Imaginary momentum and virtual particles

[cabrera]

There is a type of exchange of particles which is generalised by a type of potential:

\frac{e^{-\alpha\r}}{R} This potential is used to explain the exchange of bounded particles (e.g a poin between neutron and proton) between two possible configurations. The potential comes from the fact that the momentum of the bounded particles is imaginary.

for instance: p=i sqrt(mp*E) I have problem understanding the meaning of a imaginary momentum.

Could the forum explain me what an imaginary momentum means in QM?

 

[cabrera]

Ah, yes. I've forgot to mention that this particles whose interaction is defined by the Yukawa potential, \frac{e^{\alpha r}{r}, are referenced as virtual particles. Could we use the definition that any particle that has a imaginary momentum is an imaginary particle?

 

[fzero]

The momentum of the exchanged particles is not imaginary. Instead, the 4-momentum of the exchanged particles does not satisfy the mass-shell condition $p_\mu p^\mu = m^2$. This is evident in the Fourier-transform relation between the Yukawa potential and the corresponding Feynman amplitude for scalar exchange, c.f. http://en.wikipedia.org/wiki/Yukawa_potential. In that expression, all values of momenta are integrated over. For a real particle, the allowed values of momenta would be constrained in terms of the energy via the mass-shell condition. Since the exchanged particle does not satisfy this constraint, it is called a virtual particle, or sometimes "off-shell".