Group properties of osp(2/1;C) gauge transformations

Group properties of osp(2/1;C) gauge transformations

Given an explicit construction of the grade star hermitian adjoint representation of osp(2/1;C) graded Lie algebra, we consider the Baker-Campbell-Hausdorff formula and find reality conditions for the Grassmann-odd trans-formation parameters that multiply the pair of odd generators of the graded Lie...

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1. Verfasser: Ilyenko, K.
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Veröffentlicht: Національний науковий центр «Харківський фізико-технічний інститут» НАН України 2007
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Quantum field theory
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spelling Ilyenko, K.
2017-01-06T17:24:43Z
2017-01-06T17:24:43Z
2007
Group properties of osp(2/1;C) gauge transformations/ K. Ilyenko // Вопросы атомной науки и техники. — 2007. — № 3. — С. 22-27. — Бібліогр.: 17 назв. — англ.
1562-6016
PACS: 11.10.Ef, 11.15.-q
https://nasplib.isofts.kiev.ua/handle/123456789/110897
Given an explicit construction of the grade star hermitian adjoint representation of osp(2/1;C) graded Lie algebra, we consider the Baker-Campbell-Hausdorff formula and find reality conditions for the Grassmann-odd trans-formation parameters that multiply the pair of odd generators of the graded Lie algebra. Utilization of su(2)-spinors clarifies the nature of Grassmann-odd transformation parameters and allows one an investigation of the corresponding infinitesimal gauge transformations. We also explore the action of a corresponding group element on an appropriately graded representation space and find that a proper (graded) generalization of hermitian conjugation is consistent with a natural generalization of Dirac adjoint. A corresponding generalization of a unitary transformation is discussed.
Виходячи з явної конструкції градуйованого узагальнено-ермітового приєднаного уявлення osp(2/1;C) градуйованої алгебри Лі, розглянуто формулу Бейкера-Кемпбелла-Хаусдорффа та знайдено умови дійсності, що накладаються на грасманово-непарні параметри, які є множниками пари непарних генераторів градуйованої алгебри Лі при експоненціюванні. Використання формалізму su(2)-спінорів прояснює природу грасманово-непарних параметрів та суттєво полегшує дослідження відповідних інфінітезимальних калібрувальних перетворень. Також вивчено дію загального групового елементу на придатному просторі уявлення та перевірено, що відповідне (градуйоване) узагальнення ермітівського спряження погоджується із природним узагальненням дираківського спряження. Обговорюються придатні узагальнення унітарного перетворення відповідного векторного простору.
На основе явной конструкции градуированного обобщенно-эрмитового присоединенного представления osp(2/1;C) градуированной алгебры Ли рассмотрена формула Бейкера-Кэмпбелла-Хаусдорффа и найдены условия вещественности, налагаемые на грассманово-нечетные параметры, которые являются множителями пары нечетных генераторов градуированной алгебры Ли при экспоненцировании. Использование формализма su(2)-спиноров поясняет природу грассманово-нечетных параметров и существенно облегчает исследование соответствующих инфинитезимальных калибровочных преобразований. Также изучено действие общего группового элемента на подходящем пространстве представления и проверено, что соответствующее (градуированное) обобщение эрмитового сопряжения согласуется с естественным обобщением дираковского сопряжения. Обсуждается подходящее обобщение унитарного преобразования соответствующего векторного пространства.
I am grateful to Drs. T.S. Tsou and V. Pidstrigach for an interest to this work and to Dr. V. Gorkavyi and Prof. Yu.P. Stepanovsky for numerous helpful discussions.
en
Національний науковий центр «Харківський фізико-технічний інститут» НАН України
Вопросы атомной науки и техники
Quantum field theory
Group properties of osp(2/1;C) gauge transformations
Групові властивості osp(2/1;C) калібрувальних перетворень
Групповые свойства osp(2/1;C) калибровочных преобразований
Article
published earlier
institution Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection DSpace DC
title Group properties of osp(2/1;C) gauge transformations
spellingShingle Group properties of osp(2/1;C) gauge transformations
Ilyenko, K.
