# What is the general definition of symmetry in quantum mechanics

Consider a quantum system with Hilbert space $$\mathcal{H}$$ and Hamiltonian $$H$$. Let $$G$$ be a Lie group and $$U$$ a unitary representation of $$G$$ on $$H$$. What are the most general conditions that $$H$$, $$G$$ and $$U$$ must fulfil so that $$G$$ can be considered a symmetry of the system?

I know, for example, that $$G$$ is a symmetry if $$[U(g), H] = 0\quad\forall g\in G\quad,$$ but this is not the most general condition. In a relativistic QFT the representations of Lorentz boosts do not commute with the Hamiltonian. In the case of Lorentz symmetry the criterium seems to be that $$H$$ must transform as the time component of a four-vector, as explained, for instance, in https://physics.stackexchange.com/a/568141/197448. But how does this generalise to arbitrary groups? Is there a broader notion of symmetry in QM for which both Lorentz symmetry in QFT and $$[U(g), H]=0$$ are just special examples?

For example, would it be valid to say that the system is symmetric under $$G$$ if $$H$$ is a linear combination of the generators of the representation $$U$$ (so that it transforms as a vector in the adjoint representation of $$G$$)? This would cover the case of Lorentz boosts in QFT if we take $$G$$ to be the Poincare group. It would also cover the cases where $$H$$ commutes with $$U(g)$$ if we let $$G$$ be the product of the time translations and some other group of transformations which do not involve time.

• Commented Jul 3, 2021 at 16:19
• While perhaps not useful in practice, I would say you can call $G$ a symmetry if there exists some $U: G \rightarrow \mathrm{End}(\mathcal{H})$ which is both unitary and faithful. Commented Jul 3, 2021 at 17:46
• @ConnorBehan That's what I would call a faithful representation of $G$. But to decide if $G$ is a symmetry the Hamiltonian $H$ should come into play somehow, right? By your definition a system of a particle in a potential would be rotationally symmetric even if the potential is not rotationally symmetric. Commented Jul 3, 2021 at 18:36
• It's what everyone would call a faithful representation of $G$ :). But the idea of symmetry is general enough that it shouldn't depend on locality or a Hamiltonian. The connection with conserved quantities is where I think $H$ needs to come into play. Commented Jul 3, 2021 at 19:12
• And I'm not seeing what you mean about rotational symmetry. We can consider a particle in a 2D box where a basis for $\mathcal{H}$ is labelled by integers $n_1$ and $n_2$ with $n_i$ and $-n_i$ identified. In that case, independent sign flips and $n_1 \leftrightarrow n_2$ represent the dihedral group faithfully as expected. But what is a permutation of these states that respects the group law of $SO(2)$? Commented Jul 3, 2021 at 19:15