A demonstration that electroweak theory can violate parity automatically (leptonic case)

We bring to light an electroweak model which has been reappearing in the literature under various guises. In this model, weak isospin is shown to act automatically on states of only a single chirality (left). This is achieved by building the model exclusively from the raising and lowering operators of the Clifford algebra Cl(4). That is, states constructed from these ladder operators mimic the behaviour of leftand right-handed electrons and neutrinos under unitary ladder operator symmetry. This ladder operator symmetry is found to be generated uniquely by su(2)L and u(1)Y . Crucially, the model demonstrates how parity can be maximally violated, without the usual step of introducing extra gauge and extra Higgs bosons, or ad hoc projectors.


Introduction
Violation of parity in nature was a finding that few could have possibly foreseen.When Lee and Yang proposed parity violation in their Nobel prize winning paper, 6 they noted that up until then, parity symmetry in the weak interactions had largely been assumed without question.][40][41][42][43][44] The aim of this review paper is to bring to a wider audience the simple mathematical system underlying electroweak theory, which has been making regular appearances in the literature.It may also introduce many readers to the importance of Clifford algebras in particle physics, beyond their implementation in the Dirac equation.

Cl(4) in Terms of Ladder Operators
This construction relies only on the Clifford algebra Cl (4).For the purposes of this discussion, readers may think of Cl(4) simply as the space of 4×4 complex matrices.
One way to generate Cl(4) is by the set of four matrices, where the usual Pauli matrices are given by σ x = 0 1 1 0 , σ y = 0 −i i 0 , and σ z = 1 0 0 −1 , while I represents the 2×2 identity.These four γ-matrices obey the relations Readers should note that there is indeed no relevant signature for Cl(4), seeing as how the algebra is over the field C. The Clifford algebra Cl(4) can be built up from products of the four γ j , as depicted in Fig. 1.Now, it is straightforward to see that the 4-C-dimensional generating space, spanned by the γ j , may be rewritten in terms of a new basis, where and β † 1 , β † 2 are the usual Hermitian conjugates of these.The two β j may be viewed as lowering operators, and the β † j as raising operators, which obey the following anticommutation relations, It is then clear that the 16-C-dimensional algebra Cl(4) may be rewritten entirely in terms of these four ladder operators.

Ladder Symmetries
Given a Clifford algebra generated entirely from ladder operators, we would now like to know what symmetries preserve this generating ladder system.To this end, we will then consider only those transformations which satisfy the following properties.
(1) We will be interested in transformations, G, on our ladder operators, O, of the form GOG −1 .These can easily be seen to preserve the anticommutation relations (5).The group element G will be taken to be G = e −iφ k g k , where φ k ∈ C and g k is allowed to be any element in Cl(4).(2) Furthermore, we require that the set of lowering operators be closed under these transformations, and likewise for the set of raising operators, as in Fig.  Fig. 2. The Clifford algebra Cl(4) may be rewritten entirely in terms of the ladder operators β i and β † j .We will be interested in transformations under which the lowering operators are closed, and likewise for the raising operators.
More precisely, we require that for given b (3) As a final condition, we will also require that any raising operator and lowering operator, which are the Hermitian conjugates of each other before a transformation are also the Hermitian conjugates of each other after the transformation.
In other words, we require that these transformations commute with Hermitian conjugation, Imposing these three conditions on a generic element φ k g k ∈ Cl(4) leaves us with only four nontrivial solutions over R: Readers may confirm that the three T i generate SU (2) and that N , a U (1) generator, commutes with all three of the T i .At the group level, this symmetry may be seen to be SU (2) × U (1)/Z 2 ≃ U (2).Hence, instead of specifying the three conditions above, we could have arrived at this symmetry more expediently by simply stating that we are interested in the unitary symmetries of these ladder operators.

States
Now that we have identified the SU (2) and U (1) symmetry generators for this Clifford algebraic system, we would like to establish a set of states on which these symmetries act.Cl(4) has just a single irreducible representation, which acts on a 4-Cdimensional space of states.The usual way to write this set of states is as a 4-Cdimensional column vector.An alternate way to write this set of states is as a special subspace within the 4 × 4 C matrices.
For various purposes, it will be useful for us to build these states as a special 4-Cdimensional subspace within the 4 × 4 C matrices.As we have shown earlier, Cl(4) can be written entirely in terms of ladder operators.Hence, so can any subspace of Cl(4).We will then find that this set of states can be written purely in terms of our ladder operators, β i and β † j .As a result, transformations on the ladder operators can now be seen to induce transformations on the states.Now, this special subspace of the Clifford algebra is otherwise known as a minimal one-sided ideal.Incidentally, minimal one-sided ideals of Clifford algebras provide one way of defining spinors. 45In constructing our one-sided ideal, we will need to make the inconsequential choice between building it as a left ideal, or a right ideal.Here, we decide rather arbitrarily to construct a minimal right ideal.
Given an algebra, A, a right ideal , B, is a subalgebra of A whereby b a is in B for all b in B, and for any a in A. That is, no matter which a we multiply onto b , the new product, b ′ ≡ b a must be in the subspace B. Intuitively, an ideal can be thought of as an algebra's stable subspace, in that it persists, no matter which algebraic element is multiplied onto it.Finally, a minimal right ideal is a right ideal which contains no other right ideals other than {0} and itself.
The construction of minimal right ideals within complex Clifford algebras, Cl(n), is quite straightforward. 5,45We begin by building the nilpotent object Ω, which is defined for Cl(4) to be the product of our lowering operators, Ω ≡ β 2 β 1 .From our nilpotent Ω, we construct what we will know as our (formal) vacuum state, Ω † Ω.The object Ω † Ω is a vacuum state only in the formal sense, as it will not represent the zero-particle state.Instead for us, it will represent the sterile right-handed neutrino.
Finally, our space of states is given simply by right-multiplying each and every element of Cl(4) onto the projector Ω † Ω; that is, S ≡ Ω † ΩCl(4).This leaves us with a 4-C-dimensional space, S, spanned by the states It is clear that our minimal right ideal then naturally exhibits the structure of a Fock space.a

