PAULI THEOREM IN THE DESCRIPTION OF n-DIMENSIONAL SPINORS IN THE CLIFFORD ALGEBRA FORMALISM.pdf

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Theoretical and Mathematical Physics, 175(1): 454474 (2013)

PAULI THEOREM IN THE DESCRIPTION OF n-DIMENSIONAL

SPINORS IN THE CLIFFORD ALGEBRA FORMALISM

D. S. Shirokov

We discuss a generalized Pauli theorem and its possible applications for describing n-dimensional(Dirac,

Weyl, Ma jorana, and MajoranaWeyl) spinors in the Clifford algebra formalism. We give the explicit

form of elements that realize generalizations of Dirac, charge, and Majorana conjugations in the case of

arbitrary space dimensions and signatures, using the notion of the Clifford algebra additional signature

to describe conjugations. We show that the additional signature can take only certain values despite its

dependence on the matrix representation.

Keywords: Pauli theorem, Clifford algebra, Dirac conjugation, charge conjugation, Majorana conjuga-

tion, MajoranaWeyl spinor, Clifford algebra additional signature

1. Introduction

In 1936, Pauli proved [1] the so-called fundamental theorem on Dirac gamma matrices. This theorem

states that any two tuples of four square complex matrices of fourth order that anticommute and whose

squares equal the identity matrix or the identity matrix with the minus sign are related by a similarity trans-

formation; moreover, the similarity matrix is unique up to a nonzero complex scalar multiplication. This

theorem plays an important role in the study of various questions arising in field theory (see, e.g., [2], [3]). In

particular, it is used to prove that the Dirac equation is Lorentz invariant, to describe the relation between

spinor and orthogonal groups, and to introduce the notion of the Majorana spinor.

There are some well-known statements generalizing the Pauli theorem in a certain sense to the case

of an arbitrary dimension. Namely, it is easy to show using representation theory methods [4], [5] that aClifford algebra has a unique irreducible representation (up to equivalence) in the even-dimensional case

and two irreducible representations in the odd-dimensional case. These statements are used in various

problems in mathematical physics, in particular, in superstring theory (see [5][19]).

In an earlier work, we proposed statements generalizing the Pauli theorem [20]. Namely, we considered

the more general question (not always reducing to studying representations) of a relation of two tuples

of Clifford algebra elements satisfying the defining anticommutation relations. We gave generalizations of

even and odd dimensions over the real and complex fields to the Clifford algebra case. We showed that

in the odd real case, there are four variants (six in the complex case) for relations between two tuples of

elements satisfying the Clifford algebra anticommutation relations. Unlike the Pauli theorem in the case

of the four-dimensional Minkowski space, where the relation is realized by a similarity transformation, in

the case of an arbitrary odd dimension (e.g., in the three-dimensional case), the two tuples are relatedby a similarity transformation up to multiplication by the Clifford algebra element 1...n1...n, which can

take four different values in the real-valued case (six in the complex case; see Theorems 5 and 6 below).

Steklov Mathematical Institute, RAS, Moscow, Russia, e-mail: [email protected].

Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 175, No. 1, pp. 1134, April, 2013. Original

article submitted June 18, 2012; revised November 2, 2012.

454 0040-5779/13/1751-0454

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Moreover, in all cases (both even and odd dimensions), we gave explicit algorithms for computing the

element realizing this relation.

Here, we also indicate possible applications of the established theorems in various problems in math-

ematical physics. It makes sense to list several directions related to applications of the generalized Pauli

theorem (GPT).

As the first direction, we can mention studying the n-dimensional Dirac equation, in particular, the

question of the invariance of the equation under pseudoorthogonal coordinate transformations (in a spe-cial case, Lorentz transformations). Currently, the three-dimensional Dirac equation is actively used for

graphene. Hence, the Dirac equation is interesting in the case of not only even but also odd dimensions.

The local GPT is used to study the DiracMaxwell and DiracYangMills equation systems [21].

The second application is connected with studying relations between spinor and orthogonal groups.

We propose an alternative proof that the spinor group is a two-sheeted covering of the orthogonal group

in the case of an arbitrary dimension using the GPT (without the CartanDieudonne theorem used in the

standard expositions). In addition, we propose explicit algorithms for computing spinor group elements

corresponding to orthogonal group elements under the two-sheeted covering.

The third possible application, on which we linger in this paper, arises in studying n-dimensional

spinors. We describe the elements realizing generalizations of Dirac, Majorana, and charge spinor conjuga-

tions in the case of arbitrary space dimensions and signatures. We note that in the case of even dimensions,

we consider two analogues for each type of conjugation. In connection with the question of the existence

of Dirac, Weyl, Majorana, and MajoranaWeyl spinors in the Clifford algebra formalism in the case of

arbitrary space dimensions and signatures, questions arise that are related to GPT applications in super-

symmetry theory (we note the classic works on supersymmetry and supergravity by Sherk, Gliozzy, and

Olive [9] and by Kugo and Townsend [8], modern reviews [5], [6], and other works [10][19]). Here, we use

the GPT to study questions of this sort.

We note that we use the technique of Clifford algebras of arbitrary dimensions and signatures over the

real and complex fields. This technique seems more natural and convenient for us (e.g., compared with the

matrix technique) for studying the questions listed above. For describing spinors, an essential role is played

by the Clifford algebra structure, which has the CartanBott eight-periodicity (see Theorem 2 below).

In Sec. 2, we define a Clifford algebra and consider some related notions that we need in the subsequent

exposition. In addition, we formulate well-known theorems on the Clifford algebra center and on the

isomorphism of Clifford and matrix algebras. Further, in Sec. 3, we formulate statements generalizing the

Pauli theorem to the case of real and complex Clifford algebras of arbitrary dimensions. These theorems

play a key role in the subsequent exposition.

In Sec. 4, we briefly review n-dimensional Dirac and Weyl spinors in the Clifford algebra formalism.

In Secs. 5 and 6, we discuss the consistency of operations on Clifford algebra elements and on matrices.

We active use the GPT in studying this question. We present new results (Theorems 7 and 8). Theorem 7

is a corollary of the GPT (Theorems 4, 5, and 6). In Theorem 8, we propose explicit formulas for relating

matrix operations to operations in the Clifford algebra (for which we use the introduced notion of the Clifford

algebra additional signature, which depends on the matrix representation). Based on these formulas, wepropose explicit formulas for elements realizing generalizations of Dirac, Majorana, and charge conjugations.

