Generalized Elko Theory

TL;DR

This paper generalizes Elko theory using totally antisymmetric spinor fields, compares it with Dirac-based theories, and explores potential links to matroids, qubits, and surreal numbers, opening new avenues in theoretical physics.

Contribution

It introduces a generalized Elko theory with totally antisymmetric spinor fields and discusses its connections to other mathematical structures, expanding the theoretical framework.

Findings

Comparison with Dirac-based theories highlights differences.

Potential links to matroids, qubits, and surreal numbers are proposed.

The formalism broadens the scope of Elko theory.

Abstract

By using a totally antisymmetric spinor field we generalize Elko theory. We compare our proposed theory with traditional totally antisymmetric spinor field theory based on the Dirac equation. As an application of our formalism we comment about the possibility to link our generalized Elko theory with matroids, qubits and surreal numbers.

Equations

$$\left[{\gamma }_{ab}^{\mu }{\hat{p}}_{\mu }{\delta }_A^B+i{m}_0{\delta }_{ab}{\epsilon }_A^B\right]{\psi }_B^b=0.$$

$${\gamma }^{\mu }{\gamma }^{\nu }+{\gamma }^{\nu }{\gamma }^{\mu }=2\epsilon {\eta }^{\mu \nu },$$

$$\left[{\gamma }_{ab}^{\mu }{\hat{p}}_{\mu }+m_0{\delta }_{ab}\right]{\psi }^b=0,$$

$$\left[{\hat{p}}^{\mu }{\hat{p}}_{\mu }+m_0^2\right]{\psi }_A^a=0$$

$$\left[{\hat{p}}^{\mu }{\hat{p}}_{\mu }+m_0^2\right]{\psi }^a=0,$$

$$\left[{\gamma }_{ab}^{\hat{\mu }}{\hat{p}}_{\hat{\mu }}{\delta }_A^B+i{m}_0{\delta }_{ab}{\epsilon }_A^B\right]{\psi }_B^b=0,$$

$$\left[{\hat{p}}^{\hat{\mu }}{\hat{p}}_{\hat{\mu }}+m_0^2\right]{\psi }_B^a=0.$$

$${}^{\ast }{\psi }^{aA}\equiv {\epsilon }^{AB}{\psi }_B^a.$$

$${\gamma }_{ab}^{\mu }{\hat{p}}_{\mu }{\psi }_A^b+i{m}_0{\delta }_{ab}{}^{\ast }{\psi }_A^b=0.$$

$${\gamma }_{ab}^{\mu }{\hat{p}}_{\mu }{}^{\ast }{\psi }_A^b-i{m}_0{\delta }_{ab}{\psi }_A^b=0.$$

$${}^{\ast }{\psi }^{a{A}_1...A_l}=\frac{1}{l!}{\epsilon }^{A_1...A_l{B}_1...B_l}{\psi }_{B_1...B_l}^a.$$

$$\left[\frac{1}{l!}{\gamma }_{ab}^{\mu }{\hat{p}}_{\mu }{\delta }_{A_1...A_l}^{B_1...B_l}+\frac{i}{l!}{m}_0{\delta }_{ab}{\epsilon }_{A_1...A_l}^{B_1...B_l}\right]{\psi }_{B_1...B_l}^a=0,$$

$${\delta }_{A_1...A_l}^{B_1...B_l}=\frac{1}{l!}{\epsilon }_{C_1...C_l}^{B_1...B_l}{\epsilon }_{A_1...A_l}^{C_1...C_l}.$$

$$\begin{array}[]{c}\left[\frac{1}{l!}\gamma_{a}^{\mu~{}c}\hat{p}_{\mu}\delta_{A_{1}...A_{l}}^{C_{1}...C_{l}}-\frac{i}{l!}m_{0}\delta_{a}^{c}\varepsilon_{A_{1}...A_{l}}^{~{}~{}~{}~{}~{}~{}C_{1}...C_{l}}\right]\times\\ \\ \times\left[\frac{1}{l!}\gamma_{cb}^{\nu}\hat{p}_{\nu}\delta_{C_{1}...C_{l}}^{B_{1}...B_{l}}+\frac{i}{l!}m_{0}\delta_{cb}\varepsilon_{C_{1}...C_{l}}^{~{}~{}~{}~{}~{}~{}B_{1}...B_{l}}\right]\psi_{B_{1}...B_{l}}^{a}=0,\end{array}$$

