Concept of gauge theory ?

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It is said that : electrodynamics is a gauge theory for U(1) gauge group . what is its physical concept?
Mathematically it mean that the field is invariant under transformation under components of U(1) group, that we can show them with e[itex]^{i\theta}[/itex] and we can consider them as a phase angle . so does it mean that if we product a field with a phase angle , the physics of problem doesn't change ? if it is true so what about teleparallel gravity that consider gravity as a gauge theory for translation group ? is it mean that the physics of problem doesn't change with small translation of spacetime points ? nad if is it true so it should be a gauge group for electrodynamics too.
Thanks
 
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  • #2
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The physical concept is that symmetries under these groups are realized as conserved currents. The symmetry group being U(1) as you stated corresponds to an invariance of the action under "alteration" by an element of U(1). This symmetry can be local too, where at each spacetime point the assigned element of U(1) can be arbitrary without affecting physical measurements. However, the realization of this symmetry is, as I said initially, by conserved currents. For U(1) this symmetry corresponds to conserved electric charge.

The invariance under translation and rotations etc . . . (the (relativistic) Lorentz group, and the (non relativistic) Galilei group) do correspond to conserved currents also. They are conservation of momentum, angular momentum, etc . . . These are physical quantities that can be measured.
 
  • #3


Yes, physically a gauge invariance means that there is a gauge transformation (that is an element of a group) which leaves the Lagrangian unaffected. As the Lagrangian is the fundamental mathematical object to the descritpion of reality, reality doesn't change by this gauge transformation.
You can see it simply on classical electrodynamics. As [itex]\nabla\cdot B=0[/itex] and we are in 3D euclidean space, we introduce a vector potential A such that [itex]B=\nabla\times A[/itex], and therefore the last equation is satisfied trivially. To work with A as our fundamental object seems rather logical, since it has no constraints whatsoever, but A has more information than we need. This is simply because if instead of A you use [itex]A+\nabla\alpha[/itex] where [itex]\alpha [/itex] is a function, then the first equation is still satisfied, for any [itex]\alpha[/itex]. As this is the only true physical equation, the physics stays exactly the same with this transformation. This excess of information in our A is a freedom we may use whatever the way we like, it's a gauge freedom. A way of vanishing this freedom is to do a gauge fixing, by imposing further constraints in the potential A.
But the fun part is not to do it, and this leads to some non trivial and amazing results. Check for 'gauge theory' on wikipedia.
 
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  • #4
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Ok. but i don't understand the relation between invariance of Lagrangian under action of transformation with an element of U(1) and conservation of current. in my point of view the U(1) elements are just phase angle.
 
  • #5
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The relation is through a mathematical relation discovered by every (male, or female) physicist's crush, Emmy Noether. This theorem shows that for locally and global symmetries alike in an action, there exists a conserved current. Here is the wiki article, however the theorems can be found in various mechanics and field theory textbooks alike.

Noether's[/PLAIN] [Broken] Theorem
 
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  • #6
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Ok. but i don't understand the relation between invariance of Lagrangian under action of transformation with an element of U(1) and conservation of current. in my point of view the U(1) elements are just phase angle.
there are two gauge invariance.one is global gauge invariance and other is local gauge invariance.electrodynamics arises as U(1) gauge theory from local gauge invariance.Don't confuse between the two.
 

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