Elko Spinor Fields and Massive Magnetic Like Monopoles

Abstract

In this paper we recall that by construction Elko spinor fields of λ and ρ types satisfy a coupled system of first order partial differential equations (csfopde) that once interacted leads to Klein-Gordon equations for the λ and ρ type fields. Since the csfopde is the basic one and since the Klein-Gordon equations for λ and ρ possess solutions that are not solutions of the csfopde for λ and ρ we infer that it is legitimate to attribute to those fields mass dimension 3/2 (as is the case of Dirac spinor fields) and not mass dimension 1 as previously suggested in recent literature (see list of references). A proof of this fact is offered by deriving the csfopde for the λ and ρ from a Lagrangian where these fields have indeed mass dimension 3/2. Taking seriously the view that Elko spinor fields due to its special properties given by their bilinear invariants may be the description of some kind of particles in the real world a question then arises: what is the physical meaning of these fields? Here we proposed that the fields λ and ρ serve the purpose of building the fields $\mathcal {K}\in \mathcal {C}\ell ^{0} (M,\eta )\otimes {\mathbb {R}}_{1,3}^{0}$ and $\mathcal {M}\in \sec \mathcal {C}\ell ^{0}(M,\eta ) \otimes {\mathbb {R}}_{1,3}^{0}$ (see (37)). These fields are electrically neutral but carry magnetic like charges which permit them to couple to a $su(2)\simeq {\textit {spin}}_{3,0}\subset {\mathbb {R}}_{3,0}^{0}$ valued potential $\mathcal {A}\in \sec {\textstyle \bigwedge \nolimits ^{1}} T^{\ast }M\otimes {\mathbb {R}}_{3,0}^{0}$. If the field $\mathcal {A}$ is of short range the particles described by the $\mathcal {K}$ and $\mathcal {M}$ fields may be interacting and forming condensates of zero spin particles analogous to dark matter, in the sense that they do not couple with the electromagnetic field (generated by charged particles) and are thus invisible. Also, since according to our view the Elko spinor fields as well as the $\mathcal {K}$ and $\mathcal {M}$ fields are of mass dimension 3/2 we show how to calculate the correct propagators for the $\mathcal {K}$ and $\mathcal {M}$ fields. We discuss also the main difference between Elko and Majorana spinor fields, which are kindred since both belong to class five in Lounesto classification of spinor fields. Most of our presentation uses the representation of spinor fields in the Clifford bundle formalism, which makes very clear the meaning of all calculations.

Appendices

Appendix A: A New Representation of the Parity Operator Acting on Dirac Spinor Fields

Let $\mathbf {\langle }{\mathring {\boldsymbol {e}}}_{\mu }=\frac {\partial } {\partial \mathtt {x}^{\mu }}\mathbf {\rangle }$ and $\mathbf {\langle }{\boldsymbol {e}}_{\mu }=\frac {\partial }{\partial x^{\mu }}\mathbf {\rangle }$ be two arbitrary orthonormal frames for TM and let ${\Sigma }_{0}=\mathbf {\langle }{\Gamma }^{\mu }=d\mathtt {x}^{\mu }\mathbf {\rangle }$ and Σ=〈γ ^μ = dx ^μ 〉 be the respective dual frames. Of course, eͦ₀ and e ₀ are inertial reference frames [31] and we suppose now that e ₀ is moving relative to eͦ₀ with 3-velocity v = (v ¹, v ², v ³), i.e.,

$$ {\boldsymbol{e}}_{0}=\frac{1}{\sqrt{1-v^{2}}}{\mathring{\boldsymbol{e}}}_{0}- {\textstyle\sum\limits_{i=1}^{3}} \frac{v^{i}}{\sqrt{1-v^{2}}}{\mathring{\boldsymbol{e}}}_{i} $$

(99)

Let ${\Xi }_{u_{0}}$ and Ξ _u be the spinorial frames associated with Σ₀ and Σ. Consider a Dirac particle at rest in the inertial frame eͦ₀ (take as a fiducial frame). The triplet (ψ ₀, Σ₀, Ξ₀) is the representative of the wave function of our particle in (Σ₀, Ξ₀) and of course, its representative in (Σ, Ξ) is (ψ, Σ, Ξ). Now,

$$ \psi=u\psi_{0} $$

(100)

where u describes in the spinor space the boost sending Γ^μ to γ ^μ, i.e.,$\gamma ^{\mu }=u\ {\Gamma }^{\mu }u^{-1}={\uplambda }_{\nu }^{\mu }{\Gamma }^{\nu }$. Now, the representative of the parity operator in (Σ₀, Ξ₀) is ${\mathcal {P}}_{u_{0}}$ and in (Σ, Ξ) is ${\mathcal {P}}_{u};$ We have according to our dictionary ((9)) that

$$ {\mathcal{P}}_{u}\psi=\gamma^{0}\psi\gamma^{0},\ \ {\mathcal{P}}_{u_{0} }\psi_{0}={\Gamma}^{0}\psi_{0}{\Gamma}^{0}, $$

