Question about KG with negative mass^2

[ChrisVer]

Well can someone review this?

KG equation:
$\square \Phi + m^{2} \Phi =0, ~~ m^{2} <0 \Rightarrow m=i \mu$
would lead to the form:
$\square \Phi = \mu^{2} \Phi$.

I'm trying to think if applying the same solution as in KG can also happen here...
Also for on-shell particles, I seem to be getting the "same" equation as we do for normal positive masses:
$\int d^{4}k [k^{2}- \mu^{2}] \tilde{\Phi}(k) e^{ikx}=0$
and so $k^{2} = \mu^{2}$

 

[Avodyne]

What is your question??

And what is the context? QFT? Classical field theory? First-quantized relativistic wave equation?

 

[ChrisVer]

My question is that in a normal KG equation, you have solutions:

$e^{i (E t- \vec{k} \vec{x})}$
Where $E^{2} - k^{2} = m^{2}$
and these are oscillating solutions... Now if I let $m^2 <0$ then it means that $E,k \in C$, is that right?
as such the solutions become exponentials...:/ however I was expecting hyperbolic solutions...

I think I'm talking about Classical FT...

 

[Orodruin]

$\cosh x = (e^x + e^{-x})/2$, $\sinh x = (e^x - e^{-x})/2$ ...

 

[ChrisVer]

and how can someone deduce from that the instability of the field? because it explodes exponentially? although I am a bit confused about x in $\phi(x)$ and what it actually means... eg some people say that it's unstable because if you make some displacement x--> x+dx then it won't remain in the same state...however $\phi$ should exist in the whole space, no?