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  Special question about particle-particle oscillations

+ 2 like - 0 dislike

Suppose we have two scalar fields $\varphi, \kappa$. Next, suppose there is a region in space where they are mix with each other, i.e., we have a lagrangian
\tag 1
L_{\text{int}} = A \varphi \kappa
By taking into account their kinetic term, we have following EOMS:
\left(\omega^{2} + \partial_{\mathbf{r}}^2 - \begin{pmatrix}0 & A \\ A & 0\end{pmatrix}\right)\begin{pmatrix}\varphi\\ \kappa\end{pmatrix} = 0
It gives rise to particle oscillations. 

Next, suppose we have a beam of $\varphi$ particles propagating along $z$ axis. After entering the domain (say, at $z=0$) in which there is the interaction $(1)$ it begins to oscillate into $\kappa$ particle. I want to calculate the probability of oscillation at $z>0$. It turns out that it is proportional to
P_{\varphi\to\kappa}\sim |e^{-ik_{+}z}-e^{-ik_{-}z}|, \quad k_{\pm} = \sqrt{\omega^2 \mp A} 
It turns out that for $|A|>\omega$ one of the momenta $k_{+}$, $k_{-}$ becomes imaginary, and the probability doesn't behave as oscillating function, but instead is exponentially amplified or damped.

What is the physical reason for this?

asked May 26, 2017 in Theoretical Physics by NAME_XXX (1,060 points) [ revision history ]
edited May 26, 2017 by NAME_XXX

1 Answer

+ 1 like - 0 dislike

I haven't done any calculations and I do not know if your equations reflect some physics, but I reasoned in the following way: both fields are "twins" - the same frequency, the same interaction. We may consider them as components of a non relativistic spin 1/2 field, $\omega^2$ being energy $E$. Then the interaction constant $A$ may be considered roughly as a "potential barrier". When the component $\varphi$ collides with it, the component $\kappa$ may appear due to interaction with the barrier. The difference $\omega^2 - A$ may cause propagating waves inside the barrier (real valued wave vector $\bf{k}$), or reflecting waves (evanescent ones inside the barrier)). (I guess the reflecting waves exist in any case, you must admit them.) The physical solutions is chosen by the physically dictated setup - no amplified wave.

answered Feb 2, 2023 by Vladimir Kalitvianski (102 points) [ revision history ]
edited Feb 3, 2023 by Vladimir Kalitvianski

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