I’m glad you’re enjoying my attempts…

I have a few reasons for rescaling the annihilation and creation operators the way I did. First, as you mention, it’s nice to retain the property that hitting a normalized state with a creation operator gives a normalized state. Second, the master equation seems to look simpler in terms of rescaled annihilation and creation operators if we do what I did, instead of the more symmetrical-looking

Third, some things about coherent states seem to work better.

But I’ll have to keep doing calculations to know for sure what’s the best setup.

]]>If we want to keep the symmetry we can use

If I were wanting to build self-adjoint Hamiltonians I would probably want to do this, since then and it’s easy to tell which combinations of and are self-adjoint.

But in stochastic mechanics we want infinitesimal stochastic Hamiltonians, with

for all using my usual definition of the angle bracket:

for

In this situation it’s very nice that the creation operators obey a simple rule:

which implies

I have chosen my rescaled annihilation, creation and number operators

so they still obey these rules:

which implies

along with simple commutation relations.

Using powers of annihilation and creation operators makes things very complicated, and I don’t think it fits the idea of ‘rescaling’ that’s built into the classical limit.

]]>so that (quantum mechanic analogy with [x,p]):

and

to obtain higher degree derivative

and

is it possible with these commutators to obtain annihilation of N particles? ]]>

where is the number operator for a single particle, and

so on.

A dead end.

I don’t like only the loss of the symmetry in the and operator definition.

I am thinking that the classical limit can be obtained using a product of creation, or annihilation:

and

in the limit of great n, could be possible that is obtained the classical limit? Or the Poisson bracket?

The operator number is:

but the commutation rules are very complex.

For pure states, we have that so that the creation operator maps pure states to pure states (i.e., in this case from a state having probability one of having things to one having probability one of having things). However, the annihilation operator satisfies so it typically maps a pure state to a non-normalized state (i.e., the coefficient is typically greater than one). So, it makes sense to rescale the annihilation operator by the inverse of the “expected” maximum “system size.” Then, both operators will typically map pure states to pure states (and hence mixed states to mixed states), as long as the number of things being annihilated at any given time is less than . If we want the possibility of an infinite “system size,” then we should take the limit .

At any rate, whether that makes any sense or not, I am looking forward to the next post on this to see where you’re going with it.

]]>Martino wrote:

If I can ask, why are you posing ? As far as I remember, even in QM, it’s usually 1.

I so often set in my thinking on quantum mechanics that I don’t know whether the ‘usual’ commutator of annihilation and creation operators is 1 or

But I know that to get Poisson brackets in the classical limit I need a bunch of quantities whose commutators are proportional to (perhaps plus higher-order corrections).

We could define quantities and as linear combinations of and , and work with those. However, in the stochastic context I don’t know what those quantities mean! Since the Hamiltonian for the master equation is built using creation and annihilation operators, and my goal is to study the large-number limit of the master equation, I’d rather work with those.

It’s a bit bizarre that we don’t know what and mean in the stochastic context! This is something I should correct sometime… I think I can sense yet another shoe dropping!

However, at present I have no understanding of these quantities in stochastic mechanics, and I have another goal in mind, so I’ll postpone thinking about them for a while. Thanks for reminding me to do it someday.

]]>Also, your definition of A and C on reminds me more of the use of x and p in QM. In that case, it would be natural to multiply p times , since it’s how it’s defined in usual QM. However, you lose , which I suppose it was the reason you defined them like that.

]]>