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.

Another paper by the same authors that is useful in understanding my methods is Bohmian Mechanics and Quantum Field Theory, quant-ph/0303156 by Detlef Duerr, Sheldon Goldstein, Roderich Tumulka, and Nino Zanghi. Now as far as I can tell, there is no paper that combines these two views as I have, that is, Bohmian mechanics, fundamental density matrices, and quantum field theory. I don’t see how it is possible to do elementary particles without defining some sort of relationship to field theory and Feynman diagrams, but as far as I’ve found, I’m pretty much alone in looking at the density matrix version of field theory.

The techncial reason for the loneliness out here is that field theory is always written in state vector / spinor form. So to put it into density matrix form, one must have tools that allow one to quickly and efficiently translate between density matrix descriptions of a quantum state and state vector descriptions of the same state. These are the tools I’ve been using, and they are elegant and powerful.

Eventually I will write a blog post giving my version of Bohmian mechanics which will, in fact, treat the density matrix as fundamental, and will unite the density matrix with the particle position in the same mathematical object. These are things that I’ve worked on for years but have not stressed because I think that interpretations of quantum mechanics are too much like religion to allow one to communicate the ideas easily. It’s better to stick with calculations. The reader doesn’t need to know how I see the ontology of quantum mechanics in order to understand how I make calculations.

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