Imaginary momentum and virtual particles

1. Apr 13, 2013

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?

2. Apr 13, 2013

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?

3. Apr 13, 2013

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".