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

I believe that the above is an accurate description of the physical situation before the experiment has been run. More precisely, I believe that the probability density exists, at least in the future. There are two points here. First, note that I said “probability density”, and not “wave function”. The reason for this distinction is that I believe in the density matrix (as a real object), which directly contains the probability density, rather than the wave function, which must have its magnitude taken, and squared, to obtain a probability density. The second point is that I believe that the wave function exists only when the experiment is considered in the future.

Wave Collapse

After the experiment is completed, the wave function collapses. With the prior drawing, a possible result of the experiment is:
2 slit particle experiment result (1 particle)
where the particle was a little lucky and arrived in the upper lobe of the interference pattern. I see this second drawing as a representation of reality that is also correct, but only when one considers the experiment after it has been run, and then, of course, only if the result of the experiment was as shown.

 Events

In relativity, events are labeled by four coordinates, 3 space and 1 time. The two different descriptions of the 2-slit experiment amount to two different descriptions of the same collection of events. One spatial coordinate defines the target, which is an extended object in 1 dimension. The time coordinate defines the time at which the experiment is run which can be thought of as having no dimension. This means that I believe in two different descriptions of what relativity treats as the same event, which one might think of as a violation of my principle that there is only one unique reality.

Event Overloading

But the two representations of the 2-slit experiment are not used at the same time. The wave description is what we know works correctly before the experiment is run. On the other hand, the particle description works correctly after the experiment has completed, but it is a retrospective description only. The action of running the experiment is related to our having to transform our description of the experiment from a general description, a probability wave, to a specific result, a particle description.

But the process of running the experiment is not entirely divorced from spacetime. Before we run the experiment, the experiment is in our future, afterwards it is in our past. So the choice of representation depends on the time position of us, the observer.

So for me, the Born probability rule amounts to a description of the result of the passage of experienced time for the observer. One can interpret this as the quantum mechanical equivalent of the passage of proper time in special relativity, to which we will return shortly.

As is mentioned in the Elitzur lecture, one cannot describe the passage of time in terms of space and time, it must be outside of space and time. This means that each event in spacetime must receive an extra coordinate, a coordinate which describes whether the event is in the future or the past of the observer. And since that extra coordinate defines the time experienced by an observer, we may very well suppose that the extra coordinate is related to proper time. Before I learned Clifford algebra, this sort of logic was how I got started working on physics, rewriting relativity to swap the positions of proper time and coordinate time, which amounts to a Wick rotation. My first write-up on the subject is The Proper Time Geometry.

Contemporaneity

To make the theory be independent of the observer, let us assume that several observers agree on the unique representation of all events in reality (both past and future). For this to be the case they must agree on the definition of “now” and so agree on whether the event requires a wave or particle description. (Such a set of observers violates the “no preferred reference frame” principle of special relativity, but is a generic attribute of Bohmian mechanics. See “The Unidivided Universe” for details.)

In other words, we will assume that these observers agree on what is “now”; they agree exactly on the age of the universe. If observers are carrying on a conversation about the experiment, this is something that must be approximately satisfied. In fact, it’s generally believed (at least among physicists) that it’s not possible to arrange 2-way communication between people who are even a few years in each other’s future or past. A community of observers, who wish to participate in a back and forth discussion about some particular event, must be contemporaneous so that the event is either in all of their futures or in all of their pasts. Violating this restriction would be a violation of causality as the observers could then communicate information from the future into the past.

The requirement of contemporaneity requires the inclusion of a 5th coordinate to describe events.

So this defines another mathematics problem. It is impossible for us to define the rate of change of “contemporaneity” in terms of the usual measurements of physics. We have an intuitive understanding of it, but it’s not something that can be measured in furlongs or fortnights. Despite this, if we assume that the universe operates this way, we can make assumptions about how the process takes place. In particular, the Born probability postulate does not define the details of the process but it does define the result.

Bohmian Mechanics

Bohmian mechanics associates with each quantum states two mathematical objects, the wave function and a particle track. In our view, these two mathematical objects are separated by having different coordinates in the 5th dimension, contemporaneity. This opens up questions. Under the intuitive assumption that contemporaneity changes continuously, we should be able to morph a wave function into a particle density in a continuous manner. And this means that particle waves and tracks need to be described in a mathematical language that admits intermediate conditions.