$$\left[\frac{1}{l!}{\gamma }_a^{\mu c}{\gamma }_{cb}^{\nu }{\hat{p}}_{\mu }{\hat{p}}_{\nu }{\delta }_{A_1...A_l}^{B_1...B}+\frac{\epsilon }{l!}{m}_0^2{\delta }_{A_1...A_l}^{B_1...B_l}\right]{\psi }_{B_1...B_l}^a=0$$

$$\left[{\hat{p}}^{\mu }{\hat{p}}_{\mu }+m_0^2\right]{\psi }_{A_1...A_l}^a=0.$$

$$\left[{\gamma }_{ab}^{\mu }{\hat{p}}_{\mu }+m_0{\delta }_{ab}\right]{\psi }_{{\alpha }_1...{\alpha }_l}^b=0$$

$${\hat{p}}^{{\alpha }_1}{\psi }_{{\alpha }_1...{\alpha }_l}^b=0$$

$${\gamma }^{{\alpha }_1}{\psi }_{{\alpha }_1...{\alpha }_l}^b=0.$$

$$C_b^a{\psi }_A^b=e^{i\theta }{\psi }_A^a,$$

$$\psi_{1}^{a}\longrightarrow\lambda=\left(\begin{array}[]{c}-\sigma^{2}\psi_{L}^{\ast}\\ \psi_{L}\end{array}\right)$$

$$\psi_{2}^{a}\longrightarrow\rho=\left(\begin{array}[]{c}\psi_{R}\\ \sigma^{2}\psi_{R}^{\ast}\end{array}\right).$$

$$C_b^a{\psi }_{A_1...A_l}^b=e^{i\theta }{\psi }_{A_1...A_l}^a.$$

$${\psi }_{A_1...[A_l}^a{\psi }_{B_1...B_l]}^a=0,$$

$${\psi }_{A_1...A_l}^a={\Omega }_{a_1...a_l}^a{\epsilon }_{i_1...i_l}{\psi }_{A_1}^{a_1{i}_1}...{\psi }_{A_l}^{a_l{i}_l},$$

$$\mid \psi >=\sum _{{\hat{A}}_1,{\hat{A}}_2,...,{\hat{A}}_N=0}^1{Q}_{{\hat{A}}_1{\hat{A}}_2...{\hat{A}}_N}\mid {\hat{A}}_1{\hat{A}}_2...{\hat{A}}_N>,$$

$${\mathcal{Q}}_{A_1...A_l}={\epsilon }_{i_1...i_l}{\omega }_{A_1}^{i_1}...{\omega }_{A_l}^{i_l},$$

$$x=\frac{p}{2^q},$$

$$-1=(-),(+)=+1,$$

$$-2=(--),-\frac{1}{2}=(-+),(+-)=+\frac{1}{2},(++)=+2,$$

Full text

GENERALIZED ELKO THEORY

J. A. Nieto 111nieto@uas.edu.mx, janieto1@asu.edu

Facultad de Ciencias Físico-Matemáticas de la Universidad Autónoma de Sinaloa, 80010, Culiacán Sinaloa, México.

Abstract

By using a totally antisymmetric spinor field we generalize Elko theory. We compare our proposed theory with traditional totally antisymmetric spinor field theory based on the Dirac equation. As an application of our formalism we comment about the possibility to link our generalized Elko theory with matroids, qubits and surreal numbers.

Keywords: Elko equation, matroids, qubits and surreal numbers.