(101)

or

$$ {\mathcal{P}}_{u}\mathbf{\Psi}=\gamma^{0}\mathcal{R}\mathbf{\Psi},\ \ {\mathcal{P}}_{u_{0}}\mathbf{\Psi}_{0}={\Gamma}^{0}\mathcal{R}\mathbf{\Psi }_{0}, $$

(102)

where ψ and ψ ₀ are Dirac ideal real spinor fields²³

$$ \mathbf{\Psi}=\psi\frac{1}{2}\left(1+\gamma^{0}\right),\ \mathbf{\Psi}_{0} =\psi_{0}\frac{1}{2}\left(1+{\Gamma}^{0}\right), $$

(103)

and if the momentum of our particle is the covector field $\boldsymbol {p} =\mathring {p}_{\mu }{\Gamma }^{\mu }=p_{\mu }\gamma ^{\mu }$ with (pͦ₀, pͦ₁, pͦ₂, pͦ₃):=(m, 0) and (p ₀, p ₁, ppͦ₂, p ₃):=(E, p) (and of course p _μ = ${\uplambda }_{\mu }^{\nu }\mathring {p}_{\mu }={\uplambda }_{\mu }^{0} \mathring {p}_{0}$) $\mathcal {R}$ an the operator such that if ψ = ϕ(p)e ^ipx then

$$ \mathcal{R}\psi=\phi(\mathbf{p})e^{-i{\boldsymbol{p}}_{\mu}{\boldsymbol{x}}^{\mu} }=\phi(\mathbf{p})e^{-i({\boldsymbol{p}}_{0}{\boldsymbol{x}}^{0}-p_{i}x^{i})}. $$

(104)

Also $u\mathcal {R=R}u$ and clearly $\mathcal {R}\psi _{0}=\psi _{0}$. Now,

$$ u{\mathcal{P}}_{u_{0}}u^{-1}u\mathbf{\Psi}_{0}=u{\Gamma}^{0}\mathcal{R} \mathbf{\Psi}_{0}=u{\Gamma}^{0}u^{-1}\mathcal{R}u\mathbf{\Psi}_{0}=\gamma ^{0}\mathcal{R}\mathbf{\Psi,} $$

(105)

from where it follows that

$$ {\mathcal{P}}_{u}=u{\mathcal{P}}_{u_{0}}u^{-1}. $$

(106)

Now we rewrite ${\mathcal {P}}_{u}\mathbf {\Psi }=\gamma ^{0}\mathcal {R} \mathbf {\Psi }$ as

$$\begin{array}{*{20}l} {\mathcal{P}}_{u}\mathbf{\Psi} & =\frac{\mathring{p}_{0}}{m}u{\Gamma} ^{0}\mathcal{R}\mathbf{\Psi}_{0}=\frac{\mathring{p}_{0}}{m}u{\Gamma}^{0} u^{-1}u\mathbf{\Psi}_{0},\\ & =\frac{\mathring{p}_{0}}{m}{\uplambda}_{\mu}^{0}\gamma^{\mu}\mathbf{\Psi} =\frac{1}{m}p_{\mu}\gamma^{\mu}\mathbf{\Psi}. \end{array} $$

(107)

We conclude that the parity operator in an arbitrary orthonormal and spin frames (Σ, Ξ) acting on a Dirac ideal spinor field ψ is

$$ \mathcal{P}={\mathcal{P}}_{u}=\frac{1}{m}p_{\mu}\gamma^{\mu}. $$

(108)

Of course, when applied to covariant spinor fields $\boldsymbol {\psi }:M\rightarrow \mathbb {C}^{4}$ the operator $\mathcal {P}$ is represented by

$$ \boldsymbol{P}=\frac{1}{m}p_{\mu}{\boldsymbol{\gamma}}^{\mu}. $$

(109)

A derivation of this result using covariant spinor fields (and which can be easy generalized for arbitrary higher spin fields) has been obtained in [36].

Appendix B: Correct Value for the Fourier Transform of $\mathcal {G}(\mathbf {p})$

According to the theory of elko spinor fields as originally developed in [1] (see also [2–6]) the evaluation of the anticommutator of an elko spinor field with its canonical momentum gives

$$ \{\uplambda(\mathbf{x,}t),{\Pi}\left(\mathbf{x},t\right.\}=i\delta(\mathbf{x}-\mathbf{x} ^{\prime})\mathbb{I}\ +i\int\frac{d^{3}p}{(2\pi)^{3}}e^{i\mathbf{p}\cdot (\mathbf{x}-\mathbf{x}^{\prime})}\mathcal{G}(\mathbf{p}), $$

