Light-front coordinates and the vacuum.

[Jezuz]

What is the relation between a vacuum state in light-front quantization and a vacuum in the equal time formulation?

For example, I quantize a free field at equal light-front time and make a mode expansion. The resulting creation and annihilation operators can then be used to define the vacuum state. How do I then translate this vacuum state to a state in a regular Lorentz frame?

I guess that for light-front coordinates to be useful one must translate the state vector of the system to the corresponding state in the light-front frame and use that. If that is true there should be some correspondence which I can use.

I am currently working on the Wigner function for a field in a plane wave (the Volkov solution) and it turns out that I can solve most of the integrals if I use light-front coordinates.

 

[Maaneli]

What is the relation between a vacuum state in light-front quantization and a vacuum in the equal time formulation?

For example, I quantize a free field at equal light-front time and make a mode expansion. The resulting creation and annihilation operators can then be used to define the vacuum state. How do I then translate this vacuum state to a state in a regular Lorentz frame?

I guess that for light-front coordinates to be useful one must translate the state vector of the system to the corresponding state in the light-front frame and use that. If that is true there should be some correspondence which I can use.

I am currently working on the Wigner function for a field in a plane wave (the Volkov solution) and it turns out that I can solve most of the integrals if I use light-front coordinates.

The quantum vacuum in light-front coordinates is trivial. See here for more details:


http://arxiv.org/abs/hep-ph/9505259

 

[tom.stoer]

I know Matthias from my time in Erlangen, Germany. We had a group working on light cone quantization in 1+1 and 3+1 dimensions. The reason for the triviality of the light cone vacuum is the dispersion relation which is E ~ 1/p where E and p are the light front energy and momentum, respectively. Therefore a Dirac sea electron and a real electron are separate by a "momentum-gap".

One can understand this by interpreting the light from vacuum as an infinite boost of the usual vacuum. Due to the singular limit the non-trivial portion of the vacuum vanishes and only the trivial vacuum survives.

Years ago I tried to calculate the quark condensate in 1+1 dim. light cone QCD. It works, but at a certain point one has to make sense of a singular expression; if I remember correctly it was

\int dp \Theta^2(-p) \delta(p) \ldots = -\frac{1}{3}\int dp \partial_p\Theta^3(-p) \ldots = -\frac{1}{3}\int dp \partial_p\Theta(-p) \ldots = -\frac{1}{3}\int dp \delta(p) \ldots

with Theta being the step function which projects on negative p. Of course this is nonsense mathematically but if you do that you recover the correct numerical factor 1/3 or so which was derived independently w/o light front coordinates.

Are you still interested in light front coordinates? My impression was that they work only in a limited sense perturbatively but fail in the IR.