Pacs numbers: 04.20.Gz, 04.60.-Ds, 11.30.Ly

June, 2019

It is known that the Dirac equation [1] can be considered as the ‘square root’ of the Klein-Gordon equation. Surprisingly, in 2005 Ahluwalia and Grumiller [2]-[3] (see also [4]-[7]) proved that such a ‘square root’ it is not unique. In fact, assuming a more general helicity eigenvalues they proved that an alternative and different field equation emerges which can also be reduced to the Klein-Gordon equation. They called their formalism Elko theory [2]-[3] (see also Refs [8]-[22]). It turns out that the physical states of the Elko equation can also be identified with $\frac{1}{2}$-spin fermions, but with the property that such a fermions are non-charged. One of the interesting aspects of the Elko fermions is that they provide one of the most interesting candidate for dark matter [23].

In this note we propose a generalized Elko equation which also implies the Klein-Gordon equation. The physical states of our proposed equation are $l$-rank totally antisymmetric tensor spinor fields of the form $\psi_{aA_{1}...A_{l}}(x^{\mu})$, which are different from the common totally antisymmetric spinor fields $\psi_{a\mu_{1}...\mu_{l}}(x^{\mu})$ satisfying the Dirac equation [24]-[26]. An interesting feature of our formalism arises when one requires that the field $\psi_{aA_{1}...A_{l}}$ satisfies the Grassmann-Plücker relations. In this case $\psi_{aA_{1}...A_{l}}$ becomes a decomposable tensor and we mention that this result establishes a connection no only with matroid theory but also with qubits and surreal number theory.

Let us start considering the fundamental equation in which the Elko theory [2]-[3] is based, namely

[TABLE]

Here, the indices $\mu,\nu$ etc. run from [math] to $3$, the indices $A,B$ run from $1$ to $2$ and the spinorial indices $a,b$ run from $1$ to $4$. Further, the $\gamma$-matrices satisfy the Clifford algebra

[TABLE]

with $\eta^{\mu\nu}=diag(-1,1,1,1)$ and $\varepsilon=\pm 1$. Moreover, $\hat{p}_{\mu}=-i\partial_{\mu}$ and $\varepsilon^{AB}$ is the totally antisymmetric $\varepsilon$-symbol, with $\varepsilon^{12}=1=-\varepsilon^{21}$.

In contrast to the usual Dirac equation [1]

[TABLE]

equation (1) requires $8$-component complex spinor field $\psi_{A}^{a}$ rather than $4$-component $\psi^{a}$ which is the case in equation (3). Furthermore, the quantities $\delta_{A}^{B}$ and $\varepsilon_{A}^{~{}~{}B}$ in (1) establish that $\psi_{A}^{a}$ is not eigenspinor of the $\gamma^{\mu}\hat{p}_{\mu}$ operator as $\psi^{a}$ in the equation (3). In spite of these key differences, one can prove that both field equations (1) and (3) imply the Klein-Gordon equations

[TABLE]

and

[TABLE]

respectively. Therefore, both (1) and (3) can be understood as the ‘square root’ of the Klein-Gordon equation. This proves that such a ‘square root’ is not unique.

There are at least two important remarks that emerge from (1). One related to the ‘background’ space associated with the indices $\mu,\nu$…etc. and the other associated with an ‘internal’ space associated with the indices $A,B$…etc. In the first case, higher dimensional background space, with the indices $\hat{\mu},\hat{\nu}$ …etc. replacing the indices $\mu,\nu$ etc. and running from [math] to $d-1$, generalizes (1) in the form

[TABLE]

which implies the $d$-dimensional Klein-Gordon equation

[TABLE]

Of course, one must also generalize $\gamma$-matrices $\gamma^{\mu}\rightarrow\gamma^{\hat{\mu}}$ in the Clifford algebra (2) and the spinor field $\psi_{B}^{a}$ must increase its components as a power of $2^{\frac{d}{2}}$.