(110)

with

$$ \mathcal{G}(\mathbf{p}):=\gamma^{5}\gamma^{\mu}n_{\mu}(\mathbf{p}),~~~ $$

(111)

where the spacelike p-dependent field n = n _μ (p)e _μ is

$$\begin{array}{*{20}l} (n_{0}(\mathbf{p}),n_{1}(\mathbf{p}),n_{2}(\mathbf{p}),n_{3}(\mathbf{p} )):=(0,\mathbf{n}(p)),\\ \mathbf{p}=(p\cos\theta,p\sin\theta\cos\varphi,p\sin\theta\sin\varphi) \end{array} $$

(112)

$$\begin{array}{*{20}l} \mathbf{n(p):=}\frac{1}{\sin\theta}\frac{\partial}{\partial\varphi}\left(\frac{\mathbf{p}}{\left\vert \mathbf{p}\right\vert }\right) =(-\sin \varphi,\cos\varphi,0)\\ =\left(-\tau(1+\tau^{2})^{-1/2},\tau(1+\tau^{2})^{-1/2},0\right) ,~~~\tau=p_{y}/p_{x}. \end{array} $$

(113)

Putting Δ = x − x′ it is²⁴

$$\begin{array}{*{20}l} \mathcal{G}(\mathbf{\Delta}) & =\int\frac{d^{3}p}{(2\pi)^{3}} e^{i\mathbf{p\cdot}(x-x^{\prime})}\mathcal{G}(\mathbf{p})\\ & =-\gamma^{5}\gamma^{1}P(\mathbf{\Delta})+\gamma^{5}\gamma^{2} Q(\mathbf{\Delta}). \end{array} $$

(114)

with

$$ P(\mathbf{\Delta})=-\frac{i}{2\pi}\delta({\Delta}_{z})\frac{{\Delta}_{y}} {({{\Delta}_{x}^{2}}+{{\Delta}_{y}^{2}})^{\frac{3}{2}}},~~~Q(\mathbf{\Delta})=\frac {i}{2\pi}\delta({\Delta}_{z})\frac{{\Delta}_{x}}{({{\Delta}_{x}^{2}}+{\Delta}_{y} ^{2})^{\frac{3}{2}}} $$

(115)

Remark 17

In [10] the integral in (114) has been evaluated for the case when Δ lies in the direction of one of the spatial axes e _i = ∂/∂x ⁱ of an arbitrary inertial reference frame e ₀ = ∂/∂x ⁰ (where (x ⁰ ,x ¹ ,x ² ,x ³ ) are coordinates in Einstein-Lorentz-Poincar〉 gauge naturally adapted to e ₀ [31, 33]). We note that the evaluation of each one of the integrals in [10] is correct, but they do not express the values of the Fourier transform $\mathcal {G}(\mathbf {x-x}^{\prime })$ for the particular values of Δ used in the calculations of those integrals. It is not licit to fix a priori two of the components of Δ as being null to calculate the integral $(2\pi )^{-3} {\textstyle \int } d^{3}pe^{i\mathbf {p}\cdot (x-x^{\prime })}\mathcal {G}(\mathbf {p})$ for this procedure excludes the singular behavior in the sense of distributions of the Fourier integral. So, it is wrong the statement in [10] that elko theory as constructed originally in [1] is local²⁵.

1.1 Plane of Nonlocality and Breakdown of Lorentz Invariance

When Δ _z ≠ 0, $\mathcal {G}(\mathbf {x-x}^{\prime })$ is null the anticommutator is local and thus there exists in the elko theory as constructed in [1, 5] an infinity number of –locality directions”. On the other hand $\mathcal {G}(\mathbf {x-x}^{\prime })$ is a distribution with support in Δ _z =0. So, the directions Δ = (Δ _x , Δ _y , 0) are nonlocal in each arbitrary inertial reference frame e ₀ chosen to evaluate $\mathcal {G}(\mathbf {x-x}^{\prime })$. Recall that given an inertial (coordinate) reference frame frame e ₀ = ∂/∂x ⁰ in Minkowski spacetime there exists [11] and infinity of triples of vector fields $\{\mathbf {e} _{1}^{(k)}=\partial /\partial x_{(k)}^{1},\mathbf {e}_{1}^{(k)}=\partial /\partial x_{(k)}^{2},\mathbf {e}_{1}^{(k)}=\partial /\partial x_{(k)}^{3}\}$ (with $(x^{0},x_{(k)}^{1},x_{(k)}^{2},x_{(k)}^{3})$, coordinates in Einstein-Lorentz-Poincar〉 gauge [31] naturally adapted to e ₀ differing by a spatial rotation) which constitutes a global section of the frame bundle. So, the labels x, y and z directions in inertial reference frame e ₀ are arbitrary (a mere convention) and thus without any physical significance. This means that the theory as constructed in [1] breaks in each inertial reference frame rotational invariance and since in different inertial references frames there are different (x, y) planes the theory breaks also Lorentz invariance. This odd feature (according to our view) of the theory of elko spinor fields as constructed originally in [1] was eventually the main reason that lead us to the investigation described in this paper. However if this odd effect will be observed in experiments we must agree that the original version of elko theory is indeed a science breakthrough. The one who lives will know.