On the bilinear covariants associated to mass dimension one spinors

Abstract

In this paper we approach the issue of Clifford algebra basis deformation, allowing for bilinear covariants associated to Elko spinors which satisfy the Fierz-Pauli-Kofink identities. We present a complete analysis of covariance, taking into account the involved dual structure associated to Elko spinors. Moreover, the possible generalizations to the recently presented new dual structure are performed.

Eur. Phys. J. C (2016) 76:563DOI 10.1140/epjc/s10052-016-4408-y

Regular Article - Theoretical Physics

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Web End = On the bilinear covariants associated to mass dimension one spinors

J. M. Hoff da Silva1,a, C. H. Coronado Villalobos1,b, R. J. Bueno Rogerio1,c, E. Scatena2,d

1 UNESP, DFQ, Campus de Guaratinguet, Avenida Dr. Ariberto Pereira da Cunha, 333, CEP 12516-410 Guaratinguet, SP, Brazil

2 Universidade Federal de Santa Catarina-CEE, Rua Pomerode, 710, CEP 89065-300 Blumenau, SC, Brazil Received: 18 August 2016 / Accepted: 26 September 2016 / Published online: 20 October 2016 The Author(s) 2016. This article is published with open access at Springerlink.com

Abstract In this paper we approach the issue of Clifford algebra basis deformation, allowing for bilinear covariants associated to Elko spinors which satisfy the FierzPauli Konk identities. We present a complete analysis of covariance, taking into account the involved dual structure associated to Elko spinors. Moreover, the possible generalizations to the recently presented new dual structure are performed.

1 Introduction

The so-called Elko spinors are a new set of spinors with a complex and interesting structure on their own. Historically, they were proposed by Ahluwalia and Grumiller when studying the properties of the Majorana spinor. Similar to them, Elko spinors are eigenspinors of the charge conjugation operator, C, but they have dual helicity and can take positive (self-conjugate) and negative (anti-self-conjugate) eigenvalues of C, while the Majorana ones take only the positive value and carry single helicity.

From the physical point of view, Elko spinors are constructed to be invisible to the other particles (i.e., they do not couple with the elds of the Standard Model, except for the Higgs boson), becoming a natural candidate to dark matter [1]. Mathematically, the dual-helicity peculiarity forces one to a redenition of the dual spinor structure, as can be found in [1]. This idiosyncrasy reects itself when constructing the spin sums for the Elko spinors, which break Lorentz symmetry. However, the spin sums are invariant under transformations of the very special relativity (VSR) [24]. There are several areas in which Elko spinors have been studied, from accelerator physics [58] to cosmology [919]. In par

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d e-mail: mailto:[email protected]

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ticular, the appreciation of new dual structures brings about interesting possibilities within the algebraic scope [20,21].

As is well known, much of the physics associated to spinor elds is unveiled from its bilinear covariants for the simple reason that single fermions are not directly experienced. In this context it is indeed important to pay special attention to the subtleties of Clifford algebra when associating real numbers to the bilinear covariants [22]. It may sound as a secondary issue, but in fact the opposite is true. In two outstanding papers in the 1990s [23,24], Crawford worked out several important formalizations concerning the bispinor algebra. Among these, a rigorous procedure to obtain real bilinear covariants was developed. The general aim of this paper is to make use of this procedure to envisage what (if any) bilinear covariants are real when dealing with mass dimension one spinors. It is our hope that the results to be shown here may shed some light on the observables associated to these spinors. With suitable, but important, changes we take advantage of the formalism developed in [23] in order to study the bilinear covariants associated to the Elko spinor case. After a complete analysis, including the right observance of the FierzPauliKonk (FPK) [2527] relations, we arrive at the subset of real bilinear covariants.

Quite recently, new possibilities concerning the eld adjoint possibilities were thoroughly investigated in [28,29]. These formalizations may lead to a local and full Lorentz spin 1/2 eld also endowed with mass dimension one, evading, thus, the Weinberg no-go theorem [30]. We also have investigated the bilinears to this case, and to some extent the aforementioned program may be applied, leading to similar conclusions.

This paper is organized as follows: in Sect. 2 we introduce the bilinears covariant and proceed with the Clifford algebra basis deformation; then, in Sect. 3, we analyze its covariant structure. Both these sections are related to Elko spinors as objects whose spin sums break Lorentz symmetry. The natural background symmetry is then encoded

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into the orthochronous proper Lorentz subgroup, leaving for Appendix B an analogous investigation taking into account VSR Elko spinors. In Sect. 4, we conclude with some remarks about the results we found.