The second case, it arises from the observation that the combination $\varepsilon^{AB}\psi_{B}^{a}$ can be interpreted as the dual ${}^{\ast}\psi^{aA}$ of $\psi_{B}^{a}$. In fact, one may define

[TABLE]

This means that (1) can also be written as

[TABLE]

So, one has discovered that (1) or (9) combines $\psi_{A}^{a}$ and ${}^{\ast}\psi_{A}^{a}$ in a particular way. An interesting observation, that we have not seen in the literature, it is that since ${}^{\ast\ast}\psi_{A}^{a}=-\psi_{A}^{a}$ one finds that the equation (9) implies

[TABLE]

Looking (8) and knowing that duality is central concept in totally antisymmetric tensors one is motivated to introduce the spinorial field $\psi_{B_{1}...B_{l}}^{a}$ and its dual

[TABLE]

Here, $\varepsilon^{A_{1}...A_{l}B_{1}...B_{l}}$ is the totally antisymmetric Levi-Civita symbol ($\varepsilon$-symbol) and the indices $A,B$ run, in this case, from $1$ to $2l$, with $l$ a natural number. Now one may ask; what is it the Elko equation for the field $\psi_{B_{1}...B_{l}}^{a}$?

Our proposed generalized Elko equation is

[TABLE]

where $\delta_{A_{1}...A_{l}}^{B_{1}...B_{l}}$ is the generalized Kronecker delta which is related to $\varepsilon_{A_{1}...A_{l}}^{~{}~{}~{}~{}~{}~{}B_{1}...B_{l}}$ by

[TABLE]

Considering the computation

[TABLE]

and using (13) one obtains that (14) implies the equation

[TABLE]

which in virtue of (2) leads to the Klein-Gordon equation

[TABLE]

The factor $\varepsilon$ in (15) is due to the permutation order in the definition (13), that is, according to (13) $\varepsilon=+1$ if $l$ is even and $\varepsilon=-1$ if $l$ is odd.

Just as $\psi_{A}^{a}$ is not eigenspinor of the $\gamma^{\mu}\hat{p}_{\mu}$ operator as $\psi^{a}$ in the equation (3) our proposed equation (12) also implies that the field $\psi_{A_{1}...A_{l}}^{a}$ is not eigenspinor of the $\gamma^{\mu}\hat{p}_{\mu}$ operator in contrast to the conventional assumption of the totally antisymmetric spinor field in which the field $\psi_{\mu_{1}...\mu_{l}}^{a}$ is required to satisfy the Dirac type equation

[TABLE]

and the two constrains;

[TABLE]

and

[TABLE]

In order to reduce the number of degrees of freedom of $\psi_{A}^{a}$ from $8$-complex components to $4$-complex components in Elko theory a kind of Majorana condition for the physical states $\psi_{A}^{a}$ is imposed, namely [2]-[3]

[TABLE]

where $C$ denotes a charge conjugation operator and $e^{i\theta}$ is a phase factor. If one chooses $C_{~{}~{}b}^{a}\psi_{A}^{b}=\psi_{A}^{a}$ one obtains the the self-conjugate Elko spinor or Majorana spinor, while if one requires the condition $C_{~{}~{}b}^{a}\psi_{A}^{b}=-\psi_{A}^{a}$ one gets the anti-self-conjugate Elko spinor, which is different than the conventional Majorana choice. The solution of this conditions, in the Weyl representation, can be written as

[TABLE]

and

[TABLE]

In our generalized Elko equation this analysis is more complicated since $\psi_{A_{1}...A_{l}}^{a}$ contains very much degrees of freedom. In fact, one obtains that $\psi_{A_{1}...A_{l}}^{a}$ contains $\frac{4((2l)!)}{(l!)(l!)}$-complex components. Nevertheless, one may write the analogue of (18) for $\psi_{A_{1}...A_{l}}^{a}$, namely

[TABLE]

An additional possible constraints arises from the observation that the constraints (18) and (19) can not be imposed in the field $\psi_{A_{1}...A_{l}}^{a}$ because the indices $A_{1},...,A_{l}$ are not spacetime indices as $\alpha_{1}...\alpha_{l}$ in $\psi_{\alpha_{1}...\alpha_{l}}^{b}$. So, recalling that for a totally antisymmetric object one can consider the Grassmann-Plücker relations one is motivated to assume such a relations for each value of the index $a$ in $\psi_{A_{1}...A_{l}}^{a}$. This means that one may assume that $\psi_{A_{1}...A_{l}}^{a}$ satisfies the relations:

