FQXi Essay Contest: The Nature of Time

The Foundational Questions Institute, FQXi, is running an essay contest, with the subject of the Nature of Time. Up to 21 prizes are to be awarded, with amounts ranging from $1000 to $10,000 per prize. But probably more important are the bragging rights. You have until December 1, 2008 to submit an essay. Your essay, assuming it’s “serious”, will be displayed on their website from soon after you turn it in until the contest concludes in mid December.

Right now, the contributions look fairly weak so who knows, maybe a little typing now will get you a little money, and some bragging rights, in early 2009. Of course I’ve typed up a contribution, and of course it has to do with the nature of time, as is suggested by the density matrix formulation of quantum mechanics.

I included some short comments on the subject of non Hermitian density matrices. These have something to do with raising and lowering operators and it’s worth typing up a quick blog post on the subject. Surely somebody is going to learn something.

First, rather than use bra-ket notation, which is slow and painful to format due to combined inadequacies of both wordpress and Roger Cortesi’s LaTex engine, (which last I use to create LaTex images), I will use operator notation. And for this subject, there’s no reason to have to worry about dependency on time, so let’s use wave functions with no time dependence.

Let f(x), g(x), and h(x) be three orthonormal wave functions. One converts these to density matrices by multiplication by the complex conjugate, with another copy of the space-time coordinate, i.e.:
\rho_f(x,x') = f(x)f^*(x')
This raises the issue of “what do you get when you replace the “g” in the above with “g”? That would be:
R_f^g = g(x)f^*(x')
This is the raising operator that converts f to g and zeroes everything else.

This may not be obvious. Let w(x) be an arbitrary function of x. Define (R_f^g\; w )\; (x) as a function of x as follows:
(R_f^g\; w)\;(x)= \int R_f^g(x,x')\;w(x')\;dx'
=\int g(x)\;f^*(x')\;w(x')\;dx'
Then (R_f^g\;f)\;(x) = g(x) , and (R_f^g\;g)\;(x) = (R_f^g\;h)\;(x) = 0 , so R_f^g is the raising operator as defined.

More general raising operators can be defined similarly. For instance, if we wish to have an operator that raises f to g, and g to h, then we define it as:
R^+ = g(x)f^*(x') + h(x)g^*(x').
In other words, we can make all sorts of raising and or lowering operators this way.

Non Hermitian Density Matrices
And what about “non Hermitian density matrices?” Let u(x) and v(x) be two normalized but not orthogonal wave functions. Define the “raising” operator from v to u as:
R_v^u(x,x') = v(x)u^*(x')
Density matrices are usually normalized so that \rho \;\rho = \rho . This requires that the above be divided by the constant k where
k = \int u^*(x')\;v(x)\;dx'
which is nonzero under the assumption that u and v are not orthogonal. Note that if u and v are orthogonal, as they would be for a raising operator, it is impossible to normalize the “non Hermitian density matrix” because this would require division by zero. But we can think of raising and lowering operators as the limits of non Hermitian density operators.

So R_v^u(x,x')/k can be thought of as a sort of normalized raising operator for non orthogonal states. But it meets all the requirements of a density matrix except one; it cannot be made from a wave function.

Note that the usual density matrices are Hermitian, replacing x with x’ changes the values to their complex conjugates. Making the same change with R_v^u(x,x')/k does not result in a simple complex conjugate so these objects are not Hermitian.

Wave functions from Density Matrices

Given a (pure) density matrix of the usual sort \rho(x,x') , it’s easy to define a wave function from the same object. Simply pick a point b where the density matrix is not identically zero, and then \rho(x,b) is a wave function that will create the density matrix \rho(x,x') . Alternatively, one could use instead \rho^*(b,x) . In this sense, the usual density matrices are two-sided. They give the same wave function, up to a complex multiple.

Translating this back into the matrix language, one obtains a state vector ket from a density matrix by taking any non-zero column. One can get a state vector bra by taking any non-zero row. For the usual Hermitian density matrix, the row and column one obtains these ways are complex multiples of each other.

