==== Front
Acta Crystallogr A Found Adv
Acta Crystallogr A Found Adv
Acta Cryst. A
Acta Crystallographica. Section A, Foundations and Advances
2053-2733
International Union of Crystallography

37272370
ib5117
10.1107/S2053273323003303
ACSAD7
S2053273323003303
Short Communications
A note on the wedge reversion antisymmetry operation and 51 types of physical quantities in arbitrary dimensions
Wedge reversion antisymmetry
https://orcid.org/0000-0001-7952-2466
Fabrykiewicz Piotr abc*
a Institute of Crystallography, RWTH Aachen University, Jägerstraße 17–19, D-52066 Aachen, Germany
b Jülich Centre for Neutron Science at Heinz Maier-Leibnitz Zentrum, Forschungszentrum Jülich GmbH, Lichtenbergstraße 1, D-85747 Garching, Germany
c Faculty of Physics, University of Warsaw, Pasteura 5, PL 02-093 Warsaw, Poland
Billinge S. J. L. Editor
Columbia University, USA
Correspondence e-mail: piotr.fabrykiewicz@frm2.tum.de
01 7 2023
05 6 2023
05 6 2023
79 Pt 4 a230400 381384
06 2 2023
11 4 2023
© Piotr Fabrykiewicz 2023
2023
https://creativecommons.org/licenses/by/4.0/ This is an open-access article distributed under the terms of the Creative Commons Attribution (CC-BY) Licence, which permits unrestricted use, distribution, and reproduction in any medium, provided the original authors and source are cited.
A full version of this article is available from Crystallography Journals Online.
It is shown that there are 51 types of physical quantities in arbitrary dimensions with distinct transformations by wedge reversion symmetry. In the paper by Gopalan [(2020). Acta Cryst. A76, 318–327] only 41 types were enumerated.

The paper by Gopalan [(2020). Acta Cryst. A76, 318–327] presented an enumeration of the 41 physical quantity types in non-relativistic 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.

multivectors
wedge reversion
antisymmetry
Clifford algebra
==== Body
pmcThe paper by Gopalan (2020 ▸) presented an enumeration of the 41 types of physical quantities in non-relativistic 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 involution (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 three-dimensional vector-like 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 ▸).

Supplementary Material

Splitting of multivector types. DOI: 10.1107/S2053273323003303/ib5117sup1.pdf

Thanks are due to Radosław Przeniosło and Izabela Sosnowska (University of Warsaw) for inspiring discussions. Open access funding enabled and organized by Projekt DEAL.

Table 1 An example of the action of , and 1′ antisymmetries on several multivectors

The action of antisymmetries and on S+V′ and S+V′+V gives mixed results, while the action of the product antisymmetry 1′ can be odd or mixed.

 			1′	
S	+	−	−	
Even	Odd	Odd	
 	 	 	 	
V′	−	+	−	
Odd	Even	Odd	
 	 	 	 	
V	−	−	+	
Odd	Odd	Even	
 	 	 	 	
S+V′	+−	−+	−−	
Mixed	Mixed	Odd	
 	 	 	 	
S+V′+V	+−−	−+−	−−+	
Mixed	Mixed	Mixed	

Table 2 Splitting of multivector types, with the left-hand side displaying the number, stabilizer subgroup, label and action of , and antisymmetries as given by Gopalan (2020 ▸), and the right-hand side displaying the number, stabilizer subgroup, label and action of all antisymmetries as presented in this work

Considering the action of all antisymetries leads to splitting of multivector types. The last three rows describe the new labels of the X, Y and Z multivector types, without splitting. An extended version of this table with grades and examples of multivectors is available in the supporting information.

Work of Gopalan (2020 ▸)	 	This paper	
 	 	 	Action of	 	 	 	 	Action of	
No.	SS	Label				 	No.	SS	Label					1′		1′†	
16		SB′(S′, B)	e	m	m		16a		SB′	e	m	m	o	m	m	o	
16b	S′SB′B	e	m	m	m	m	m	m	
 	 	 	 	 	 	 	 	 	 	 	 	 	 	 	 	 	
