A mentioning about virtual particle problem in my other thread just reminded me of some thoughts, which I now succeeded putting together.(adsbygoogle = window.adsbygoogle || []).push({});

When calculating cross sections in QFT, we encounter terms like this

[tex]

\langle 0|a_{\textbf{k}'} a_{\textbf{p}'} a^{\dagger}_{\textbf{p}_1} a^{\dagger}_{\textbf{p}_2} a'_{\textbf{k}_1} a^{'\dagger}_{\textbf{k}_2} a_{\textbf{p}_3} a_{\textbf{p}_4} a^{\dagger}_{\textbf{k}} a^{\dagger}_{\textbf{p}}|0\rangle,

[/tex]

where we start with some excitation, operate on it with annihilation operators, create some intermediate particle, annihilate the intermediate particle, create two final particles, and then consider the inner product with some fixed outcome.

As it turns out, the calculations lead into some expressions that contain propagator of the intermediate particle, and the energy-momentum in the propagator is not on shell. This is usually interpreted as sign of the intermediate particle being off shell, but this doesn't fully make sense to me.

Firstly, the calculation is quite abstract. I don't see any obvious reason to interpret the resulting propagator as actually describing the propagation of the intermediate particle.

Secondly, isn't it impossible to create off shell particles with the usual creation operators? I mean, if you operate on vacuum with [itex]a^{'\dagger}_{\textbf{p}}[/itex], you get a particle whose energy momentum is [itex](\sqrt{|\textbf{p}|^2 + (m')^2}, \textbf{p})[/itex], right? Here [itex]m'[/itex] is the constant mass, characteristic to the particular particle type.

So, what I hear everyone explaining, is this:

"Since the energy-momentum is conserved in the vertexes, the intermediate particle is clearly off shell. This, however, is not problematic, since it is a virtual particle that exists only for short period of time, and cannot be observed directly."

But why not like this:

"Since the intermediate particle is on shell, clearly the energy-momentum is not conserved in the vertexes. This, however, is not problematic, since the state, at which energy-momentum conservation is violated, exists only for a short period of time, and cannot be observed directly."?

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# Virtual particle energy-momentum

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