Notes on Bohmian Mechanics
I am reading "A survey of Bohmian Mechanics" :
Authors: K. Berndl, M. Daumer, D. Dürr (LMU Munich), S. Goldstein (Rutgers), N. Zanghi (Genova)
Journal-ref: Nuovo Cim. B110 (1995) 737-750
This web page contains my notes and thoughts - I will be writing while I am reading. So this page will grow with time.
First of all: why I got interested in the subject? Mostly because the idea has some similarity with my own EEQT, namely that fo a more complete desription of quantum systems we need to use both classical degrees of freedom AND quantum wave function. Therefore, EEQT is dualistic and Bohmian mechanics is dualistic. This dualistic nature of Bohmian mechanics is stressed explicitly in "Quantum Equilibrium and the Origin of Absolute Uncertainty":
The conventional wisdom that the wave function provides a complete description of a
quantum system is certainly an attractive possibility: other things being
equal, monism—the view that there is but one kind of reality—is perhaps
more alluring than pluralism. But the problem of the origin of quantum
randomness, described at the beginning of Section 1, already suggests that
other things are not, in fact, equal.
Moreover, wave function monism suffers from another serious defect, to which the
problem of randomness is closely related: Schrodinger’s evolution tends to produce
spreading over configuration space, so that the wave function $\psi$ of a
macroscopic system will typically evolve to one supported by distinct, and
vastly different, macroscopic configurations, to a grotesque macroscopic
superposition, even if $\psi$ were originally quite prosaic. This is
precisely what happens during a measurement, over the course of which the
wave function describing the measurement process will become a superposition of
components corresponding to the various apparatus readings to which the
quantum formalism assigns nonvanishing probability. And the difficulty with this
conception, of a world completely described by such an exotic \wf,
is not even so much that it is extravagantly bizarre, but rather that this
conception—or better our place in it, as well as that of the random
events which the quantum formalism is supposed to govern—is exceedingly
obscure.\footnote{What we have just described is often presented more
colorfully as the paradox of Schrodinger’s cat\recite{cat paper}.}
What has just been said supports, not the impossibility of wave function monism, but
rather its incompatibility with the Schrodinger evolution. And the allure of wave function
monism is so strong that most interpretations of quantum mechanics in fact involve the
abrogation of Schrodinger’s equation. This abrogation is often merely implicit and,
indeed, is often presented as if it were compatible with the quantum
dynamics. This is the case, for example, when the measurement postulates,
regarded as embodying “collapse of the wave packet,'’ are simply combined
with Schrodinger’s equation in the formulation of quantum theory. The “measurement problem'’
is merely an expression of this inconsistency. (…)The theory of GRW modifies Schrodinger’s equation by the incorporation of a random
“quantum jump,'’ to a macroscopically localized wave function. As an explanation of
the origin of quantum randomness it is thus not very illuminating,
accounting, as it does, for the randomness in a rather ad hoc manner,
essentially by fiat. Nonetheless this theory should be commended for its
precision, and for the light it sheds on the relationship between Lorentz
invariance and nonlocality (see\recite{Bellj}).
A related, but more serious, objection to proposals for the modification of
Schrodinger’s equation is the following: The quantum evolution embodies a deep
mathematical beauty, which proclaims “Do not tamper! Don’t degrade my
integrity!'’ Thus, in view of the fact that (the relativistic extension of)
Schrodinger’s equation, or, better, the quantum theory, in which it plays so
prominent a role, has been verified to a remarkable—and
unprecedented—degree, these proposals for the modification of the quantum
dynamics appear at best dubious, based as they are on purely conceptual,
philosophical considerations.
But is wave function monism really so compelling a conception that we must
struggle to retain it in the face of the formidable difficulties it entails?
Certainly not! In fact, we shall argue that even if there were no such
difficulties, even in the case of “other things being equal,'’ a
strong case can be made for the superiority of pluralism.
But back to "A survey of Bohmian Mechanics". The abstract starts with:
"Bohmian mechanics is the most naively obvious embedding imaginable of Schrodinger’s equation into a completely coherent physical theory."
I have doubts about "coplete coherence" of the resulting theory. My guess is that Bohmian mechanics, and understood and as presented today is not a coherent theory. But in order to be able to prove this hypothesis I need to study Bohmian mechanics in some details.
Next the following paragraph comes:
Suppose that when we talk about the wave function of a system of N
particles, we seriously mean what our language conveys, i.e., suppose we insist that
“particles'’ means particles. If so, then the wave function cannot provide a
complete description of the state of the system; we must also specify its
most important feature, the positions of the particles themselves!
Now, why only "positions" rather than "positions and momenta"? Probably the authors were trying to answer my natural question, but they did not come with a right answer. Notice that there is a difference between the way Bohmian mechanics is presented by Bohm himself and the way it is being presented by the authors. In "Undivided Universe" (Chapter 3.1, point 4.) Bohm and Hiley state explicitly:
4. The particle momentum is restricted to p= gradient S.
But Berndt et al. skip this and prefer not mention it (even if it is an automatic results of their postulates). The question is why is the momentum constrained? Of course, once the momentum is constrained, the resulting dynamics is non-Newtonian, that is clear. But why it must be non-Newtonian? What would happen if we would like to remove the constraint and replace it by some dynamical equation? Did anyone try it in the past? Personally I did not like constraints that are postulated without giving any reason. We do not really understand our theories if we can’t explain our constraints.
Berndl et al. propose then:
Suppose, in fact, that the complete description of the quantum system—its state—is given by(Q,psi)where Q = (Q_1,…,Q_N) is in R^3N with Q_k the positions of particles, and psi=psi(q) is the wave function.
Now, here we have a problem. We more or less know what are positions, how to observe positions, even with some errors. We can use photographs, cloud chambers, microscopes and telescopes to register what we think are positions. But how do we observe a "wave function"? What kind of microscope or telescope we are supposed to use to even guess what psi can be? And why only Q instead of (Q,P)? We can observe positions and we can observe velocities. Why are we choosing here positions only?
Then we shall have a theory once we specify the law of motion for the state (Q,psi). The simplest possibility is that this motion is given by first-order equations—so that (Q,psi) is indeed the state in the sense that its present specification determines the future.
Well, if the equations of motion are supposed to be deterministic, how are we going to account for apparent randomness observed in Nature? Lack of knowledge? But what is "knowledge"? How "knowledge" is modelled in Bohmian mechanics?
We already have an evolution equation for psi, i.e., Schrodinger’s equation,
