Daily Archives: February 3, 2008

Contemporaneity and Density Matrices

I’m continuing to watch the Perimeter Institute lectures and watching the latest one, Dynamic Time: The “Missing Link” in the Search for a Unified Theory? by Avshalom Elitzur, has influenced me to type up my version of the ontology of quantum mechanics.

As Elitzur states, the central conundrum is to ask the question, “what is the past, the present and the future”. The intuitive answer to this is that only the present is real. The future is just our expectations, and the past is just our memory. The standard physics answer to the question is that physics must be written as a theory on space time. The past and future exist just as surely as left and right, or up and down, and the notion of “now” is either a figment of our imagination, or an arbitrary choice of the time coordinate origin.

My view on this is in the middle. I believe that the past and future do exist, but that the present should be treated as a preferred time. In this, I part ways with relativity which denies preferred things. And analogously, I part ways with gauge theory, which denies a preferred gauge. As a matter of philosophy, I believe in the existence of a unique (i.e. exactly 1) world which must therefore have a unique representation in mathematics. If we are unable to distinguish between two different representations, it is only because of the inadequacy of our understanding and not due to the world being several different things at once.

These ideas are difficult to explain because the word “time” is overloaded in our language. To get the point across, let us consider the 2-slit experiment with a single quantum particle, and examine the target region from two different perspectives. This experiment is simpler than the “2-hole experiment” because the symmetry in the direction of the slit reduces one of the dimensions. Instead of three dimensions of space and one time dimension, the 2-slit experiment has only two space dimensions. Furthermore, we concentrate our attention on the target (which is the only part of the experiment which we can “measure” anyway). This reduces the problem to one spatial dimension. We look at only a single time slice, so our pictures are only 1+0 dimensional.

The wave function shows interference. It might look like this:
2 slit experiment wave function
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Fundamental Bohmian Density Matrices

I’ve been listening to Perimeter Institute lectures and came upon one by Roderich Tumulka that was interesting enough that I looked up his papers. They are voluminous. One that applies more or less directly to the area I play in is this one: On the Role of Density Matrices in Bohmian Mechanics quant-ph/0311127, by Detlef Dürr, Sheldon Goldstein, Roderich Tumulka, and Nino Zanghì and published as Foundations of Physics 35 (2005) 449-467.

A density matrix is relevant in yet another way: in a modified version of Bohmian mechanics in which the particles are guided not by a wave function but by a density matrix. Let us call this W-Bohmian mechanics. Whereas in the conventional version of Bohmian mechanics the wave function (of the universe) is something real, as an objective component of the state of the universe at a given time, in W-Bohmian mechanics instead of a wave function (of the universe) we may have only a density matrix. This density matrix does not arise in any way from an analysis of the theory, but is built into the fundamental postulates of W-Bohmian mechanics. It is a fundamental density matrix, W_{fund}, in contrast to the four other density matrices we have discussed, which were derived objects, derived from \psi and Q.

This is one of only a very few references I’ve seen in the literature to the possibility that the density matrix is fundamental. Since this is one of my assumptions in my view of elementary particles, it is gratifying to see it in print.

They approach the subject very thinly. One can find how one defines a density matrix from a state vector in most quantum mechanics text books, but as far as I know, it is only in my book on density matrices that one can find a demonstration of how one defines a state vector from the density matrix. As part of my NYR to be more professional about physics, I’m rewriting the book to remove the inside jokes, make the cover art actually look like a physics book instead of a heavy equipment operation guide, etc. So if, like some of my friends, you think a physics book with a picture of a ditch witch on the front is cool, you’d better order a copy soon because it is not going to be available for very much longer (and I’m not going to buy you one).

The new book will be in paperback format 8.5″ x 11″ (Misner Thorne & Wheeler inspired my layout for both this and the paperback), and will be cheaper at around $15 to $20.

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Filed under Bohmian Mechanics, physics