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My understanding of a symmetry is this: apply an operation (e. g. parity inversion) to a system. If it behaves the same afterwards, it is symmetric under that operation.

Now, quite often I see statements like this:

Isospin is regarded as a symmetry of the strong interaction under the action of the Lie group SU(2), the two states being the up flavour and down flavour. [...] In simple terms, [the] energy operator for the strong interaction gives the same result when an up quark and an otherwise identical down quark are swapped around.

(from https://de.wikipedia.org/wiki/Isospin)

  • How can the strong interaction "have a symmetry"? An interaction is not a one-time operation like parity inversion. Is the meaning of this that any strong interacting process does not affect the isospin? Or that reversal of all isospins in a system does not change the behavior of the strong interaction?
  • I also don't see how in the specific example from above a down quark is suddenly "otherwise identical" to an up quark except for its isospin. Wouldn't up and down quarks always differ by mass and electric charge?
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  • $\begingroup$ In the case of isospin you can change a proton for a neutron and vice versa without affecting the result of the interaction. Would you not call this a "symmetry"? $\endgroup$ – ZeroTheHero Sep 23 '18 at 14:14
  • $\begingroup$ Thanks, that makes sense! So it is meant that flipping all isospins does not change the behavior the strong interaction? As another example, would electric charge be a symmetry of the electromagnetic force? Conjugating all charges would still leave to electrons repelling each other etc. $\endgroup$ – jmb Sep 23 '18 at 14:22
  • $\begingroup$ The "otherwise identical" wording of the article is unfortunate, as I think it has mislead you in this case. Particles can (and usually are) symmetrical /invarient/indistinguisible in respect of some properties/operations, but by no means for all operations, is what should have been stressed. $\endgroup$ – user207480 Sep 23 '18 at 14:31
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A general way of understanding what a symmetry means for physicists is to think of an operation that generates new solutions to the equations of motion from previous known solutions. In classical mechanics, for example, if you take a two-body problem in which the potential energy that governs the interaction between the two particles is central (only depending on the distance between them), you can take a known solution (for instance, the one in which the center of mass of the system lies on the origin of your coordinate frame) and translate it by a constant distance, generating a new solution (a solution in which the center of mass is not in the origin of your coordinate system). If you had an interaction which depended on the absolute value of the position of those particles with respect to your coordinate system, however, translation would not, in general, take the system to a new possible solution; the evolution of the system would be essentially different.

That intuition is readily applicable to field theory, where the role of the equations of motion is played by the field equations (Maxwell's equation in the case of electromagnetism, or Yang-Mills' equations in the case of quantum chromodynamics). So what is meant by "a symmetry of the interaction" is that if you have a field configuration that solves your equation of motion, and you swap the flavours of the particles involved, you still get a solution to the equation of motion.

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  • $\begingroup$ Thanks, this is insightful. So in the isopsin example from my question, the "operation that generates new solutions" is a 180° rotation in isospin space and the "new solution" is that e. g. proton and neutron still interact the same way under the strong force? $\endgroup$ – jmb Sep 23 '18 at 15:52
  • $\begingroup$ Would, by this logic, the electric charge be a symmetry of the electromagnetic force? Conjugating all charges would still leave to electrons repelling each other etc. But what then is different in the equations of motion? $\endgroup$ – jmb Sep 23 '18 at 15:54
  • $\begingroup$ "Thanks, this is insightful. So in the isopsin example from my question, the "operation that generates new solutions" is a 180° rotation in isospin space and the "new solution" is that e. g. proton and neutron still interact the same way under the strong force?" Yes, you could think of it that way. The thing with the electric charge is a bit more subtle, because it involves the interaction of the electromagnetic field (which is itself described in terms of a neutral boson, the photon) with another field which is the one that possesses charge (the Dirac Field in case of an electron). $\endgroup$ – Bruno De Souza Leão Sep 23 '18 at 16:59
  • $\begingroup$ Having said that, you can still say that charge conjugation (the operation that swaps particles for antiparticles in the Dirac Field) is a symmetry of the theory, and so it is true that the interaction between the Dirac Field and the electromagnetic field is invariant under charge conjugation. (One minor detail is that you have to add time reversal too, but that's not so relevant here.) $\endgroup$ – Bruno De Souza Leão Sep 23 '18 at 17:05

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