Relativistic density operators: Dirac dynamics, open quantum systems and non-standard neutrino interactions

Abstract

In the present work we propose the correct covariant density operator formalism for half spin particles as an extension of the standard Dirac equation. We discuss the possibility of extending said formalism to the description of relativistic open quantum systems yet we arrive at the unavoidable conclusion that this is simply unfeasible due to the nature of Lorentz symmetry, even so, we show how a Lindblad-type eq. can be recovered in the non-relativistic limit of our theory. Finally making use of the new formalism presented here we describe the time evolution of a beam of solar neutrinos and obtain certain constraints congruent with experimental measurements, on the intensity of possible non-standard interactions affecting the physics of flavour oscillation.

Appendix A: Consistency and reality conditions for the Lagrangian density functional of the DvN equation

From the Lagrangian density of the total system (31), we obtain the eqs. of motion, for the Dirac field these are

$$\begin{aligned} \frac{\partial \mathscr {L}}{\partial \bar{\varPsi }}-\partial _{\mu }\left( \frac{\partial \mathscr {L}}{\partial (\partial _{\mu }\bar{\varPsi })}\right)= & {} \left( i\gamma ^{\mu }\overrightarrow{\partial }_{\mu }-\gamma ^{\mu }A_{\mu }-m\right) \varPsi +\frac{\partial \mathscr {L}_{\text {S}}}{\partial \bar{\varPsi }}-\partial _{\mu }\left( \frac{\partial \mathscr {L}_{\text {S}}}{\partial (\partial _{\mu }\bar{\varPsi })}\right) =0, \end{aligned}$$

(A.1)

$$\begin{aligned} \frac{\partial \mathscr {L}}{\partial \varPsi }-\partial _{\mu }\left( \frac{\partial \mathscr {L}}{\partial (\partial _{\mu }\varPsi )}\right)= & {} -\bar{\varPsi }\left( i\gamma ^{\mu }\overleftarrow{\partial }_{\mu }+\gamma ^{\mu }A_{\mu }+m\right) +\frac{\partial \mathscr {L}_{\text {S}}}{\partial \varPsi }-\partial _{\mu }\left( \frac{\partial \mathscr {L}_{\text {S}}}{\partial (\partial _{\mu }\varPsi )}\right) =0. \end{aligned}$$

(A.2)

In order to recover the DvN eq. with the source term (14) and to ensure the consistency of (A.1) and (A.2), that is, that they be complex conjugates one of the other, the Lagrangian contribution of the source term must satisfy the following conditions

$$\begin{aligned}&\frac{\partial \mathscr {L}_{\text {S}}}{\partial \bar{\varPsi }}-\partial _{\mu }\left( \frac{\partial \mathscr {L}_{\text {S}}}{\partial (\partial _{\mu }\bar{\varPsi })}\right) =-\bar{\mathcal {F}}\varPsi , \end{aligned}$$

(A.3)

$$\begin{aligned}&\frac{\partial \mathscr {L}_{\text {S}}}{\partial \varPsi }-\partial _{\mu }\left( \frac{\partial \mathscr {L}_{\text {S}}}{\partial (\partial _{\mu }\varPsi )}\right) =-\bar{\varPsi }\gamma ^0\bar{\mathcal {F}}^{\dag }\gamma ^0, \end{aligned}$$

(A.4)

if we multiply (A.3) by $\bar{\varPsi }$ from the left and (A.4) by $\varPsi$ from the right, we can rewrite them as

$$\begin{aligned}&-\bar{\varPsi }\frac{\partial \mathscr {L}_{\text {S}}}{\partial \bar{\varPsi }}+\bar{\varPsi }\partial _{\mu }\left( \frac{\partial \mathscr {L}_{\text {S}}}{\partial (\partial _{\mu }\bar{\varPsi })}\right) =\bar{\varPsi }\bar{\mathcal {F}}\varPsi , \end{aligned}$$

(A.5)

$$\begin{aligned}&-\frac{\partial \mathscr {L}_{\text {S}}}{\partial \varPsi }\varPsi +\partial _{\mu }\left( \frac{\partial \mathscr {L}_{\text {S}}}{\partial (\partial _{\mu }\varPsi )}\right) \varPsi =(\bar{\varPsi }\bar{\mathcal {F}}\varPsi )^{\dag }, \end{aligned}$$

(A.6)

which immediately implies

$$\begin{aligned} \frac{\partial \mathscr {L}_{\text {S}}}{\partial \varPsi }\varPsi -\partial _{\mu }\left( \frac{\partial \mathscr {L}_{\text {S}}}{\partial (\partial _{\mu }\varPsi )}\right) \varPsi =\left( \frac{\partial \mathscr {L}_{\text {S}}}{\partial \bar{\varPsi }}\right) ^{\dag }\gamma ^0\varPsi -\partial _{\mu }\left( \frac{\partial \mathscr {L}_{\text {S}}}{\partial (\partial _{\mu }\bar{\varPsi })}\right) ^{\dag }\gamma ^0\varPsi . \end{aligned}$$

(A.7)

And simplifying the previous eq. in terms of Wirtinger derivatives it takes the form

$$\begin{aligned} \frac{\partial \text {Im}\{\mathscr {L}_{\text {S}}\}}{\partial \text {Im}\{\varPsi _s\}}+i\frac{\partial \text {Im}\{\mathscr {L}_{\text {S}}\}}{\partial \text {Re}\{\varPsi _s\}}-\partial _{\mu }\left( \frac{\partial \text {Im}\{\mathscr {L}_{\text {S}}\}}{\partial (\partial _{\mu }\text {Im}\{\varPsi _s\})}+i\frac{\partial \text {Im}\{\mathscr {L}_{\text {S}}\}}{\partial (\partial _{\mu }\text {Re}\{\varPsi _s\})}\right) =0. \end{aligned}$$

(A.8)

Therefore if the eqs. (A.3), (A.4) are valid, the only necessary condition $\mathscr {L}_{\text {S}}$ must satisfy, in order to recover the DvN equation, is that its imaginary part must be solution to the differential eq. (A.8). In particular, it can be seen that $\text {Im}\{\mathscr {L}_{\text {S}}\}=0$ is a trivial solution, which tells us that it is always possible to take the Lagrangian density of the total system as as a real functional and still obtain the desired formalism.