In Secs. 7, 8, and 9, we prove theorems on the respective analogues of Dirac, Majorana, and charge

conjugations in the case of arbitrary space dimensions and signatures. We also recall well-known statements

about the realization ofn-dimensional Majorana and MajoranaWeyl spinors (computations are done in

the Clifford algebra formalism). We prove a theorem stating that the Clifford algebra additional signature

can take a limited number of values despite its dependence on the matrix representation (Theorem 12).

We mention that certain aspects of n-dimensional spinors are represented (sometimes in a different

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formalism) in the literature. We tried to give a more complete mathematical description of Weyl, Majorana,

and MajoranaWeyln-dimensional spinors in the case of arbitrary dimensions and signatures. We discuss

our view of the questions posed and present several new results (Theorems 79, 11, 12, and 14). In

Theorems 9, 11, and 14 we give explicit formulas for the Clifford algebra elements A, B, andC, which

depend explicitly on the Clifford algebra matrix representation. For that, we use the notion of the Clifford

algebra additional signatures (k, l) and (r, s).

The key role in our considerations is played by the GPT (Theorems 46), which we use to relateoperations on matrices to operations on Clifford algebra elements.

2. Mathematical technique of Clifford algebras

Clifford algebra (the original name was geometric algebra) was discovered by the British mathematician

William Clifford [22] in 1878 as an algebra combining properties of Grassmann algebra [23] and Hamiltons

quaternions [24]. The further development of Clifford algebras was associated with a series of famous

mathematicians and physicists (R. Lipschitz, T. Valen, E. Cartan, E. Witt, C. Chevalley [25], M. Riesz [26],

and others). The discovery in 1928 of the Dirac equation for the electron [27], to which Clifford algebra is

directly related, proved essential for studying Clifford algebras. The Dirac equation is written using four

complex-valued matrices (Dirac gamma matrices) that satisfy the same defining relations as the Clifford

algebra generatorsC(1, 3). The connection of Clifford algebra with spinors attracted the attention ofmany physicists and mathematicians to the Clifford algebra theory. Clifford algebras are currently used in

many areas of modern mathematics and physics, for example, in field theory [28], [29], robotechnics, signal

and image processing, chemistry, celestial mechanics, electrodynamics, computing, electrodynamics, and

geometry among others.

There are several (equivalent) definitions of Clifford algebras known in the literature.1 In the Clifford

algebra that we consider below, we use a basis of a special form labeled by ordered multi-indices. We stress

that the generators and basis introduced below are fixed (they do not change). Such a definition is closer

to Cliffords original definition.

LetEbe a vector space over the field F of real numbers R or complex numbers C. Letn be a natural

number, and let the dimension ofEbe dim E= 2n. We assume that there is a basis

e, ea, ea1a2 , . . . , e1...n, a1 < a2


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field and byCC(p, q) C(p, q) in the case of the complex field.2 When our argumentation is applicable toboth cases, we writeCF(p, q), implying that F = R or F = C.

The elementsea are called Clifford algebra generators,3 and the elemente is called the Clifford algebra

identity. The number pair (p, q) is called the signatureof the Clifford algebraCF(p, q). We note that thenumber p qalso is often called the signature.

Any elementUof the Clifford algebraCF(p, q) can be expanded in the basis

U=ue +uaea +

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Parity conjugation. The operation of parity conjugation U U is such that it multiplies oddelements by1 and does not change even elements. In particular, we have (ea) =ea. For an elementU C(p, q), we have U =nk=0(1)kkU. The equalitiesU =U, (U V) =UV, (U+V) =U+V,and (U) =U hold.

Operations of projecting onto subspaces CFk

(p, q) and the trace operation. We introduce a

notation for linear operations of projecting onto the subspaces of rank-k elements:

Uk =kU=

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3. Generalized Pauli theorem

In [20], we proposed theorems generalizing the so-called fundamental Pauli theorem on gamma matrices.

We used the Clifford algebra formalism there. We formulate the proved theorems.

LetIdenote the set of multi-indices A of length from 0 to n:I ={, 1, . . . , n , 12, 13, . . . , 1, . . . , n},where is the empty multi-index. Let

Ieven={A I:|A| is even}, Iodd ={A I:|A| is odd}.

We set A= a1...ak =ak a1 = (A)1 and a = abb = (a)1.

Theorem 4. LetCF(p, q) be a real or complex algebra of even dimension n= p +q. Let two tuplesof Clifford algebra elementsa anda, a = 1, 2, . . . , n, satisfy the relations

ab +ba = 2abe, ab +ba = 2abe.

Then the two tuples generate Clifford algebra bases,4 and there exists a unique (up to nonzero real or

complex scalar multiplication) invertible Clifford algebra element T CF(p, q) such that

a =T1aT a= 1, . . . , n . (3)

Moreover, such an element T has the form T =

A AF A, where any element of{A, A Ieven} if

1...n =1...n or of{A, A Iodd} if1...n =1...n such that

A AF A= 0 can be taken asF.

Theorem 5. LetCR(p, q) be a real Clifford algebra of odd dimension n = p+q. Let two tuples ofClifford algebra elementsa anda, a = 1, 2, . . . , n, satisfy the relations

ab +ba = 2abe, ab +ba = 2abe.

Then in the case of the Clifford algebraCR(p, q) of signaturep q 1 (mod 4), the elements1...n and1...n either take valuese1...n, and the corresponding tuples generate a Clifford algebra basis, or they takevaluese, and the tuples do not generate a basis. Then cases14 listed below are realized.

In the case of the Clifford algebraCR(p, q) of signaturepq 3 (mod 4), the elements1...n and1...n always take valuese1...n, and the corresponding tuples always generate a Clifford algebra basis.Then only cases1 and2 are realized.

There exists a unique (up to an invertible Clifford algebra center element multiplication) invertible

Clifford algebra elementT such that

1. a =T1aT

a= 1, . . . , n

1...n =1...n,

2. a =T1aT a= 1, . . . , n 1...n =1...n,

3. a =e1...nT1aT a= 1, . . . , n 1...n =e1...n1...n,

4. a =e1...nT1aT a= 1, . . . , n 1...n =e1...n1...n.

(4)

4In other words, all possible products a1 ak =a1...ak = A and A form two bases in CF(p, q).

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We note that all four cases can be uniformly written as

a = (1...n1...n)T1aT.

Moreover, in the case of a real Clifford algebra of signaturep q 1 (mod 4), the element T whoseexistence is asserted in all four cases in the theorem has the form

T =

AIeven

AF A, (5)

where an element of the set{A +B , A, B Ieven} can always be taken asF.In the case of a real Clifford algebra of signaturep q3 (mod 4), the element Twhose existence is

asserted in cases1 and2 in the theorem has form (5). Moreover, an element of the set{A, A Ieven}such that the convolution constructed with it

AIeven

AF A= 0 can be taken asF.

