I have been reading Sakurai's Modern Quantum Mechanics and I notice that every proposition that book has depends primarily on the Hamiltonian formalism of classical mechanics. Even the time evolution of a quantum system is determined by the Hamiltonian (since the Hamiltonian is the generator of time). This seems quite odd since even though we are doing new physics (compared to classical mechanics) we can not get rid of the basic tools of old physics.
I want to ask whether the Hamiltonian formalism is the best that we can do? It is pretty obvious for physics that the limitations of our theory is the limitations of our axioms. What is currently the best formalism (the one which can cover as many regions of physics as possible) for physics, and what are some ongoing research on this field and related reading material?
To narrow the question down: Can you give examples of useful formalisms in mathematical physics, which can accomplish what Hamiltonian formalism can't?
This is precisely what I am asking for:
Are you asking if this operator can be something else than 𝐻?