Quantum field theory
title_short Group properties of osp(2/1;C) gauge transformations
title_full Group properties of osp(2/1;C) gauge transformations
title_fullStr Group properties of osp(2/1;C) gauge transformations
title_full_unstemmed Group properties of osp(2/1;C) gauge transformations
title_sort group properties of osp(2/1;c) gauge transformations
author Ilyenko, K.
author_facet Ilyenko, K.
topic Quantum field theory
topic_facet Quantum field theory
publishDate 2007
language English
container_title Вопросы атомной науки и техники
publisher Національний науковий центр «Харківський фізико-технічний інститут» НАН України
format Article
title_alt Групові властивості osp(2/1;C) калібрувальних перетворень
Групповые свойства osp(2/1;C) калибровочных преобразований
description Given an explicit construction of the grade star hermitian adjoint representation of osp(2/1;C) graded Lie algebra, we consider the Baker-Campbell-Hausdorff formula and find reality conditions for the Grassmann-odd trans-formation parameters that multiply the pair of odd generators of the graded Lie algebra. Utilization of su(2)-spinors clarifies the nature of Grassmann-odd transformation parameters and allows one an investigation of the corresponding infinitesimal gauge transformations. We also explore the action of a corresponding group element on an appropriately graded representation space and find that a proper (graded) generalization of hermitian conjugation is consistent with a natural generalization of Dirac adjoint. A corresponding generalization of a unitary transformation is discussed. Виходячи з явної конструкції градуйованого узагальнено-ермітового приєднаного уявлення osp(2/1;C) градуйованої алгебри Лі, розглянуто формулу Бейкера-Кемпбелла-Хаусдорффа та знайдено умови дійсності, що накладаються на грасманово-непарні параметри, які є множниками пари непарних генераторів градуйованої алгебри Лі при експоненціюванні. Використання формалізму su(2)-спінорів прояснює природу грасманово-непарних параметрів та суттєво полегшує дослідження відповідних інфінітезимальних калібрувальних перетворень. Також вивчено дію загального групового елементу на придатному просторі уявлення та перевірено, що відповідне (градуйоване) узагальнення ермітівського спряження погоджується із природним узагальненням дираківського спряження. Обговорюються придатні узагальнення унітарного перетворення відповідного векторного простору. На основе явной конструкции градуированного обобщенно-эрмитового присоединенного представления osp(2/1;C) градуированной алгебры Ли рассмотрена формула Бейкера-Кэмпбелла-Хаусдорффа и найдены условия вещественности, налагаемые на грассманово-нечетные параметры, которые являются множителями пары нечетных генераторов градуированной алгебры Ли при экспоненцировании. Использование формализма su(2)-спиноров поясняет природу грассманово-нечетных параметров и существенно облегчает исследование соответствующих инфинитезимальных калибровочных преобразований. Также изучено действие общего группового элемента на подходящем пространстве представления и проверено, что соответствующее (градуированное) обобщение эрмитового сопряжения согласуется с естественным обобщением дираковского сопряжения. Обсуждается подходящее обобщение унитарного преобразования соответствующего векторного пространства.
issn 1562-6016
url https://nasplib.isofts.kiev.ua/handle/123456789/110897
citation_txt Group properties of osp(2/1;C) gauge transformations/ K. Ilyenko // Вопросы атомной науки и техники. — 2007. — № 3. — С. 22-27. — Бібліогр.: 17 назв. — англ.