Theorem 6. LetC(p, q) be a complex Clifford algebra of odd dimension n = p + q. Let two tuples ofClifford algebra elementsa anda, a = 1, 2, . . . , n, satisfy the relations

a

b

+

b

a

= 2

ab

e,

a

b

+

b

a

= 2

ab

e.

Then in the case of the Clifford algebraC(p, q) of signaturep q1 (mod 4), the possible values for theelements1...n and1...n aree1...n if the corresponding tuples generate Clifford algebra bases ande ifthe tuples do not generate a basis. Then cases14 in the theorem are realized.

In the case of a Clifford algebraC(p, q) of signaturepq 3 (mod 4), the possible values for theelements1...n and1...n areie1...n if the corresponding tuples generate Clifford algebra bases andieifthe tuples do not generate a basis. Then cases1, 2, 5, and6 in the theorem are realized.

There exists a unique (up to an invertible Clifford algebra center element multiplication) invertible

Clifford algebra elementT such that

1. a =T1aT

a= 1, . . . , n

1...n =1...n.

2. a =T1aT a= 1, . . . , n 1...n =1...n,

3. a =e1...nT1aT a= 1, . . . , n 1...n =e1...n1...n,

4. a =e1...nT1aT a= 1, . . . , n 1...n =e1...n1...n,

5. a =ie1...nT1aT a= 1, . . . , n 1...n =ie1...n1...n,

6. a =ie1...nT1aT a= 1, . . . , n 1...n =ie1...n1...n.

(6)

We note that all six cases can be uniformly written as

a

= (1...n

1...n)T1

a

T.

Moreover, the elementTwhose existence is asserted in all six cases in the theorem has the form

AIeven

AF A,

where an element of the set{A +B , A, B Ieven} can always be taken asF.

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We note that the proposed theorems can be reformulated in terms of matrices using Theorems 2 and 3.

In this paper, we omit this because the resulting formulations are cumbersome.

4. Dirac and Weyl spinors in the Clifford algebra formalism

Spinors were considered first by Cartan in 1913 and rediscovered by Dirac in 1928. Since then, a series

of approaches has been developed, and a series of spinor realizations have been proposed, including the case

of an arbitrary space dimension.

The discovery of the Dirac equation in 1928 [27] attracted the attention of many physicists and math-

ematicians to studying spinors. In 1930, Juvet [31] and Sauter [32] considered the spinor as an element of a

left ideal in a matrix algebra. In 1947, Riesz first interpreted spinors as elements of a left ideal in a Clifford

algebra [26]. This approach is most convenient when consideringn-dimensional spinors and studying their

properties (see, e.g., [30], [33]). We also mention the now classic works of Rashevsky [34] and Rumer [35].

We note that the notion of spinor has two faces. The first face (algebraic) is that the spinor (in the

simplest case) is an element of a certain minimal left ideal (see below), i.e., essentially, simply a column, if

we use the matrix formalism. The second face is related to the fact that the spinor is in fact a tuple (column)

of functions depending on the point in space and is multiplied from the left by a spinor group element under

orthogonal transformations with a matrixP O(p, q). Our subsequent considerations almost always relateto only the first, the algebraic face. The second face must be taken into account when we consider the

spinor not as an abstract, simply algebraic object but as the unknown in the Dirac equation.

Dirac spinors. We consider the real Clifford algebraCR(p, q) and a primitive idempotent t2 = t,t CR(p, q). There is a corresponding minimal left ideal (spinor space) I(t) =CR(p, q)t, generated by theidempotentt. We call the elements I(t) spinors (in the Clifford algebra formalism).

An irreducible representation of the Clifford algebraC(p, q) End(I) is injective when pq=1 (mod 4) (see Theorem 2). In the case where pq 1 (mod 4), the Clifford algebra is no longersimple and is the direct sum of two prime ideals. In this case, we consider the left ideal (double spinor

space)I I, constructed on the idempotents t andt. We thus obtain an injective representation in the case

pq1 ( mod 4). But it is reducible and is the direct sum of two irreducible representations. Each of theseirreducible representations is called a half-spinor, and the corresponding left ideals are called half-spinorspaces.

We consider the complex Clifford algebraC(p, q) analogously (see Theorem 3). In the case wherenis even, we consider the spinor representation. Elements of the left ideal are calledDirac spinors. In the

case where n is odd, we consider either double spinorsor half-spinors. In what follows, we consider only

complex Clifford algebras.

Weyl spinors. We consider the set of Dirac spinors, realized in the complex Clifford algebra (as a

set of the left ideal elements) EDirac={I(t)}. We consider the chiral operator(or pseudoscalar)

=e

1...n

, p q0, 1 (mod 4),ie1...n, p q2, 3 (mod 4).

It is easy to verify that = 1 = . Moreover, for all a = 1, 2, . . . , n,{, ea} = ea +ea = 0 ifn iseven, and [, ea] =ea ea= 0 ifn is odd.

We define two operators

PL =e

2 , PR=

e +

2 .

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They are orthogonal idempotents (projectors) because (PR)2 = PR, (PL)

2 = PL, and PRPL = PLPR = 0.

We note that in the case wheren is odd, the operatorsPL and PRbelong to the center ofC(p, q) and inducea decomposition of the Clifford algebra into the direct sum of two ideals: C(p, q) = PRC(p, q) PLC(p, q).

We consider a complex Clifford algebraC(p, q) of even dimension n = p + q. Then leftand right Weylspinors(or chiral spinors) are defined as

ELWeyl={EDirac|PL= }, ERWeyl={EDirac|PR= }.

Hence, Weyl spinors are eigenspinors of the operators PL and PR. We note that we can rewrite the

conditions on spinors equivalently as PL = = and PR = = . We note thatEDirac = ERWeyl ELWeyl, i.e., we have = L+R, L = PL, and R= PR for any EDirac.

5. Consistency of matrix operations and operations in Clifford

algebras

We consider matrix representations of complex Clifford algebras (see Theorem 2)

:C(p, q)Mat(2n/2,C), n is even,

Mat(2(n1)/2,C) Mat(2(n1)/2,C), n is odd.

In the case where n is odd, Clifford algebra elements are represented as block-diagonal matrices of size

2(n+1)/2. We leta =(ea) denote matrices corresponding to the Clifford algebra generators in the matrix

representation.