[TABLE]

for each fixed value of the index $a$. Here, the bracket $[A_{l}B_{1}...B_{l}]$ means totally antisymmetric. It is well known that a totally antisymmetric tensor satisfies the Grassmann-Plücker relations if and only if it is decomposable. Thus, one finds that due to (24) the field $\psi_{A_{1}...A_{l}}^{a}$ is decomposable and therefore can be written as

[TABLE]

where the only non-vanishing terms of $\Omega_{a_{1}...a_{l}}^{a}$ are $\Omega_{1...1}^{1}=\Omega_{2...2}^{2}=\Omega_{3...3}^{3}=\Omega_{4...4}^{4}=1$. One recognizes in (25) a kind of generalization of the Plücker coordinates for $\psi_{A}^{ai}$. Hence, (25) determines one to one correspondence between the fields $\psi_{A_{1}...A_{l}}^{a}$ and $\psi_{A}^{ai}$. It turns out that the Plücker coordinates are a key concept in one of the possible definitions of realizable oriented matroids [27] (see also Refs. [28]-[36] and references therein). In fact, in this mathematical scenario one can identify $\psi_{A_{1}...A_{l}}^{a}$ with the complex version of the chirotopes which are called phirotopes (see Ref. [32] and references therein) This connection establishes a bridge between our generalized Elko theory and the mathematical structure of oriented matroids which can provide with a rich number of possible interesting new physical routes. In particular, the duality mathematical notion is a central concept in oriented matroid theory, in the sense that every oriented matroid has a dual, and therefore the duality expression (11) may also emerges as a key concept in our proposed generalized Elko theory. It turns out that recently we have found [36] that the Plücker coordinates is also a key concept in qubits, twistors and surreal numbers. This means that the expression (25) leads also to consider our generalized Elko theory in the context of qubit theory and surreal numbers. Let us briefly explain this observation. In qubit theory a general a complex state $\mid\psi>\in C^{2^{N}}$ is expressed as (see Ref. [37] and references therein)

[TABLE]

where the states $\mid\hat{A}_{1}\hat{A}_{2}...\hat{A}_{N}>=\mid\hat{A}_{1}>\otimes\mid\hat{A}_{2}>...\otimes\mid\hat{A}_{N}>$ correspond to a standard basis of the $N$-qubit. It turns out that, in a particular subclass of $N$-qubit entanglement, the Hilbert space can be broken into the form $C^{2^{N}}=C^{L}\otimes C^{l}$, with $L=2^{N-n}$ and $l=2^{n}$. Such a partition allows a geometric interpretation in terms of the complex Grassmannian variety $Gr(L,l)$ of $l$-planes in $C^{L}$ via the Plücker embedding. The idea is to associate the first $N-n$ and the last $n$ indices of $Q_{\hat{A}_{1}\hat{A}_{2}...\hat{A}_{N}}$ with a $L\times l$ matrix $\mathbf{\omega}_{A}^{i}$. This can be interpreted as the coordinates of the Grassmannian $Gr(L,l)$ of $l$-planes in $C^{L}$. Using the matrix $\mathbf{\omega}_{A}^{i}$ one can define the Plücker coordinates

[TABLE]

which one can associate with (25) by making the identification $\mathbf{\omega}_{A}^{i}\rightarrow\psi_{A}^{ai}$. Just as in qubit theory the transformation $\omega\rightarrow S\omega$, with $S\in GL(l,C)$, the Plücker coordinates transform as $\mathcal{Q}\rightarrow Det(S)\mathcal{Q}$ (see Ref. [37] for details) one discovers that under $\psi_{A}^{ai}\rightarrow S_{j}^{i}\psi_{A}^{aj}$ the field $\psi_{A_{1}...A_{l}}^{a}$ transform as $\psi_{A_{1}...A_{l}}^{a}\rightarrow Det(S)\psi_{A_{1}...A_{l}}^{a}$.

Surprisingly, the rebits (the real part of the complex qubits) can be connected with the fascinating subject of surreal numbers and therefore one can also associate the Plücker coordinates of $\psi_{A_{1}...A_{l}}^{a}$ with the surreal numbers. There are two equivalent ways for defining surreal numbers; The Conway [38]-[39] and Gonshor [40] formalisms. Here, although we shall focus in the Gonshor approach, let us just mention that using Conway algorithm one finds that at the ‘$j$-day’ one obtains $2^{j+1}-1$ numbers all of which are of form

[TABLE]

where $p$ and $q$ are integers numbers. Of course, the numbers (28) are dyadic rationals which are dense in the real $R$.

