GENERAL PROBLEMS OF THEORETICAL PHYSICS

YU.V. KHOROSHKOV 4, Petrovs’kyi Str., apt. 40, Kyiv 03087, Ukraine (e-mail: [email protected]) MIRROR SYMMETRY AS A BASIS FOR CONSTRUCTING PACS 02.20.Sv, 02.40.-k, 11.30.Er A SPACE-TIME CONTINUUM

By mirroring a one-dimensional oriented set in a complex space specially created on the basis of a symmetry, a mirror 푛-dimensional space with 푛 > 1 has been constructed. The geometry of the resulting space is described by the Clifford algebra. On the basis of the algebra of hyperbolic hypercomplex , a pseudo-Euclidean space has been constructed with the metric of the . The conditions for a function of a hyperbolic hypercomplex argument to be analytic (ℎ-analyticity) are obtained. The conditions implicitly contain the Maxwell equations for the 4-potential in a free space. K e y w o r d s: mirror transformation, Clifford algebra, hyperbolic hypercomplex numbers, Minkowski space.

1. Introduction ematical Physics) 1 testifies to the outlook of this di- Within the last decades, there emerged the necessity rection, and the scope of application permanently ex- in the search for the physical ideas and a mathemat- tends from mechanics and signal processing to chem- ical apparatus that would be capable of describing istry and biology. Concerning physical applications, the variety of physical phenomena from unique po- it should be noted that even the formulation of tra- sitions. For instance, a tendency is observed to re- ditional physical problems in the language of the consider the classical space-time concepts in favor Clifford algebra leads more often to a simplification of their treatment using various methods of alge- of mathematical calculations and/or new unexpected bra. In particular, the binary geometrophysics (the results. relational theory) [1], the algebraic theory of space- As an example, the analysis of the equations of time on the basis of [2, 3], and the al- motion for a material point in an inhomogeneous gebraic geometry on the basis of the Clifford al- anisotropic space in the Clifford algebra basis par- gebra [4, 5, 12], quaternions being examples of the tially agrees with the Einstein equations. Moreover, latter. the very principle of construction of equations is an The geometrical Clifford algebra pretends to play alternative to variational methods [18]. The equa- the role of the unified language in mathematical tions of motion for a classical particle with spin in physics [17] owing to a powerful mathematical appa- an electromagnetic field become substantially sim- ratus of all known complex and hypercomplex num- pler in the Clifford basis. In particular, instead of bers, which are naturally included into this alge- the nonlinear equations of perturbation theory, lin- bra. Regular conferences and numerous publications ear equations are to be solved [19], and the analysis in the framework of ICCA (International Conference of the specific features in the neutron-nucleus inter- on Clifford Algebras and their Applications in Math- 1 The 10-th ICCA was held in Tartu on 4–8 August, 2014. The ICCA proceedings are published in the ICCA journal “Ad- ○c YU.V. KHOROSHKOV, 2015 vances in Applied Clifford Algebras”. 468 ISSN 2071-0194. Ukr. J. Phys. 2015. Vol. 60, No. 5 Mirror Symmetry action, as well as the possibility for neutron-nucleus molecules to exist, on the basis of the Dirac equation [22] can be carried out, by using the Clifford algebra language 2. In turn, the Clifford algebra is connected with spinors [10]. The latter are specific geometric objects sometimes called “semivectors”, because only spin- tensors of rank 푛 > 1 are associated with observ- ables. Such a role of spinors and the spinor space in the geometry is similar to that played by the wave functions in quantum mechanics and requires an ad- ditional analysis. Since the Clifford algebra pretends to be a unified language of mathematical physics, the issue concerning its interrelation with fundamental laws of the nature arises as well. The principles of symmetry that form the basis of nature’s laws [6, 20] have been substantially cor- rected recently. Recent researches in the string the- Fig. 1. Action of the symmetry operators on the basis vectors ory [7, 8] unexpectedly drew a close attention to ex- of a coordinate system tended capabilities of the mirror symmetry. In those researches, two sets, 푋 and 푋′, were connected with research, which is planned to be presented in the each other as mirror pairs with the use of an auxil- future. iary space. The properties of the pairs obtained com- pletely depend on the properties of this space at 2. Construction special points and are not confined to the right-to- of an Auxiliary Space on the Basis left substitution, as in the conventional geometri- of the Complex-Plane Symmetry cal mirror symmetry. In this work, the idea of mir- According to E.´ Cartan [9], symmetry is defined as ror symmetry with the help of an auxiliary space the operation S of geometrical mirroring with respect is used to construct, from a 1-dimensional oriented to a hyperplane 푀 that passes through the coordinate set, a with dimensionality 푛 > 1 in origin, in the direction of an anisotropic vector 푠 or- the mirror space. As an auxiliary space, the com- thogonal to this hyperplane. Let us define the opera- plex space specially created on the basis of a sym- tions of mirror symmetry ±푠1 and ±푠2 in the coordi- metry is used, in which the role of singular points nate system on the plane that change the directions is played by certain planes (the planes of “mirrors”), of the basis vectors and the orientation of the coordi- relative to which the geometrical mirror symmetry nate system in accordance with the diagram depicted is obeyed. An analogy between the obtained space in Fig. 1. For an arbitrary point 푃 = (푥, 푦) on the and the properties of complex numbers is consistently complex plane, the symmetry operation in the vector drawn, and the algebra, geometry, and physical prop- basis 1 → (1, 0) and 2 → (0, 푖) can be presented by erties of the obtained vector space are analyzed. For the expression the presentation of the material to be logic and com- prehensive, both the known and original results are (︁ 푥 )︁ (︁ 푥 )︁ (︁푖푦)︁ (︁푥 + 푖푦)︁ (︁푧)︁ discussed. This work should be considered as a first (푠1 + 푠2) 푖푦 = −푖푦 + 푥 = 푥 − 푖푦 = 푧¯ , part – the substantiation of the method – of a wider (1)

