=Paper=
{{Paper
|id=Vol-3091/paper06
|storemode=property
|title=Natural non-group symmetry in modern applications
|pdfUrl=https://ceur-ws.org/Vol-3091/paper06.pdf
|volume=Vol-3091
|authors=Mikhail Kharinov
}}
==Natural non-group symmetry in modern applications==
https://ceur-ws.org/Vol-3091/paper06.pdf
Natural non‐group symmetry in modern applications
Mikhail Kharinov 1
1
St. Petersburg Federal Research Center of the Russian Academy of Sciences (SPC RAS), 39, 14th Line V.O., St.
Petersburg, 199178, Russia
Abstract
Intuitively perceived symmetry is formalized for effective application in physics, mathematics,
and engineering. In this regard, several scientific research directions are indicated, which are
expressed by three generalizations: a) the concept of symmetry using the example of
normalized Hadamard matrices; b) cross vector product for the cases of three arguments and
seven-dimensional space, c) Lorentz transformations for doubling the spacetime dimension.
To generalize and formalize the concept of symmetry, the preservation of the symmetry of
matrices under permutations of rows (columns) is studied. It is shown that the set of symmetry-
preserving permutations does not constitute a group. For the development of the octonion
toolkit and the best generalization of the vector product, based on symmetry considerations,
the decomposition of the triple product of octonions into the sum of a triple anticommutator, a
triple commutator (generalized vector product) and an associator is deduced. To begin the
generalization of Lorentz transformations Lorentz boost is recorded in terms of quaternions so
that the treated expressions retain their meaning in the octonionic space. To speed up the
assimilation of the research results, the paper proposes some elementary information on the
three listed topics, which it is desirable to place in reference books, as well as bring to the
attention of students in general education courses at technical universities.
Keywords 1
Symmetrixes, symmetry, cross product, Lorentz boosts, quaternions
1. Introduction
More than 30 years ago the author has been starting research in mathematical physics with the
formalization of natural (intuitively perceived) symmetry and published out the results in the USSR
patent [1] for a game series. Later, this type of games was reinvented in Japan and was called
"Symmetrixes" [2]. It was conceived, starting with games, then to publish the results related to the
properties of spacetime symmetry. However, to our surprise, at the moment we found that some early
scientific results partially retained their novelty. So, in this paper we reveal the scientific background
of the mentioned games and offer the ideas of its further development and utilization.
In Section 2, the additive decomposition of an operator into self-adjoint and skew-symmetric parts
is generalized, the notions of permutation and reassignment are refined, as well as the notion of
permutational matrix symmetry is introduced. Section 3 deals with the development of the apparatus of
hypercomplex numbers by generalizing the cross vector product. Section 4 describes the progress of
work on the generalization of the Lorentz transformations. In the Conclusion some corrections and
additions to commonly used reference books, as well as classic textbooks on the related topic, are
discussed using specific examples.
Proceedings of MIP Computing-V 2022: V International Scientific Workshop on Modeling, Information Processing and Computing,
January 25, 2022, Krasnoyarsk, Russia
EMAIL: khar@iias.spb.su (Mikhail Kharinov)
ORCID: 0000-0002-5166-1381 (Mikhail Kharinov)
© 2022 Copyright for this paper by its authors.
Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0).
CEUR Workshop Proceedings (CEUR-WS.org)
2. Permutable symmetric truncated normalized Hadamard matrices
A truncated (down-sized) normalized Hadamard matrix (NHM) is understood as a normalized
Hadamard matrix [3] without the row and the column of only 1 that are deleted. NHMs arise in linear
algebra when generalizing the additive decomposition of a linear operator into symmetric (self-adjoint)
and skewsymmetric parts (Figure 1, [1,4]).
Expansion of a linear operator A 1 2 7 4 5 5 1 6 3
2 4 4 5 2 7 7 7 7
A
3 7 2 1 6 3 6 1 5
A A
2
4 1 3 3 7 6 5 3 1
A A A
5 3 5 6 3 2 4 4 2
2
6 6 1 2 1 4 2 2 4
7 5 6 7 4 1 3 5 6
Figure 1: Permutable symmetric truncated NHMs
Black and white cells in Figure 1 denote 1 and, 1 respectively. At the top left in Figure 1, the
well-known additive decomposition of a linear operator is written out. It is described by the truncated
1 1 NHM, that contains the single 1 , and shown at the bottom left. The next, really black-and-white
truncated 3 3 NHM describes the decomposition of a linear operator into four symmetric-
skewsymmetric parts, which either change or retain their sign under the action of each of the two
operations, for example, Hermitian conjugation " " and the operator inversion " ". This 3 3 matrix
1
preserves symmetry for any row permutation and is therefore always symmetric. The next black-and-
white symmetric truncated 7 7 NHM with numbered rows describes the additive operator expansion
into an octet of symmetric-skewsymmetric parts for three commuting operations of Hermitian
conjugation type, which form an abelian group of the self-inverse operations. The extreme right table
lists eight digital columns obtained by permutations of the rows of the truncated 7 7 NHM that
preserve its symmetry. Cyclic repetition of any of eight specified permutations preserves the matrix
symmetry. Each column from the columns, isolated by bold lines, gives, respectively, two, three, and
six additional symmetric matrices. A total of 28 symmetric matrices are obtained.
