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Microcausality and quantization of the fermionic Myers–Pospelov model

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We study the fermionic sector of the Myers and Pospelov theory with a general background n. The spacelike case without temporal component is well defined and no new ingredients came about, apart from the explicit Lorentz invariance violation. The lightlike case is ill defined and physically discarded. However, the other case where a nonvanishing temporal component of the background is present, the theory is physically consistent. We show that new modes appear as a consequence of higher time derivatives. We quantize the timelike theory and calculate the microcausality violation which turns out to occur near the light cone.

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J.L. acknowledges support from DICYT Grant No. 041131LS (USACH) and FONDECYT-Chile Grant No. 1100777. C.M.R. acknowledges partial support from DICYT (USACH) and Dirección de Investigación de la Universidad del Bío-Bío (DIUBB) Grant No. 123809 3/R.

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Correspondence to Carlos M. Reyes.

Appendix: General solutions and dispersion relations

Appendix: General solutions and dispersion relations

In this appendix we will characterize the general solutions and dispersion relations of the equation of motion for the general fermionic Myers and Pospelov theory. This characterization is not essential for the understanding of the body of the work apart from some particular aspects concerning the dispersion relation. However, we include it for the sake of completeness.

Consider the equation of motion


where a μ p μ g 1 n μ (np)2 and b μ g 2 n μ (np)2 are four-vectors which will help us to clear up the notation.

Let us define the following matrices:


do not confuse the \(\hat{h}\) operator here with the Dirac Hamiltonian \(\hat{h}_{D}\) in the text. These operators satisfy the relations,

$$ \begin{aligned} &[ \hat{ M},\hat{h} ]=0, \\ &({\hat{M}}+2m) {\hat{M}} = a^2 - b^2 - m^2 - \hat{h}. \end{aligned} $$

This means that the solutions of the equation of motion can be expressed in terms of the eigenvectors of \(\hat{h}\).

By noticing that


the general dispersion relation is given by


or by Eq. (4) in terms of p. In the case b μ=0, we have the simplified dispersion relation


Now, we will calculate the solutions of the equations of motion. As we pointed out above, we can find these solutions among the eigenvectors ψ i satisfying


for the eigenvalues h i . Then, let us find those eigenvectors. To do so, we notice that the \(\hat{h}\) operator can be written in terms of a rank two antisymmetric tensor, T μν a μ b ν a ν b μ , that is,


with and the convention ε 0123=1.

From this tensor, we define two orthogonal three-vectors,


and thus


where \(\hat{e}_{1}\), \(\hat{e}_{2} \) and \(\hat{e}_{3}\) are three orthonormal space vectors on the direction of u, v and w, respectively. The norm of these vectors are,

$$ \begin{aligned} u &= \sqrt{ \bigl(a^0 \bigr)^2(\mathbf{b})^2 + \bigl(b^0 \bigr)^2(\mathbf{a})^2 - 2 \bigl(a^0b^0 \bigr) (\mathbf{a}\cdot\mathbf{b})}, \\ v &= \sqrt{ (\mathbf{a})^2(\mathbf{b})^2 -(\mathbf{a}\cdot\mathbf{b})^2}. \end{aligned} $$

Note that


The negative values of T 2 correspond to real eigenvalues for \(\hat{h}\) and the positive ones correspond to purely imaginary eigenvalues. By making use of the analogy with the electromagnetic tensor F we will call the T 2<0 “electric” case and T 2>0 the “magnetic” case.

Now, we define the rotation and boost generators in the spinor representation,


where the spatial indices are referring to the e basis defined above. Then, the \(\hat{h}\) operator turns out to be


Performing a boost transformation on the eigenspinor in the \(\hat{e}_{3}\) direction


the \(\hat{h}\) operator transforms as


Because −1<tanhη<1, we can distinguish two cases. For u>v we set \(\tanh\eta= \frac{v}{u}\) so that


However, for v>u, we can set \(\tanh\eta= \frac{u}{v}\) such that


Since the eigenvalues of and are \(\pm \frac{1}{2}\) and \(\pm\frac{i}{2}\), respectively, we have \(h_{i}=2\varepsilon_{i} \sqrt{u^{2}-v^{2}}\) for u>v, and \(h_{i}=2i\varepsilon_{i} \sqrt{v^{2}-u^{2}}\) for v>u as we expected. The convention here is ε 1=+1 and ε 2=−1.

The eigenspinors in the chiral representation for u>v can be written as


with (uσ)ξ i =−ε i i .

Notice that these eigenvectors have the property, in the e basis,


However, for v>u, the eigenspinors have the form,


with (vσ)χ i =ε i i . Similarly, the eigenspinors have the property, in the e basis,


The constants α i , β i , δ i , γ i reflect the fact that the eigenspinors are twofold degenerate.

Now, we are ready to find the solutions of the equations of motion in terms of the spinors ψ i . Performing the same transformation on \(\hat{M}\) we obtain after some algebra:

In the electric case (u>v), we can set, by choosing appropriately the parameter η, \(a^{\prime}_{3}=b^{\prime}_{3}=0\) and find

$$ \begin{aligned} a_0^\prime&=a_0\sqrt{1-\frac{v^2}{u^2}}=a_0 \frac{\vert h\vert }{2u}, \\ b_0^\prime&=b_0\sqrt{1-\frac{v^2}{u^2}}=b_0 \frac{\vert h\vert}{2u}. \end{aligned} $$

In the other hand, in the magnetic case (v>u), we can set \(a_{0}^{\prime}=b_{0}^{\prime}=0\) and find

$$ \begin{aligned} a_3^\prime&=-a_0\sqrt{\frac{v^2}{u^2}-1}= a_0\frac{\vert h\vert}{2u}, \\ b_3^\prime&=-b_0\sqrt{\frac{v^2}{u^2}-1}= b_0\frac{\vert h \vert}{2u}. \end{aligned} $$

where \(\vert h \vert\equiv\sqrt{ \vert u^{2}-v^{2} \vert}\) and where we have considered that the 2-direction is perpendicular to a and b. Hence, in the electric case the equations of motion, \(\hat{M}^{\prime}\psi_{i}=0\), are


where we have used (A.21). This equation fixes the constants in Eq. (A.20)


where is a normalization constant. The equations of motion, Mψ i =0 in the magnetic case are


where we have used property (A.23). This implies that the constants in Eq. (A.22) are


where is another normalization constant.

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Lopez-Sarrion, J., Reyes, C.M. Microcausality and quantization of the fermionic Myers–Pospelov model. Eur. Phys. J. C 72, 2150 (2012).

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  • Dispersion Relation
  • Fermionic Sector
  • Retarded Green Function
  • Modify Dispersion Relation
  • Lorentz Invariance Violation