Regarding real field Klein Gordon Equations

Question

Here are 2 doubts:

1. If we change the sign of the mass term in the free massive KG Lagrangian to get:

   $L = \frac{1}{2}\partial^\mu\phi\partial_\mu\phi + \frac{1}{2}m^2\phi^2$,

   What would be the $physical$ implications of this change? (aside from on shell condition not being satisfied)?

2. Let $\phi^{(1)}$ and $\phi^{(2)}$ be 2 real scalar fields with the Lagrangian:

   $L = \frac{1}{2}\sum_{i}\partial^\mu\phi^{(i)}\partial_\mu\phi^{(i)} - \frac{1}{2}m^2 \sum_{i,j,k} \phi^{(i)} M_{ij}M_{jk} \phi^{(k)}$ .

   where $M_{11} = \lambda$, $M_{12} = 1$, $M_{21} = 1$, $M_{22} = 0$ AND $\lambda >> 1$.

   What is the mass ratio of the 2 particles in the theory?

   EDIT: As for the 2nd question, I found that $L$ should simplify to $\frac{1}{2}\sum_{i}\partial^\mu\phi^{(i)}\partial_\mu\phi^{(i)} - \frac{1}{2} m^2(\lambda \phi^{(1)} + \phi^{(2)})^2$. I tried continuing from the viewpoint that we can consider the superposition $\phi^{(3)} = \lambda \phi^{(1)} + \phi^{(2)}$ and try to eliminate cross-terms, thereby getting a simple sum of non-interacting Lagrangians, but that didn't work. (Now I'm out of my depths)

Thanks in advance

Answer 1

Here we will only address the first question. A tachyonic negative mass-square term corresponds to a potential term $V$ that is unbounded from below. This drives the value $|\phi| \to \infty$. In other words, a physical instability.

Answer 2

For the second part of the question, note that the kinetic term for $N$ scalar fields $\phi_i$ is invariant under and O(N) rotation of the fields $\phi_i \to O^j_i \phi_j$:

$$ (\partial \phi_i) (\partial \phi_i) \to (\partial \phi_j O^j_i) (\partial \phi_k O^k_i) = O^j_i O^k_i (\partial \phi_j)(\partial \phi_k),$$

and using $O^j_i O^k_i = \delta^{jk}$. But the mass term changes:

$$ \phi_i (M^2)_{ij} \phi_j \to \phi_k (O^k_i (M^2)_{ij} O^l_j) \phi_l.$$

Since the mass matrix is symmetric you can choose an $O^k_i$ to diagonalise it:

$$ O^k_i (M^2)_{ij} O^l_j = m_k^2 \delta_{kl}, $$

(no sum on $k$ on the right hand side) and the eigenvalues $m_i^2$ are the physical masses and the eigenvectors are the physical mass eigenstates.