2 In the Internet, one should pay attention to A.A. Ketsaris’s where 푖 is the imaginary unit, and 푧¯ means the com- lectures (http://toe-physics.org/ru/lectures.htm) or to the plex conjugate value. As a result of the mirror sym- works by R. Dahm, D. Hestenes, and N.G. Marchuk (appli- metry (1), the complex plane in the basis (1, 푖) is cations to the field theory), W.E. Baylis, B. Jancewicz, and mapped in the 3-dimensional basis (1, 0), (0, 1), (푖, 푖) P. Puska (electrodynamics), and D. Hestenes and D.S. Shi- rokov (Clifford algebra). as a plane that passes through the axis of imaginary ISSN 2071-0194. Ukr. J. Phys. 2015. Vol. 60, No. 5 469 Yu.V. Khoroshkov

where 푧1 ∙ 푧¯2 stands for the scalar and [푧1 × 푧¯2] for the vector product of the vectors 푧1 and 푧¯2. The real and imaginary parts of Eq. (2) are presented in the symmetric and antisymmetric forms, 1 Re(푧1푧2) = (푧1 ∙ 푧¯2) = (푧1푧¯2 + 푧2푧¯1), 2 (3) 푖 Im(푧 푧 ) = [푧 × 푧¯ ] = − (푧 푧¯ − 푧 푧¯ ). 1 2 1 2 2 1 2 2 1 Below, it will be shown that expressions (2) and (3) are true for hypercomplex numbers as well.

Fig. 2. Auxiliary complex space of mirror images 3. Construction of a Coordinate System in the Mirror Space The coordinate system is constructed in the mirror space by applying the operation of mirror symmetry to the basis vector 휙+ in the plane (휙+, 휙−) of the ++ space ℰ2 . For this purpose, let us write down the operator of mirror symmetry in the general form as follows: 훼 훼 S(훼) = cos 푠 + sin 푠 , (4) Fig. 3. Construction of an orthogonal coordinate system in 2 1 2 2 the mirror space where the angle 훼/2 is reckoned counterclockwise coordinates along the bisector of the orthogonal axes from the basis vector 휙+, and the angle 훼 itself is (1, 0) and (0, 1) (Fig. 2). The axis of imaginary coor- an angle between the new basis vectors in the mir- dinates is represented as a combination of two oppo- ror space and the mirror image of the vector 휙+. The sitely oriented imaginary axes with the common point factor 1/2 follows from the properties of mirror im- 0, which are orthogonal to the plane of basis vectors ages. It should be noted, first of all, that, owing to (1, 0) and (0, 1). Let us denote those orthogonal vec- this factor, the symmetry transformation within the angular interval 0 ≤ 훼/2 < 2휋 covers the mirror space tors as 휙+ = (1, 0) and 휙− = (0, 1). The complexifi- cation of the real-valued Euclidean plane by a combi- twice. Therefore, the transformations within the in- nation of symmetric imaginary axes generates a space tervals 0 ≤ 훼/2 < 휋 and 휋 ≤ 훼/2 < 2휋 have to be ℰ++ of two orthogonal complex planes in the bases analyzed separately. 2 In the mirror space, let us construct an orthogonal (휙+, 푖휙+) and (휙−, 푖휙−), respectively. The planes in- tersect each other along the imaginary axes, possess basis coordinate system on a plane corresponding to one common point 0, and determine, in the general the angles 훼 = 0, 휋/2, 휋, and 3휋/4. Substituting ++ those values into Eq. (4), we obtain, in the mirror case, a complex vector 훾 = (휉, 휂¯). If ℰ2 is consid- ered as a 3-dimensional space, any of its points 푃 has space, two pairs of oppositely oriented basis vectors: real-valued coordinates: 푃 = (푥1, 푥2, ±푦), where the (︁1)︁ 1 (︁1)︁ coordinate ±푦 corresponds to a combination of the 휙+ = S(0)휙+ = , 휓+ = S(휋/2)휙+ = √ , 0 2 1 imaginary axes, and the vector 훾 has the coordinates (5) 휉 = 푥1 + 푖푦 and 휂¯ = 푥2 − 푖푦. The representation of a (︁0)︁ 1 (︁−1)︁ complex by a 2-dimensional complex vector 휙− = S(휋)휙+ = , 휓− = S(3휋/4)휙+ = √ . 1 2 1 (푧,푧¯) has its logic substantiation. Really, the product of two complex numbers can be expressed in the form The operations of mirror symmetry S(훼) with re- ++ spect to the selected planes in the space ℰ2 map 푧1푧2 = (푧1 ∙ 푧¯2) + 푖[푧1 × 푧¯2], (2) (Fig. 3) the initial 1-dimensional vector space 푅1 onto

