In special relativity we have the relation that for a free particle(adsbygoogle = window.adsbygoogle || []).push({});

[tex] E^2 = \vec p^2 + m_0^2[/tex]

and that also hold in relativistic free field theories (free Klein-Gordon etc) where one can show that we have a completeness relation

[tex] 1 = \int \frac{d^3 \vec p}{(2\pi)^3} \frac{1}{2E_{\vec p}} |\vec p\rangle \langle \vec p|.[/tex]

Now in proving the so called 'Kallen-Lehman representation' one starts from a completeness relation of the form

[tex] 1 = |\Omega\rangle \langle \Omega| + \int \frac{d^3 \vec p}{(2\pi)^3} \frac{1}{2E_{\vec p}} |\vec p\rangle \langle \vec p|_{\text{one particle}} + \text{two particle} + \ldots[/tex]

where it is stated that

[tex] E^2 = \vec p^2 + m^2[/tex]

where m is the physical mass of the particle. This leads to the result that the two point correlation function (as a function of the momentum) has a pole at the physical mass of the particle.

But isn't this really just a definition of the physical mass of the particle? Do we have any justification to say that E in the coefficient [itex]1/2E[/itex] for the completeness relation above

actually obeys

[tex] E^2 = \vec p^2 + m^2[/tex]

or do we just postulate it? If so what are the empirical reasons that we call this the physical mass; how do we know that we actually measure m and not m_0?

Thanks to anyone that can shed some light on this.

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# Physical mass in a quantum field theory

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