According to the Pauli theorem, all other matrix representations in the case where n is even can be

derived from the initial representation in the form a =T1aT, where the invertible matrix T is defined

up to a nonzero constant. In the case wheren is odd, all matrix representations can be derived from the

initial one as a =T1aT, where the matrix T is defined up to multiplication by an invertible elementof the center Z = 1+J, where J = diag(1, 1, . . . , 1, 1, . . . , 1) is the diagonal matrix with an equalnumber of 1 and

1 on the diagonal.

We consider the operation of Hermitian conjugation of a Clifford algebra element, which is defined by

(see [36])

U =

e1...pUe1...p, pis odd,

e1...pUe1...p, p is even,

U =

ep+1...nUep+1...n, qis even,

ep+1...nUep+1...n, qis odd.

(7)

We can always choose a matrix representation (see [36]) such that the (ea) = a are unitary:

(a

)

= (a

)1

=aa

a

. (8)

In what follows, we consider only matrix representations for which (8) holds. We note that relation (8) is

equivalent to ((ea)) =((ea)) and hence to (U) = (U)U C(p, q), i.e., the Hermitian conjugationof a matrix is consistent with Hermitian conjugation of a Clifford algebra element. We also note that (8)

implies that the firstp matricesa are Hermitian and the last qare skew-Hermitian.

We consider various conjugation operations (see Sec. 2) of the elements of a Clifford algebra C(p, q):U, U, U, U, U, UT,

U.

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We here consider two new operations on Clifford algebra elements: the operation of transposing a

Clifford algebra elementand the operation of complex matrix conjugating a Clifford algebra element,

UT =1(((U))T),

U =1((U)), (9)

which depend on the choice of the matrix representation . Here,(U) is the matrix complex conjugate of

(U).We consider a tuple ea of generators of a Clifford algebraC(p, q). We note that (ea) =ea, (ea) =

ea, ea =ea, and (ea) =ea. We consider tuples of elements obtained from the initial tupleea using oneof the conjugation operations:

a =(ea), (ea)T, ea. (10)

We note that the tuples considered satisfy the defining relations of the Clifford algebra ab+ba = 2abe.

The following theorem holds.

Theorem 7. The tuplesa introduced in (10) generate new bases of the Clifford algebraC(p, q).In the case wheren is even, for the considered tuplesea anda, there always exists a unique(up to

multiplication by a complex constant) invertible elementT such that

a =T1eaT, a= 1, . . . , n .

Moreover,T Ceven(p, q) if1...n = e1...n, andT Codd(p, q) if1...n =e1...n. The element T has theform

A

AeBeA for someeB.

In the case wheren is odd, for the considered tuplesea anda, there always exists a unique (up to

multiplication by an invertible element of the center) invertible elementT such that

a =T1eaT, a= 1, . . . , n (in the case1...n =e1...n),

a =T1eaT, a= 1, . . . , n (in the case1...n =e1...n).

In both cases, T Ceven(p, q) (or multiplying bye1...n, we obtain anotherT Codd(p, q)). Moreover,the elementThas the form

AIeven

AeBeA for someeB.

Proof. The theorem is a consequence of Theorems 46.

6. Clifford algebra additional signatures

We note that we can always choose matrix representations such that in addition to (8), they satisfy

the conditions

(a)T =a, a =a, (11)

i.e., matricesa are real or purely imaginary (and hence symmetric or skew-symmetric because of unitarity).

We introduce the Clifford algebra additional signatures

(k, l), 0k n, 0ln, k+ l=n,

(r, s), 0rn, 0sn, r+s=n,

which depend on the Clifford algebra matrix representation . Here, k is the number of symmetric

matrices among the a, andlis the number of skew-symmetric matrices among the a. Analogously,r is

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the number of real matrices among the a, and sis the number of purely imaginary matrices among the

a.

The introduced quantities depend on the matrix representation. In what follows, we sometimes omit

the index . We note that despite the dependence on the matrix representation, we find constraints on

possible values of the parameters k and l(see Theorem 12 below).

The parametersk,l,r, ands can be insufficient to completely recover the necessary information about

the chosen matrix representation of the Clifford algebra. Therefore, we propose starting with two other

independent parameters [kp] and [kq], which are the respective numbers of symmetric matrices in the

firstp matricesa and of symmetric matrices in the last qmatricesa. We note that [kp] k, [kp]p,[kq]k, and [kq]q. These two parameters (together with the fixed independent parameters p and qofthe matrix representation) suffice to recover the parameters k, l, r, s, [lp], and [lq] unambiguously.

The last two of these parameters are the respective numbers of skew-symmetric matrices in the first p

matricesa and of skew-symmetric matrices in the last qmatrices a. For these parameters, we have the

formulas

[lp]=p [kp], [lq]=q [kq],

k= [kp]+ [kq], l= [lp]+ [lq]=p+q [kp] [kq]=n k,

r= [kp]+ [lq]= [kp]+ q [kq], s= [lp]+ [kq]=p [kp]+ [kq].

Finally, the dimensionn of the Clifford algebra is thus represented as the sum of four numbers

n= [kpr]+ [lps]+ [kqs]+ [lqr],

where [kpr] = [kp], [lps] = [lp], [kqs] = [kq], and [lqr] = [lq]. Here, the indices r and s reflect

realness and imaginariness of the corresponding matricesa, which follow automatically from unitarity and

(skew-)symmetry.

For example, in the case (p, q) = (n, 0), we have [kq] = [lq] = 0, k = r = [kp], and l =

s = [lp] = p[kp], i.e., in this case, all symmetric matrices representing generators are real, and allskew-symmetric ones are imaginary.

In the case (p, q) = (0, n), we have [kp]= [lp]= 0, k=s= [kq], andl=r= [lq]=q [kq],i.e., in this case, all symmetric matrices representing generators are imaginary, and all skew-symmetric ones

are real.

Under the stated assumptions, we have a =ea for all tuples (10). According to the GPT, theelements Tdiscussed above can then be found among elements of the form

eAeBeA, i.e., among elements

of the basis{eA}.Let eb1 , . . . , ebk denote the generators ea for which the matrices a are symmetric. Analogously, we

haveec1 , . . . , ecl for skew-symmetric matrices,ed1 , . . . , edr for real ones, andef1 , . . . , efs for purely imaginary

ones.

We can write explicit formulas for the operations UT and

U. Namely, we have the following result.

Theorem 8. We consider a complex Clifford algebraC(p, q) and a matrix representationsatisfyingconditions (8) and (11). Then we have the following formulas for the operations of transposing and of

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taking the complex matrix conjugation(9) of a Clifford algebra element:

UT =

eb1...bkUeb1...bk , k is odd,

eb1...bkUeb1...bk , k is even,

UT

=

ec1...clUec1...cl , l is even,

ec1...clUec1...cl , l is odd,

(12)

U =

ed1...drU ed1...dr , r is odd,

ed1...drUed1...dr , r is even,

U =

ef1...fsUef1...fs, s is even,

ef1...fsUef1...fs , s is odd.