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a 2-dimensional vector one in the mirror space, the or- consider the vector 푒¯푖 to be a conjugate value of the thogonal coordinate axes of which are presented as a vector 푒푖. Such approach allows us to consider trans- bundle of two oppositely oriented basis vectors with formations in the spin space within the angular inter- a common zero point (the coordinate origin) (right val of 0 ≤ 훼 < 2휋 only, provided that the conjugation panel in Fig. 3). The obtained system of basis vectors operation is introduced as an independent basic op- agrees well with the concept of arbitrary choice for eration. Hence, formally, the mirror symmetry oper- the direction and the orientation of a coordinate sys- ations (5) performed the operation S: 푅1 ↦→ 푅2(2) in tem. However, it can be considered only as a possible the auxiliary complex space, where 푅2(2) denotes a 2- realization of the coordinate system, because the co- dimensional vector space in the mirror space with the ordinates of arbitrary point have an uncertainty con- bundle of oppositely oriented basis vectors. In turn, nected with the sign of numerical values. A possibility 푅2(2) is a geometric image of the algebra of matrix for the basis vector to have the opposite orientation operators 푒푖, which will be used below to construct can be formalized by introducing the vector-operator the Euclidean vector space 퐸2. of symmetry 푒, for which the oppositely oriented vec- tor pairs are eigenvectors, and its eigenvalues ±1 are 4. Algebra and Geometry responsible for that or another orientation of the ba- of 퐸2 in the Mirror Space sis vector. Since the eigenvectors are determined by In order to construct an Euclidean space, we must expressions (5), it is easy to obtain the matrix repre- define a scalar product. Since vectors are presented sentation of the vector-operators in the form by matrices, their product is noncommutative in the (︁1 0 )︁ (︁0 1)︁ general case. Therefore, formally, we may write 푒1 = 0 −1 , 푒2 = 1 0 , (6) 1 1 2 푒1푒2 = (푒1푒2 + 푒2푒1) + (푒1푒2 − 푒2푒1). (9) 푒푖휙푖± = ±1휙푖±, 푒푖푒푖 = 푒푖 = 1, 2 2 where 1 is a diagonal matrix, which can be considered The symmetric defines the internal, or as the identity symmetry operator. The right panel of scalar, product of vectors, Fig. 3 can be considered as a geometric image of basis vector-operators. 1 푒 ∙ 푒 = (푒 푒 + 푒 푒 ), (10) From Eq. (4), we obtain the following relation for 1 2 2 1 2 2 1 the S(훼) transformations in the interval of angles and the antisymmetric form defines their external 2휋 ≤ 훼 < 4휋: product, S(2휋 ≤ 훼 < 4휋) = −S(0 ≤ 훼 < 2휋). (7) 1 푒 = 푒 ∧ 푒 = (푒 푒 − 푒 푒 ). (11) It allows us to obtain two more pairs of orthogonal 12 1 2 2 1 2 2 1 ¯ basis vectors in the mirror space, 휙¯± and 휓±, which are connected with vectors (5) by the relations The condition of vector orthogonality is 푒1 ∙ 푒2 = 0, so that Eqs. (9) and (10) yield ¯ 휙¯± = −휙±; 휓± = −휓±. (8) It is evident that the basis vectors (8) define the 푒1푒2 = −푒2푒1, 푒12 = 푒1푒2, 푒21 = −푒12. (12) vector-operators 푒¯푖 = −푒푖. Expressions (7) and (8) il- From Eq. (12), it follows that the orthogonal vec- lustrate a well-known problem of sign uncertainty for tors anticommute with each other. The external prod- transformations in the spinor space 3. However, it can uct generates a bivector 푒12, which is a simple vec- be considered from another perspective. Really, tak- tor product for orthogonal vectors and changes its ing advantage of the analogy with the vector repre- sign if the order of multipliers (the order of in- sentation of a together with its con- dices) changes. The latter property is the property of jugate value in the auxiliary complex space, we will an antisymmetric form for orthogonal vectors rather than that of the noncommutativity of a matrix prod- 3 A description of this problem can be found both in textbooks uct. The bivector 푒12 also determines the symmetry (see, e.g., lecture 24 in: M.M. Postnikov, Lectures in Geom- of the coordinate system, because it is a combination etry. Semester II. (Mir, Moscow, 1983) and of symmetries. To find it, let us use an expression for special literature (see, e.g., p. 33 in [15]). ISSN 2071-0194. Ukr. J. Phys. 2015. Vol. 60, No. 5 471 Yu.V. Khoroshkov