2 Bilinear analysis

Let be a given spinor eld belonging to a section of the vector bundle PSpine1,3 (M)

inition leads, ultimately, to a new normalization, culminating in a basis deformation satisfying the FPK identities.

Let us make these assertions more clear by explicitly showing the mentioned problem. In order to guarantee the sequential readability of the paper we leave for Appendix A a brief, but self contained, overview on the spinorial formal structure of Lorentz breaking Elko elds. Taking advantage of what was there dened, we use as an example the spinor S{,+}( p) and its dual given by, respectively,

S{,+}( p) =

C4 where stands for the entire representation space D(1/2,0) D(0,1/2), or a given

sector of such. The bilinear covariants associated to , as usual, read

= 0, = 00123, J = 0 ,

K = 0i0123 , S =

1

20i ,

i sin(/2)ei/2 i cos(/2)ei/2

cos(/2)ei/2 sin(/2)ei/2

(4)

and1

S

{,+}( p) = +

( i sin(/2)ei/2 i cos(/2)ei/2 cos(/2)ei/2 sin(/2)ei/2 ).(5)

We stress once again that the dual structure associated to Elko spinors is obtained in a very judicious fashion in [29], leaving no space for modications, an exception being made for the generalizations found in Ref. [1,29]. Using (4) and (5), as a direct calculation shows, Eq. (1) gives

= 2m, (6)

= 0, (7)

J0 = 0,J1 = 2im cos cos ,

J2 = 2im cos sin ,

J3 = 2im sin , (8)

K0 = 0,K1 = 2m sin ,

K2 = 2m cos ,K3 = 0, (9)

and

S01 = 2im sin cos ,

S02 = 2im sin sin ,

S03 = 2im cos ,

S12 = S13 = S23 = 0. (10) As can be veried, the above bilinear covariants do not obey the FPK equations. More specically, the relation containing S is not satised. In view of this problem, we revisited the formulation performed in [23] in order to nd an appropriate Clifford basis upon which the bilinear covariants can be constructed, leading to the right verication of the FPK

1 The explicit form of the boost factor, , is dened in (A7).

(1)

where the Dirac matrices are written in the chiral (or Weyl) representation,

0 =

O I

I O

, i =

O i

i O

. (2)

On general grounds, it is always expected that one can associate (1) to physical observables. For instance, in the usual case, bearing in mind the relativistic description of the electron, is the invariant length, J is associated to the current density, K is the spin projection in the momentum direction, and S is the momentum electromagnetic density. The bilinear covariants, as is well known, obey the so-called FierzPauli Konk (FPK) identities, given by [31]

J2 = 2 + 2, JK K J = S

S/A

h ( p) = [ ( p)S/Ah( p)]0. This redef-

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Eur. Phys. J. C (2016) 76 :563 Page 3 of 7 563

relations. The price to be paid is that only a subset of bilinear covariants comprises real quantities.

2.1 Deformation of the Clifford algebra basis

As is well known, the very constitutive relation of the Clifford algebra is given by

{, } = 2gI, , = 0, 1, 2, . . . , N 1, (11) where g is an N = 2n even-dimensional space-time met

ric, which in Cartesian coordinates has the form diag(1, 1,

. . . , 1). The generators of the Clifford algebra are then the

identity I and the vectors , usually represented as square matrices. The standard approach dictates the complementation of the Clifford algebra basis, in order to guarantee real bilinear covariants. This complement is performed by the composition of the vector basis, used as building blocks [23],

12NM

1 M!

12N NM+1 NM+2 N .

(12)

As is easy to see, the lowest M value is 2 (the smallest combination), nevertheless, it runs in the range M = 2, 3, . . . , N.

In this respect, the elements that form the (real) Clifford algebra basis are

{ i} {I, ,

12...N2, . . . ,

,

}, (13)

where

12NM .

In view of the new elements appearing in the denition of the Elko dual, it is necessary to adapt the Clifford algebra basis complementation. As shown previously, it is absolutely necessary for the right appreciation of the FPK relations. We shall stress that for the Dirac spinorial case, the set (13) is suitably deformed (by a slightly different normalization) in order to provide real bilinear covariants. We shall pursue something similar here, and we will be successful in correcting the problem related to the FPK relations. This notwithstanding, only a subset of bilinear covariants ends up as real in the Elko spinorial case.