(13)

Proof. The proof is analogous to the proof of formulas (7) (see [36]).

We note that formulas (12) and (13) are similar to Hermitian conjugation formulas (7), but they dependon the matrix conjugation explicitly. Depending on k,l,r, ands, there are always two equivalent formulas

for transposition and two equivalent formulas for complex matrix conjugation.

7. Generalizing Dirac conjugation

In this section, we introduce the analogues of Dirac conjugation in the case of an arbitrary space

dimension and signature. In the case where the dimension n is even, we consider two different Dirac

conjugations. In the case where n is odd, we consider only one Dirac conjugation.

We consider a complex Clifford algebra. Using considerations in Sec. 6, we conclude that according to

the GPT (Theorem 7), among the basis elements{eA}, there is an element A such that

(ea) =A1 eaA. (14)

Moreover, both elements A always exist when n is even, and only one element A+ or A exists when n

is odd. We can rewrite these formulas in the form U = A1+ UA+ and U = A

1 U

A. In the next

theorem, we give explicit formulas for A. We note that in the literature, the elemente1...p is used as the

element realizing Dirac conjugation for even n. We consider all possible cases in the next theorem.

Theorem 9. The element A+ always exists except in the case wherep is even andq is odd, and the

elementA always exists except in the case wherep is odd andqis even. For these elements, we have

A+ =

1e1...p, p andqare odd,

2ep+1...n, p andqare even,

Z1e1...p =Z2ep+1...n, p is odd andqis even,

(15)

A =

1ep+1...n, p andqare odd,

2e1...p, p andqare even,

Z1e1...p =Z2ep+1...n, p is even andq is odd,

(16)

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wherei are arbitrary nonzero complex constants andZi are arbitrary invertible elements of the center of

the Clifford algebraC(p, q). Furthermore, we can always choosei andZi such that the elementsA+ andA satisfy the relations

A+ = A+ = A

1+ =A

+, A = A

= A

1 =A

. (17)

Proof. According to Theorem 7, the element A+ exists if (e1)

(en) = e1...n, i.e., when qis even,

and the element A exists if (e1) (en) =e1...n, i.e., when q is odd. Explicit expressions for the

elements A+ and A can be derived from (7). Choosing the number 1 or i as depending on p and q, we

can verify properties (17).

In the case of different signatures, we can introduce the Dirac conjugation operations

D+ =(A+)1 = (A+)

1, except whenp is even and qis odd, (18)

D =(A)1 = (A)

1, except when p is odd and q is even. (19)

For example, in the case of the Clifford algebra C(1, 3) with the generatorse0,e1,e2, ande3, we obtaintwo Dirac conjugations,D+ =e0 andD =e123, the first of which coincides with the standard one.

We consider the following two arrays of elements called bilinear covariants (Dirac bilinear forms):jA =

DeA, where A is an arbitrary multi-index of length from 0 to n. We can verify that under

orthogonal coordinate transformations x x = px, where P =p O(p, q), bilinear covariantschange (if spinors change by the rule S, whereSis the spinor group element) as tensors or almost astensors (the minus sign enters the transformation formula): (j1...k )

= p11 pkkj1...k or (j1...k ) =p11 pkkj1...k depending onk .

8. Generalizing Majorana conjugation and a theorem on the

Clifford algebra additional signature

We consider the complex Clifford algebraC(p, q) and the operation of transposing Clifford algebraelement (9). We note that this operation depends on the choice of the matrix representation.

According to the GPT, there exist elements C such that

(ea)T =C1 eaC. (20)

Moreover, in the case where n is even, both elements C always exist, and in the case where n is odd, only

one of the elements C+ andC exists. We can also rewrite the formulas in the form UT =C1+ U

C+ and

UT =C1 UC.

We formulate the following theorems about the elementsC. Formulas (21) and (22) are well known.

We also give the explicit form ofC(formulas (23) and (24)) and expressions for the constants depending

on the values of the additional signature (k, l) (see (25) and (26)).

Theorem 10. The element C+ exists when n= 3 (mod 4), and the element C exists when n=1 (mod 4). For these elements, we have the formulas

(C)T =C,

CC = e, (21)

where

+ =

+1, n0, 1, 2 (mod 8),1, n4, 5, 6 (mod 8),

=

+1, n0, 6, 7 (mod8),1, n2, 3, 4 (mod 8).

(22)

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Theorem 11. In the case where the matrix representations satisfy(8) and(11), we have the explicit

form of elementsC

C+ =

1eb1...bk , k andl are odd,

2ec1...cl , k andl are even,

Z1ebl...bk =Z2ec1...cl , k is odd andl is even,

(23)

C =

1ec1...cl , k andl are odd,

2eb1...bk , k andl are even,

Z1ebl...bk =Z2e

c1...cl , k is even andl is odd,

(24)

wherei are arbitrary nonzero complex constants andZi are invertible elements of the center of the Clifford

algebraC(p, q). For, we also obtain the values

+ =

(

1)k(k1)/2, k andl are odd,

(1)l(l+1)/2, k andl are even,(1)k(k1)/2 = (1)l(l+1)/2, k is odd andl is even,

(25)

=

(1)l(l+1)/2, k andl are odd,(1)k(k1)/2, k andl are even,(1)k(k1)/2 = (1)l(l+1)/2, k is even andl is odd.

(26)

Proof of Theorems 10 and 11. The explicit formulas for the elementsC in the case of the stated

matrix representations are obtained from formulas (12).

For evenn, we obtainea =CTC1 eaC(C1 )T from (20), i.e.,C(C1 )T =e, because the obtainedexpression commutes with all elements. Because Det(C) = 1 (we understand the determinant of a Clifford

algebra element as the determinant of the corresponding matrix element; see [37]), we have = 1, 1.Moreover,CC = e, and we hence obtain

C = (CT)

= (C) =C1 .

Next, for an arbitrary ordered multi-index A of lengthk , we have

(eA)T = (1)k(1)k(k1)/2C1 eAC,

(CeA)T = (1)k(1)k(k1)/2(CeA).

We note that because the tupleeA forms a basis in the Clifford algebra, it follows that the tuple of symmetric

and skew-symmetric matrices CeA also forms a basis. The elements are represented as complex squarematrices of size 2n/2. The number of skew-symmetric matrices among them must then be (1/2)2n/2(2n/21).On the other hand, their number is equal to

nk=0

1

2(1 (1)k(1)k(k1)/2)Ckn.