the mirror image of the vector 푥 in the basis (푒1, 푒2) in will be referred to as an aggregate. The C2 algebra the direction of the unit vector 푠 [10]: 푥′ = −푠푥푠. Let contains all three known systems of complex numbers 푥 = 푒푖 and let the mirroring be performed twice: first with the basis pairs (1, 푖), (1, 푒), and (1, 휖), which have at 푠 = 푒2, and then at 푠 = 푒1. We obtain their analogs in C2: ∙ ordinary complex numbers 푎+푖푏, 푖2 = −1 1 → 1; ′ 푒푖 = 푒1푒2푒푖푒2푒1 = −푒푖, (13) 푖 → 푒12; 2 ∙ binary numbers 푎 + 푒푏, 푒 = 1 1 → 1; 푒 → 푒1, 푒2; where properties (12) were taken into account. Since 2 ∙ dual numbers 푎+휖푏, 휖 = 0 1 → 1; 휖 → (푒1 +푒12). the even images generate rotations, Eq. (13) corre- According to the properties of their absolute val- sponds to a rotation of the basis coordinate system 2 ∘ ues |푧| = 푧푧¯, which are defined by the expressions by an angle of 180 . In the case of 푅2(2), this oper- |푧|2 = 푎2+푏2, |푧|2 = 푎2−푏2, and |푧|2 = 푎2, those num- ation corresponds to the swapping of the basis vec- bers are sometimes called elliptic, hyperbolic, and tors with the subscripts “+” and “−”. The bivector parabolic complex numbers, respectively [11]. 푒12 determines the oriented area of a parallelogram But the capabilities of C2 are not restricted to that. constructed on the vectors 푒1 and 푒2. Therefore, the If we formally change from the basis (휙 , 휙 ) to rotation axis in 푅 is considered to be the axis of 1± 2± 2(2) the basis (푖휙 , 푖휙 ) in 푅+(+) in the case of C , bivector coordinates. Then, similarly to Eq. (8), we 1± 2± 2(2) 2 have we will obtain a new basis (푖푒1, 푖푒2, 푒12) in the right- handed coordinate system and a basis (푖푒1, 푖푒2, 푒21) 2 in the left-handed one. Let us introduce new no- 푒12휙12± = ±푖휙12±, 푒12 = −1. (14) tations for those vectors: (푖, 푗, 푘) for the right- +(+) handed coordinate system and (푖*, 푗*, 푘*) for the left- The complexification 푅2(2) → 푅2(2) as a result of in- troducing a bundle of imaginary rotation coordinate handed one. The Clifford algebras (15) constructed axes becomes possible owing to the extension of the on those basis vectors form the systems of elliptic symmetry concept to the case of a plane by including hypercomplex numbers, which are known as quater- ∘ nions. Historically, things so happened that the ma- the ±180 -rotations. The bivector 푒21 = −푒12 has the opposite orientation and changes the orientation jority of authors use the system with the left-handed orientation of basis vectors. By expressing a quater- of the system of basis vectors 푒1, 푒2, and 푒21, as a whole, to 푅+(+). If this system of symmetry vector- nion in terms of the scalar and vector parts, 푄(푎, 푢), 2(2) it is easy to verify with the help of Table for the mul- operators is appended by the operator of identical tiplication of basis vectors that the product of two symmetry, 1, we obtain a system of linearly indepen- quaternions, 푄(푎, 푢)푄(푏, 푣), looks like dent vectors, on which the Clifford algebra C2 over the field of real or complex numbers is built, 푄푅(푎푏 − 푢 ∙ 푣, 푎푣 + 푏푢 − 푢 × 푣) (16a) A = 푎01 + 푎1푒 + 푎2푒 + 푎12푒 . (15) 1 2 12 for the right-handed coordinate system and Expression (15), where 푎0 is a scalar, 푎1 and 푎2 are 푄 (푎푏 − 푢 ∙ 푣, 푎푣 + 푏푢 + 푢 × 푣) (16b) vector components, and 푎12 is a bivector component, 퐿

Multiplication tables of for the left-handed one, where 푢∙푣 denotes the scalar basis vectors for right- and left-handed product of vectors, and 푢 × 푣 stands for their vector orientations of the coordinate system product. The difference in expressions (16) consists in different signs before the vector product. * * * 푖 푗 푘 푖 푗 푘 It should be noted that, by their properties, the basis vectors of quaternions 푄푅 and 푄퐿 are bivec- 푖 −1 −푘 푗 푖* −1 푘* −푗* tors and form a 3-dimensional space of bivectors 퐵3 with the right- and left-handed orientations. A neces- 푗 푘 −1 −푖 푗* 푘* −1 푖 sity to consider the orientations of a coordinate sys- −푘 −푗 푖 −1 푘* 푗* −푖* −1 tem stems from the fact that the invariant of a com- plex or is formed in the general

