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The following is taken from pages 99 and 100 of Warren Siegel's textbook 'Fields'.

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?

*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.(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?

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