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A A question from Warren Siegel's textbook 'Fields'

  1. Apr 18, 2016 #1
    The following is taken from pages 99 and 100 of Warren Siegel's textbook 'Fields'.


    We begin by reviewing some general properties of symmetries, including as an example the symmetry group of nonrelativistic physics. Symmetries are the result of a redundant, but useful, description of a theory. (Note that here we refer to symmetries of a theory, not of a solution to the theory.) For example, translation invariance says that only differences in position are measurable, not absolute position: We can’t measure the position of the “origin”. There are three ways to deal with this:

    (1) Keep this invariance, and the corresponding redundant variables, which allows all positions to be treated equally.

    (2) Choose an origin; i.e., make a “choice of coordinates”. For example, place an object at the origin; i.e., choose the position of an object at a certain time to be the origin. We can use translational invariance to fix the position of any one object at a given time, but not the rest: The differences in position are “translationally invariant”. For example, for N particles there are 3N coordinates describing the particles, but still only 3 translations: The particles interact in the same 3-dimensional space.

    (3) Work only in terms of the differences of positions themselves as the variables, allowing a symmetric treatment of objects in terms of translationally invariant variables. However, this would require applying constraints on the variables: In the above example, there are 3N(N− 1)/2 differences, of which only 3(N− 1) are independent.



    I have a couple of queries about this excerpt from the textbook. Firstly, what is the symmetry group of nonrelativistic physics (as in the first line of the excerpt)? Is it the symmetry group of translations? What then about the symmetry group of rotations?
     
    Last edited: Apr 18, 2016
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  3. Apr 19, 2016 #2

    vanhees71

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    The symmetry group of non-relativistic (Galilei-Newton) spacetime is the Galilei group, which is generated by

    -time translations
    -space translations
    -rotations
    -boosts

    The latter is the change of space-time coordinates from one inertial frame to another, i.e.,
    $$t'=t, \quad \vec{x}'=\vec{x}-\vec{v} t,$$
    where ##\vec{v}## is the constant velocity of the observer at rest in the frame ##\Sigma'## relative to the observer at rest in the frame ##\Sigma##.
     
  4. Apr 19, 2016 #3
    Thanks so much for the answer.

    I have been trying to understand the next two sentences.

    I am trying to understand the meaning of the phrase redundant, but useful, description of a theory.

    To use the example of the gauge invariance of the potential upon solving the source-free half of Maxwell’s equations, am I wrong in saying that the redundant, but useful description of the theory of Maxwell's equations is in the use of the potential (and not the fields, which don't directly give rise to the gauge invariance)?

    What would be the redundant description that gives rise to the translational invariance of the laws of nature? Is it the position of the particles or the differences in the positions of the particles?
     
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