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case as a of the initial and conjugate The complete Clifford algebra on C3 includes 8 basis numbers. In this work, the conjugate number is in- vectors: 0 troduced with the help of symmetry operations that ∙ the scalar (0-vector) 푎 1 = 퐴푆, 1 2 3 change the orientation of coordinate systems. Really, ∙ the vector 푎 푒1 + 푎 푒2 + 푎 푒3 = 퐴푉 , 23 31 12 considering aggregate (15) as a hypercomplex num- ∙ the bivector 푎 푒23 + 푎 푒31 + 푎 푒12 = 퐴퐵, 123 123 ber written in the general form in the 4-dimensional ∙ the 3-vector 푎 푒123 = 푎 푖 = 퐴푃 푆. space, the conjugate aggregate A¯ is constructed in The basis 3-vector or pseudo-scalar 푖 = 푒123 com- the basis mutes with all vectors and bivectors, and its proper- ties are determined by the expressions 1, −푒1, −푒2, −푒12, (17) 2 −1 −1 푖 = −1, 푖 = 푒321 = −푒123, 푖푖 = 1, 푖 = 푖1. (20) and the invariant or fundamental quadratic form (FQF), which is defined as |A|2 = AA¯ , looks like The geometric image of a 3-vector is the volume of a parallelepiped constructed on the vectors 푒푖. The |A|2 = (푎0)2 + (푎12)2 − (푎1)2 − (푎2)2. (18) multiplication table for the basis vectors can be ob- tained rather easily, if we take into account that the This indefinite FQF corresponds to the pseudo- even permutations of subscripts result in a multiplier (2) Euclidean space 퐻4 . It is easy to verify that, of +1, and odd ones in −1. The C3 algebra includes for quaternions, the FQF is positive definite and all operations of the ordinary in the corresponds to the Euclidean space 퐻4. A necessity description of classical mechanics [4]. However, the to be attentive to the orientation of the systems properties of C3 are not confined to that. Let us form of basis vectors and bivectors substantially grows, a system of hyperbolic hypercomplex numbers H and while changing from the 2-dimensional space to a a conjugate one, H¯ , as follows: 3-dimensional one, which is considered in the next 0 1 2 3 section. Really, the 3-dimensional space formally in- H = 푎 푒0 + 푎 푒1 + 푎 푒2 + 푎 푒3, cludes 3 oriented planes, in which the orthogonal po- (21) ¯ 0 1 2 3 lar basis vectors are determined, and their orientation H = 푎 푒0 − 푎 푒1 − 푎 푒2 − 푎 푒3, is given by the corresponding bivector. Therefore, where 푒 = 1. The FQF |H|2 = HH¯ looks like a common orientation of the basis of polar vectors 0 is closely related to the orientation of basis bivec- |H|2 = (푎0)2 − (푎1)2 − (푎2)2 − (푎3)2. (22) tors. Conventionally, the bases with the right-hand orientation are used by default, if any special reasons It corresponds to the metric of the pseudo-Euclidean (3) do not force this rule to be violated. In this case, a Minkowski space 푀4 . In order to prove that H ∈ special remark is made. (3) ∈ 푀4 , let us determine the metric tensor 푔훼훽 using the scalar product of basis vectors 푒훼 (훼 = 0, 1, 2, 3) 5. Algebra and Geometry analogously to that for complex numbers (Eq. (3)): of 퐸3 in the Mirror Space 1 To obtain the third vector of the Clifford basis, let us 푔훼훽 = 푒훼 ∙ 푒¯훽 = (푒훼푒¯훽 + 푒훽푒¯훼) (23) perform the mirror symmetry operation (4) for the 2 basis vector 휙+ in the plane (휙+, 푖휙−). In view of With the help of the equality 푒0푒푛 = 푒푛푒0 (푛 = the orientation of this plane (Fig. 2), a 3-dimensional = 1, 2, 3), it is easy to check that 푔훼훽 has a Clifford basis with the right-hand orientation isob- diagonal form with the signature diag(푔훼훽) = tained, if the angle 훼 = −휋/2 or −3휋/4. In this case, = (+1, −1, −1, −1). In other words, a hyperbolic hy- the eigenvectors 휙3± and the basis vector 푒3 look like percomplex number H is a vector in the pseudo- (︂ )︂ 1 1 Euclidean Minkowski space 푀 (3) with the orthogonal 휙3+ = S3(−휋/2)휙+ = √ ; 4 2 푖 basis vectors 푒훼 (훼 = 0, 1, 2, 3). The hypercomplex (19) number H can be regarded as a hyperbolic quaternion (︂ )︂ (︂ )︂ 1 −1 0 −푖 or H-quaternion, which, unlike ordinary quaternions, 휙3− = S3(−3휋/4)휙+ = √ ; 푒3 = . 2 푖 푖 0 does not form a field and a , although ISSN 2071-0194. Ukr. J. Phys. 2015. Vol. 60, No. 5 473 Yu.V. Khoroshkov

