Hamiltonian Algebras: What Do We Mean By "Generated"?

[charlesworth]

I keep hearing jargon like "the algebra generated by the hamiltonian", and I'd like to get to the bottom of it.

Given a set of hamiltonians, does the "algebra that they generate" refer to the unitaries that they generate and their subsequent combinations? Or does it refer to the hamiltonians themselves along with their nested commutators or something?

A rigorous reference to any of this would be greatly appreciated, whether it be a textbook or a paper.

 

[strangerep]

I keep hearing jargon like "the algebra generated by the hamiltonian", and I'd like to get to the bottom of it.

Given a set of hamiltonians, does the "algebra that they generate" refer to the unitaries that they generate and their subsequent combinations? Or does it refer to the hamiltonians themselves along with their nested commutators or something?

You might get a better answer if you post an actual quote in context, and give
the reference.

(When I hear this phrase, I tend to think "universal enveloping algebra" by default,
but that might or might not be what's intended...)

 

[xepma]

To me it's also not really clear what you mean charles. You could mean the geometric structure associated with Hamilton's equations in classical mechanics. The algebra structure is coming from the Poisson bracket.

But you could also mean the C* algebras -- which are sometimes called Hamiltonian algebras. In the context of quantum mechanics this algebra usually refers to the algebra of linear operators acting on a Hilbert space.