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.