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last_indexed 2025-11-26T00:10:43Z
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fulltext GROUP PROPERTIES OF OSP(2/1;C) GAUGE TRANSFORMATIONS K. Ilyenko1,2 1Institute for Radiophysics and Electronics of NAS of Ukraine, Kharkiv, Ukraine; e-mail: kost@ire.kharkov.ua; 2Kharkiv Humanitarian Pedagogical Institute Given an explicit construction of the grade star hermitian adjoint representation of graded Lie alge- bra, we consider the Baker-Campbell-Hausdorff formula and find reality conditions for the Grassmann-odd trans- formation parameters that multiply the pair of odd generators of the graded Lie algebra. Utilization of su -spinors clarifies the nature of Grassmann-odd transformation parameters and allows one an investigation of the correspond- ing infinitesimal gauge transformations. We also explore the action of a corresponding group element on an appro- priately graded representation space and find that a proper (graded) generalization of hermitian conjugation is con- sistent with a natural generalization of Dirac adjoint. A corresponding generalization of a unitary transformation is discussed. );1/2(osp C )2( PACS: 11.10.Ef, 11.15.-q 1. INTRODUCTION A natural extension of the Lie algebras, which un- derlie the modern gauge theory, are graded Lie algebras introduced and studied to some extent in the articles [1- 3]. In this paper we explore how one can utilize a - graded extension osp of the compact Lie alge- bra for the purposes of defining a meaningful gauge theory of the Yang-Mills type (see, e.g., [4-6]). The form of defining relations (proposed in [5] and refined in [6]) utilizes the Pauli matrices and strongly suggests a relation to spinors. Exponentiating the algebra, we observe the necessity of introduction of anticommuting (Grassmann-odd) spinors, which multi- ply the odd generators of the graded Lie algebra. Fi- nally, we study some of infinitesimal properties of the composition law of group transformations and consider a generalization of the Dirac adjoint, thus, making preparations for an investigation of the gauge invariance of the proposed field strength for such a gauge theory, [5-6]. 2Z );1/2( C );1/2( C )2(su osp 2. GRADED LIE ALGEBRA OSP(2/1;C) The algebra osp is a graded extension of algebra by a pair of odd generators, , which anticommute with one another and commute with the three even generators, T , of . It is customary to assign a degree, , to the even ( deg = 0) and odd ( = 1) generators. We use the square brackets to denote the commutator and the curly ones to denote the anticommutator. The defining relations have the form, [3,5-7]: );1/2( C a αTdeg )2(su Aτ aT )2(su Aτdeg .)( 2 }{ ;)( 2 1][;][ aAB a BA B B AaAacabcba Ti, ττ τ, τTT iε , TT σ σ = == (1) Summation is assumed over all repeated indices. Low- ercase Roman indices from the beginning of the alpha- bet run from 1 to 3; uppercase Roman indices run over 1 and 2; = ( ), ( 1) and (∈ 1) are the three dimensional identity matrix and the Levi-Civita totally antisymmetric sym- bols in three and two dimensions, respectively; the ma- trices [ ( ] are just the usual Pauli matrices: abδ =∈12 Aa )σ ABa ) abδ = σ = δ baab δδ = =ABa ) (σ C Ab ∈)(σ abcε =BA a ) =abcε ab (σδ AB∈ = abδ 12 B( (σ ABa ) CB ab . 01 10 , 0 0 , 10 01 )( ; 10 01 , 0 0 , 01 10 )(                     − −      − =               −      −      = i i i i AB a B Aa σ σ We use the Levi-Civita symbols in two dimensions to raise and lower uppercase Roman indices paying atten- tion to their antisymmetric properties: 1|||| 01 10 |||| −Σ−=∈=      − =∈≡Σ AB AB . Note that, as concerned to these indices, we are working with two-component spinors and adopt conventions of the book [8]. We shall follow those conventions as more suitable for our purposes even when complex con- jugation of spinor and Grassmann quantities is in- volved. It turns out that not all of the osp algebra generators are hermitian. A proper generalization of the hermitian conjugation is denoted by ( ‡ ): on the even generators the operation coincides with ordinary hermi- tian conjugation ( ) while the odd ones obey more complicated relations. Following the papers [9-10], we shall call them the grade star hermiticity conditions: );1/2( C + ∓ττ ±=± ‡ , (2) where . 