invariant under space inversion. A notorious feature of this Lagrangian is that an ordinary right-handed field χk R ,k = 1, 2, is not coupled to the same gauge field as the left-handed field χk L . The right-handed field χk R behaves as a "singlet" with respect to the SU(2) gauge field a µīk , which the left-handed field χk L is coupled to. Instead, χk R behaves as an SU (2) doublet with respect to another gauge field, namely, d µīk .
Lagrangian (48) contains, amongst others, a term χ † iLσ µ a µīk χk L . As already mentioned, a µīk is a 2 × 2 matrix in the spinor indicesᾱ,β = 1, 2. If, in particular, a µīk is a diagonal matrix in indicesᾱ,β, then the term χ † iLσ µ a µīk χk L is just like the interactive Lagrangian for weak interactions, a µīk being the weak SU(2) gauge fields W µīk .
However, Lagrangian (48) contains other terms besides the one describing the usual weak interaction. It is beyond the scope of this Letter to investigate the physical meaning of those extra terms. One possibility is that due to certain superselection rules, the fields c µ i k and b µ i k vanish. Another possibility is that Eq. (48) predicts the existence of new interactions. From now on, we will only consider the part of L int that contains the fields a µ i k and d µ i k . We adopt an interpretation that the group SU(2) of R γrγs ∈ SU(2) that acts on a generalized spinor Φ from the right and that "mixes" either indicesī,k or i, k is the gauge group of weak interactions. Because we know from experiments that right-handed ordinary fields do not interact under the weak SU(2) gauge fields, we will assume d µīk = 0. In addition, let us also assume a µ i k = 0. Then, the Lagrangian (48) contains the following part: Here, we can adopt an interpretation that χk L are spinor fields describing ordinary particles coupled to the ordinary weak SU(2) gauge fields, whereas χ k R are mirror fields describing mirror particles that are coupled to the mirror gauge fields d µ i k . Notice the difference between the barred and underlined indices. In other words, spinor fields of the 1st and 2nd ideals describe ordinary particles (and antiparticles), whereas spinor fields of the 3rd and 4th ideals describe mirror particles (and mirror antiparticles).
Putting it into a more compact notation, we write (52) as where '♮' denotes mirror fields, so that χ L ≡ χk L and χ ♮ R ≡ χ k R . Here, W µ and W ♮ µ are SU(2) gauge fields, and thus they are 2 × 2 matrices in indicesī,k and i, k, respectively, i.e., W µ ≡ a µīk and W ♮ µ ≡ d µ i k . The action (38) then contains the following part: where In Eq. (53), we identified the weak gauge field W µ with the diagonal part (with respect to hidden indicesᾱ,β) of a µīk and the mirror gauge field W ♮ µ with the diagonal part of d µ i k . Remember that the latter gauge fields, in addition to being matrices in the "internal" indicesī,k and i, k, are also matrices in the spinor indicesᾱ,β = 1, 2 (that, for simplicity, are not explicitly displayed). On the other hand, if we take i = k and allow forᾱ =β, then a µ i k and d µ i k correspond to the ordinary spin connections 2 . The action (54), which besides the ordinary particles described by ψ L and the ordinary gauge fields W µ contains the corresponding mirror partners ψ ♮ R and W ♮ µ , encodes the proposal by Kobzarev et al. [2]. Later, a model essentially equivalent to that given by the action (54) was proposed by Foot et al. [5]. In this paper, we have shown how such a system with ordinary and mirror particles coupled to the corresponding gauge fields can be described within a framework based on the algebraic spinors and their behavior under space inversion.

Discussion and conclusion
The observation that space inversion P and time reversal T transform a spinor of one left ideal of Cl(1, 3) into a spinor of another left ideal brings a novel insight on the invariance of nature under P and T that, to my knowledge, has not yet been explicitly pointed out. The idea that the invariance of weak interactions under P and T can be restored by introducing mirror particles has been considered by Lee and Yang [1] and later in Refs. [2]- [5]. In this letter, we provide a theoretical basis for such a model that comes naturally from the wellknown theory of algebraic spinors, i.e., the objects embedded within the Clifford algebra that operates on them.
Electron spin was first postulated based on experimental evidence in atomic spectra and was later explained by the Dirac equation, which provided a deep theoretical insight. We have a somewhat analogous situation with mirror particles that were postulated to restore left-right symmetry. In this paper, we have shown that the generalized spinors, which encompass all four left ideals, describe not only the ordinary spin and weak isospin but also the ordinary and mirror particles. The corresponding SU(2) gauge fields are incorporated in the generalized spin connection. One further step that should be done is to include the U(1) gauge field and to show precisely how the electric charge and the electromagnetic field occur within this Clifford algebra-based theory. To include the color SU(3) as well, an extension of Cl (1,3) or its complex version Cl(4, C) to a suitable higher dimensional Clifford algebra, such as Cl (8, C), would be necessary. Ackonwledgement This work was supported by the Ministry of High Education, Science and Technology of Slovenia.
2 This can be shown by observing that for i = k, the generalized spin connection G µ αi βk is just the ordinary spin connection Γ µ α β = 1 2 ω µ ab [γ a , γ b ]; then, by taking into account Eqs. (44), one finds that in Eq. (46), the part with a µ i k and d µ i k , for i = k, corresponds to the contribution of the ordinary spin connection.