0 푛 some reservations should be made. Nevertheless, H- A(휙 푒0, 퐴 푒푛), calculating the divergence of the sec- quaternions with the Minkowski space structure have ond equality in Eq. (26), and substituting the re- to be studied further in physical problems. First at- sult into the first equality in Eq. (26), we obtain the tempts to construct the algebraic theory of space-time Maxwell equation for the scalar potential 휙. An al- and matter on the basis of 2-dimensional hyperbolic ternative way can also be used. Namely, we calculate numbers have already been done [13,14]. In this work, 0 0 we only mark some features of differential operations ∇¯ A¯ = (휕0휙 +div A⃗ )푒0−휕0A⃗ −grad 휙 +푖 rot A⃗ , (28) over hyperbolic hypercomplex numbers. 푛 As in the case of ordinary complex numbers, the where A⃗ = 퐴 푒푛. Expression (28) contains the scalar 0 0 coordinates of 4-vector are considered to be functions 퐴푆 = 휕0휙 + div A⃗ , the vector E = −휕0A⃗ − grad 휙 , 훼 0 1 2 3 of 4 variables: 푎 (푥 , 푥 , 푥 , and 푥 ). Let us introduce and the bivector 푖H = 푖 rot A⃗ . If we put 퐴푆 = 0 a space-time 4-gradient as a covariant vector in the (the Lorenz gauge condition) and associate E and H 0 푛 reciprocal basis (푒 , 푒 ), with the vectors of electric and magnetic fields, re- spectively, then expression (28) will define a complex 훼 0 푛 1 휕 ∇ = 푒 휕훼 = ∇(푒 휕0, 푒 휕푛), 휕0 = , (24) Riemann–Silberstein vector R = E+푖H. It should be 푐 휕푡 emphasized that the introduction of the 4-potential A¯ where the summation over repeated indices is im- conjugation is associated with the necessity to equal- plied. Let us formulate the analyticity conditions for ize the orientations of basis coordinate systems in ex- the function G(H) of a hyperbolic hypercomplex ar- pression (28). At last, the equation ∇R = 0 contains gument, similar to Cauchy–Riemann ones for ordi- the Maxwell equation for the vector R in the free nary complex numbers. For this purpose, let us con- space (휖 = 휇 = 1). struct the invariant of a hypercomplex number as its combination in the initial and conjugate bases (the 6. Coordinate Transformation right- and left-hand orientations of the coordinate ++ in ℰ2 and in the Mirror Space system). For the 4-gradient, the relation between the ++ initial and reciprocal bases is given by the metric ten- The complex Euclidean space ℰ2 with the mirroring 훽 operation is, in essence, an eigenvector space of sym- sor 푒훼 = 푔훼훽푒 . Therefore, the 4-gradient in the ini- tial basis is defined by the expression metry operators 푒푖. Therefore, any vector 훾 in this space can be decomposed into any pair (휙푖+, 휙푖−) of ∇¯ = ∇(푒0휕0, 푒¯푛휕푛). (25) orthogonal eigenvectors 푒푖,

푖 푖 The analyticity condition for the function G(H) 훾푖 = 휉 휙푖+ + 휂 휙푖−, (29a) 0 is written in the form of the equation ∇¯ G(푎 푒0, 푛 푎 푒푛) = 0. Direct calculations and zeroing the com- and the influence of 푒푖 on 훾 is described by the ex- ponents of a scalar, a vector, and a bivector give pression

휕 푎0 = div ⃗푎; 휕 ⃗푎 = grad 푎0; rot ⃗푎 = 0, (26) 푖 푖 0 0 푒푖훾 = 휉 휙푖+ + (−휂 )휙푖−, (29b) where the notation ⃗푎 = 푎푛푒 is used. In the simple 푛 where 휉푖 and 휂푖 are the representation of 훾 in the case of binary numbers, it follows from Eq. (26) that corresponding basis. In Section 3, the mirror images ++ 휕푎0 휕푎1 휕푎1 휕푎0 in ℰ2 that form a of improper rotations were 0 = 1 ; 0 = 1 . (27) considered. In this section, we consider eigenrotations 휕푥 휕푥 휕푥 휕푥 ++ of the coordinate system V in ℰ2 and in the mirror Expression (27) corresponds to the condition of hy- space. Subjecting both Eqs. (29) to some transfor- perbolic analyticity or ℎ-analyticity for the binary mation V, it is easy to obtain a relation between the ++ numbers [16]. Therefore, we adopt that conditions transformation in ℰ2 and the basis vectors 푒푖 in the (26) are the extension of ℎ-analyticity onto the form case of hypercomplex hyperbolic numbers. Defining ′ ′ 푖 ′ 푖 ′ the function G(H) as a vector of 4-potential 푒푖훾 = 휉 휙푖+ + (−휂 )휙푖−, (30) 474 ISSN 2071-0194. Ukr. J. Phys. 2015. Vol. 60, No. 5 Mirror Symmetry