21 τττ i±=± OF ATOMIC SCIENCE AND TECHNOLOGY. 2007, N3 (1), p. 22-27. 22 Let us consider complex-valued matrices:       = D A M 0 0 even and ,       = 0 0 odd C B M where, for the purposes of this paper, and C are 2 3 rectangular blocks and and are 3 3 and 2 2 square blocks representing the division of a repre- sentation space into even and odd parts. On these matri- ces the grade star hermiticity condition reads B × × A D ×         = + + D AM 0 0‡ even and .         −= + + 0 0‡ odd B CM We shall also use multiplication of algebra generators by scalars. Such an operation must take into account that Grassmann-odd scalars anticommute with the odd algebra generators while commute with complex num- bers and the even algebra generators, [11]. The follow- ing construction possesses all of these properties. Let be a scalar and deg be its degree (0 or 1 depending on whether it is Grassmann-even or Grassmann-odd, re- spectively). Then multiplication by is defined as fol- lows: a a a aMaM eveneven= and . aMaM a odd deg odd )1(−= 3. THE GROUP PROPERTY Given a Lie algebra one can turn over to a Lie group by exponentiating the generators multiplied by trans- formation parameters. This, in a usual fashion, gives us the gauge transformations. In the case of a graded Lie algebra we are faced with a problem: anticommutators seem to rule out the application of the Baker-Campbell- Hausdorff formula, which is necessary to prove that subsequent transformations do not leave the group manifold. This problem is solved via introduction of Grassmann-odd parameters (cf., [1]). In the case under consideration these are Grassmann-odd - spinors , , etc., which multiply the odd generators. They are included on equal footing with ordinary (Grass- mann-even) parameters multiplying the even gen- erators (hopefully, there will not be confusion about use the same kernel letter, , to denote a Grassmann-even transformation parameter and the Levi-Civita totally antisymmetric symbol in three dimensions). By defini- tion, , , etc. satisfy )2(su Aξ Aθ Aξ aε ε Aθ .0],[ ;0],[ = = BA Aa θξ θε { ;0},{}, == BABA θθξξ Then, the necessary relations can be given in terms of commutators only: aAB aBABA B B A A Ti ))(( 2 ],[ ][}{ σθξθξτθτξ +−= , where and are convenient shorthand nota- tions. This result was obtained using anticommutator for odd generators in definition (1). Using a fundamental fact of spinor algebra, ∈ , one can calculate =}{ BAθξ −( /)ABθξ 2/)( ABBA θξθξ + 2 =][ BAθξ BAθξ +∈CDAB +∈∈ DBAC 0=∈∈ BCAD C C ∈= )( 2 1 θξ a )σ BA ][ θξ AA i [ 2 ,[ θτξ −≡ ]B Bτθ ,[ ,[ a a a a T T θε εκ aθε ),(U θε )… =M ) expexp M ),( ξκU ),( θεU ′′ a aTi ε( + Aθ … k N +( Ti a aε …AB a)](σaa εε =′ …A B) Z A θθ =′ AB . From symmetry of ( in the uppercase indices, it then follows that AB aABaBA aABaBABB T Ti )](, )( 2 ] }{ σθξ σθξτ −= (3) and, in particular, the commutator [ van- ishes identically. One can also calculate ,A Aτθ ,)( 2 1] ;] ][ B B Aa Aa A A c abc ba b b TiT τσθετ εεκ = = where is again a Grassmann-odd trans- formation parameter. B Aa A )(σ Group elements are obtained by exponentiating the algebra )](exp[ A A a aTi τθε += (4) and the Baker-Campbell-Hausdorff formula, ],[ 2 1exp( +++= NMNMN (5) may be applied to determine motion in the parameter space under a (left) multiplication with a group element : ),(),( θεξκ UU= . (6) 4. INFINITESIMAL TRANSFORMATIONS AND REALITY CONDITIONS Let us examine expression (5) restricting ourselves by taking into account the first non-trivial contribution – the two-fold commutator . Writing and , we have ],[ NM a aT( + Aθ)Aτ =N i ε Aτ +++= ],[ 2 1) NMNMA Aτθ , where dots denote the sum of linear combinations of - fold ( k > 2, ) commutators of M and [12]. Substituting expressions for , and using (3), we obtain after some algebra Z∈k M N BAbca cb a ,[ 4 1 2 1 θξεεκκ +−+ ; bB b B b AA i )(( 4 σξεθκξ −++ (7) Here again dots denote the contribution from the sum of linear combinations of -fold ( > 2, ) commu- tators. The first three summands in the first row of for- mula (7) reflect the non-commutative character of the k k ∈k 23 proper Lie subalgebra, , of osp the last one being contribution from the odd part of the algebra. The last summand in the second row of the formula is obviously a Grassmann-odd quantity, and it reflects the non-commutative property of the even and odd parts of the graded Lie algebra. )2(su ,ξ θ a A∈− a aTε + aε )2 =2 2 );1/2( C ) ξσ C C B a )2( Aξ )]Aτ a Aξ 122 ββ−= In the view of intended applications, contribution from Grassmann-odd part of the algebra into the law of composition of Grassmann-even parameters needs to be investigated in more detail. First, let us calculate that ()( )](,[2 σθθσξ θσξ σθξ TaT ABC C B aABA ABaBA Σ−Σ= ∈= (8) where we employed some self-evident matrix notations; the superscript denotes transposition. Comparing the result (8) and a description of su -spinors of 3D Euclidean space in the book [13, p. 48], one imme- diately realizes that the last term of the first equation in system (7) is, in general, a complex vector of 3D Euclidean space, e.g. it transforms like a vector un- der transformations. Second, the representation (8) tells us that components of this vector vanish if as required by a property of a one-parameter subgroup of transformations (6). Finally, this vector also has all components equal to zero if . This shows that the inverse of the group element has the form )( T SO(3) = Aθ ),θε Aξ (U = Aθ− (exp[),(1 AiU θθε −=− . If one intends, as customarily done in a meaningful Yang-Mills theory, to treat , , etc. as real-valued Grassmann-even transformation parameters, then it is necessary to impose some conditions on the su - spinors , , etc. in order to ensure that (8) will be a real 3D Euclidean vector. Such a condition must be compatible with transformation properties of the corre- sponding space of -spinors, , and take into account that its members are also Grassmann-odd quan- tities. In fact, this condition should involve a passage from an su -spinor to its conjugate and, thus, rely on the definition of an anti-involution in the space of spi- nors (see, e.g. [13, p. 100]). Let us observe first that for a Grassmann algebra on one generator the last term in the first relation in (7) vanishes identically. This is a somewhat trivial situation. The next non-trivial one arises when all -spinors under consideration take values in a Grassmann algebra on two odd generators, and : 0, (see, e.g. [14, p. 7]). We shall employ lowercase Roman indices from the middle of the alphabet running over 1 and 2 to enumerate the decompositions of various quantities in the corresponding basis of the Grassmann algebra. De- composing and into this basis one obtains κ 1β )2( Aξ ( 2β Aθ )2 β Aξ (su )2 β Bθ (su =2 11β β iA i A βξξ = and , jB j B βθθ = where and are ordinary (commuting), su - spinors of 3D Euclidean space, and summation over repeated indices is assumed. In this case we can write A i ξ B j θ )2( )(2)](,[ 2121 21 ξσθθσξββσθξ aTaT AB aBA Σ−Σ= . (9) Now we shall impose some additional conditions on -spinors and , etc. to ensure that (9) gives a real Grassmann-even 3D Euclidean vector. One way of doing so in a manner preserving all the spinor transformations properties is to define )2(su A i ξ A j θ ' 2 ' 1 B B AA iC ξξ = , ' 2 ' 1 B B AA iC θθ = , etc., (10) where the ‘charge conjugation’ matrix (C =CC I− ) is given by CCCC B A B A ==      − == |||| 01 10 |||| ' ' . (11) In (10) and (11) a bar over the spinors in the left-hand sides of the relations and primes over the indices denote complex conjugation. We again adhere to Penrose's notations whenever spinors are concerned, [8]. The charge conjugation matrix, , is responsible for invariant preservation of spinor properties (for details see, e.g. the review article [15, p. 72-73]; also compare with the treatment in [13, p. 100]). Note that the defini- tions (10) are essentially the proper generalization of reality conditions from numbers to spinors. As also seen from those definitions, each Grassmann-odd su - spinor , , etc. is defined by a single ordinary, i.e. complex-valued Grassmann-even, su -spinor. For the sake of notations denoting 'B AC )2( Aξ Aθ )2( AA 2 ξη = and , BB 2 θϑ = respectively, we write ).( 2121 ησϑϑση ξσθθσξυ aTTaTT aTaTa CCi Σ−Σ= Σ−Σ≡ On comparison with [13, p. 50], one can check that is indeed a real 3D Euclidean vector. In components it reads: aυ ).( ; );( 2'22'21'11'1 3 1'22'12'11'2 2 2'11'21'22'1 1 ηϑϑηηϑϑηυ ηϑϑηηϑϑηυ ηϑϑηηϑϑηυ +−−= −−+= −+−= i i These are obviously real quantities and the vector vanishes if and only if aυ AA ϑη ±= as expected. 5. ACTION ON A REPRESENTATION SPACE Having formulated meaningful reality conditions, we are in position to explore action of the group ele- ment (4) on a suitable vector space. 24 First, let us observe that because of definition of the matrix by its Taylor's expansion, the fact that the generators and are ‘block’ and ‘off-block’ di- agonal (see [6]), respectively, their multiplication prop- erties and those of the Grassmann-even and Grassmann- odd transformation parameters, it is easy to see that any matrix U has a specific decomposition U aT Aτ     = DC BAU , (12) U where is a ( ) sub-matrix, is a ( ) sub- matrix, is a ( r ) sub-matrix and is a ( ) sub-matrix. Following nomenclature of the book [14], we shall call the matrix U a ( ) super-matrix; in the adjoint representation under consideration of ma- trices and , is a ( ) sub-matrix, is a ( 3 ) sub-matrix, C is a ( ) sub-matrix and is a ( ) sub-matrix. Moreover, the sub-matrices and have contributions only from an even number of s’ multipliers and, hence, only even multipliers of Grassmann-odd transformation parameters s’ are pre- sent there. Thus, elements of those sub-matrices are in the even subspace, , of the complex Grassmann algebra (for more details see the book [14, p. 10-11]). The sub-matrices and C by an analogues argument include an odd number of s’ and s’ multipliers and, hence, are in the odd subspace, , of the complex Grassmann algebra. Therefore, any such a super-matrix is an even super-matrix and by the results of the previous section such matrices form a supergroup. Fur- thermore, by construction any such a super-matrix is invertible. A C aT 2 pr × q× Aτ A C B B r× θ 1BL ps × B θ D s qs × D A qp // 33× 2× C 2× 2 D τ U 3 × 0LB τ Second, consider even super-column Ψ [( ) super-matrices] and super-row Φ [(1 ) super-matrices] vectors: ×qp / sr /×1/0 0/       = 2 1 Ψ Ψ Ψ and Φ , (13) ),( 21 ΦΦ= where and Φ are (1 ) and ( ) sub-matrices, and Φ are (1 ) and ( ) sub-matrices, re- spectively. The elements of Ψ and Φ are Grassmann- even and those of Ψ and Φ are Grassmann-odd enti- ties. Action of even super-matrices U on such even super-column(-row) vectors transform them again into even super-column(-row) vectors. 