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Combining the two expressions, we obtain

= cosn

4 sinn

4 =

2sin

4(n 1)

,

which was to be proved.

For odd n, we can obtain the same formulas (21) using the fact that in the case of the stated matrix

representations, the elements C can always be found among the basis eA elements (see formulas (23)

and (24)). We obtain formulas (25) and (26) (for even n) by combining formulas (23), (24), and (21).

Next, choosing a concrete matrix representation (see, e.g., [36]), we obtain the values for depending on

n(mod 8) in the case where n is odd.

We note that although the elements C depend on the matrix representation, their symmetricity (21)

is determined by only the space dimension n given by (22).

The following result is new. Although the additional signature parameters k and l (see Sec. 6) are

determined by the choice of matrix representation, their values cannot be arbitrary and take certain specific

values depending on the space dimensionn.

Theorem 12. We consider the complex Clifford algebrasC(p, q) and a matrix representation satisfy-ing(8) and(11). Depending on the Clifford algebra dimensionn = p +q, we then have possible values for

the additional signature(k, l) listed in Table1.

Table 1n( mod 8) (k (mod 4), l(mod 4))

0 (0, 0), (1, 3)

1 (1, 0)

2 (1, 1), (2, 0)3 (2, 1)

4 (3, 1), (2, 2)

5 (3, 2)

6 (3, 3), (0, 2)

7 (0, 3)

Proof. The proof is obtained by combining explicit formulas (25) and (26) for depending onk and

l and formulas (22) depending on n (mod 8).

As an example, we consider the Clifford algebrasC

(p, q) of dimension n= 2 and different signatures

(p, q) = (2, 0), (1, 1), and (0, 2). The following three Pauli matrices anticommute and when squared give

+1: 0 ii 0

,

0 1

1 0

,

1 0

0 1

.

As1 and2, we can take any two of these three matrices (multiplying by the imaginary uniti, if necessary).

Hence, only the signatures (k, l) = (2, 0) and (1, 1) are realized. Possible values for additional signatures of

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dimensionsn = 1, 2, 3, 4 Clifford algebras are listed in Table 2.

Table 2

(p, q) ([kp], [kq]) (k, l) (r, s)

(1, 0) (1, 0) (1, 0) (1, 0)

(0, 1) (0, 1) (1, 0) (0, 1)

(2, 0) (2, 0) (2, 0) (2, 0)

(1, 0) (1, 1) (1, 1)

(1, 1) (1, 1) (2, 0) (1, 1)

(1, 0) (1, 1) (2, 0)

(0, 1) (1, 1) (0, 2)

(0, 2) (0, 2) (2, 0) (0, 2)

(0, 1) (1, 1) (1, 1)

(3, 0) (2, 0) (2, 1) (2, 1)

(2, 1) (1, 1) (2, 1) (1, 2)

(2, 0) (2, 1) (3, 0)

(1, 2) (1, 1) (2, 1) (2, 1)

(0, 2) (2, 1) (0, 3)

(0, 3) (0, 2) (2, 1) (1, 2)

(p, q) ([kp], [kq]) (k, l) (r, s)

(4, 0) (3, 0) (3, 1) (3, 1)

(2, 0) (2, 2) (2, 2)

(3, 1) (3, 0) (3, 1) (4, 0)

(2, 1) (3, 1) (2, 2)

(2, 0) (2, 2) (3, 1)

(1, 1) (2, 2) (1, 3)

(2, 2) (2, 1) (3, 1) (3, 1)

(1, 2) (3, 1) (1, 3)

(2, 0) (2, 2) (4, 0)

(1, 1) (2, 2) (2, 2)

(0, 2) (2, 2) (0, 4)

(1, 3) (1, 2) (3, 1) (2, 2)

(0, 3) (3, 1) (0, 4)

(1, 1) (2, 2) (3, 1)

(0, 2) (2, 2) (1, 3)

(0, 4) (0, 3) (3, 1) (1, 3)

(0, 2) (2, 2) (2, 2)

We indicate the algorithm we used for filling in this table using only the statement of Theorem 12 and

formulas in Sec. 6. Considering the Clifford algebra C(p, q) of dimensionn = p + q, we first use Theorem 12and find all possible values for the parameters kand l. We then decompose the parameterkin all possible

ways into pairs ([kp], [lq]) such that k= [kp]+ [kq], 0[kp] p, and 0[kq]q. Next, using theparameters ([kp], [lq]), we recover the additional signature (r, s) by the formulas r= [kp]+ q [kq]ands=n r.

As an example, we consider the Clifford algebra C(1, 3). The standard Dirac representation correspondsto the case where (k, l) = (2, 2) and (r, s) = (3, 1). The Majorana representation corresponds to the

case where (k, l) = (3, 1) and (r, s) = (0, 4). The two remaining cases in Table 2 are also realized. We

obtain the case where (k, l) = (2, 2) and (r, s) = (1, 3) from the Dirac representation by multiplying

the matrices corresponding to the generatorse1 ande2 by the imaginary unit i and exchanging them. The

case (k, l) = (3, 1) and (r, s) = (2, 2) can be obtained from the Majorana representation by multiplying

the matrices corresponding to the generatorse1 and e2 by the imaginary uniti and exchanging them.

We call one of the following conjugations (every time when the corresponding C exists) the Majorana

conjugation:

M+ =T(C+)1 = (C+)1, n = 3 (mod 4),M =T(C)

1 = (C)1, n= 1 (mod 4).

As an example, we consider the case of the Clifford algebra C(1, 3) and the standard matrix represen-tation using the Dirac gamma matrices0, 1, 2, and 3. It is easy to verify that the matrices 0 and2

are symmetric and that the matrices 1 and 3 are skew-symmetric. We hence have k = l = 2. Therefore,

C+ = e13, C= e02, M+ =Te13, andM =Te02. In this case, we have CT =C, i.e., =1.

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9. Generalization of charge conjugation, Majorana and

MajoranaWeyl spinors in the Clifford algebra

We now consider operation (9) of complex matrix conjugationUof the elements of the Clifford algebra

C(p, q). This operation depends on the choice of the matrix representation. We should not confuse it withthe operation of complex conjugationea =ea of Clifford algebra elements.

According to the GPT (see Theorem 7), we haveea =B1 eaB. (27)

Moreover, in the case where n is even, both elements B exist, and in the case where n is odd, only one of

B+ andB exist. We can rewrite these formulas in the forms

U =B1+ UB+ and

U =B1 U

B.

We formulate theorems about the elements B. We note that formulas (28) and (29) are well known.