The rst two bilinears arising from the Clifford algebra basis are

h( p)Ih( p), (14)

J

12NM h( p)]

= (1)M(M1)/2

h( p) ( p)

h( p)h( p), (15)

where

h( p) = [ ( p)h( p)]0 is identied as the Elko

dual spinor.2 The operator ( p) is responsible to change the spinor helicity (or rather the type of Elko spinor), here labeled by h. All the details concerning such an operator can

2 We once again refer the reader to Appendix A for the denition of the dual, as well as its necessity.

be found in Ref. [29]. The requirement = leads auto

matically to 0 = ( p) 0 ( p) since 2( p) = 1. This

constraint is readily satised, in such a way that (14) is real.3 By the same token, one should require J = J. This imposition leads to the constraint 0 = ( p) 0 ( p) which,

however, cannot be fullled. This is an important point. In fact, the counterpart associated to Dirac spinors is simply 10 0 = , a constraint naturally achieved. The new

dual structure, therefore, has forced a new interpretation of the bilinear covariants. First of all, as a matter of fact, for mass dimension one spinors, J cannot be associated to the conserved current. Obviously, in order to have J = 0

it is necessary to use the Dirac equation. This truism has lead to interesting algebraic possibilities [20], but the point to be emphasized here is that there is no problem in having a complex quantity related to the bilinear J. The additional important consequence of an imaginary J is that in order to satisfy the FPK equations K or S must also be imaginary.

Notice that, instead of the usual Crawford deformation, here we do not arrive at an entire real bilinear set. In fact, in trying to implement the full reality condition it is mandatory to change the building block of the Clifford basis . It would inevitably lead, however, to a change in the constitutive algebraic relation of the Clifford algebra (11). Therefore, this change must be excluded. It is important to emphasize, moreover, that even if we were willing to accept a modication of (11), the resulting constraint needed to get a real set of bilinears cannot be fullled.

Having said that, we may proceed deforming the usual basis in order to redene bilinear covariants which satisfy the FPK identities. Making use of Eq. (12) and considering that the norm for the Elko spinor is real we have

[

h( p)

12NM ( p)h( p).(16)

It can be readily veried that the following redenition of

12NM is appropriate to ensure K to be a real quantity:

12NM

= (i M(M1)/2/M!) ( p) 1N NM+1 N ( p).(17)

With the redenition above, one is able to dene the bispinor Clifford algebra basis as in (13), but with the gammas given by (17). As an example, consider the four-dimensional space-time. In this case the basis is given, accordingly, by

3 Notice that it must be proved that the bilinears composed with ( p)s are covariant. We shall investigate this issue in the next section.

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M = 4

= i ( p)5 ( p), (18)

M = 3

= ( p)5 ( p), (19) M = 2

=

i2 ( p) ( p), (20)

where 5 = i0123. Now, with the real Clifford algebra

basis at hand, it is possible to construct the bilinear forms, given by

I E =

h( p)Ih( p),

JE =

h( p)h( p),

E = i

h( p)5h( p), (21)

KE =

1

2m S( )(S{+}( p)

S{+}( p) + S{+}( p)

h( p) ( p)5 ( p)h( p),

h( p) ( p) ( p)h( p).

From the above construction, after a simple but tedious checking process, one ensures that the slight modications of the bilinear covariants are enough to guarantee that the FPK identities (3) are satised. After all, we arrive at and K as real non-null quantities. As remarked in the Introduction, it is our hope that these quantities shall be considered in the determination of possible experimental outcomes of the Elko construction.

3 Covariant structure

So far we have worked out quantities dened as (21) claiming that they must be faced as bilinear covariants. While they are bilinear quantities, their covariant structure must be evinced. The whole issue is related to the (necessary) presence of the ( p) operator.

Suppose that the Elko spinor belongs to a linear representation of the symmetry group in question, in such a way that, seen in another frame, the eld undergoes a transformation

S/Ah( p[prime]) = S( )S/Ah( p). (22) There is a Dirac-like operator that annihilates Elko spinors [29] (not related to the eld dynamics) given by

( p ( p) m)S/Ah( p) = 0, (23) from which we shall investigate the covariance. Applying the transformation (22) for the elds in Eq. (23), we nd

( p ( p) m) S/Ah( p[prime]) = 0. (24) The momentum can be written as p i and the partial

derivative transforms usually as = . Therefore, in

order to ensure covariance of Eq. (23) the following behavior of the Dirac matrices and the ( p) operator, respectively, is necessary:

= S( )S1( ) , (25) ( p) = S( ) ( p)S1( ). (26)

Equation (25) is the usual requirement to be posed to the gamma matrices in order to achieve a covariant Dirac equation. The requirement (26) is the new ingredient of the Elko theory, which must be investigated.

