# Quantummechanics without Hamiltonians

1. Mar 11, 2005

### eljose

Let,s suppose we have a classical equation:

$$F(x,y,dy/dx,.....)=0$$

but this can not be derived from a Lagrangian or a Hamiltonian..then how would we quantizy it?..

Another question given the Lagrangian with position and velocities...could we obtain a quantization of it,i mean obtain a quantummechanics based on position and velocities instead of in momenta and positions..

2. Mar 11, 2005

### dextercioby

Yes,to the second question.

What does F represent...?

Daniel.

3. Mar 11, 2005

### BlackBaron

I suppose F just represent ANY function (or maybe, if you have more than one degree of freedom, a set of functions) of the postion, the velocities (and, maybe, the acceleration), then, F = 0, are just the equations of motion.
I suppose the question could be slighty rephrased:
If one has some system whose classical equations of motion can't be derived from a Lagrangian or a Hamiltonian, how do you quantize it?
(is that right, eljose?)
I think there are indeed some classical systems that don't have a Lagrangian or a Hamiltonian. If that's the case, I can't think a way to formulate the quantum equivalent.
Now I wander, could such a system really have a quantum equivalent? It would be nice to have some example. Because at that point, one really has to wanders if it makes physical sense to describe such a system with quantum mechanics (remember that one actually needs a physical interpretation, this ain't just mathematics).
What do you people think?

4. Mar 11, 2005

### dextercioby

What systems do not have Lagrangians (and implicitely Hamiltonians)...?Give an example.

Daniel.

5. Mar 11, 2005

### BlackBaron

I'm not sure.
I remember once some guys discussing with a teacher about the subject (I wasn't paying much attention, though ), the exact point was if a cellular automata could be described through a Hamiltonian, I think the conclusion was it wasn't (this is a good chance to ask you people what do you think about it).
It does have some sort of "equations of motion" (the rules of evolution), but I can't think a way those can be derived from a Lagrangian/Hamiltonian.
That particular example does serve to show my previous point, even supposing it has some kind of "Lagrangian", does it really makes sense to quantize it? (i.e. to look for its "quantum equation of motion"?).

6. Mar 16, 2005

### eljose

i think the equation of transport in statistical mechanics..

A question of interest is when dealing with gravity we know the Lagrangian,and this is a polynomial in the velocities (i mean in the variables $$\frac{dg_{ab}}{dt}$$) so we could apply perhaps? a formula to quantizy it by:

$$L\phi(x)=\lambda_n\phi(x)$$ with the lambda a kind of eigenvalue..would be that correct?...

Last edited: Mar 16, 2005