We also present the explicit form of the elements B (see (30) and (31)).

Theorem 13. The element B+ exists in the cases where pq= 3 (mod 4), and the element Bexists in the cases wherep q= 1 (mod 4). For these elements, we have the formulas

BT = B, BB= e, (28)

where

+ =

+1, p q0, 1, 2 (mod 8),1, p q4, 5, 6 (mod 8),

=

+1, p q0, 6, 7 (mod 8),1, p q2, 3, 4 (mod 8).

(29)

Theorem 14. In the case of matrix representations satisfying conditions (8) and (11), we have the

explicit forms of the elementsB

B+=

1ed1...dr , r ands are odd,

2ef1...fs

, r ands are even,

Z1edl...dr =Z2ef1...fs , r is odd ands is even,

(30)

B=

1ef1...fs , r ands are odd,

2ed1...dr , r ands are even,

Z1edl...dr =Z2ef1...fs , r is even ands is odd,

(31)

wherei are arbitrary nonzero complex constants andZi are invertible elements of the center of the Clifford

algebraC(p, q).We have the following relations between the elements(in the cases where each triple of elements exists):

B+= (A1+ )

C+, B+ = (A1 )

C, B = (A1 )

C+, B = (A1+ )

C. (32)

Proof of Theorems 13 and 14. The explicit formulas for the elementsB in the case of the stated

matrix representations follow from formulas (13).

In the case where n is even, we have ea =B1

ea

B =

B1 B

1 e

aBB, i.e., B

B= e. Hence,

R. It follows from Det(B) = 1 that = 1, 1. Moreover, (B)B = e and BT = (B) =

(B1 )

=B.

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In the case where n is odd, we can obtain the same formulas (28) using the fact that in the case of

the stated matrix representations, the elements B can always be found among the basis elements eA (see

formulas (30) and (31)).

The elements B can be expressed in terms of the elements A andC. For example, n3 (mod 4)except when p is even and q is odd. From U = A1+ U

A+ and UT = C1+ U

C+, we obtain

U =

C1+ (A1+ U

A+)C+ = C

1+ A

+U(A

1+ )

C+ and hence B+ = (A1+ )

C+. The three other formulas in the

theorem statement are proved analogously.We now obtain the explicit form for the coefficients . In the case whereA+and C+ exist, we compute

+ = BT+B

1+ = ((A

1+ )

C+)T((A1+ )

C+)1 =CT+(A

1+ )

TC1+ AT+ =

=+C+C1+ A

1+ C+C

1+ A

+ = +A

1+ A

+.

In the case where A and C exist, we compute analogously and obtain + = A1 A

. In the case

where A and C+ exist, we obtain = +A1 A

. In the case where A+ and C exist, we obtain

= A1+ A

+ . Knowing the values (depending onn) and the values A (depending on the parities

ofpandq), from these four formulas (which include all possible cases), we obtain the constants depending

on p and qand given by (29).

We note that although the elementsB depend on the matrix representation, their symmetricity (28)

is determined only by the space signature p q(see (29)).We consider the two operations ofcharge conjugation

ch+ =B+ = B+, p q= 3 (mod 4),

ch =B = B, p q= 1 (mod 4).

Theorem 15. The following definitions of charge conjugations(when the corresponding elementsC

andA exist) are equivalent:

ch+ =C+(D+)T, ch+ =C(D)T,

ch =C+(D)T, ch =C(

D+)T.(33)

Proof. Taking relations (32) between the elements A, B, andC into account, we obtain

ch+ =B+ = (A1+ )

C+ =C+(A

1+ )

TC1+ C+ =C+(

A1+ )T =C+(

D+)T.

We can obtain the three other formulas analogously.

As an example, we consider the Clifford algebraC(1, 3) and the standard matrix representation usingDirac gamma matrices. Because the matrices 0,1, and3 are real and the matrix2 is purely imaginary,

we have r = 3, s = 1, + =1, and = 1 (because pq= 6 (mod 8)) and B+ = e013, B = e2,ch+ =e013

, and ch =e2

.

Majorana spinorsand pseudo-Majorana spinors(often simply called Majorana spinors) are defined as

EM ={EDirac|ch =}, EpsM={EDirac|ch+ =}.

Finally, we quote the well-known statement on the realization of Majorana and MajoranaWeyl spinors

in the case of arbitrary space dimensions and signatures.

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Proposition 1. Majorana spinors are realized in the case wherep q0, 6, 7 (mod 8), and pseudo-Majorana spinors are realized in the case wherep q0, 1, 2 (mod 8).

Proof. For pseudo-Majorana spinors, we obtain

=B+ , B1+ =

=B+=+B1+ , (1 +)= 0, + = 1.

Analogously, for Majorana spinors, we obtain = 1. We next use formulas (29).

We note that in the cases where p q0, 1, 2 (mod 8), the Clifford algebraCR(p, q) is isomorphic tothe algebra of real matrices, and in the cases where p q0, 6, 7 (mod 8), the same can be said about theClifford algebraCR(q, p) (see Theorem 2).

Left and right MajoranaWeyl spinorsare defined as

ELMW ={ELWeyl|ch+ =}, ERMW ={ERWeyl|ch =},

ELpsMW ={ELWeyl|ch =}, ERpsMW ={ERWeyl|ch+ =}.

Proposition 2. MajoranaWeyl spinors are realized in the casep q= 0 (mod 8).

Proof. Weyl spinors are realized only in the case where n is even. Moreover, = e1...n for p q=0 (mod 4) and = ie1...n for pq = 2 (mod 4) (see Sec. 5). We show that for pq = 2 (mod 4),MajoranaWeyl spinors are not realized. By Proposition 1, we then obtain only the case where p q =0 (mod 8). Indeed, let = ie1...n. Then the left Weyl spinor condition ie1...n = for the actionof complex matrix conjugation becomes the right Weyl spinor condition ie1...n = (indeed, we have

B1+ (ie1...n)B+ = andB+ =), and we obtain a contradiction. We can compute analogously

for the element B using B =.

We stress the connections of the results obtained here to other results. In the literature, it is customary

to pay the greatest attention to the case of signatures (1, n1), while we here presented the spinor formalismfor arbitrary (p, q) signatures. In [5], [7], [9] (only in the case of signatures(1, n

1)), the properties of three

elements were studied, each of which corresponds to one element of each form A, B, and C in thispaper. In [8], the case of an arbitrary signature (p, q) and also one element of each form (A, B, and

C) were considered for the first time. In [6], [18], two elements were considered for each form, and the

basic properties of these elements were proved. We gave the explicit forms of the elements A, B, and

C depending on the matrix representation here (see Theorems 9, 11, and 14).