Leptons Under su(2) L and u(1) Y
Now that we have constructed our 4-C-dimensional space of states, we would like to identify which particles these states represent.Of course, this is achieved by characterizing how the states transform under a given symmetry.
In keeping with the transformations of ladder operators, where the g k here are given by Eqs.(8).It so happens that our generators (8) have the property g k S = 0, which explains the final equality of Eq. (10).Under the SU (2) generators T i , we find that the states Ω † Ω and Ω † Ωβ † 1 β † 2 each behave as singlets, whereas the states Ω † Ωβ † 1 and Ω † Ωβ † 2 transform into each other as a doublet.For this reason, we will then identify Ω † Ω and Ω  Defining weak hypercharge as Y ≡ −1/2N then assigns to our vacuum state, Ω † Ω, a hypercharge of zero, to our left-handed states, Ω † Ωβ † 1 and Ω † Ωβ † 2 , each a hypercharge of −1/2, and to our final right-handed state, Ω † Ωβ † 1 β † 2 , a hypercharge of −1.Here, we have defined hypercharge according to the conventions of Schwartz. 48ith their behaviour under the ladder symmetries su(2) L and u(1) Y , we may then identify Ω † Ω as a right-handed neutrino, Ω † Ωβ † 1 as a left-handed neutrino, Ω † Ωβ † 2 as a left-handed electron, and Ω † Ωβ † 1 β † 2 as a right-handed electron.Our states then take the matrix form where V R , V L , E − L , E − R are complex coefficients (see Fig. 3).

Charge Quantization
It is a feature of this model that the U (1) generator should be proportional to N , the number operator for the system, N = j=1,2 β j β † j .Number operators, of course, count the number of raising operator excitations in each state, and must therefore take on only integer values.Hence, we find a straightforward explanation for the quantization of hypercharge Y on leptons.
This same number-operator-quantization was a central feature in the case of electric charge, Q, as described in earlier work. 5,33There, we examined the Clifford algebra Cl(6) and found ladder symmetries corresponding uniquely to su(3) c and u(1) em .This time, the ladder symmetries acted on minimal left ideals, which were then found to behave as does a generation of quarks and leptons.Readers are encouraged to see the earlier work of Barducci et al. [2][3][4] who showed similar Clifford algebraic extensions.Woit also identified further standard model structure in his paper, 1 via (projective) geometrical methods.

Relation to Supermultiplets
Readers familiar with supersymmetry will notice a striking resemblance between the minimal right ideal constructed here, and the supermultiplets of certain N = 2 extended SUSY models.However, it bears emphasizing that the irreducible representation constructed here is not inherently supersymmetric.In particular, the ladder operators β i and β † i do not carry spin indices, and hence do not map bosons to fermions and vice versa.
The 4-C-dimensional Cl(4) irreducible representation constructed for this paper is a special case of the more general 2 n C-dimensional Cl(2n) irreducible representation.In the general case, the Clifford algebra's 2n-dimensional generating space is partitioned into two maximal totally isotropic subspaces, according to a bilinear form given by the anticommutator.The elements of this MTIS are then used to construct an idempotent (or alternately, nilpotent) object, and subsequently a minimal one-sided ideal.The construction may be most familiar in the case of Spin(10) and SU (5) grand unified theories, where the internal degrees of freedom of quarks and leptons may be described by the irreducible representation of the Clifford algebra Cl(10) (for further details, see Refs. 5, 45-47).

Summary and Outlook
2][3][4][5] Here we have demonstrated how the Clifford algebra Cl(4) can be constructed purely from ladder operators.These ladder operators are subject to a unitary symmetry which turns out to be generated by su(2) and u(1).We identify these symmetries as weak isospin and hypercharge.
These ladder operators furthermore allow us to build a set of states in the form of a Fock space.Under su(2) and u(1), these states are found to behave as does the set of left-and right-handed electrons and neutrinos.In particular, the u(1) generator automatically assigns those eigenvalues corresponding to weak hypercharge.The su(2) generators are seen to act nontrivially on only half of the Fock states, which we thereby identify as left-handed.This is achieved without the aid of extra gauge bosons, Higgs bosons, or ad hoc projectors.
Observations like the one detailed here continue to inform ongoing research along the same lines. 5,22,33,49In particular, we continue to work towards understanding where this specific Clifford algebraic structure comes from.Of the infinite number of Clifford algebras available, why would nature choose Cl(4)?And given this Clifford algebra, why would nature choose unitary ladder symmetries over spin groups?Such choices might again seem puzzling and arbitrary, however, we hold to the expectation that nature will hand over these lessons in time.

Fig. 3 .
Fig.3.Fock space structure of leptonic states.These states transform under su(2) L and u(1) Y : symmetries determined by the ladder operators of the system.It is the hierarchy induced by the β † j which allow one chirality to be chosen over another.Readers should note that we did not need to rely on extra Higgs bosons, gauge bosons, nor ad hoc projectors here for su(2) to act on only a single chirality.