Light-Cone Gauge: Benefits for Quantum Relativistic Particles & Strings

[wam_mi]

Hi there,

How is the light-cone gauge useful for a quantum relativistic particle and for a quantum relativistic string? Why is it important that the light-cone gauge is used? Does it actually simplify the mathematics a lot?

Thanks

 

[tom.stoer]

Our group studied 1+1 dim QCD in light cone coordinates 15 years ago; you should look for papers from "F. Lenz" and "M. Thies" - unfortunately not in arxiv.

The main difference in light cone coordinates is the following

p = (p₀, p₁, p₂, ...) => (p₊, p_-, p₂, ...)

The dispersion relation for massive particles reads:

p₊ = (... + m²) / 2p_-

where ... means the perpendicular components.

Due to this dispersion relation for the light cone energy p₊ the Dirac sea becomes trivial; there is a unique solution for p₊ for given light cone momentum p_- ; that means that a sea-particle cannot be excited w/o violation of the light cone momentum. Therefore a "mixing" of quarks and anti-quarks is forbidden = the vacuum structure is trivial.

Nevertheless you can find non-vanishing condensates, but in a different formalism.

The light cone gauge for a gauge field A then means

A_- = 0

It is comparable to the axial gauge, that means
- one polarization is eliminated completely
- a "Coulomb potential" arises due to invertion of the Gauss constraint
Usually one derives a Hamiltonian framework (one has to check for Poincare covariance)

In some sense the light cone frame is related to the infinite momentum frame.

As far as I know the light cone approach is not widely used in QCD; the trivial vacuum structure seems to be a benefit when one starts with the calculations, but there are other difficulties like non-dynamical fermionic degrees of freedom (in 1+1 dim. you can see that one component of the spinor is non-dynamical as there is no light cone time derivative) and the non-local light cone energy operator.

I have no idea how this relates to string theory.

 

[Entropia]

for your question! The light-cone gauge is a powerful tool in studying quantum relativistic particles and strings. It is a specific choice of gauge (a mathematical framework used to simplify equations and calculations) that has many benefits in these areas.

Firstly, the light-cone gauge allows for the decoupling of transverse degrees of freedom in the equations of motion for particles and strings. This means that the equations become simpler and easier to solve, as they no longer involve complicated interactions between different dimensions. This is particularly useful in string theory, where the equations can become very complex due to the multiple dimensions involved.

Secondly, the light-cone gauge has a special symmetry known as "light-cone Lorentz invariance". This symmetry makes the equations of motion for particles and strings more elegant and easier to work with, as it allows for certain simplifications and symmetries to be applied. This is especially important in quantum field theory, where symmetries play a crucial role in understanding the behavior of particles.

Moreover, the light-cone gauge has been shown to have a close connection to the concept of "conformal invariance", which is a fundamental symmetry in string theory. This connection allows for a deep understanding of the underlying structure of strings and their interactions.

Overall, the use of the light-cone gauge greatly simplifies the mathematics involved in studying quantum relativistic particles and strings. It allows for a more elegant and powerful approach to these topics, and has played a crucial role in many important developments in theoretical physics. So, yes, it does simplify the mathematics a lot and is an important tool in these areas of research.