- #1

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I worked for hours on this simple commutator of real scalar fields in qft:

[tex]\left[\Phi\left(x\right),\Phi\left(y\right) \right] = i\Delta\left( x-y \right)[/tex]

where

[tex]\Delta\left(x\right) = \frac{1}{i}\int {\frac{{d^4 p}}

{{\left( {2\pi } \right)^3 }}\delta \left( {p^2 - m^2 } \right)\operatorname{sgn} \left( {p^0 } \right)e^{ - ip \cdot x} }[/tex]

The task is to solve this integral and to look at the cases [tex]x^2 = 0[/tex] and [tex]m\rightarrow 0[/tex].

But, what I get in my calculations is that for space like x the commutator is zero whereas the integral diverges for time-like x. Normally there should be a Bessel function involved in the end and I am just totally confused now. But maybe I should show the steps I took:

i) Spacelike x

The expression is Lorentz-invariant. For space-like x we go into a reference frame where [tex]x_0[/tex] is zero, then

[tex]\Delta \left( x \right) = \frac{1}

{i}\int {\frac{{d^3 \vec p}}

{{\left( {2\pi } \right)^3 }}\frac{1}

{{2\sqrt {\vec p^2 + m^2 } }}\left( {e^{ - i\vec p \cdot \vec x} - e^{ - i\vec p \cdot \vec x} } \right)}

[/tex]

and this is zero. What argument fails here? Calculating just one term of the bracket leads to a modified Bessel function. I followed Weinberg, p. 202 here - its strange...

ii) Timelike x

Then we go into a reference frame where [tex]\vex x = 0[/tex] and we get

[tex]\begin{gathered}

\Delta \left( x \right) = \frac{1}

{i}\int {\frac{{d^3 \vec p}}

{{\left( {2\pi } \right)^3 }}\frac{1}

{{2\sqrt {\vec p^2 + m^2 } }}\left( {e^{ - iEx_0 } - e^{iEx_0 } } \right)} \hfill \\

= 4\pi \frac{1}

{i}\int {\frac{{dp}}

{{\left( {2\pi } \right)^3 }}\frac{{p^2 }}

{{2\sqrt {p^2 + m^2 } }}\left( {e^{ - iEx_0 } - e^{iEx_0 } } \right)} \hfill \\

\end{gathered}

[/tex]

This integral diverges quadratically.

So there has to be something wrong, otherwise this task wouldn't make any sense... I hope somebody can help me here, I am totally lost with this task. I worked hours on that problem and I don't know what to do any more.

A big thanks in advance to everybody!

Blue2script