19		V′B′(S′, T′)	m	e	m		19a		V′B′	m	e	m	m	m	o	o	
19b	S′V′B′T′	m	e	m	m	m	m	m	
 	 	 	 	 	 	 	 	 	 	 	 	 	 	 	 	 	
22		SV′(S′, V)	m	m	e		22a		SV′	m	m	e	m	o	m	o	
22b	S′SV′V	m	m	e	m	m	m	m	
 	 	 	 	 	 	 	 	 	 	 	 	 	 	 	 	 	
25		V′B(S′, T)	m	m	m		25a		V′B	m	m	m	e	o	o	m	
25b	S′V′BT	m	m	m	e	m	m	m	
 	 	 	 	 	 	 	 	 	 	 	 	 	 	 	 	 	
28	1′	VB′(S′, T)	m	m	m		28a	1′	VB′	m	m	m	o	e	o	m	
28b	S′VB′T	m	m	m	m	e	m	m	
 	 	 	 	 	 	 	 	 	 	 	 	 	 	 	 	 	
31		ST′(S′, T)	m	m	m		31a		ST′	m	m	m	o	o	e	m	
31b	S′ST′T	m	m	m	m	m	e	m	
 	 	 	 	 	 	 	 	 	 	 	 	 	 	 	 	 	
38	1	W	m	m	m		38a	1	SVB′T′	m	m	m	o	m	m	m	
38b	SV′BT′	m	m	m	m	o	m	m	
38c	V′VB′B	m	m	m	m	m	o	m	
38d	SV′B′T	m	m	m	m	m	m	o	
38e	S′SV′VB′BT′T	m	m	m	m	m	m	m	
 	 	 	 	 	 	 	 	 	 	 	 	 	 	 	 	 	
39	1	X	m	m	o	→	39	1	B′BT′T	m	m	o	m	m	m	m	
 	 	 	 	 	 	 	 	 	 	 	 	 	 	 	 	 	
40	1	Y	m	o	m	→	40	1	SVBT	m	o	m	m	m	m	m	
 	 	 	 	 	 	 	 	 	 	 	 	 	 	 	 	 	
41	1	Z	o	m	m	→	41	1	V′VT′T	o	m	m	m	m	m	m	

Table 3 Classification of extended multivector types for physical properties using the same notation as in Table 1 of Gopalan (2020 ▸)

The actions of all seven generalized inversions and grades are given explicitly. Entries in bold in columns 1 and 2 are the eight principal multivector types. Note that the last column contains sums (not products) of principal multivectors, but the ‘+’ signs are omitted.

 	 	 	 	Action of	 	 	
New No.	Old No.	SS	Label					1′		1′†	Grades	Multivectors (omitting ‘+’ signs)	
1	1		S′	e	e	e	e	e	e	e	4g	S′	
 	 	 	 	 	 	 	 	 	 	 	 	 	
2	2		V′	o	e	e	e	o	o	o	4g+1	V′	
3	3	S′V′	m	e	e	e	m	m	m	4g, 4g′+1	S′V′	
 	 	 	 	 	 	 	 	 	 	 	 	 	
4	4		B′	e	e	o	o	e	o	o	4g+2	B′	
5	5	S′B′	e	e	m	m	e	m	m	4g, 4g′+2	S′B′	
 	 	 	 	 	 	 	 	 	 	 	 	 	
6	6		T′	o	e	o	o	o	e	e	4g+3	T′	
7	7	S′T′	m	e	m	m	m	e	e	4g, 4g′+3	S′T′	
 	 	 	 	 	 	 	 	 	 	 	 	 	
8	8		S	e	o	e	o	o	e	o	4g	S	
9	9	S′S	e	m	e	m	m	e	m	4g, 4g′	S′S	
 	 	 	 	 	 	 	 	 	 	 	 	 	
10	10	1′	V	o	o	e	o	e	o	e	4g+1	V	
11	11	S′V	m	m	e	m	e	m	e	4g, 4g′+1	S′V	
 	 	 	 	 	 	 	 	 	 	 	 	 	
12	12		B	e	o	o	e	o	o	e	4g+2	B	
13	13	S′B	e	m	m	e	m	m	e	4g, 4g′+2	S′B	
 	 	 	 	 	 	 	 	 	 	 	 	 	
14	14		T	o	o	o	e	e	e	o	4g+3	T	
15	15	S′T	m	m	m	e	e	e	m	4g, 4g′+3	S′T	
 	 	 	 	 	 	 	 	 	 	 	 	 	
16	16a		SB′	e	m	m	o	m	m	o	4g, 4g′+2	SB′	
17	17	SB	e	o	m	m	o	m	m	4g, 4g′+2	SB	
18	18	B′B	e	m	o	m	m	o	m	4g+2, 4g′+2	B′B	
19	16b	S′SB′B	e	m	m	m	m	m	m	Three or four out of: 4g, 4g′, 4g′′+2, 4g′′′+2	SB′B, S′B′B, S′SB, S′SB′, S′SB′B	
 	 	 	 	 	 	 	 	 	 	 	 	 	
20	19a		V′B′	m	e	m	m	m	o	o	4g+1, 4g′+2	V′B′	
21	20	V′T′	o	e	m	m	o	m	m	4g+1, 4g′+3	V′T′	
22	21	B′T′	m	e	o	o	m	m	m	4g+2, 4g′+3	B′T′	
23	19b	S′V′B′T′	m	e	m	m	m	m	m	Three or four out of: 4g, 4g′+1, 4g′′+2, 4g′′′+3	V′B′T′, S′B′T′, S′V′T′, S′V′B′, S′V′B′T′	
 	 	 	 	 	 	 	 	 	 	 	 	 	
