User talk:67.198.37.16/Draft:ELKO Theory

This page was created in respond to the Mass dimension one fermions AfD, as a possible replacement for that article that is a bit more grounded with concrete claims. A primary impediment is that the current WP articles on Majorana spinors and related topics are inadequate; they don't cover the quantized field theory, and do not characterize the field operator. It seems like this would be needed to flesh out the ELKO development presented below.

In mathematics and physics, Elko theory is the exploration of the properties and solutions of a twisted form of the Dirac equation, which couples together a pair of Dirac spinors. The solution to this coupled equation has several unusual properties. One of these is that the Lagrangian for it has the form of a scalar field Lagrangian (the Klein–Gordon Lagrangian), thus implying that the field itself carries a mass dimension of one (as opposed to a dimension of 3/2 as is conventional for Dirac fermions). A second important property is that the twisted field can couple to normal matter with a coupling constant that is suppressed by a factor of the Planck scale, thus making it a natural candidate for dark matter. It is particularly unusual, as it appears to be a minimalist extension of the Dirac operator, historically overlooked for uncertain reasons.

Although the word Elko is often written in mixed case, it was originally the all upper-case acronym ELKO, short for Eigenspinoren des LadungsKonjugationsOperators, German for "eigenspinors of the charge conjugation operator".

Definition

The Elko equation is a twisted form of the Dirac equation. It ties together a pair of Dirac spinors, thus forming a single eight-component spinor. It may be written as

$\left(i\gamma ^{\mu }\partial _{\mu }\delta _{A}^{B}-im{\varepsilon _{A}}^{B}\right)\psi _{B}=0$

where $A,B=1,2$ identify the two spinor components, and where ${\varepsilon ^{1}}_{2}=1=-{\varepsilon ^{2}}_{1}$ is the totally antisymmetric tensor. It can be contrasted to the conventional Dirac equation

$\left(i\gamma ^{\mu }\partial _{\mu }-m\right)\psi =0$

as well as to the Majorana equation, which can be written as

$\left(i\gamma ^{\mu }\partial _{\mu }-m{\mathsf {C}}\right)\psi =0$

where ${\mathsf {C}}=-\eta \gamma ^{0}CK=\eta C{\overline {\psi }}^{T}$ is the charge conjugation operator, $\gamma ^{0}=\beta$ a gamma matrix, $C$ the charge conjugation matrix, $\eta$ an arbitrary phase factor, and $K$ the complex conjugation operator $K(x+iy)=x-iy.$

Comparing these, one sees that the Elko equation entangles a pair of Dirac spinors. It also differs from the Majorana equation, which reduces to the Dirac equation when the spinor is self-conjugate: ${\mathsf {C}}\psi =\psi ~,$ that is, when the spinor is taken to be the positive eigenvector of the charge conjugation operator.

Physicality

The physical interpretation of the peculiar mass term can be identified by observing that the Elko operator squares to the Klein–Gordon operator. Explicitly,

$\left(i\gamma ^{\mu }\partial _{\mu }\delta _{A}^{B}+im{\epsilon _{A}}^{B}\right)\left(i\gamma ^{\nu }\partial _{\nu }\delta _{B}^{C}-im{\epsilon _{B}}^{C}\right)=-\left[\square +m^{2}\right]\delta _{A}^{C}$

after noting that ${\epsilon _{A}}^{B}{\epsilon _{B}}^{C}=-\delta _{A}^{C}~.$ The resulting mass term in the Klein–Gordon operator is then real and positive. This is comparable to the conventional "squaring" of the Dirac operator:

$\left(i\gamma ^{\mu }\partial _{\mu }+m\right)\left(i\gamma ^{\nu }\partial _{\nu }-m\right)=-\left[\square +m^{2}\right]$

The flipped sign on the mass term for both the Dirac and the Elko operators can be most easily understood in terms of the Lorentz covariance of each part, individually. The product is consistently covariant only when the sign on the mass term is flipped, as shown.

Solutions

The Elko equation has solutions that are related to, but independent from solutions to the Dirac and Majorana equations. The two spinors $\psi _{1},\psi _{2}$ must be taken to be linearly independent of one-another; f they are not, then the mass term necessarily vanishes. It suffices to take these two to be the two eigenstates ${\mathsf {C}}\psi ^{(\pm )}=\pm \psi ^{(\pm )}$ of the charge conjugation operator ${\mathsf {C}}~.$ That is, setting $\psi _{1}=\psi ^{(+)}$ and $\psi _{2}=\psi ^{(-)}$ will solve the Elko equation. The spinor $\psi ^{(+)}$ is a solution to the Dirac equation, but $\psi ^{(-)}$ is not.

Charge conjugation

Explicit eigenstates of the charge conjugation operator can be given in the Weyl basis. In this basis, the charge conjugation operator takes the form

${\mathsf {C}}=-\eta \gamma ^{0}CK={\begin{bmatrix}0&\eta \omega K\\-\eta \omega K&0\end{bmatrix}}$

where $\omega =i\sigma ^{2}$ is the symplectic form (it defines the symplectic group, which is a double-cover of the Lorentz group) and $\sigma ^{2}$ is the second Pauli matrix. The charge conjugation operator has eigenstates ${\mathsf {C}}\psi ^{(\pm )}=\pm \psi ^{(\pm )}$ given by

$\psi _{L}^{(\pm )}={\begin{pmatrix}\psi _{L}\\\mp \eta \omega \psi _{L}^{*}\end{pmatrix}}$

and

$\psi _{R}^{(\pm )}={\begin{pmatrix}\pm \eta \omega \psi _{R}^{*}\\\psi _{R}\end{pmatrix}}$

where $\psi _{L,R}$ are two-component left and right-chiral Weyl spinors. Since charge conjugation also takes the complex conjugate, ${\mathsf {C}}i=-i{\mathsf {C}}~,$ that the charge conjugation eigenstates can be taken to be linear combinations $a\psi ^{(\pm )}+ib\psi ^{(\mp )}$ for $a,b$ real.

Properties

The field $\psi _{B}$ can be shown to have a number of properties.

• The field is a fermion, in that each of the two Dirac spinors making up the Elko field transform in the conventional fermionic fashion under Lorentz transforms.
• It can be written as an eigenstate of the charge conjugation operator.
• In four dimensions, the field has a mass dimension one, for exactly the same reason that the scalar Klein–Gordon field has mass dimension one: it appears in the Lagrangian with a mass term of $m^{2}|\psi |^{2}$ and, since the integral of the Lagrangian, the action must be dimensionless, the field itself must carry one mass dimension. This is in sharp contrast to the Dirac mass term $m{\overline {\psi }}\psi$ which implies that the Dirac field carries mass dimension 3/2 (in four-dimensional spacetime). In this case, the fields are referred to as the mass dimension one fermions.
• In four dimensions (that is, in Minkowski spacetime), a renormalizable Lagrangian can be written with both a quartic interaction, and a coupling to the Higgs field of the Standard Model. They can also be coupled to ordinary fermions; this coupling is suppressed by a factor of the Planck scale, and so the coupling is extremely weak, making them a natural candidate for dark matter.

History

Mass dimension one fermions were first proposed in 2004 by Dharam Vir Ahluwalia and Daniel Grumiller as a dark matter candidate.

References

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