In both the approach of chapters 2 and 4 a central property was the use of linear indepen-dence between irreducible representation spaces. Naturally this is the core investigation of our research, and as such the appearance of similar features in the two constructions should not come as a complete surprise. Additionally, we identified interesting features in chapter 2, from the perspective of understanding how representation spaces arise and how they compare with the Standard Model, and used ideas gained from these features to identify mechanisms by which we may obtain a set of irreducible representation spaces that are interesting in comparison with modern particle theory. This necessarily implies the existence of many points of similarity between the two approaches. In this section we will discuss merits and advantages of the different approaches in understanding the simultaneous realisation of multiple representation spaces.
The requirement of linear independence in chapter 2 was the origin behind the sub-space identified with Higgs doublet transformations not having an independent complex conjugate subspace. Similarly, in the induction process of chapter 4 linear independence implied existence of the H−spaces, which we argued showed similar features to the
Stan-4.7 Inducing Vs. Embedding Representations 61
dard Model implementation of the Higgs doublet representation in interactions. We found two distinct properties under map composition between these “Higgs-like” representations of chapters 2 and 4. In chapter 2 the subspace identified with the Higgs doublet represen-tation had the potential to include a non-zero projection on subspaces describing fermion transformations. As we commented before, this non-zero intersection implies the interac-tion strengths between the Higgs doublet and different generainterac-tions of fermions could be incorporated into how these representations appear as subspaces within a matrix algebra.
Clearly, the explicit embedding of the Standard Model yielded insights into how to form invariants between representation spaces within M(8,C) and, as all interactions of fields are described in terms of invariants, therefore gave insight into how interactions would appear in such a construction. This feature arose precisely because we demanded the existence of three generations of fermionic representation spaces in chapter 2.
On the other hand, in chapter 4 we found representation spaces, i.e. the H−spaces, which necessitated the inclusion of parameter α and β to distinguish map compositions with respective conjugate representations. As discussed, this could potentially describe interactions with matter-antimatter asymmetry, or an asymmetry between interactions of up- and down-type particles,14a feature we did not see in the explicit embedding of chapter 2. Both of these features are required to yield the mass spectrum seen in the Standard Model Yukawa interactions. This makes it clear that in the study of simultaneous real-izations of representation spaces, both the embedding and induction of these “Higgs-like”
representation spaces yielded interesting insight into the encoding of interaction strengths.
This presents the idea that the explicit embedding of Standard Model representation spaces is useful for understanding what relationships between representation spaces can be imposed by the inclusion of linear independence. Inducing representations instead indicates what sets of linearly independent representation spaces are consistent with certain underlying structures, i.e. like the SU(2) and SU(3) decompositions of C⊗O in chapter 4. To exemplify this, note that in chapter 2 there was no fundamental reason, apart from matching with Standard Model representations, to not include higher representations of the gauge groups, i.e. like the 6-dimensional representation of SU(3). If in chapter 4 we had included maps M from 3∗ to 3, these maps would have transformed as M →U M UT when 3(∗) → U(∗)3(∗), and thus contained the 6 representation of SU(3). However, such maps do not form a Lie algebra describing the redundancy of the direct sum decomposition of M(C⊗O), i.e. principle 2.15 So for the particular setup we presented in section 4.2, the appearance of the 6 representation of SU(3) is not possible. This shows that the set of irreducible representation spaces that may arise when inducing these spaces in the matrix algebra is heavily dependent on the principles by which the induction occurs. This is of course to be expected. Indeed we wanted certain features out of our representation spaces, as discussed in chapter 3. Even so, it demonstrates certain relationships which are
14We reiterate our disclaimer here that in principle we cannot make definite statements about inter-action structures without an inter-action. Our statements originate from an interpretation of features of map composition as a strong indication of interaction structures.
15Indeed because mapsM : 3(∗)→3 have different right and left ideals the set of such maps do not even form a matrix subalgebra, much less a Lie algebra.
uncovered when inducing representation spaces. We believe more work in this direction could lead to a deeper understanding of how one may simultaneously incorporate different representation spaces within an algebra of linear maps.
Apart from linear independence between irreducible representation spaces, it is inter-esting to consider how the different groups acted on the irreducible representation spaces in these approaches to studying the simultaneous realization of gauge representations. In both chapters we achieved the fundamental transformations of SU(2)×SU(3) through em-ploying bi-representations, with SU(3) acting from the left and SU(2) acting from the right. However, as chapter 2 was just an embedding of Standard Model representations, it offered no explanation for the origin of adjoint and fundamental representations. This resulted in the use of ad-hoc projectors which ruined the simplicity of the construction. In contrast, when inducing the direct sum decomposition in chapter 4, adjoint representations were seen as describing endofunctions preserving C⊗O|C,H, while fundamental represen-tations were endofunctions ofC⊗Owhich mapped C⊗O|C to C⊗O|H. Here the idea of projectors can have the natural origin as describing subspaces of different decompositions transforming under different gauge groups. As such the induction of irreducible repre-sentation spaces provided a mechanism by which the different adjoint and fundamental representations could both be seen as endofunctions obeying different properties. Further, it also provided a mechanism through which it was evident we could not have SU(3) trans-formations of SU(2) or visa-versa, something which needed to be imposed additionally in chapter 2. Such a natural inclusion of different transformations could help understand how to construct theories which have only the types of irreducible representations seen in the Standard Model. The ambition of such work is not only to have vector-adjoint represen-tation and spinor-fundamental represenrepresen-tations, i.e. the matching of gauge and Lorentz representations seen in the standard model. For example, understanding how represen-tations of these groups combine for different constructions could help uncover structures that result in chiral discrimination of gauge groups, deepening our understanding of this phenomenon.
While the approaches of embedding and inducing representation spaces are quite dif-ferent, it is clear that they are both advantageous, and have each helped uncover different structures associated to representations in M(8,C). All of this is based on the idea of in-corporating linear independence of finite-dimensional spaces as a useful tool for the study of irreducible representation spaces in the context of particle theory. A tool that provides relationships between and restrictions on sets of irreducible representation spaces. Formu-lating a deeper understanding of how these relationships originate from the choice of base space and setup could offer insights into what types of structures should be present in ap-proaches to unifying particle theory. We conclude this comparison with the emphasis that we are not implying our work, or generalizations, could directly yield the Standard Model’s particle content. Rather we suggest that understanding how to incorporate relationships between irreducible representation spaces may guide future approaches to unification in particle theory.