# Variant of Bohm Mechanics for Other Observables

1. Jan 7, 2015

### stevendaryl

Staff Emeritus
This seems like an obvious thing to ask, so I assume it has been asked and answered, but I'd like to know what is special about position in the Bohm-DeBroglie interpretation of quantum mechanics (called "B-DB" in the following).

Here's a way of thinking about B-DB: Because not all observables commute, it's not consistent to assume that they all have definite (but unknown) values at all times. So one way out is to treat one complete, commuting set of observables specially, and assume that they have definite values at all times. Other, noncommuting observables simply don't have a value until measured. For nonrelativistic quantum mechanics (ignoring spin), the most obvious choice for a special observable is position, since macroscopically it appears that objects have definite positions at all times. If we assume a special initial distribution for positions, and assume a special deterministic equation of motion, then we can have a theory where position has a definite value at all times, and this theory makes the same probabilistic predictions as orthodox quantum mechanics.

So my question is whether you could have done the same thing with any other observable--say, momentum. Could we develop a variant of B-DB in which momentum has a definite value at all times, for example? The only thing that seems different about position (and I'm not sure whether this is key to B-DB working, or just a nice feature) is that typically, interactions are position-dependent (using a potential of the form $V(x)$), rather than momentum-dependent.

Assuming (which might be contrary to fact) that there are variants of B-DB for every possible complete commuting set of observables, then that leads to a kind of double Many-Worlds theory: There is one world for each choice of a set of observables, and for each initial valuation of those observables.

Maybe this idea, if worked out in detail, would turn out to be equivalent to (or a special case of) the consistent-histories interpretation of QM (I can't remember who advocated that--Hartle, or Omnes, or ...?)

2. Jan 7, 2015

3. Jan 7, 2015

### stevendaryl

Staff Emeritus
Thanks. That's a very interesting paper. But if I understand their argument, there is nothing special about position--it's really that momentum is special (nonrelativistically) in that it only appears quadratically in the hamiltonian. The claim seems to be something along the lines of: You can't use one observable, Q, as the hidden variable unless its conjugate, P, appears only quadratically in the hamiltonian. So for non-quadratic potentials, you can't take p as the hidden variable. For harmonic oscillator, p or x are equally good.

On the other hand, if you try to include relativistic effects by considering higher-order terms in p, it seems that that would imply that Bohmian mechanics would no longer work.

4. Jan 7, 2015

### Demystifier

There is something special about position in Bohmian mechanics. To understand what, see e.g. Sec. 2 of
http://lanl.arxiv.org/abs/1112.2034
especially
- the first paragraph of page 5, and
- the paragraph around Eq. (7).

5. Jan 9, 2015