So the generalization of Hermitian density matrices to non Hermitian density matrices amounts to allowing the left and right side wave functions to be different.

Density matrices and Complex Phases

When one converts a wave function / state vector to a density operator / matrix, the arbitrary complex phase of the wave function / state vector is lost. Density matrices still keep complex numbers, but now they’re expressed as the difference in phase between two points in space-time, the incoming and outgoing states.

When one moves from density matrices to non Hermitian density matrices, one ends up with differences in spin between incoming and outgoing states. In that sense, non Hermitian density operators are a natural generalization of density operators.

5 Comments

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5 responses to “FQXi Essay Contest: The Nature of Time

  1. HI Carl. I’ve been thinking about FQXi too. Perhaps they will enjoy an essay based upon the simple and self-evident axiom that R = ct, scale of the Universe expands by some multiple of t.

  2. carlbrannen

    Louise,

    That’s a great idea! They want you to write something that isn’t published already, so you may have to think of a new way of looking at it.

    For instance, what does R = ct say about the nature of time as far as an individual’s experience of time goes?

  3. carlbrannen

    Matti Pitkanen is writing an essay for the contest. So far, it hasn’t been accepted because it is too long! I expect he will shorten it to the required 10 pages.

    Until then, you can read the full length version here: About the Nature of Time.

    “The naive identification of the experienced time and geometric time involves well-known problems.
    Physicist is troubled by the reversibility of the geometric time contra irreversibility of experienced
    time, by the conflict between determinism of Schroedinger equation and non-determinism of state function reduction, and by the poor understanding of the origin of the arrow of geometric time. In biology the second law of thermodynamics might be violated in its standard form for short time intervals. Neuroscientist knows that the moment of sensory experience has a finite duration, does not understand what memories really are, and is bothered by the Libet’s puzzling finding that neural activity seems to precede conscious decision. These problems are discussed in the framework of Topological Geometrodynamics (TGD) and TGD inspired theory of consciousness constructed as a generalization of quantum measurement theory.”

    P.S. I also think that neural activity precedes conscious decision making is significant. I believe it implies a coupling of the seat of decision making, the soul, to the brain through a chaotic mechanism that requires about a second to grow to a large size. So the will is wired to the body about a second late.

  4. Narendra Nath

    What happens in the brain as the chain of thoughts continue. That is in the wakeful state. What transformation takes during the dream and the deep sleep stage? How one may assign a role here to the consciousness that is always there in all the three stages? Consciousness gets differentiated into a different form called the sub-conscious mind. In what way the cosmic consciousness interacts with an individual consciousness! There has been a report attributed to Prof. Eccles of Oxford, a Nobel prize awardee neurologist. According to him, the Supplementary Motor Area (SMA) of the brain has a non-physical shield/sheath through which the interactions of the outside affects the neurons of the SMA, keeping them active. Such interactions get recorded in that sheath that does not die with the death of the body. What possible role such a sheath may have with the individual consciousness that may survive the death of the body with which it had the association?
    The reactions / comments are welcome for a dialogue on this topic as it apparently connects science with consciousness, latter being a non-physical entity! Can time have reversibility and /or non-linearity in such interactions. If so, the concept of time will assume a mysterious role in the birth, life and death of an individual!

  5. Narendra Nath

    Time may have reversal or a non-linear behavior during the human thought processes. There are intuitive and inspirational thoughts that are beyond the confine of an individual human brain. There is knownn interaction between SMA of the brain with the outside, as noted through the activity of the neurons in SMA ( Supplementary Motor Area ) by Prof. Eccles, the Oxford University Neurologist, a Nobel Laureate. He expressed a belief that such interactions are duly recorded in a covering shield over the SMA that is non-physical in nature. It survives the death of a body while retaining the history of such interactions with the outside of the human body.
    Perhaps one can well invoke here the role of the non-physical consciousness that permits individual’s interaction with the cosmic consciousness, an experience the individual concerned can experience and remember. The technique of Yoga / meditation permits enhancement of such capability, as the fourth state of consciousness besides the three known states of wakeful, dream and deep sleep!

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