arXiv subjects

Craig McRae

Publications and source records attributed to Craig McRae.

6 recordsLinked to original sources

arXiv2026

Parity Oddness of $\operatorname{Spin}(1,4)$ Dirac Mass Terms

In the representation theory of Lorentzian orthogonal groups there are well known arguments as to why the parity operator $\mathcal{P}$ and the time reversal operator $\mathcal{T}$ should be realized as linear and anti-linear operators respectively (Wigner 1932). In this paper it is shown that the only operators satisfying the requisite properties for the spinor representation of the de Sitter group $\operatorname{SO}(1,4)$ lead to fermion self-couplings which are necessarily parity odd, ruling out standard Dirac mass terms for theories with fermions obeying $\operatorname{Spin}(1,4)$ symmetry.

math-ph↗

arXiv2025

The Anti-Unitarity of Time Reversal & Co-representations of Lorentzian Pin Groups

In the representation theory of Lorentzian orthogonal groups, there are well known arguments as to why the parity inversion operator $\mathcal{P}$ and the time reversal operator $\mathcal{T}$, should be realized as linear and anti-linear operators respectively (Wigner 1932). Despite this, standard constructions of double covers of the Lorentzian orthogonal groups naturally build time reversal operators in such a manner that they are linear, and the anti-linearity is put in ad-hoc after the fact. This article introduces a viewpoint naturally incorporating the anti-linearity into the construction of these double covers, through what Wigner called co-representations, a kind of semi-linear representation. It is shown how the standard spinoral double covers of the Lorentz group -- $\operatorname{Pin}(1,3)$ and $\operatorname{Pin}(3,1)$ -- may be naturally centrally extended for this purpose, and the relationship between the $\mathcal{C}$, $\mathcal{P}$, and $\mathcal{T}$ operators is discussed. Additionally a mapping is constructed demonstrating an interesting equivalence between Majorana and Weyl spinors. Finally a co-representation is built for the de Sitter group $\operatorname{Pin}(1,4)$, and it is demonstrated how a theory with this symmetry has no truly scalar fermion mass terms.

math-ph↗

arXiv2025

Sporadic Isogenies to the Quaternionic Orthogonal Groups $\operatorname{SO}^*(2n)$

For small dimensional Lie algebra's there are many so-called accidental isomorphisms which give rise to double covers of special orthogonal groups - Spin groups - which happen to coincide with groups already belonging to another classification. The well known catalog of these sporadic isogenies given by Dr. Paul Garrett keeps track of these facts for the complex orthogonal groups $\operatorname{SO}(n, \mathbb{C})$, and those real forms which are manifestly real: $\operatorname{SO}(p,q)$, while avoiding the quaternionic real forms, $\operatorname{SO}^*(2n)$. This article serves to complement and complete the existing catalog by presenting the sporadic isogenies to the first four quaternionic orthogonal groups in one place, with contemporary proofs of those group and algebra homomorphisms. A review of the definition of $\operatorname{SO}^*(2n)$ is presented in a modern notation, emphasizing the concept of `quaternion reversion' as a useful, albeit redundant, second conjugation upon quaternions, helping to explicate the self-conjugacy of quaternionic representations. Lastly, the triality of $\operatorname{SO}^*(8)$ is explored in a similar manner to work done by the author previously.

math.RT↗

arXiv2025

Exploring Triality Explicitly: Convenient bases for $\mathrm{SO}(8)$, $\mathrm{Spin}(1, 7)$, and $\mathrm{G}_2$

The property of triality only appears in one linear simple Lie algebra: $D_4$, a.k.a. $\mathfrak{so}(8, \mathbb{C})$. Though often explored in abstract, it is desirable to have an explicit realization of the concept since there are no other linear examples to gain intuition from. In this paper several convenient representations and bases are constructed in order to facilitate the exploration of the three fold symmetry known as the triality of representations. In particular the three $8$ dimensional representations for the Euclidean and Lorentzian real forms of $\mathfrak{so}(8,\mathbb{C})$ are constructed, and the maps between representations are given in each case, respectively. It is also seen explicitly how $\mathfrak{g}_2 \subset \mathfrak{so}(8, \mathbb{R})$ arises as the intersection of non-conjugate $\mathfrak{spin}(7,\mathbb{R})$ sub-algebras, and also as the stabilizer of the outer automorphism group $\mathrm{Out}(\mathfrak{so}(8,\mathbb{R}))$. It is argued that $\mathfrak{spin}(1,7)$ is in some sense the more natural stage for triality to play out upon, and it is shown that triality can be seen to be simply the multiplication of bases by third roots of unity, just as dualities are often the application of second roots of unity upon Lie algebra bases. Once these are understood a short discussion is had about obstacles to a theory of triality which attempt to explain the three generations of matter via some form of triality.

math.RT↗

arXiv2025

A Convenient Representation Theory of Lorentzian Pseudo-Tensors: $\mathcal{P}$ and $\mathcal{T}$ in $\operatorname{O}(1,3)$

A novel approach to the finite dimensional representation theory of the entire Lorentz group $\operatorname{O}(1,3)$ is presented. It is shown how the entire Lorentz group may be understood as a semi-direct product between its identity component and the Klein four group of spacetime reflections: $\operatorname{O}(1,3) = \operatorname{SO}^+(1,3) \rtimes \operatorname{K}_4$. This gives way to a convenient classification of tensors transforming under $\operatorname{O}(1,3)$, namely that there are four representations of $\operatorname{O}(1,3)$ for each representation of $\operatorname{SO}^+(1,3)$, and it is shown how the representation theory of the Klein group $\operatorname{K}_4$ allows for simple book keeping of the spacetime reflection properties of general Lorentzian tensors, and combinations thereof, with several examples given. There is a brief discussion of the time reversal of the electromagnetic field, concluding in agreement with standard texts such as Jackson, and works by Malament.

math-ph↗

arXiv2023

A novel approach to the global analysis of proton form factors in elastic electron-proton scattering

We present a novel method for the global analysis of elastic electron-proton scattering when combining data sets from experiments with different overall normalization uncertainties. The method is a modification of one employed by the NNPDF collaboration in the fitting of parton distribution data. This method is an alternative to the 'penalty trick' method traditionally employed in global fits to proton electric and magnetic form factors, while avoiding the biases inherent in that approach. We discuss issues that arise when extending the method to nonlinear models. For data with $Q^2>1$ GeV$^2$ we find relatively minor differences to traditional model fits when the normalization uncertainties from different experiments are correctly accounted for. We discuss implications of this method for the well-known discrepancy between the form factor ratio $G_E/G_M$ extracted from the Rosenbluth and polarization transfer techniques.

hep-ph↗