Gauge redundancy and Discrete space time

[Prathyush]

hi,
Zee in QFT in nut shell says
"The most unsatisfying feature of field theory is the present formulation of gauge theories. Gauge symmetry does not relate 2 different physical states but the same physical state. We have this strange language with redundancy which we cannot live without"
He also says "We even know how to avoid this redundancy from the start at the price of a discrete space time"
(closing words pg 456)
Does anyone know about the theory of discrete space he is talking about.
Also any comments on Gauge redundancy will be very helpful.

 

[krishna mohan]

Hi...

I am not sure I understand what Zee means...why does he think that the gauge freedom is unsatisfying?

In my mind, i generally draw an analogy between gauge symmetry and rotations...Just as some systems are rotationally invariant, so the Lagrangians in QFT are invariant under some rotation-like objects..

I do not feel uncomfortable thinking that a rotation in space will take an s orbital of an atom to itself...as the s orbital is spherically symmetric..

So I wonder why Zee has this opinion..

I do not know about the discrete spacetime bit...

By the way, when you quote from a book, please mention the edition and publisher too...just in case...

 

[Prathyush]

thanks for replying,
i almost gave up, ;)

I am not sure I understand what Zee means...why does he think that the gauge freedom is unsatisfying?

In my mind, i generally draw an analogy between gauge symmetry and rotations...Just as some systems are rotationally invariant, so the Lagrangians in QFT are invariant under some rotation-like objects..

I do not feel uncomfortable thinking that a rotation in space will take an s orbital of an atom to itself...as the s orbital is spherically symmetric..

So I wonder why Zee has this opinion..

I do not know about the discrete spacetime bit...

By the way, when you quote from a book, please mention the edition and publisher too...just in case...

I think Zee is disturbed by the fact rotational invariance corresponds to 2 different physical position. It so happens that a sphere looks the same in all directions, and cube looks same along it axis and so on. Rotation actually does something physical happening to the system.

One cannot change the gauge group by any physical transformation. The gauge freedom is however is more like the potential being invariant under a addition of a constant(in the case of gauge it is the gauge transformation). It is different from rotational invariance in the sense that choice of of a gauge does not in itself say anything physical about the system. It is just an artifact of the formalism.

The point Zee is making is that we have language in terms of gauge invariance that we cannot do without. I hope I'm clear in communicating the difference.

 

[Physics Monkey]

Hi Prathyush,

I can only guess what Zee is referring to with the discrete spacetime comment, but for what it's worth I'll share my opinion. My guess is that he is referring to lattice gauge theory where it is indeed possible to formulate the theory entirely in terms of gauge invariant operators like Wilson lines. The lattice gets around many of the rather subtle issues surrounding attempts to define Wilson lines as a complete set of observables in the continuum. The Wilson line formulation is nevertheless unusual compared to, say, the Ising model because the physical Wilson line variables are non-local and constrained.

I may write again to say something more about gauge redundancy later.

Hope this helps.

 

[krishna mohan]

The analogy between rotation and gauge invariance, I think, is much stronger than what I imagined in the first place...

Consider the universe to be filled with cubes of all sizes...now imagine rotating each cube by an angle around an axis passing through its centre...in such a way that all the rotation axes are parallel and all the cubes are rotated by the same angle...the old system and the new system are identical ...this is like global gauge invariance...

Imagine now that you have cylinders instead of cubes...all with axes parallel to each other..now, we rotate the cylinders around their long axes, with each cylinder rotated through a different angle...the system does not change...like local gauge invariance..


The symmetry is even bigger if we have all spheres as we can rotate each sphere in whatever way we want about an axis through its centre..and nothing changes..

Now...we can make out a rotation in our world because there are things like cubes and cricket bats which are not completely round...

As far as I comprehend, this is very similar to what happens in spontaneous symmetry breaking...a particular scalar particle(Higgs) acquires vacuum expectation value...because of this particle, the symmetry is broken...we now have different particles like electron and neutrino having different masses ..if the symmetry was not broken, electron could be rotated to a neutrino and hence they had to have the same mass...

Now we are not allowed to do the gauge rotations on the physical particle fields... except for the electromagnetic gauge symmetry, which is not broken if the scalar is neutral...


Anyway, my current understanding is very shallow...as you say, Zee must be calling for a formalism where you don't have this extra freedom hanging around...