where the transformed quantities are defined by the the axis 휙+ is described, similarly to Eq. (35), and expressions the corresponding transformation matrix looks like (︃ )︃ 푒′ = V푒 V−1훾′ = V훾, 훾 훾 푖 푖 cos 2 푖 sin 2 (31) V(훾/2) = 훾 훾 . (36) ′ ′ 푖 sin 2 cos 2 휙푖+ = V휙푖+휙푖− = V휙푖−. Let us confine the consideration here to unimodular The transformation in the planes (휙+, 푖휙+) and transformations V with the determinant equal to +1. (휙−, 푖휙−) is a common transformation of a pair of Rotations will be analyzed in 5 planes of ℰ++, which complex numbers and is described, in the general 2 form, by the expression are defined by the pairs of basis vectors (휙+, 휙−), (휙+, 푖휙+), (휙−, 푖휙−), (휙+, 푖휙−), and (휙−, 푖휙+). The (︂ )︂ 푘 exp(푖훿/2) 0 notation 푖 for the imaginary unit means that the V(푘, 훿/2) = . (37) 0 푘−1 exp(−푖훿/2) imaginary coordinate axis 푖휙− is oriented oppositely to the imaginary axis 푖휙+ (Fig. 2). The counterclock- A transformation of this type is a homothety (comp- wise transformation of the basis in the real plane ression-expansion) with a rotation in the complex (휙+, 휙−) is described by the expression plane. 훼 훼 To analyze the basis transformations in the mirror 휙′ = cos 휙 + sin 휙 , 푖+ + − space, let us enumerate the basis vectors 푒푖 so that the 2 2 (32) ′ 훼 훼 components of the vector r = (푥, 푦, 푧) in the mirror 휙푖− = − sin 휙+ + cos 휙−. 2 2 space, r = 푥푒1 + 푦푒2 + 푧푒3, would correspond to the matrix representation The matrix describing the basis rotation by the angle (︂ )︂ 훼/2 around the bundle of imaginary axes looks like 푧 푥 − 푖푦 r = . (38) (︃ )︃ 푥 + 푖푦 −푧 cos 훼 sin 훼 V(훼/2) = 2 2 . (33) 훼 훼 Matrices (33) and (35)–(37) of the coordinate trans- − sin 2 cos 2 formation can be expanded in the Clifford basis The transformation of the basis in the plane (1, 푒푖푒푗), and the rotation in the mirror space can be (휙+, 푖휙−) by the angle 훽/2 around the axis 휙− to- expressed in terms of quaternionic variables Q. For ward the positive direction of the axis 푖휙− (clockwise) instance, rotation (33) will obtain the form looks like 훼 훼 ′ 훽 훽 V(훼/2) = Q(훼/2) = cos 1 + sin 푒3푒1. (39) 휙푖+ = cos 휙+ + sin (푖휙−), 2 2 2 2 (34) ′ 훽 훽 Since 푒′ = V푒 V−1, and since 푒 anticommutes with (푖휙−) = − sin 휙+ + cos (푖휙−). 푖 푖 31 2 2 (푒1, 푒3) and commutes with 푒2, the transformation of This direction of a rotation was chosen for the angle a vector r in the mirror space reads 훽 to be positively defined. By multiplying the second ′ r = Q(훼)(푥푒1 + 푧푒3) + 푦푒2. (40) equality (34) by −푖, the rotation around the axis 휙− in the plane (휙+, 푖푖휙−) is transformed into the cor- Expression (40) describes a rotation around the ba- responding rotation around the bundle of imaginary sis axis 푒2. Analogously to transformation (35), the axes in the plane (휙 , 휙 ). The matrix of this trans- + − rotation around the basis vector 푒1 takes the form formation looks like ′ (︃ )︃ r = 푥푒1 + Q(훽)(푦푒2 + 푧푒3), (41) cos 훽 푖 sin 훽 V(훽/2) = 2 2 . (35) 훽 훽 where Q(훽) = cos 훽 1 + sin 훽 푒 푒 . Expressions (35) 푖 sin 2 cos 2 2 3 and (36) are an example of the “degeneration” effect The transformation of the basis in the plane induced by a symmetry; in the specific case, this is the (휙−, 푖휙+) counterclockwise by the angle 훾/2 around symmetry of imaginary axes. A similar phenomenon