The Gonshor definition of surreal numbers is

Definition 1. A surreal number is a function $f$ from initial segment of the ordinals into the set $\{+,-\}$.

For instance, if $f$ is the function so that $f(1)=+$, $f(2)=-$, $f(3)=-$, $f(4)=+$ then $f$ is the surreal number $(++-+)$. In the Gonshor approach one obtains the sequence: $1$-day

[TABLE]

in the $2$-day

[TABLE]

and $3$-day

[TABLE]

respectively. Moreover, in Gonshor approach one finds the different numbers through the formula [40],

[TABLE]

where $\varepsilon,\varepsilon_{0},\varepsilon_{1},...,\varepsilon_{q}\in\{+,-\}$ and $\varepsilon\neq\varepsilon_{0}$. Furthermore, one has $\mid+\mid=+$ and $\mid-\mid=-$. For instance, one has

[TABLE]

A connection with $N$-qubit structure can be obtained by introducing a surreal complex numbers $\mathcal{Z}$ in the form

[TABLE]

where $\mathcal{J}_{1}$ and $\mathcal{J}_{2}$ are two surreal numbers according to the formula (32). So one may identify a complex operator $\mathcal{\hat{Z}}_{\hat{A}_{1}\hat{A}_{2}...\hat{A}_{N}}$ such that [36]

[TABLE]

In ordinary quantum mechanics, the $z$-component $\hat{J}_{z}$ of the total angular momentum $\hat{J}$ is quantized as $J_{z}=n\pm\frac{1}{2}$, with the identification of $\frac{1}{2}$-spin of the system. This means that there must exist infinite number of $\mathcal{J}$-spins, according to the formula (32). Thus, in general one must expect a particles with dyadic rational $\frac{p}{2^{q}}$-spin and in particular particles with $\frac{1}{4}$-spin (see Refs. [41] and [42]) and $\frac{1}{8}$-spin. At this respect it is tempted to propose this $\frac{p}{2^{q}}$-spin particles as a new candidate for dark matter.

Now, according to (11) the generalized Elko equation can be written as

[TABLE]

Hence, since both qubits and surreal numbers make sense in $2^{N}$-dimensions one requires that a connection with the generalized Elko equation can be established if $s=n+1$ and $2^{s}=2l$, where $s=N-n$. This means that parameters $l$ and $N$ are linked by $2^{N-n-1}=l$ and one also obtains $N=2n+1$. One of the simplest examples is a $3$-qubit system, with $Q_{\hat{A}_{1}\hat{A}_{2}\hat{A}_{3}}$ which implies the Plücker coordinates $\mathcal{Q}_{A_{1}A_{2}}=\varepsilon_{i_{1}i_{2}}\mathbf{\omega}_{A_{1}}^{i_{1}}\mathbf{\omega}_{A_{2}}^{i_{2}}$. In turn, this means that one must have the correspondence $\mathcal{Q}_{A_{1}A_{2}}\rightarrow\psi_{A_{1}A_{2}}^{a}$ and $\mathbf{\omega}_{A}^{i}\rightarrow\psi_{A}^{ai}$, with indices in the field $\psi_{A_{1}A_{2}}^{b}$, $A_{1},A_{2}$…etc running from $1$ to $4$. Therefore, in this case (35) becomes

[TABLE]

with ${}^{\ast}\psi_{A_{1}A_{2}}^{a}=\frac{1}{2}\varepsilon_{A_{1}A_{2}}^{~{}~{}~{}~{}~{}~{}A_{3}A_{4}}\psi_{A_{3}A_{4}}^{a}$.