It's remarkable, that the symmetry-preserving row permutations of 7 7 matrix in Figure 1 do not
form a group, but are only a union of cyclic subgroups of some general group of 168 permutations that
preserve the so-called "hidden symmetry" [1, 4]. The example of a matrix possessing hidden symmetry
is shown in Figure 2
7 3 5 6 1 2 4
6
5
2
7
1
4
3
Figure 2: Asymmetrical matrix with hidden symmetry
The asymmetric matrix in Figure 2 is obtained by composition of a pair of permutations from
different cyclic subgroups Figure 1.
Regarding the expected group properties of the permutable Hadamard matrices, note that the set of
their rows, together with the unit row containing only the 1 s, form an Abelian group of the self-
inverse elements with respect to termwise row multiplication.
It is verified that among the symmetric black-and-white 7 7 matrices from all the different rows,
the symmetric truncated 7 7 NHM are maximally permutable [1].
As it is easy to establish, the permutability of symmetric truncated 7 7 NHM determines the set
of rows and the matrices themselves, accurate to the reversal of the roles of black and white cells.
Moreover, to restore the septet of rows and the matrices, it is enough to use just three digital columns,
selected from 28 ones according to the specific algorithm [5].
For the convenience of checking the statements under consideration, it is useful to distinguish
between the concepts of "substitution", "permutation" and "reassignment".
As usual, substitution here refers to the reversible mapping of a set of some elements onto
1 2 3 4 2 4 1 3
3 4 1 2 4 2 3 1
themselves, for example, for digital elements: . Let's order the top row
in substitutions, as in the left one in the given example. Omitting the natural series of digits in the top
row of substitutions, we represent the product of substitutions as the product of rows. For example
3 4 1 2 3 1 2 4 1 3 4 2 . Similarly, the product of substitutions is represented as the
product of columns:
3 3 1
4 1 3
(1)
1 2 4
2 4 2.
Let's treat the product of a pair of substitutions on the left side (1) either as a permutation of the left
column of the pair, or as a reassignment of its right column, by which the column on the right side (1)
is obtained.
The meaning of the formulated permutation and reassignment definition manifests itself when
multiplying by a given column on the right or/and on the left of some table of several columns and lies
in the fact that the columns of the table can be transformed term by term, or the entire table can be
transformed with the same result. Moreover, due to the associativity of a substitution composition, when
performing multiplication on the right and on the left, the result does not depend on the order in which
the right and left multiplication is performed.
For a pair , of any symmetry-preserving permutations from the set S , it is true that their
composition also belongs to S :
S , S S (2)
Property (2) means that the complete permutability table of 28 columns for Figure 1 is preserved up
to the permutation of the columns when multiplying from the left and also from the right by any of its
columns.
Property (2) implies that cyclic permutation groups are subsets of the set S :
S k S , k 1, 2,... (3)
A specific feature of just the 7 7 black-and-white matrix in Figure 1 is that it is antisymmetric
with respect to the secondary diagonal, wherein antisymmetry notion is defined with the necessary
reservations regarding the elements occupying the secondary diagonal [1]. According to the [1], non-
trivial permutations of the columns of the truncated 7 7 NHM in Figure 1, provide to get eight such
symmetric-antisymmetric matrices of the 7th order. Are there such symmetric-antisymmetric matrices
of the 15th order, that is the current important question.
3. Twofold generalization of a cross vector product
The above-mentioned decomposition of the operator into symmetric-skewsymmetric parts turned
out to be useful in a twofold generalization of the vector product to the case of three arguments, as well
as to the seven-dimensional subspace of the eight-dimensional octonionic space [6,7].
In [7], a linear operator is considered, which is produced by conjugation of a vector argument with
subsequent multiplication by fixed vectors on the left and on the right. Then, the decomposition of the
product of three octonions into four symmetric-skewsymmetric parts is obtained using a pair of suitable
operations of the Hermitian conjugation type. One of four part turned out to be zero. The rest of the
parts are: triple anticommutator, triple commutator and associator. Triple commutator 1 2 3 is
u ,u ,u
defined in two ways by the formulaе:
u u u u u u u u u u3u2 u1
u1 , u2 , u3 1 2 3 3 2 1 1 2 3 (4)
2 2 ,
where u1 , u2 , u3 are arbitrary octonions, u2 2(u2 , i0 )i0 u2 is conjugated octonion u2 , i0 is the
multiplicative identity, i.e. a unit vector along the real axis.