24	22a		SV′	m	m	e	m	o	m	o	4g, 4g′+1	SV′	
25	23	V′V	o	m	e	m	m	o	m	4g+1, 4g′+1	V′V	
26	24	SV	m	o	e	o	m	m	m	4g, 4g′+1	SV	
27	22b	S′SV′V	m	m	e	m	m	m	m	Three or four out of: 4g, 4g′, 4g′′+1, 4g′′′+1	SV′V, S′V′V, S′SV, S′SV′, S′SV′V	
 	 	 	 	 	 	 	 	 	 	 	 	 	
28	25a		V′B	m	m	m	e	o	o	m	4g+1, 4g′+2	V′B	
29	26	BT	m	o	o	e	m	m	m	4g+2, 4g′+3	BT	
30	27	V′T	o	m	m	e	m	m	o	4g+1, 4g′+3	V′T	
31	25b	S′V′BT	m	m	m	e	m	m	m	Three or four out of: 4g, 4g′+1, 4g′′+2, 4g′′′+3	V′BT, S′BT, S′V′T, S′V′B, S′V′BT	
 	 	 	 	 	 	 	 	 	 	 	 	 	
32	28a	1′	VB′	m	m	m	o	e	o	m	4g+1, 4g′+2	VB′	
33	29	VT	o	o	m	m	e	m	m	4g+1, 4g′+3	VT	
34	30	B′T	m	m	o	m	e	m	o	4g+2, 4g′+3	B′T	
35	28b	S′VB′T	m	m	m	m	e	m	m	Three or four out of: 4g, 4g′+1, 4g′′+2, 4g′′′+3	VB′T, S′B′T, S′VT, S′VB′, S′VB′T	
 	 	 	 	 	 	 	 	 	 	 	 	 	
36	31a		ST′	m	m	m	o	o	e	m	4g, 4g′+3	ST′	
37	32	T′T	o	m	o	m	m	e	m	4g+3, 4g′+3	T′T	
38	33	ST	m	o	m	m	m	e	o	4g, 4g′+3	ST	
39	31b	S′ST′T	m	m	m	m	m	e	m	Three or four out of: 4g, 4g′, 4g′′+3, 4g′′′+3	ST′T, S′T′T, S′ST, S′ST′, S′ST′T	
 	 	 	 	 	 	 	 	 	 	 	 	 	
40	34	1′†	VB	m	o	m	m	m	o	e	4g+1, 4g′+2	VB	
41	35	BT′	m	m	o	m	o	m	e	4g+2, 4g′+3	BT′	
42	36	VT′	o	m	m	o	m	m	e	4g+1, 4g′+3	VT′	
43	37	S′VBT′	m	m	m	m	m	m	e	Three or four out of: 4g, 4g′+1, 4g′′+2, 4g′′′+3	VBT′, S′BT′, S′VT′, S′VB, S′VBT′	
 	 	 	 	 	 	 	 	 	 	 	 	 	
44	38a	1	SVB′T′	m	m	m	o	m	m	m	Three or four out of: 4g, 4g′+1, 4g′′+2, 4g′′′+3	VB′T′, SB′T′, SVT′, SVB′, SVB′T′	
45	38b	SV′BT′	m	m	m	m	o	m	m	Three or four out of: 4g, 4g′+1, 4g′′+2, 4g′′′+3	V′BT′, SBT′, SV′T′, SV′B, SV′BT′	
46	38c	V′VB′B	m	m	m	m	m	o	m	Three or four out of: 4g+1, 4g′+1, 4g′′+2, 4g′′′+2	VB′B, V′B′B, V′VB, V′VB′, V′VB′B	
47	38d	SV′B′T	m	m	m	m	m	m	o	Three or four out of: 4g, 4g′+1, 4g′′+2, 4g′′′+3	V′B′T, SB′T, SV′T, SV′B′, SV′B′T	
48	39	B′BT′T	m	m	o	m	m	m	m	Three or four out of: 4g+2, 4g′+2, 4g′′+3, 4g′′′+3	BT′T, B′T′T, B′BT, B′BT′, B′BT′T	
49	40	SVBT	m	o	m	m	m	m	m	Three or four out of: 4g, 4g′+1, 4g′′+2, 4g′′′+3	VBT, SBT, SVT, SVB, SVBT	
50	41	V′VT′T	o	m	m	m	m	m	m	Three or four out of: 4g+1, 4g′+1, 4g′′+3, 4g′′′+3	VT′T, V′T′T, V′VT, V′VT′, V′VT′T	
 	 	 	 	 	 	 	 	 	 	 	 	
51	38e	S′SV′VB′BT′T	m	m	m	m	m	m	m	Varied	All other sums of: S′SV′VB′BT′T
==== Refs
References

Gopalan, V. (2020). Acta Cryst. A76, 318–327.
Hlinka, J. (2014). Phys. Rev. Lett. 113, 165502.
Lounesto, P. (2009). Clifford Algebras and Spinors, 2nd ed. Cambridge University Press.