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Please explain gauge invariance un-mathmatically

  1. Oct 26, 2015 #1
    please explain what gauge symmetry is, gauge transformation is, gauge invariance is, and also how gauge invariance deletes the timelike polarization of a massless vector boson. without fancy math and formulas.
     
    Last edited by a moderator: Oct 28, 2015
  2. jcsd
  3. Oct 26, 2015 #2

    ChrisVer

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    Gold Member

    Gauge invariance has to do with making a transformation of your fields, going to another gauge, without changing anything physically interesting (eg changing a phase).
    This for example is known from the classical electrodynamics where you had the freedom to choose different gauges (Lorentz, Coulomb etc) and the equations that you had to solve did not change. The eg [itex]\partial_\mu A^\mu=0[/itex] (Lorentz) . If for example you had found a solution to your electrodynamic problem, let's say [itex]A^\mu[/itex], then you can also find that [itex]A^\mu + \partial^\mu K[/itex] would be a solution of the same equations (K has to be harmonic function). The gauge condition (gauge fixing) is the equation [itex]\partial_\mu A^\mu=0[/itex] which "fixes" the gauge by applying an additional condition:
    [itex] \frac{\partial \phi}{\partial t} + \vec{\nabla} \cdot \vec{A} =0[/itex] (so you can get rid of one of the components of [itex]A^\mu[/itex] by choosing an appropriate gauge /they are unphysical components and depend on your gauge choice).

    For more, you'd better take a particle physics course when you will be ready to do it.

    [Mentor's note: Edited to remove a digression referring to another thread]
     
    Last edited by a moderator: Oct 28, 2015
  4. Oct 27, 2015 #3

    atyy

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    Gauge invariance means we call the same thing by more than one name.

    As a simple example, water always falls down, or one can say that water falls from higher to lower heights. So for example, water falls from the 5th floor to the 4th floor. But what is the "5th floor"? Well, it depends. In the US, people start counting from 1, but in some other countries people start counting from 0 or the ground floor. So the 5th floor in the US is the same as the 4th floor in another counting method - in other words, we call the same thing by more than one name.

    What is the "same thing"? Whether you say water falls from "5 to 4" or from "4 to 3", what is the same is that it falls in the "-1 direction", where we take the final floor minus the initial floor to be the meaning of the word "direction". But obviously it is easier to say "5 to 4", once we have fixed that we count from 1. So gauge invariance is a matter of convenience.

    Gauge invariance can delete "degrees of freedom" because if initially there appear to be two things like "5 to 4" and "4 to 3", by saying that two things are only different names for one thing, we have in some sense deleted "degrees of freedom".
     
  5. Oct 28, 2015 #4
    thanks
     
  6. Oct 28, 2015 #5
    thank you too
     
  7. Oct 28, 2015 #6

    Nugatory

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    Staff: Mentor

    Atyy and ChrisVer have done a pretty decent job at that above....
    ... But that we can't help you with. We need the "fancy math and formulas" to even have an honest notion of what a massless vector boson is.
     
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