Bohmian mechanics is not just an “interpretation” of quantum mechanics. It is a radical revision. In this note, I’d like to point out one reason that it’s an implausible revision: Bohmian mechanics is rampantly indeterministic in a way that quantum mechanics is not.
Setup: Review of Bohmian mechanics
In Bohmian mechanics, the locations of particles are described by a point in a real manifold , called the configuration space. The trajectory of a system of particles is a curve through that manifold. The theory also includes a set of square-integrable functions on this space called wavefunctions.
A physical system in Bohmian mechanics can be characterized by a configuration space , a wavefunction space , and also a self-adjoint linear operator called the Hamiltonian. This Hamiltonian generates a one-parameter group of wavefunctions that solves the Schrödinger equation,
For a given Bohmian system , an initial condition is a pair , with and . The fundamental law of Bohmian mechanics then says that, given an initial , the trajectory of the system is a solution to the Bohmian Guidance Equation,
where is the solution to the Schrödinger equation with initial condition .
Rampant Bohmian indeterminism
Take as simple a Bohmian system as you can imagine: a free particle confined to a finite space. It turns out that the Bohmian description is rampantly indeterministic.
Let the configuration space be , which describes the possible locations of a particle on a string of finite length. Let be the set of differentiable square integrable functions . And let , the Hamiltonian for a particle free of any forces or interactions, and where .
As a dirt-simple example of indeterminism, choose the initial condition , and given by,
This wavefunction is square integrable and differentiable. (Square-integrability follows from the fact that , and differentiability is obvious.) But let’s calculate what the Guidance Equation looks like for this initial wavefunction. Since , the unitary propagator for our Hamiltonian satisfies . Therefore,
The differential equation is well-known to be indeterministic, following a much-discussed example of John Norton (2008 / animated summary). But let me make it explicit: a Bohmian particle with this initial configuration is compatible with a continuum of future trajectories all satisfying the Guidance Equation. Namely,
for any arbitrary time . (We restrict our attention to times during which the particle is in the interval , namely .) These solutions correspond to a Bohmian particle that sits at up until an arbitrary time , when it randomly begins moving.
Not the dome
As a point of comparison, recall that many have complained about the “unphysical” features of the surface of Norton’s dome, such as the infinite Gaussian curvature at the apex (e.g. here and here). No such complaining need be tolerated in the case of Bohmian mechanics. There is no surface to complain about. There is only the wavefunction , which is a perfectly boring, deterministic wavefunction from the perspective of orthodox quantum mechanics. In particular, it is the initial condition for a unique solution to the Schrödinger equation, which is defined for all times . It is only with the addition of the Bohmian Guidance Equation that a pathology occurs.
In order to avoid such pathologies, Bohmian mechanics must somehow excise this class of wavefunctions from the theory. But it’s not clear how to motivate this excision in a non-ad hoc way. And it’s even less clear whether it can be done in a way that avoids doing damage to the ordinary quantum dynamics.
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