The triple commutator 1 2 3 possesses the property of antipermutability of its arguments, is
u ,u ,u
orthogonal to each argument, and turns into an ordinary two-argument cross vector product when the
central argument is replaced by the multiplicative identity i0 . It also has other properties that are
characteristic for an ordinary two-argument cross vector product that are detailed in [7]. So, the triple
commutator 1 2 3 is exactly what it is a doubly generalized cross vector product, coinciding with
u ,u ,u
the conventional two-argument cross vector product 1 3 written in the space of quaternions or
u ,u
octonions [7]. And the expressions for 1 2 3 taken from [7] complete the prolonged search [8–11]
u ,u ,u
for most perspective generalization of cross vector product.
4. The problem to generalize Lorentz transformations
It was William Rowan Hamilton who first discovered quaternions as the spacetime [12]. Nowadays
the generalization of quaternions (octonions) and the double generalization of the cross vector product
have been invented, which provides convenient work with non-associative octonions. So, а
generalization of the Lorentz transformations suggests itself in order to clarify and develop the motion
laws. On the way to this goal, a quaternionic record of Lorentz boost was found in [13,14]. It turns out
that the Lorentz boost is decomposed into a linear combination of rotation and orthogonal multiplicative
transformation, expressing in twofold ways by the formulae:
Lu aua sh nu L u aua sh un , (5)
a i0 ch n sh
where the cross denotes the Hermitian conjugation, 2 2 , n is the unit vector along the
v
th
speed, is the rapidity: c , v is the speed magnitude, c is the speed of light.
It should be noted that in [15] the Lorentz boost is expressed by the half-expression (5), but this is
done by increasing the dimension of the vector space.
It is noteworthy that quaternionic expressions of Lorentz boost, as well as the rotation expression
V u bub, b, b 1
do not depend on the multiplication order and retain their meaning in octonions,
that exemplify the possible generalization of Lorentz transformations. At first glance, this is quite
sufficient for the eight-dimensional generalization of the Lorentz transformations VL as a superposition
of rotation V and Lorentz boost L . This may be true, but other options should also be considered for
comparison.
In four-dimensional space, both the rotation V and the Lorentz boost L are the elementary
transformations that modify some two-dimensional plane and do not change the orthogonal vectors in
another two-dimensional plane.
In both quaternions and octonions, Lorentz boost (5) describes the stretching by a certain number of
times of one vector and the contraction by the same number of times of another vector that belong to
the same complex plane while preserving purely spatial vectors that are orthogonal to this complex
plane. The transformation of the rotation
V u bub, b, b 1
in octonions is more complicated than
the Lorentz boost, because it modifies the rest six-dimensional subspace of pure spatial vectors, while
maintaining the complex plane defined by the rotational axis.
Lorentz boost L differs significantly from the rotation V in that it has a full quartet of basis
eigenvectors, while the rotation V through a nontrivial angle preserves only the directions of the
rotational and the time axes. For this reason, the general Lorentz transformations in the form of a
composition VL of rotation V and boost L are subdivided into boost-like transformations with a
quartet of basis vectors, and the rest ones, referring to as rotation-like. The most interesting is that the
composition L1 L2 of Lorentz boosts L1 and L2 is always a boost-like transformation [14].
The eigenvectors for the composition L1 L2 of the Lorentz boosts together with the corresponding
eigenvalues are listed in Table 1.
Table 1
Eigenvectors for the composition of Lorentz boosts L1 L2
Notation Eigenvector Eigenvalue
c i0 d exp exp
0
c1 i0 d exp exp
1 2 2 1
coth n1 , n 2 coth coth n1 , n 2 coth
c2 i0 n1 2 2 n
2
2 2 1
1 n1 , n 2 1 n1 , n 2
2 2
c3 n1 , n2 1
In Table 1 n1 and n2 are the unit spatial vectors along the considered intersecting velocities, such
that 1 1 2 2 and 1 0 2 0 . The cross vector product 1 2 is directed along
n ,n n ,n 1 n ,i n ,i 0 n ,n
the Wigner rotational axis [16], so that
n1 , n2 1 n1 , n2
2
. The spatial part of the eigenvectors
c0 and c1 depends on the eigenvalue and, up to the sign, coincides with the unit vector d that is
defined as a function of eigenvalue in the form [14]:
n1 sinh 1 n2 sinh 2
d 2 2
1 2 (6)
cosh cosh
2 2 .
d exp d exp
The spatial parts and of the eigenvectors c0 and c1 are obtained by
exp exp
substituting in (6) the values by and , respectively. The scalar parameter is
defined in accordance with well-known cosine rule:
cosh 1 cosh 2 n1 , n2 sinh 1 sinh 2
cosh (7)
2 2 2 2 2 .
and the scalar parameters 1 and 2 are the rapidities, such that the velocities v1 , v 2 divided by scalar
speed of light c are expressed as v1 c n1 tanh 1 , v 2 c n2 tanh 2 . Note that (7) refers to the half
hyperbolic angles 2 , 1 2 and 2 2 , while the well-known velocity addition is expressed via
holistic hyperbolic angles , 1 and 2 [17,18].