1Ψ 1 p× 2 1×r 1 1 2Ψ 2 q× 2 ×s 1 For the sake of argument let U be (1 ) ma- trices [see (12)], the actual size can be easily treated the same way, and let also Ψ be a (1 ) even super- column vector as regarded to the linear transformations defined below. The entries Ψ and Ψ themselves could be, for example, Dirac bispinors. Consider a lin- ear transformation 1/11/ × 1 2 /0× 1 1/ UΨ DΨCΨ BΨAΨ Ψ Ψ Ψ ≡      + + =      ′ ′ ≡′ 21 21 2 1 . (14) Taking transposition of each line in (14) (it acts on ’s) and complex conjugate as well as denoting the Dirac conjugates as iΨ i′Ψ , we obtain: ,** **),( *)**,*(),( 21 212121       −= +−=′′ DB CAΨΨ DΨCΨBΨAΨΨΨ where the Grassmann character of the involved quanti- ties has been taken into account. Recall that for any super-matrix partitioned as in (12) the super- transpose is defined by       −− −= TTU TUT st DB CAU deg deg )1( )1( ; (15) for even super-column(-row) vectors this implies: ),( 21 TTst ΨΨΨ = and Φ . (16)         − = T T st Φ Φ 2 1 Given definitions imply ** ** **          =     −=      − st DC BA DB CA DB CA , (17) i.e. if, as in (14), Ψ hen UΨ=' t ‡UΨΨ =′ , (18) thus, generalizing the corresponding result in the Yang- Mills gauge theory. It also follows from (17) that grade star hermitian conjugation, denoted earlier by ‡ , can be interpreted as complex conjugated of super-transpose. 6. DISCUSSION AND OUTLOOK The result (18) calls for a study of an analogue of ‘unitary’ property for matrices U given by (4). A first suggestion would be to have U , however, a direct calculation shows that this does not hold. A direct calculation shows that 1−=U‡ aa TT =‡)( and B B AA Ci ττ ' ‡)( −= . (19) The later equation in (19) is just a re-statement of her- miticity of the even generators of the graded Lie algebra while the former one, to the best of the author’s knowledge, for the first time exhibits a strong connection between compact graded Lie algebras on one side and Euclidean spinors on the other (cf. (2)). This same property of ’s prevents one from having . Traced back to basic definitions, the prob- lem lies in the very definition of the super-transpose, in particular, ( but );1/2(osp C 1‡ −=UU Ψ Aτ Ψstst ≠) ΨΨ stststst =)))((( , which also motivates our spinor approach to the treat- ment of graded Lie algebras. In the view of this spinor connection, it is understandable that a generalization of unitary property for a matrix U given by (4) cannot 25 have the form U . In the theory of the Dirac equation we have 1‡ −=U ϕΠ+ ϕS 1 −− Π S );1 C inv)‡ = ϕϕϕ ≡ γ S =′ϕ + = ΠSS /2(osp Π D ( ‡‡‡ ΨΠΨ    +    2 1 Ψ Ψ 3×    D , where the matrix , which numerically coincides with in the chiral and Dirac representations of -matrices but not in the Ma- jorana representation, defines the invariant real-valued internal product on the (symplectic) space of Dirac bispinors , [15, p. 49-50; 16, p. 85]. Then, for a linear transformation, , preserving the internal product, we have (cf., e.g., [17, p. 32]) instead of unitary property for . Π ‡‡) 0γ ϕ = I 0 )ΠΨ    I 0 =‡ 1 = D−    1 0 1 It is, therefore, natural to define an internal unitary- symplectic product on the space given by even super- vectors (13), where the adjoint representation of the graded Lie algebra acts, as inv‡ΠΨΨ (20) with the matrices given by     = DΠ 0 . (21) Here is the square three-dimensional identity matrix while hermiticity property of the square two- dimensional matrix remains to be determined. To clarify this issue, let us consider the grade star hermitian conjugated of (20) with account for (15): I ( ‡‡ =Ψ , where = DΠ 0‡ . Using (13) and (16), we infer    − ‡)(Ψ , where is (1 ) sub-matrix and Ψ is (1 ) sub- matrix in terms of definition (13). Denoting Ψ , we can preserve the invariant property of the internal product (20) by requiring Ψ 2 2× (~ = Ψ ΠΨΨΠ ~‡ . This can be accomplished if the matrix is anti- hermitian D D =+ . Taking into account the described analogy with the Dirac bispinors, one is lead to the following choice: = 0 1iD , (22) where the matrix numerically coincides with . 1σi Thus, it is necessary to check that for a matrix U given by (4) 11‡ −−= ΠΠUU holds, where (21) and (22) define the matrix . Π ACKNOWLEDGMENTS I am grateful to Drs. T.S. Tsou and V. Pidstrigach for an interest to this work and to Dr. V. Gorkavyi and Prof. Yu.P. Stepanovsky for numerous helpful discus- sions. REFERENCES 1. F.A. Berezin, G.I. Kac. Lie groups with commuting and anticommuting parameters //Math. USSR – Sb. 1971, v. 11, p. 311-325. 