Question about KG with negative mass^2

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In summary, the conversation discusses the application of the Klein-Gordon equation (KG) in various contexts and the behavior of its solutions. The speaker is trying to determine if the same solutions for normal positive masses can also be applied for negative masses in the KG equation. They also mention the use of hyperbolic solutions and the instability of the field when considering the whole space.
  • #1
ChrisVer
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Well can someone review this?

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

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:
[itex] \int d^{4}k [k^{2}- \mu^{2}] \tilde{\Phi}(k) e^{ikx}=0 [/itex]
and so [itex] k^{2} = \mu^{2} [/itex]
 
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  • #2
What is your question??

And what is the context? QFT? Classical field theory? First-quantized relativistic wave equation?
 
  • #3
My question is that in a normal KG equation, you have solutions:

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

I think I'm talking about Classical FT...
 
  • #4
##\cosh x = (e^x + e^{-x})/2##, ##\sinh x = (e^x - e^{-x})/2## ...
 
  • #5
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 [itex] \phi(x) [/itex] 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 [itex]\phi[/itex] should exist in the whole space, no?
 

1. What is negative mass squared in the context of KG?

Negative mass squared in the context of KG (Klein-Gordon) refers to a theoretical concept in physics where a particle's mass is represented by a negative value. This is a hypothetical scenario that is still being studied and has not been observed in nature.

2. How is negative mass squared calculated in KG?

Negative mass squared is calculated by taking the square root of a negative value in the KG equation. This value is then used to determine the mass of a particle in the theory, but it should be noted that this is a theoretical concept and has not been confirmed in experiments.

3. Can negative mass squared exist in our universe?

Currently, there is no evidence to support the existence of negative mass squared in our universe. It is a theoretical concept that is being studied and explored by scientists, but it has not been observed or confirmed in experiments.

4. What are the implications of negative mass squared in physics?

The existence of negative mass squared would have significant implications in the field of physics, challenging our current understanding of concepts such as gravity, energy, and momentum. It could potentially lead to new theories and discoveries in the field.

5. How does negative mass squared affect the behavior of particles in KG?

The presence of negative mass squared in the KG equation can result in particles exhibiting unusual behaviors and properties such as negative kinetic energy and imaginary momentum. This is still a theoretical concept and requires further research to fully understand its potential effects on particle behavior.

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