ISSN 2071-0194. Ukr. J. Phys. 2015. Vol. 60, No. 5 475 Yu.V. Khoroshkov

takes place in quantum mechanics, where the energy sor 푔훼훽 of an orthogonal basis in the Minkowski space degeneration of the spin states can be eliminated with (3) 푀4 for hyperbolic hypercomplex numbers. For this the help of a magnetic field owing to a decrease in space, the ℎ-analyticity conditions were obtained for the symmetry. Therefore, let us describe the rota- functions of a hypercomplex argument, which are sim- tion around the basis vector 푒3 by expression (37). ilar to the Cauchy–Riemann conditions for complex Then, putting 푘 = 1 and 훿 = 훾, the corresponding numbers. Moreover, the conditions of ℎ-analyticity quaternion looks like Q(훾) = cos 훾1 + sin 훾푒1푒2. If implicitly include the Maxwell equations for a 4-po- 푘 ̸= 1, no analogs of the transformations of a 3- tential in the free space. dimensional space are known. However, in the space 3. In the vector interpretation of hypercomplex with the Minkowski metric, the variable 푘 is responsi- numbers, the important role is played by the orien- ble for the boost transformation in the special theory tation of a coordinate system. A change in the ori- of relativity. It should be noted that the quaternion entation of the system of basis vectors for hypercom- in expressions (40) and (41) can act on the coordi- plex numbers corresponds to the operation of complex nates in the parentheses not only from left to right, number conjugation. The construction of the invari- but also from right to left. In the latter case, the signs ant for hypercomplex numbers–for complex numbers, of angles in the transformation formulas have to be this is the absolute value composed of conjugated changed to the opposite ones. numbers–corresponds to the composition of numbers in the left- and right-hand oriented coordinate sys- 7. Discussion of the Results tems. Just those reasons served as a basis to obtain the conditions of ℎ-analyticity. The consideration of 1. Two- and three-dimensional spaces obtained by a coordinate system orientation also made it possible mirroring in the auxiliary complex space have a basis to distinguish between right- and left-handed quater- of the vector Clifford algebra in the mirror space. In nions, thereby emphasizing the vector character of turn, the basis vectors 푒푖 of the Clifford algebra are the Clifford algebra. symmetry operators describing two opposite orienta- tions of a basis vector. These properties of the mir- The author is thankful to Prof. V.I. Vysotskyi for ror images emphasize a direct relationship between his valuable remarks and discussion, which made it the Clifford algebra and the fundamental principles of possible to reevaluate the work from another view- ++ symmetry. The auxiliary complex space ℰ2 , which point. was constructed with the use of symmetry operations and in which the mirroring operations are performed, 1. Yu.S. Vladimirov, Relational Theory of Space-Time and is, in essence, a space of eigenvectors for the sym- Interactions, Parts 1 and 2 (Moscow State Univ. Publ. House, Moscow, 1996–1998) (in Russian). metry vector-operators 푒푖 of the Clifford basis. This space has the Euclidean metric with orthogonal ba- 2. A.P. Efremov, Quaternion Spaces, Frames, and Physical sis vectors and does not fit the classical definition of Fields (RUDN Publ. House, Moscow, 2005) (in Russian). spin-space [21], which has an antisymmetric metric on 3. A.V. Berezin, Yu.A. Kurochkin, and E.A. Tolkachev, Quaternions in Relativistic Physics (Editorial URSS, isotropic basis vectors. At the same time, the prop- Moscow, 2003) (in Russian). ++ erties of ℰ2 have a certain similarity with those of 4. D. Hestenes, New Foundations for Classical Mechanics the classical spin-space. This statement concerns the (Kluwer Academic, New York, 2002). identity of the expressions for coordinate transforma- 5. D. Hestenes, Clifford Algebra to Geometric Calculus tions and their sign ambiguity. However, the intro- (D. Reidel, Dordrecht, 1987). duction of the Clifford basis conjugation operation at 6. H. Weyl, Symmetry (Princeton Univ. Press, Princeton, NJ, the stage of the basis construction allowed us, in a 1956). definite sense, to solve the problem of transformation 7. Sh.-T. Yau and S. Nadis, The Shape of Inner Space: String ambiguity. Theory and the Geometry of the Universe’s Hidden Di- mensions (Basic Books, New York, 2010), p. 151. 2. The consistent application of the vector formal- 8. A.N. Kapustin and D.O. Orlov, Usp. Matem. Nauk 59, ism of complex numbers in the Clifford algebra made N 5, 101 (2004). it possible to redefine the scalar product for the Clif- 9. E.´ Cartan, Le¸conssur la Th´eoriedes Spineurs (Hermann, ford basis and to correctly introduce the metric ten- Paris, 1938).

476 ISSN 2071-0194. Ukr. J. Phys. 2015. Vol. 60, No. 5 10. P.K. Rashevsky, Usp. Matem. Nauk 10, No. 2, 3 (1955). 21. Yu.B. Rumer, Spinor Analysis (ONTI, Moscow, 1936) (in 11. I.M. Yaglom, Complex Numbers in Geometry (Academic Russian). Press, New York, 1968). 22. V.I. Vysotskii and M.V. Vysotskyy, Eur. Phys. J. A 44, 279 (2010). 12. E. Hitzer, SICE J. Control Meas. 4, 1 (2011). Received 27.05.14. 13. D.G. Pavlov and S.S. Kokarev, Giperkompl. Chisla Geom. Translated from Ukrainian by O.I. Voitenko Fiz. 7, No. 1, 78 (2010). Ю.В. Хорошков 14. D.G. Pavlov and S.S. Kokarev, Giperkompl. Chisla Geom. ДЗЕРКАЛЬНА Fiz. 7, No. 2, 11 (2010). СИМЕТРIЯ ЯК ОСНОВА ПОБУДОВИ 15. R. Penrose and W. Rindler, Spinors and Space-Time, ПРОСТОРОВО-ЧАСОВОГО КОНТИНУУМУ Vol. 1: Two-Spinor Calculus and Relativistic Fields (Cam- Р е з ю м е bridge Univ. Press, Cambridge, 1987). За допомогою дзеркального вiдображення 1-вимiрної орiєн- 16. M.A. Lavrent’ev and B.V. Shabat, Problems in Hydrody- тованої множини у спецiально створеному на основi симе- namics and Their Mathematical Models (Nauka, Moscow, трiї комплексному просторi будується в задзеркаллi простiр 1977) (in Russian). розмiрностi 푛 > 1. Геометрiя отриманого простору опису- 17. D. Hestenes and G. Sobczyk, Clifford Algebra to Geomet- ється векторною алгеброю Клiффорда. На основi алгебри ric Calculus, A Unified Language for Mathematics and гiперболiчних гiперкомплексних чисел будується псевдоев- Physics (Kluwer, Dordrecht, 1984). клiдiв простiр з метрикою простору Мiнковського. Одер- 18. S.V. Terekhov, Vestn. Novgorod. Gos. Univ., No. 26, 56 жано умови аналiтичностi функцiї вiд гiперболiчного гiпер- (2004). комплексного аргументу (h-аналiтичнiсть), в яких в неяв- 19. A.L. Glebov, Teor. Mat. Fiz. 48, 340 (1986). ному виглядi мiстяться рiвняння Максвелла для 4-потен- 20. E. Wigner, Phys. Today 17, 34 (1964). цiалу у вiльному просторi.