Significance of gauge transformation any experts?

[ZealScience]

Though I've learned gauge transformation for a while, I can't figure out why it is significance in describing fields? For example, why electromagnetic tensor has to be gauge invariant? What does it physically mean?

 

[Hurkyl]

You're asking a trick question -- a gauge transformation has no physical significance!

Instead, a gauge transformation tells us what that is not physically significant. If two configurations are related by a gauge transformation, then they describe exactly the same physical state of affairs.


On the hypothesis that every component of the electromagnetic tensor has physical significance, can you now explain why the tensor should be gauge invariant?

 

[ZealScience]

You're asking a trick question -- a gauge transformation has no physical significance!

Instead, a gauge transformation tells us what that is not physically significant. If two configurations are related by a gauge transformation, then they describe exactly the same physical state of affairs.


On the hypothesis that every component of the electromagnetic tensor has physical significance, can you now explain why the tensor should be gauge invariant?

Thank you for replying, I am confused with some concepts. In what I have learnt, EM tensor is a curl of a vector potential which is gauge invariant, so I mean EM tensor is related to gauge invariance to some extent.

Can you specify on gauge transformation? Why do we need gauge theory, and when?

 

[phyzguy]

Think of it this way. At each point in spacetime, there are additional, hidden degrees of freedom. Whether these degrees of freedom are hidden dimensions that are 'curled up' to very small dimensions or whether they are some other degree of freedom is currently a matter of debate, but the degrees of freedom are there. In EM, the degree of freedom is a single complex phase, corresponding to what is called the U(1) group. So you can think of a small circle sitting at each point. Since things are completely symmetric around the circle, it doesn't matter where around the circle you call the zero point. So at each point, you can add an arbitrary phase (i.e. a number between zero and 2*pi), without changing any of the physics. This is basically re-setting the zero point around the circle at each point to a new value. This is completely analogous to doing a mechanics problem and changing x to x+K. It doesn't matter where I call my zero when I'm measuring distances - changing the zero point will not change the physics. Since none of the physics can change, everything needs to be invariant to this change of phase. So in any problem, if we change the QM wave function by adding a phase phi (which multiplies the wave function by e^(i*phi) ), and change the EM potential everywhere by grad(phi), everything comes out the same. The other forces (gravity, weak and strong interaction) have similar gauge invariances, but they are more complicated that EM.

 

[Phrak]

Thank you for replying, I am confused with some concepts. In what I have learnt, EM tensor is a curl of a vector potential which is gauge invariant, so I mean EM tensor is related to gauge invariance to some extent.

Can you specify on gauge transformation? Why do we need gauge theory, and when?

I think this is a very good question I can't answer. If you'll allow me to rephrase, and correct me if I have it wrong, "When has the gauge invariance of one discriptive equation of physical law lead to another?

 

[haushofer]

Though I've learned gauge transformation for a while, I can't figure out why it is significance in describing fields? For example, why electromagnetic tensor has to be gauge invariant? What does it physically mean?

My 2 cents:

Think about a photon. You want to describe it with something which is Lorentz invariant, right? A scalar won't do. The next simplest possibility is a vector.

But a vector in 4 dimensions contains 4 degrees of freedom (DOF's). But we measure that light has only 2 polarization directions. One DOF can be eliminated by imposing equations of motion (EOM). So we have one extra DOF left, which we don't measure. It is "eliminated" by stating that the vector has a redundant DOF which is not physical.

These redundant DOF's mean there should be something like an extra "symmetry". Between quotes, because we introduced it basically by hand by demanding that the photon is described by a Lorentz vector. This "symmetry" we call a gauge symmetry. For the photon this redundant DOF can be described by a phase, or the group called U(1).

This idea seems to be (why? we don't know!) a very fruitfull way to describe other fundamental interactions, like the weak and strong interactions, and in some degree gravity. This involves other groups like SU(N).

The idea, inspired by the electromagnetic case, is to observe that a free theory of matter has a global symmetry (at every event the same). Promoting this global symmetry to be local (changing from event to event) demands that you introduce fields to correct for the extra terms. These fields are interpreted as the fields propagating the fundamental forces, and describe the coupling of matter to these fields.

So: promoting a global symmetry of your theory enables you to describe the interaction (which depends on the symmetry!) of matter.

Apparently Nature likes symmetries, and thus gauge theories. But again, we don't know why :)