# 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. – DanielC 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. – Chiral Anomaly Jul 24 '19 at 1:42
• @DanielC, it would be interesting to see evidence that they indeed cannot be captured by this formalism – user56834 Jul 24 '19 at 4:55