Particle Tracks as Wave Functions

The problem with writing a particle track in wave function form is that wave functions have built into them Heisenberg’s uncertainty principle (HUP). [Thanks to the anonymous commenter who has corrected the spelling, but I’d rather you comment on the physics.] An assignment that beginning quantum mechanics students should be assigned is to try to write a wave function that defines the particle position within some certain limits, but that has momentum smaller than that given by the HUP. Of course this is impossible in standard quantum mechanics. To do it, one first rewrites Schroedinger’s equation into a separable equation in the variables R and P where R is the probability density, R = |\psi(x,t)|^2, and P is the momentum density, P = |\nabla\psi(x,t)|^2. One can then write a solution that violates HUP, but one cannot transform it back into a Schroedinger wave form.

Something somewhat similar is done in some versions of Bohmian mechanics; a scalar wave function is rewritten as a probability density and a phase: \psi(x,t) = \sqrt{\rho(x,t)}\exp(iS(x,t)). This is especially natural in the density matrix form where the square root is removed. In this form, position information seems to be entirely contained in the density \rho(x,t), while momentum information is contained in the phase S(x,t). This would make it seem that one could independently control these two features and define a particle defined to have a limited position and yet have limited momentum. When one actually attempts this, one finds that the momentum information is not entirely contained in the angle, but also pops out of the gradient of the density information.

To write a wave function that breaks the HUP, one must break the wave function up into its spatial density and momentum density parts. Each of these will be real functions and since one started with a complex valued function, one hasn’t added or lost any degrees of freedom. On doing this, one can modify the two parts, the momentum density and the position density, separately. But splitting the density matrix function into spatial and momentum density functions is easy – one already begins with the spatial density, and the momentum density is a derived object.

With the wave functions split (as is very much more natural in density matrix language), one can imagine how the wave function morphs into a particle density. To display this, we have to add an extra dimension to our drawings (of 1 spatial dimension and 0 time dimensions). For the 2-slit example given above, a morphing between the two modes is:
2 slit wave density morphing to result
The above morph will be familiar to fans of Bohmian mechanics because the same thing is used in statistical Bohmian mechanics, but in reverse, from the right side to the left.

Bohmian mechanics has the problem that it is possible to assume a probability density for the particle tracks that will give results incompatible with the Born probability rule. However, as time goes on, incompatibilities with the Born rule disappear due to what is a sort of decoherence. Most introductions to Bohmian mechanics will explain how this happens, for example, see “The Divided Universe”.

Contemporaneity as a Dimension

Having contemporaneity as an extra dimension knocks the total dimension count up from 4 to 5. This turns out to be convenient for my work because I require 5 dimensions in order to get a Clifford algebra (or geometric algebra) sufficiently complicated that the elementary particles can be obtained. If one sticks to only 4 dimensions, then the associated Clifford algebra is the Dirac algebra. For such an algebra, there are only 4 distinct (i.e. orthogonal) elementary particles. When modeling QED with the Dirac wave functions, the four elementary particles would be the up and down electron, plus the up and down positron. Adding a 5th dimension allows one to add the up and down neutrino and anti-neutrino. The quarks then end up as mixtures of these states. The details are in one of my old and rather out of date papers, The Geometry of Fermions.

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2 Comments

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2 responses to “Contemporaneity and Density Matrices

  1. Pingback: Pages tagged "arbitrary"

  2. I hope you will add this in your book. And also more experimental background.

    When you say ontology of Quantum Mechanics, I see a parallel with the ontology of Newtonian mechanics. I don’t know if this applies to Quantum Mechanics as well but I wanted to mention it. Physicists have been discussing Time and Space because Newton discussed them in his book in a scholium. Newton’s scholiums were nothing more than philosophical asides. Einstein took Newtonian discussions seriously, or fooled by them, and built a theory of relativity of stuff.

    But the engine under Newtonian philosophy is Kepler’s rule. In Kepler’s rule we don’t have Time, we have period. We don’t have Space, we have radius of the orbit. In this sense I believe that theoretical discussions of Time and Space are disconnected from experiment.

    I believe that something similar is happenning in Quantum Mechanics, that’s why I like the experimental approach. But when you write “To get the point across, let us consider the 2-slit experiment with a single quantum particle…” I will have to question your assumption of “a single quantum particle.” How do you justify the assumption of a single particle? I see the beam as a fluid, not as a collection of ultimate particles.

    I hope you’ll have the book ready by the summer when I’ll have time to study this stuff in detail.

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