# Is unitarity intimately "connected" to symmetry?

I have a couple of questions about unitarity and symmetries:

1. Is unitarity connected to all fundamental symmetries? Is it linked to symmetries like CPT, Lorentz, Poincaré, diffeomorphism, translational and even gauge symmetres?

2. If unitarity was broken somehow, would all these symmetries break?

• This is too vague. Can you write down a non-unitary theory which has those symmetries? Yes. Can you write down a unitary theory which does not have those symmetries? Yes. And if you think "Lorentz, Poincaré, diffeomorphism, translational" are all different symmetries, there's probably a basic textbook that you read too quickly. Apr 8 at 12:52
• There are plenty of quantum mechanical systems that have none of these symmetries... so, no. Unitarity has absolutely nothing to do with physical symmetries. What does it mean to break unitarity in QM? That you lose your inner product (under time evolution)... I don't know how much of the general structure of QM can be salvaged after that. See this: physics.stackexchange.com/questions/15858/… Apr 8 at 19:46

1. Yes, since in Quantum Mechanics unitarity is a key concept. Every symmetry is related to a group of transformations, of elements T. Now the action of this elements on the rays of QM states must be implemented through two kinds of operators acting on $$\mathcal{H}$$:
• A Unitary and Linear operator $$\hat{U}$$ such that you map T $$\rightarrow \hat{U}(T)$$
• An Anti-Unitary and Antilinear operator $$\hat{\Theta}$$ such that you map the trasformation T $$\rightarrow \hat{\Theta}(T)$$
1. If Unitarity is broken the symmetries are not broken, the symmetries are indipendent of their implementation into an Hilbert space $$\mathcal{H}$$ and symmetry breaking is a different topic. What would be broken is the implementation of the symmetry through the operators defined above.