From the generalized Elko equation (36) it is evident that the presence of the quantity ${}^{\ast}\psi_{A_{1}...A_{l}}^{a}$ (see Eq. (11)) means that the duality concept in terms of the $\varepsilon$-symbol ($\varepsilon$-duality) must be present in such an equation. In addition, we are requiring that $\psi_{A_{1}...A_{l}}^{a}$ satisfies the complex Grassmann-Plücker relations (24). Therefore, one can say that both the $\varepsilon$-duality and the Grassmann-Plücker relations are part of the mathematical structure of the generalized Elko equation. It turns out that, mathematically, the $\varepsilon$-duality can be identified with the Hodge-dual of $p$-forms, while Grassmann-Plücker relations can be associated with the Grassmannian variety $Gr(L,l)$ corresponding to $l$-planes in $C^{L}$ via the Plücker embedding. Therefore, both, mathematical concepts, the Hodge-dual and the Grassmannian variety enter in a natural way on the generalized Elko equation (36). Remarkably, these two concepts can be considered as two basic concepts in one possible way to define realizable oriented matroids (see Ref. [27] for details). Moreover, if one calls $L$ the dimension of the space where the indices $A_{1},...,A_{l}$ ‘live’ one observes from (11) that the dual field ${}^{\ast}\psi_{A_{1}...A_{l}}^{a}$ requires the relation $L=2l$ which in turn motivates to introduce the $L\times l$ matrix $\psi_{A}^{i}$ from which one can obtain $\psi_{A_{1}...A_{l}}^{a}$ according to (25). Surprisingly, the matrix $\mathbf{\psi}_{A}^{i}\equiv\psi_{A}^{ai}$ can also be obtained if one makes the identification of the first $N-n$ and the last $n$ indices of $N$-qubit system $Q_{\hat{A}_{1}\hat{A}_{2}...\hat{A}_{N}}$, with the dimensions $L$ and $l$ written in the form $L=2^{N-n}$ and $l=2^{n}$, implying the numerical formula $N=2n+1$. Again, this is linked to the fact that one can associate $N$-qubits with the complex Grassmannian variety $Gr(L,l)$ of $l$-planes in $C^{L}$ (see Ref. [37] for details). The surreal numbers enter in this scenario via the Gonshor mathematical formalism [40] in the sense that one can repeat exactly the same exercise that in the case of $N$-qubit, namely identifying the the first $j-n$ and last $n$ indices of ($j$-day)-surreal number $\mathcal{J}$ with the matrix $\mathbf{\psi}_{A}^{i}$. However, one note that an important difference emerges when the matrix $\mathbf{\psi}_{A}^{i}$ is associated with the ($j$-day)-surreal number $\mathcal{J}$. This difference refers to the fact that in the case of surreal numbers one can associate several dyadic rational type $\frac{p}{2^{q}}$ numbers to the matrix $\mathbf{\psi}_{A}^{i}$, where $p$ and $q$ are integer numbers. In this case, interesting possibility arises in the sense that just as usual spinors $\psi$ can be associated with $\frac{1}{2}$-spin one is tempted to propose that field $\mathbf{\psi}_{A}^{i}$ corresponds to particles with $\frac{m}{2^{p}}$-spin. In particular, surrreal numbers of the type $(++,...,+,+)$ correspond to integer spin (bosons), while surreal numbers of the type $(++,...,+,-)$ refers to half-integer spin (fermions). But one may have many other combinations of plus and minus signs associated to surreal numbers. For instance, a surreal number of the type $(+--)$ leaves to particles of $\frac{1}{4}$-spin, and so on. Thus, one wonders whether this motivating idea can induce an alternative route for the investigation of the origin of dark matter which, in spite of many attempts, it is known continues as an unsolved problem.

Finally, let us just mention that in Ref. [32] (see also Ref. [29]) a connection between fiber bundle oriented matroid theory [43] and super $p$-branes was established. Thus, according to the present development, one may expect that eventually a link between super $p$-branes and the generalized Elko theory can be considered and therefore one is tempting to believe that our proposed generalized Elko theory may be useful in the search for quantum gravity.

Acknowledgments

I would like to thank professor D. V. Ahluwalia, for his friendship and making me become interested in Elko theory. I would also like to thank the reviewer for helpful comments. This work was partially supported by PROFAPI-UAS/2013.