Symmetry group of a system

Given a general dynamical system (see On Physics from the beginning) consisting of a set $\Omega$, a kind of structure $\mathcal{S}$ (topology, differential structure, metric, symplectic form, all of them,...), and a dynamical law $\mathcal L$, we can consider:

• $\mbox{Sym}(\Omega)$: the bijections of $\Omega$. Is the most general set of alternative descriptions of the space. This descriptions can be very crazy, how by an alien.

• $\mbox{Aut}(\Omega,\mathcal S)$: the bijections of $\Omega$ preserving $\mathcal{S}$, i.e. automorphisms of $(\Omega,\mathcal{S})$. They are alternative descriptions of the system but preserving some "intuitive features".

• $\mathcal{G}(\Omega,\mathcal S, \mathcal L)$: the symmetry group of the dynamical system

Examples:

a). A classical Hamiltonian system consisting of a symplectic manifold $(M,\omega)$ together with a Hamiltonian $H$. $\mbox{Aut}(\Omega,\mathcal S)$ are the symplectomorphism and $\mathcal{G}(\Omega,\mathcal S, \mathcal L)$ are the Hamiltonian symmetrys.

b). The set $\Omega=\{1,2,3,...,9,0\}$ with empty structure $\mathcal S$, and dynamical law $s(t+1)=s(t)+1$, if $s(t)\neq 9$; $s(t+1)=0$, if $s(t)=9$. See xournal 230

c). A quantum mechanical system: a Hilbert space with a Hamiltonian $H$. $\mbox{Aut}(\Omega,\mathcal S)$ are the unitary transformations and the symmetry group $\mathcal{G}(\Omega,\mathcal S, \mathcal L)$ is made of the unitary transformation generated by Hermitian operators commuting with the Hamiltonian. See unitary operator#In Quantum Mechanics.

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Author of the notes: Antonio J. Pan-Collantes

antonio.pan@uca.es