short communications
A note on the wedge reversion antisymmetry operation and 51 types of physical quantities in arbitrary dimensions
^{a}Institute of Crystallography, RWTH Aachen University, Jägerstraße 17–19, D52066 Aachen, Germany, ^{b}Jülich Centre for Neutron Science at Heinz MaierLeibnitz Zentrum, Forschungszentrum Jülich GmbH, Lichtenbergstraße 1, D85747 Garching, Germany, and ^{c}Faculty of Physics, University of Warsaw, Pasteura 5, PL 02093 Warsaw, Poland
^{*}Correspondence email: piotr.fabrykiewicz@frm2.tum.de
The paper by Gopalan [(2020). Acta Cryst. A76, 318–327] presented an enumeration of the 41 physical quantity types in nonrelativistic physics, in arbitrary dimensions, based on the formalism of Clifford algebra. Gopalan considered three antisymmetries: spatial inversion, 1, time reversal, 1′, and wedge reversion, 1^{†}. A consideration of the set of all seven antisymmetries (1, 1′, 1^{†}, 1′^{†}, 1^{†}, 1′, 1′^{†}) leads to an extension of the results obtained by Gopalan. It is shown that there are 51 types of physical quantities with distinct symmetry properties in total.
Keywords: multivectors; wedge reversion; antisymmetry; Clifford algebra.
The paper by Gopalan (2020) presented an enumeration of the 41 types of physical quantities in nonrelativistic physics in arbitrary dimensions within the formalism of Clifford algebra (Lounesto, 2009). This classification is based on three antisymmetries: spatial inversion, , time reversal, , and wedge reversion, . [Note that, in Clifford algebra, spatial inversion is termed (main) grade (Lounesto, 2009).] The 41 types of multivectors representing physical quantities were derived and presented in Table 1 of Gopalan (2020). Gopalan's classification is an extension of the classification of threedimensional vectorlike physical quantities (Hlinka, 2014) to arbitrary dimensions.
The transformation of the physical properties represented by the principal multivectors S′, S, V′, V, B′, B, T′, T, or their combinations, under the antisymmetries , , were considered by Gopalan (2020). S, V, B and T denote scalar, vector, bivector and trivector, respectively. The prime symbol ′ means invariance to time reversal, . There are three different outcomes of the operation, even (e), odd (o) or mixed (m). Mixed means that it is neither even nor odd, as explained in the short example in Table 1.

These outcomes for all physical properties are shown for each multivector type in Table 1 of Gopalan (2020) in columns titled `Action of , , '. The actions of the remaining antisymmetries 1′, , and 1′^{†} were not given in Table 1 of Gopalan (2020). [In Clifford algebra, the product of spatial inversion and wedge reversion, , is termed Clifford conjugation (Lounesto, 2009).] The consideration of all seven operations leads to new results, which are given here.
When a physical quantity is represented by a sum of two or more different principal multivector types then the action of at least four antisymmetries on this quantity gives mixed results. The analysis of the action of only three antisymmetries by Gopalan (2020) does not provide a unique solution.
Let I_{1} and I_{2} be two different antisymmetries (any out of the seven). If the action of both I_{1} and I_{2} on some multivector is mixed then the action of I_{1} · I_{2} on this multivector can be even, odd or mixed. Specifying the action of only three antisymmetries (especially , and ) on a multivector, as was considered by Gopalan (2020), is not sufficient to obtain the result of the action of the remaining four antisymmetries; see a simplified example with three antisymmetries in Table 1. This has led to a clustering of different multivector types into one type in Table 1 of Gopalan (2020): the types numbered 16, 19, 22, 25, 28 and 31 should be separated into two types each and type 38 into five types. This gives in total ten new multivector types which were not given by Gopalan (2020), as shown in Table 2 for all seven antisymmetries. New labels for the X, Y, Z multivectors are proposed in Table 2 in a coherent notation, which uses four out of the eight principal multivectors. An extended version of Table 2 with grades and examples of multivectors is given in Table 3, which is the final table for these new results, with all 51 multivector types describing the action of all seven antisymmetries, given in the same layout as Table 1 of Gopalan (2020).


Supporting information
Splitting of multivector types. DOI: https://doi.org/10.1107/S2053273323003303/ib5117sup1.pdf
Acknowledgements
Thanks are due to Radosław Przeniosło and Izabela Sosnowska (University of Warsaw) for inspiring discussions. Open access funding enabled and organized by Projekt DEAL.
References
Gopalan, V. (2020). Acta Cryst. A76, 318–327. Web of Science CrossRef IUCr Journals Google Scholar
Hlinka, J. (2014). Phys. Rev. Lett. 113, 165502. Web of Science CrossRef PubMed Google Scholar
Lounesto, P. (2009). Clifford Algebras and Spinors, 2nd ed. Cambridge University Press. Google Scholar
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