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.

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Physical mass in a quantum field theory

**Physics Forums | Science Articles, Homework Help, Discussion**