Interestingly enough, from Eq. (A13), along with (25), it is possible to see that [29,32]

( p) =

1

2m ( S{+}( p[prime])

S{+}( p[prime]) + S{+}( p[prime])

S{+}

( p[prime]) A{+}( p[prime]) A{+}( p[prime]) A{+}( p[prime]) A{+}( p[prime])),

SE = i

= S{+}( p) A{+}( p)

A{+}( p)

A{+}( p)

A{+}( p))0S( )0,

and therefore

( p) = S( ) ( p)S1( ), (27)

as expected. Once having veried the right transformations, we are able to evince the bilinear quantities. Starting from , we have

E =

S/A

h ( p[prime]) S/Ah( p[prime])

= S/Ah( p)S( )S1( ) ( p)S( )0S( )S/Ah( p)

=

S/A

h ( p)S/Ah( p),

= E,

implying to be a scalar. Repeating the same procedure for the remaining bilinear forms, we obtain

J E

h( p)h( p), (Vector),

h( p) ( p)5 ( p)h( p), (Scalar),

K E det( ) i

E det( )i

h( p) ( p)5 ( p)h( p), (Vector),

S E

i2

h( p) ( p) ( p)h( p),

(Bivector).

Therefore, the nomenclature previously adopted is indeed adequate to the present case. We shall conclude pointing out that the investigation of covariance, which concerns the SIM(2), HOM(2) Lorentz subgroups, is taken into account in Appendix B. The analysis is quite analogous, and the physical statements are essentially the same.

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4 Final remarks

In this paper we have shown that it is necessary to deform the usual Clifford algebra in order to ascertain the right obser-vance of the FierzPauliKonk identities, regarding Elko spinor elds. As a result, only a subset of bilinear covariants are real. After have found the proper deformation, and having observed its dependence on the new dual operator, we study the covariance of the relevant quantities showing explicitly its behavior under a typical transformation of the relativistic group in question.

We would like to conclude this paper by returning to the point raised at the end of the introductory section. As already mentioned, in Refs. [29,30] a subtle way is described to evade Weinbergs no-go theorem, by exploring another possibility of the dual structure, this time constructing the dual with the additional requirement of Lorentz invariant spin sums.Additional mathematical support was given in [28]. In the formulation presented in [29], great care was taken in treating the new dual structure in order not to jeopardize the simplest bilinear , i.e., the spinor was required to be an eigenspinor of the new operator, say O, entering into the dual redenition.4

By assuming associativity between O and S, whose meaning

is the same of Sect. 3 (except that the full Lorentz group is the relativistic group at hand), and that Eq. (23) is still holding, we have

(i ( p) m)(S( )O)1O S/A = 0, (28) which can be recast into the form

(i S( ) ( p)O1S1( )O

mS( )(S( )O)1O ) S/A = 0. (29)

Therefore the following identications immediately hold:

S( )O1S1( )O = 1, (30)

( p) = S( ) ( p)(S( )O)1O . (31)

From (30) one sees that the new operator must transform as

O = S( )OS1( ), leading to the same transformation

rule of (26) for the operator ( p). In this vein the covariance of (23) is ensured. The whole covariant analysis of the corresponding bilinear quantities then follows immediately, leading essentially to the same results.

Acknowledgments JMHS thanks to CNPq (445385/2014-6; 304629/ 2015-4) for partial nancial support. CHCV thanks to PEC-PG and RJBR thanks to CAPES for partial nancial support.

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/

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Web End =ons.org/licenses/by/4.0/ ), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit

4 In this context, we work in an abstract way, denoting the new operators

A and B, dened in [29], by O, aiming to simplify the notation.

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Appendix A: A brief overview: the Elko formal structure

Elko spinors are eigenspinors of the charge conjugation operator C

CS/A( p) = S/A( p), (A1) where

C =

O i

i O

K, (A2)

and K takes the complex conjugates of any spinor that appears on the right side, while is the Wigner time-reversal operator in the spin 1/2 representation, which is given by [1,33]

=

0 1

1 0

. (A3)

It is worth mentioning that Elko spinors form a complete set of eigenspinors of C with positive (S) and negative (A) eigenvalues. Another important aspect of these spinors is that they are constructed in such a way that they carry dual helicity.

From the last considerations, the explicit form of Elko spinors, in an arbitrary referential, is

S{,}( p) = (p)S{,}(0) (A4) and

A{,}( p) = (p)A{,}(0). (A5) The action of the boost operator (p) over the spinors, in

the rest-frame, is given by

(p)S/A{,}(0) =

E + m

2

1 pE + m

S/A{,}(0).

boost factor

(A6)

To summarize the notation, we choose to dene the boost factor by

=

E + m

2

1 pE + m . (A7)

The four rest spinors are shaped by two self-conjugate

S{,+}(0) = +

i [+L(0)]

+L(0)

,

S{+,}(0) = +

, (A8)

and the other two are anti-self-conjugate

i [L(0)]

L(0)

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563 Page 6 of 7 Eur. Phys. J. C (2016) 76 :563

A{,+}(0) =

i [+L(0)]

+L(0)

Appendix B: VSR Elko spinors

As is extensively shown in [2,3], Elko spinors can be understood as objects carrying a linear representation of SIM(2) or HOM(2) Lorentz subgroups. In this vein, it would be important to explore a possible Clifford algebra basis deformation in this case, if necessary.

