Solving the Massless Klein Gordon Equation

[bartadam]

I am being really thick here

I have this wave equation, the massless klien gordon equation

\partial_{\mu}\partial^{\mu}\phi(x)=0

where the summation over \mu is over 0,1,2,3

the general solution is a superposition of plane waves yes? i.e

\phi(x)=\int d^4 p \overline{\phi}(p)exp(i p_{\mu}x^{\mu})

where \overline{\phi} is the weighting function.

When you susbsitute this back into the klein gordon equation you get down two factors of p, i.e

p_{\mu}p^{\mu} which equals zero. (mass shell constraint), thus satisfying the equation of motion.

My question is, is \overline{\phi}(p) arbitrary? I don't really understand why this is so, let alone believe it.

Hope peeps understand the question.

 

[neu]

The superpostion equation you wrote is simply the Fourier transform of \phi (x).

\phi (p) are not arbitrary as \phi (p) = FT^{-1}[\phi (x)]

Have I misunderstood your question?

 

[bartadam]

yeah I understand that.

I'm being really thick here.

I want the general solution \phi(x) to the equation, ie as a superposition of plane wave solutions, but in Fourier space.

Ultimately I want to know what \overline{\phi}(p) is.

 

[bartadam]

What I mean is, if \overline{\phi}(p) is one weighting function whose Fourier transform solves the klein gordon equation and \overline{\psi}(p) is another different weighting function is the Fourier transform of \overline{\psi} also a solution.

 

[lbrits]

Well, any \phi(p) will make \phi(x) a solution. These are merely coefficients in your Fourier expansion. However, once you write the Hamiltonian in terms of \phi(p) and \pi(p), you will find that it simplifies greatly (decoupled harmonic oscillators, one for each p), and quantization is the next step.

 

[bartadam]

Well, any \phi(p)[\tex] will make \phi(x)[\tex]a solution

<br /> <br /> great as long as you're sure about that. That's what I hoped.<br /> <br /> I have guessed some expression between \phi(p)[\tex] and a whole load of other stuff and I want to test my conjecture on the computer. So, presumably I can just invent some suitable function for \phi(p)[\tex] stick it into a c-program and check it works.

 

[nrqed]

yeah I understand that.

I'm being really thick here.

I want the general solution \phi(x) to the equation, ie as a superposition of plane wave solutions, but in Fourier space.

Ultimately I want to know what \overline{\phi}(p) is.

As others said, phi(p) is arbitrary. The way to see this is the following: plane waves (with the condition on p^2) are solutions of the equation and the equation is linear, therefore arbitrary linear combinations of plane waves will satisfy the equation. Therefore phi(p) is arbitrary.

 

[strangerep]

[...]
When you susbsitute this back into the klein gordon equation you get down two factors of p, i.e
p_{\mu}p^{\mu} which equals zero. (mass shell constraint), thus satisfying the equation of motion. [...] is \overline{\phi}(p) arbitrary? [...]

Adding my $0.02 to what others have already said, the \overline{\phi}(p) is not
entirely "arbitrary", because you have imposed the mass-shell constraint p_{\mu}p^{\mu}=0.
Think of that as a "constraint hypersurface" in 4D momentum space.

I.e., \overline{\phi}(p) is undefined for values of p which are not on
the constraint hypersurface.

 

[nrqed]

Adding my $0.02 to what others have already said, the \overline{\phi}(p) is not
entirely "arbitrary", because you have imposed the mass-shell constraint p_{\mu}p^{\mu}=0.
Think of that as a "constraint hypersurface" in 4D momentum space.

I.e., \overline{\phi}(p) is undefined for values of p which are not on
the constraint hypersurface.

True but this is not a condition on the functional form \overline{\phi}(p), it's a restriction on the argument p. I mean, as long as p is a valid p, any function phi(p) is valid, right?

A different consideration arises if we consider localized wavepackets phi(x). Then there must be a condition on phi(p).

 

[strangerep]

True but this is not a condition on the functional form \overline{\phi}(p), it's a restriction on the argument p.
I mean, as long as p is a valid p, any function phi(p) is valid, right?

A different consideration arises if we consider localized wavepackets phi(x).
Then there must be a condition on phi(p).

Yes and yes.