2. L. Corwin, Y. Ne’eman, S. Stenberg. Graded Lie algebras in mathematics and physics (Bose-Fermi symmetry) //Rev. Mod. Phys. 1975, v. 47, p. 573-603. 3. V. Kac. Representations of classical Lie superalge- bras //LMS Lecture Notes. 1977, v. 676, p. 597-626. 4. Y. Ne’eman. Irreducible gauge theory of a consoli- dated Salam-Weinberg model //Phys. Lett. B. 1979, v. 81, p. 190-194. 5. R. Brooks, A. Lue. The monopole equations in topo- logical Yang-Mills //J. Math. Phys. 1996, v. 37, p. 1100-1105. 6. K. Ilyenko. Field strength for graded Yang-Mills theory //Problems of Atomic Science and Technol- ogy. 2001, v. 6, p. 74-75 (hep-th/0307230). 7. J.W. Hughes. Representations of and the metaplectic representation //J. Math. Phys. 1981, v. 22, p. 245-250. );1/2(osp C 8. J. Stewart. Advanced general relativity. Cambridge: “Cambridge University Press”, 1996, 228 p. 9. M. Scheunert, W. Nahm, Y. Rittenberg. Graded Lie algebras: generalization of hermitian representations //J. Math. Phys. 1977, v. 18, p. 146-154. 10. M. Scheunert, W. Nahm, Y. Rittenberg. Irreducible representations of the and spl graded Lie algebras //J. Math. Phys. 1977, v. 18, p. 155-162. )1,2(osp )1,2( 11. L.E. Gendenshtein, I.V. Krive. Supersymmetry in quantum mechanics //Sov. Phys. – Usp. 1985, v. 28, p. 645-666. 12. W. Magnus, A. Karras, D. Solitar. Combinatorial group theory. London: “Interscience”, 1966, 578 p. 13. E. Cartan. The theory of spinors. Paris: “Hermann”, 1966, 157 p. 14. J.F. Cornwell. Group theory in physics. London: “Academic Press”, 1989, v. 3, 628 p. 15. P.K. Rashevskij. The theory of spinors //Transl. Am. Math. Soc. II (Ser. 6) 1957, p. 3-77. 16. H.K. Dreiner, H.E. Haber, S.P. Martin. Supersym- metry. Cambridge: “CUP”, 2004, 271 p. 17. J.D. Bjorken, S.D. Drell. Relativistic quantum me- chanics. New York: “McGraw Hill”, 1975, 732 p. 26 ГРУППОВЫЕ СВОЙСТВА OSP(2/1;C) КАЛИБРОВОЧНЫХ ПРЕОБРАЗОВАНИЙ К. Ильенко На основе явной конструкции градуированного обобщенно-эрмитового присоединенного представления градуированной алгебры Ли рассмотрена формула Бейкера-Кэмпбелла-Хаусдорффа и найдены условия вещественности, налагаемые на грассманово-нечетные параметры, которые являются множителями пары нечетных генераторов градуированной алгебры Ли при экспоненцировании. Использование форма- лизма ) -спиноров поясняет природу грассманово-нечетных параметров и существенно облегчает ис- следование соответствующих инфинитезимальных калибровочных преобразований. Также изучено дейст- вие общего группового элемента на подходящем пространстве представления и проверено, что соответст- вующее (градуированное) обобщение эрмитового сопряжения согласуется с естественным обобщением ди- раковского сопряжения. Обсуждается подходящее обобщение унитарного преобразования соответствующе- го векторного пространства. );1/2(osp C 2(su ГРУПОВІ ВЛАСТИВОСТІ OSP(2/1;C) КАЛІБРУВАЛЬНИХ ПЕРЕТВОРЕНЬ К. Ільєнко Виходячи з явної конструкції градуйованого узагальнено-ермітового приєднаного уявлення градуйованої алгебри Лі, розглянуто формулу Бейкера-Кемпбелла-Хаусдорффа та знайдено умови дійсності, що накладаються на грасманово-непарні параметри, які є множниками пари непарних генераторів градуйо- ваної алгебри Лі при експоненціюванні. Використання формалізму -спінорів прояснює природу грас- маново-непарних параметрів та суттєво полегшує дослідження відповідних інфінітезимальних калібруваль- них перетворень. Також вивчено дію загального групового елементу на придатному просторі уявлення та перевірено, що відповідне (градуйоване) узагальнення ермітівського спряження погоджується із природним узагальненням дираківського спряження. Обговорюються придатні узагальнення унітарного перетворення відповідного векторного простору. );1/2(osp C )2(su 27

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