Most stupid question ever.

[coboldo]

Ok.

 

Take a bosonic propagator. Or a Green function call it as you wish.

 

Know what I mean, huh? The solution of ([] - m^2) D(x-y) = - delta(x-y).

 

 

It's something like D(x-y) = integral over d^4p of exp(ip(x-y)) times 1 / (p^2 - m^2 +- i epsilon)

 

 

Now. First of all, I don't want to bother about prefactors, "i" factors, signs, prescription (feynman, casual...)

 

 

 

The point is. I have seen this expression so many times. (and actually NEVER used it, I'm not a scientist).

 

Then suddendly I realized that... that...

 

...isn't it INFINITE?!

 

 

Ok forget what happens on the poles.

 

 

But we are talking about an integral over "p" in 4 dimensions.

 

And, the integrated function goes like 1 / p^2 for p ---> infinity.

 

Shouldn't it vanish faster than 1/p^5 for the integral to be convergent at infinity?!

 

 

 

Oh, ok, there is an oscillating phase multiplying all that. Good. I hope it does the job, with positive and negative pieces cancelling each other.

 

But I'm not really sure.

 

Someone can help me understanding the whole story?!

 

thnx in advance.

[Ronald Hyde]

Not really a stupid question but it is for sure that you are interpreting the meaning of the propagator wrongly.

 

It's not to be thought of as a wave function, so the total integral doesn't have to have any particular value, such as 1.

 

What it does is tell how the system evolves with time, so that only how it looks at time t, any given time, is what

 

is important. If the total 'wavefront', if you want to call it that, grows as [latex] 4 \pi t^2 [/latex], then for the probability to

 

total 1, which by logic it must, then the value of the function at t must be [latex]1 /t_2 4 \pi [/latex] .

 

That's for probabilities for positions, for amplitudes or momenta it would be different.

Edited August 22, 2012 by Ronald Hyde

[coboldo]

well, I tought that D(x-y) was equal to <0|phi(x)phi(y)|0>.

 

So your answer makes me even more confused.

 

Isn't my interpretation correct?

 

 

If so. Take a FREE theory. Like a filed theory for bosonic spinless massive particles. A lambda4 theory... with lambda=0.

 

Then <0|phi(x)phi(y)|0> = D(x-y) where D is the free propagator, i.e. exactly the integral above.

 

 

So... this mean value is infinity?

 

I have to stress that, I don't care about the "value" of the integral, I just would like it to be finite.

 

 

And. If it is not, then what does this mean?

 

 

It means that even free theories need to be "renormalized"?