Understand Virtual Particles in Quantum Gravity

[Careful]

The other point is if even after imposing the energy preservation we can claim that the off-shell particles are related to Heisenberg uncertainty. This is, if the probability of propagating a particle of mass $$m_a$$ and offshell 4-momentum $$(E_a,p_a)$$ is related via uncertainty to a "most probable" interval $$(\Delta t_a, \Delta x_a)$$. I think it is so because for an on shell particle such interval is infinite (you can go as far as you wish for so much time as you wish).

I think you are asking here for something like a mean lifetime or path length of an off shell particle (in scattering processes). I don't know what it is worth but I found a springer paper on the web http://www.springerlink.com/content/m27154006404kl8j/
written by two russians in 2004. However, I would never speak about this in the context of Heisenberg uncertainty since there is no self adjoint operator for them (it is not an observable quantity). It is just so that in the path integral, these intermediate lines have a certain lifetime (with a certain weight factor) over which is integrated.

 

[arivero]

Which in the Electoweak case becomes:

$$\def\pds{\kern+0.1em /\kern-0.55em \partial} \def\lts#1{\kern+0.1em /\kern-0.65em #1} \left( \pds_\mu - i\kern+0.25em /\kern-0.75em {\vec A}_\mu \cdot {\vec t}_L\right) u$$

Which can be read from the YM Lagrangian (Weinberg 21.3.11) where
t is the isospin. The real task is now to establish if this interaction can
be enough to mostly cancel the rest-mass energy...

But we are speaking of the same thing! Problem is, you call $$A_\mu$$ "interaction", I call it "Potential", or "potential energy". Your point is that even if a fermion is off-shell against its free equation, it is on shell for its interacting equation for some potential $$A_\mu$$. Now, this potential comes from nowhere, and it has a characteristic scale in units of Energy (it is a potential!). The inverse of this scale should be a characteristic time for off-shell propagation.

Now it is true that the general (sum of) non abelian gauge becomes more complicated than the naive electrostatic $$A_\mu=(V,0,0,0)$$ but the mechanisms are the same, or should be.

 

[arivero]

I would never speak about this in the context of Heisenberg uncertainty since there is no self adjoint operator for them (it is not an observable quantity).

Well this is a typical objection, isn't it? Quantum mechanics (0+1 QFT) has no time operator, and in the same way 3+1 quantum field theory has not time nor position operator at all, I guess.

 

[Careful]

Well this is a typical objection, isn't it? Quantum mechanics (0+1 QFT) has no time operator, and in the same way 3+1 quantum field theory has not time nor position operator at all, I guess.

I don't see much problem with space operators, but for time yes. Btw, typical objections are usually there for a meaningful purpose !

 

[arivero]

I don't see much problem with space operators,:

Ah, but you should! In QFT space and time have the same issues. Consider 1+1 QFT. Or consider 3+1 as usual: do you see a space operator there?

 

[Careful]

Ah, but you should! In QFT space and time have the same issues. Consider 1+1 QFT. Or consider 3+1 as usual: do you see a space operator there?

I could think about constructing a suitable Hamiltonian theory (although, granted, it doesn't exist yet for interacting theories). In relativistic quantum mechanics, we dispose of the Newton Wigner operator (which becomes singular in the massless case). Actually, I think there is nothing wrong with introducing a classical tetrad, so that I can define eg. in an invariant way an electric field strength operator for the electromagnetic theory of which I can study localization properties. Anyway, I never gave this much thought so tell me if I miss something.


Careful

 

[CarlB]

"Diagrammatica" arrived in the mail and has not yet taught me anything I didn't already know. Kind of surprising, actually. I was hoping for a more revolutionary interpretation of Feynman diagrams.