It should be noted that the expressions for the eigenvectors of Table 1 turned out to be laconic.
However, they are obtained by rather cumbersome intermediate calculations in terms of quaternions
[14]. Meanwhile, it is precisely the laconism of calculations that is usually the main advantage of
quaternions. So, the use of quaternions alone is not enough for a transparent derivation of the formulae
Table 1. Apparently, for a transparent representation of obtaining of the eigenvectors Table 1 it will be
useful to decompose L1 L2 into symmetric-skewsymmetric parts according to Figure 1 and
accompanying symmetry considerations. To generalize Lorentz transformations, it seems worth trying
to generalize the laconic formulae (5) for the Lorentz boost to the case of a composition L1 L2 of boosts
L1 and L2 , say, expressing them in terms of eigenvectors and eigenvalues from Table 1.
4
The eigenvectors from Table 1 form the basis of the considered space R . Their pairwise
pseudoscalar products i k
c , c , i , k 0, 1, 2, 3
, accounting for commutativity, are given by the
formulae (8).
c0 , c0 c0 , c2 c0 , c3 c1, c1 c1 , c2 c1 , c3 c2 , c3 0,
sinh 2
c0 , c1 2 1 2
2
,
2 2
cosh cosh 2cosh cosh 1 cosh 2
2 2 2 2 2 (8)
1 2 1 2
coth 2
coth 2
2 n1 , n2 coth coth
c2 , c2 1 2 2 2 2 ,
1 n1 , n2
2
c3 , c3 n1 , n2 2 1.
.
Using formulae (8), it is easy to expand an arbitrary vector in the basis of the eigenvectors of the
composition L1L2 of boosts L1 and L2 , and then obtain an expression for the composition L1L2 in terms
of its eigenvectors c0 c1 c2 c3 :
c exp u, c1 c1 exp u, c0 u, c2 c u, c3
L1L2u 0 c2 (9)
c0 , c1 c2 , c2 3 c3 , c3 .
Expression (9) determines the composition of the Lorentz boosts L1L2 in terms of its own
eigenvectors. Obviously, this formula can be rewritten as a linear combination of orthogonal
transformations, which are elegantly expressed in terms of quaternionic multiplication. In this case, we
obtain a generalization of the quaternionic record of the single Lorentz boost (5) to the case of a
composition of a pair of Lorentz boosts, which is useful for further generalization to the case of an
eight-dimensional octonionic space.
5. Conclusions
The current tasks formulated in the final paragraphs of the second and fourth sections are quite
capable of solving by interested senior students.
And it is didactic, that in [19], in the last paragraph of the section "Quaternions in Vector Symbolics",
Erwin Madelung refers the rotation bub, b, b 1 of the complex plane in the quaternionic space to
the class of Lorentz transformations. It would be appropriate to supplement this paragraph with the
formulae (5) for Lorentz boost, so that the beginners do not mix it with the orthogonal transformations.
When assimilating hypercomplex numbers (quaternions and octonions) according to the well-known
book [6], one should pay attention to the fact that formula (8) is written out twice in the book, but when
used for the first time the unfortunate mistake was made in this formula (the comma is omitted), which
radically changes the meaning. The definition and examples of utilization of the doubly generalized
vector product (4) can be presented as a useful application to [6]. Formal clarification (1) of difference
between the concepts of permutation and reassignment (redesignation) compensate for the rather
confused interpretation of these concepts in most reference books on mathematics and physics.
The latter examples show the need for some modernization of classical reference books and
guidelines, aids for lecturers teaching students to solve the problems of current interest. The method of
additive expansion of а linear operator and concomitant symmetry considerations Figure 1 deserve
increased attention, as well as an introduction to the vector product in terms of quaternions. This is so
because the conventional cross vector product appeals to intuition in the "left-hand rule," which prevents
its generalization. Compared to the conventional one, the quaternionic cross vector product was
invented earlier and is more promising for the effective development of scientific and, all the more sо,
engineering research. In particular, it is promising for generalizing of classical transformations of
coordinates to better understand the laws of motion. Let's take this into account in the future.
6. Acknowledgements
This research was funded within the framework of the budgetary theme 0060-2019-0011
(Fundamentals and technologies of big data for sociocyberphysical systems).
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