We also note that we discussed mathematical structures and constructions. Relating the proposed

mathematical constructions to real world ob jects (elementary particles) goes beyond the scope of this

investigation. Our goal here was to develop the theory mathematically.

Certain results regarding the existence of Weyl, Majorana, and MajoranaWeyl spinors for various

space dimensions and signatures are applied in supersymmetry theory. In the case of signaturesp q=0 (mod 8), MajoranaWeyl spinors are often taken as supercharges. In the case of signatures p

q =

0, 1, 2, 6, 7 (mod 8), Majorana and pseudo-Majorana signatures are considered. Because of considering

different types of spinors, the Poincare algebra supersymmetric extension is built differently, depending on

the space dimension and signature (see the references in [5][19]).

Acknowledgments. The author is deeply grateful to I. V. Volovich, N. G. Marchuk, and the referee

for the useful comments.

This work was supported by the Program for Supporting Leading Scientific Schools (Grant No. NSh-

2928.2012.1) and the Ministry of Science and Education of the Russian Federation (Contract No. 8215).

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REFERENCES

1. W. Pauli, Ann. Inst. H. Poincare, 6, 109136 (1936).

2. N. N. Bogolyubov and D. V. Shirkov, Quantum Fields [in Russian], Nauka, Moscow (1980); English transl.,

Wiley, New York (1980).

3. N. N. Bogolubov, A. A. Logunov, A. I. Oksak, and A. I. Todorov, General Principles of Quantum Field Theory

[in Russian], Nauka, Moscow (1987); English transl. (Math. Phys. Appl. Math., Vol. 10), Kluwer, Dordrecht

(1990).

4. M. A. Naimark and A. I. Stern,Theory of Group Representations(Grundlehren Math. Wiss., Vol. 246), Springer,

Berlin (1982).

5. P. C. West, Supergravity, brane dynamics, and string duality, in: Duality and Supersymmetric Theories

(D. I. Olive and P. C. West, eds.), Cambridge Univ. Press, Cambridge (1999), pp. 147266; arXiv:hep-th/

9811101v1 (1998).

6. Y. Tanii, Introduction to supergravities in diverse dimensions, arXiv:hep-th/9802138v1 (1998).

7. J. Scherk, Extended supersymmetry and extended supergravity theories, in: Recent Developments in Gravi-

tation Cargese 1978 (M. Levy and S. Deser, eds.), Plenum, N. Y. (1978), pp. 479517.

8. T. Kugo and P. Townsend, Nucl. Phys. B, 221, 357380 (1983).

9. F. Gliozzi, J. Sherk, and D. Olive, Nucl. Phys. B, 122, 253290 (1977).

10. B. DeWitt, Supermanifolds, Cambridge Univ. Press, Cambridge (1984).11. I. Ya. Arefeva and I. V. Volovich, Sov. Phys. Usp., 28, 694708 (1985).

12. V. S. Vladimirov and I. V. Volovich, Theor. Math. Phys., 59, 317335 (1984).

13. V. S. Vladimirov and I. V. Volovich, Theor. Math. Phys., 60, 743765 (1984).

14. J. Strathdee, Internat. J. Mod. Phys. A, 2, 273300 (1987).

15. P. G. O. Freund, Introduction to Supersymmetry, Springer, Berlin (1986).

16. P. West, Introduction to Supersymmetry and Supergravity, World Scientific, Singapore (1990).

17. A. V. Galazhinskii, Introduction to Supersymmetry[in Russian], Tomsk Polytech. Univ. Press, Tomsk (2008).

18. M. Rausch de Traubenberg, Clifford algebras in physics, in: Lectures of the 7th International Confer-

ence on Clifford Algebras and their Applications ICCA-7 (Toulouse, France, 1929 May 2005), Institut de

Mathematiques de Toulouse, Toulouse (2005); arXiv:hep-th/0506011v1 (2005).

19. F. Quevedo, Cambridge lectures on supersymmetry and extra dimensions, arXiv:1011.1491v1 [hep-th] (2010).

20. D. S. Shirokov, Dokl. Math., 84, 699701 (2011).21. N. G. Marchuk and D. S. Shirokov, Local generalized Paulis theorem, arXiv:1201.4985v1 [math-ph] (2012).

22. W. K. Clifford, Amer. J. Math., 1, 350358 (1878).

23. H. Grassmann, Die Lineale Ausdehnungslehre, ein neuer Zweig der Mathematik, Verlag von Otto Wigand,

Leipzig (1844).

24. W. R. Hamilton, Philos. Mag., 25, 489495 (1844).

25. C. Chevalley, Collected Works, Vol. 2, The Algebraic Theory of Spinors and Clifford Algebras, Springer, Berlin

(1997).

26. M. Riesz, Sur certaines notions fondamentales en theorie quantique relativiste, in: C. R. Dixieme Congres

Math. Scandinaves (Gjellerups Forlag, Copenhagen, July 1946), Jul. Gjellerups Forlag, Copenhagen (1947),

pp. 123148; Lequation de Dirac en relativite generale, in: Collected Papers(L. Garding and L. Hormander,

eds.), Springer, Berlin (1988), pp. 814832.

27. P. A. M. Dirac, Proc. Roy. Soc. London A, 117, 610624 (1928).

28. D. Hestenes and G. Sobczyk, Clifford Algebra to Geometric Calculus: A Unified Language for Mathematical

Physics, Reidel, Dordrecht (1984).

29. N. G. Marchuk, Equations of Field Theory and Clifford Algebras, RKhD, Izhevsk (2009).

30. P. Lounesto, Clifford Algebras and Spinors(London Math. Soc. Lect. Note Ser., Vol. 286), Cambridge Univ.

Press, Cambridge (2001).

31. G. Juvet, Comment. Math. Helv., 2, 225235 (1930).

32. F. Sauter, Z. Phys., 63, 803814 (1930); 64, 295303 (1930).

473


21/21

33. I. M. Benn and R. W. Tucker, An Introduction to Spinors and Geometry with Applications in Physics, Adam

Hilger, Bristol (1987).

34. P. K. Rashensky, Sov. Math. Surveys, 10, 3110 (1955).

35. Yu. B. Rumer, Spinor Analysis [in Russian], ONTI, Moscow (1936).

36. N. G. Marchuk and D. S. Shirokov, Adv. Appl. Clifford Algebr., 18, 237254 (2008).

37. D. S. Shirokov, Concepts of trace, determinant, and inverse of Clifford algebra elements, arXiv:1108.5447v1

[math-ph] (2011).

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