In order to illustrate the situation, we shall make use of the Elko spinors found in [3]. Notice that, in general, the helicity operator does not commute with the VSR boost. Therefore, one cannot freely choose the rest spinors as a basis for such an operator. This is the kernel of the subtle difference between the Elko spinors studied previously and the spinors here investigated.

For an arbitrary momentum, it is possible to see that in the VSR scope we have [3]

S{,+}( p) = m

,

A{+,}(0) =

i [L(0)]

L(0)

. (A9)

The associated components are dened by imposing them to be eigenfunctions of the helicity operator, resulting in the left-hand components given by

+L(0) = m

cos(/2)ei/2 sin(/2)ei/2

,

L(0) = m

sin(/2)ei/2 cos(/2)ei/2

(A10)

and the right-hand components presented by

+R(0) = m

i sin(/2)ei/2 i cos(/2)ei/2,

,

R(0) = m

i cos(/2)ei/2

i sin(/2)ei/2

(B1)

. (A11)

Once having addressed the formal structure for the Elko spinors, we can now dene the dual by the following general formula:

S/A

h ( p) = [ ( p)S/Ah( p)]0, (A12)

with the operator ( p) dened as

( p)

1

2m

i px ipy

m(p0pz) ei/2

i

p0pz

m ei/2

p0pzm ei/2

px +ipy

m(p0pz) ei/2

as a prototype spinor and

S

{,+}( p) = m

0 i

0 (B2)

as its dual, whose construction obeys the same previous prescription, i.e., S/Ah( p) = [ ( p)V SR S/Ah( p)]0. All the

aspects investigated in the main text have a parallel here. The only differences are the general boost of VSR, which reads

ei/2

ei/2

p0pz

m

p0pz

m

S{+}( p)S{+}( p) + S{+}( p) S{+}

( p) A{+}( p)

A{+}( p) A{+}( p)

A{+}( p)

,

(A13)

which in a matrix form reads

( p) =

i(E+p cos())ei

m 0 0

i(Ep cos())ei

m

ip sin()m 0 0

0 0 ip sin()

m

i(Ep cos())ei

m

, (A14)

ip sin()m

0 0 i(E+p cos())ei

m

ip sin() m

where 2( p) = I, and 1( p) indeed exists and is equal to

( p) itself [29]. Now, the explicit form for the dual can be readily written as

VV SR

=

m p0pz

px ipy

m(p0pz)0 0

0

m 0 0 0 0

p0pz

m 0

p0pz

,

0 0

px +ipy

m(p0pz)

m p0pz

S/A

{,+}( p) = +i[S/A{+,}( p)]0, (A15)

S/A

{+,}( p) = i[S/A{,+}( p)]0. (A16)

For more details as regards the whole construction, please see [29].

(B3)

and the ( p)V SR operator. In order to nd this last operator, it is possible to see that starting from Eq. (A13), but replacing the usual spinors by VSR Elko spinors (just as (B1)) in the composition of ( p)V SR, one arrives at

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Eur. Phys. J. C (2016) 76 :563 Page 7 of 7 563

i(px ipy)ei

m

i(px ipy)2ei

m(p0pz) imeip0pz 0 0

i(p0pz)ei

m

( p)V SR =

(px ipy)ei

m 0 0 0 0 i

(px +ipy)ei

m

i(p0pz)ei

m

i

. (B4)

0 0 i(px +ipy)2ei

m(p0pz) +

ieim p0pz

i(px +ipy)ei

m

This operator acts in VSR spinors just as the ( p) operator does in the usual Elko spinors: it changes the spinor type, obeys 2( p)V SR = 1, and has determinant equal to 1, ensur

ing the existence of the inverse.

With such ingredients, it is possible to see that all the previous steps are repeated here, namely: the FPK identities are not respected, except if the Clifford basis undergoes the same deformation (with ( p)V SR replacing ( p)); V SR and K V SR are non-null real quantities, V SR is zero, and the remaining bilinears are imaginary. Besides, as Eq. (23) still holds, the covariance structure is also the same, provided we use (B3) instead of the usual Lorentz transformation.

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