General definition of symmetry in physics? [duplicate]

I've looked at a number of questions on what symmetries are in physics, such as this one, this one and this one. However, I found the questions and answers to be not completely satisfying because they were somewhat informal in certain ways that confused me.

Let $$(S,\Phi)$$ be a dynamical system, where $$S$$ is the state space, and $$\Phi:\mathbb R\times S \to S$$ gives the time evolution of the system given an initial condition. i.e. $$s_t=\Phi(t,s_0)$$ gives the state of the system at time $$t$$ if the initial state is $$s_0$$. Then a transformation $$T:S\to S$$ is a symmetry transformation of the physical system $$(S,\Phi)$$ if and only if for all $$s_0$$ and all $$t$$, $$\Phi(t,s_0)=T^{-1}\left(\Phi(t,T(s_0))\right)$$. In other words, we can transform the system by $$T$$, let the system evolve, and transform it back by $$T^{-1}$$, and it would be as if we hadn't transformed the system at all.

$$(S,\Phi)$$ could represent any physical system, whether $$S$$ is the phase space of a few classical point-mass particles, or the quantum state of an atom.

Is this a good general definition of what is meant by "symmetry" in physics?

• I have seen people define a symmetry as something that leaves the lagrangian unchanged. However, some changes in the lagrangian (e.g. adding a constant) have no effect on the trajectory $$\Phi(t,s_0)$$ in the underlying physical system, so this seems like a bad definition.

• I am not sure whether this captures gauge symmetries, or whether gauge symmetries are different from other symmetries (as I don't know what gauge symmetries are).

• I think each theory of physics has its own definition for „symmetries” and one cannot have a definition to include them all as particular cases for your chosen S. Jul 23 '19 at 22:10
• The paper Symmetries in quantum field theory and quantum gravity goes into some depth to try to define some of the things we call "symmetries" more carefully, with emphasis on relationships between different kinds of symmetries. The context is quantum physics, which might not be the focus of this question, but it still illustrates just how deep this subject(s) can be. Jul 24 '19 at 1:42
• @DanielC, it would be interesting to see evidence that they indeed cannot be captured by this formalism Jul 24 '19 at 4:55

1 Answer

Symmetry, while precise sounding, is a term of many meanings even in physics--there may even be a standard definition but as the uneducated would perceive perhaps its not what they intuitively think and may depend on context. Symmetry is usually an transformation that leaves something invariant. But the nature of these transformations is not just restricted to oh the transforms on a manifold that leave the metric invariant or ohh... And in physics since what you ultimately want to understand is the physics, these starched suit approaches may be suffocating and miss the point altogether. For example for a field a symmetry blah blah blah is just a transformation of the field that leaves something invariant (what? well see) first what the tranformation is: Typically its a complicated thing: Think of rotating an E&M field. Whats called an active rotation. Now we check how this evolves in time and whether this changed field satisfies the equations of motion. Now to be more precise this should really be done to a field solution over spacetime--transform it, and see the transformed version satisfies the eqs of motion (or just are allowed physical histories without explicit mention of time). This is not all, though because you also have to check whether or not they give the same physics. And now you may decide whether to call symmetry just those that send sols to sols or only those that leave the physics invariant.