Density Operators and Time by Carl A Brannen

The author seems to misunderstand the basic formalism of Quantum Mechanics (QM). The density matrix description of QM is entirely equivalent to the wave function description, as can easily be seen by the definition of the density matrix. Thus using the density matrix description doesn't resolve the measurement problem in QM. What the density matrix description provides is a slightly clearer and neater presentation of the measurement process.

And the author is incorrect in stating that the measurement process in QM "must act outside of time". QM describes completely the evolution of a quantum state in time, both in between measurements, and during measurements. e.g. in non-relativistic quantum mechanics (NRQM), the density matrix of a system undergoes unitary evolution in between measurements (say from time t1 to t2):

rho(t2) = U(t2,t1) rho(t1) U*(t2,t1)

where U(t2,t1) is the time-evolution operator for the wave function from time t1 to t2.

During measurement (at time t2, say), the density matrix undergoes a non-unitary transformation R(t2):

rho(t2,after measurement) = R(t2) rho(t2) R*(t2)

The difficulty of the QM measurement problem is not the description of measurement in time (which QM is perfectly capable of doing), but the seeming incompatibility between the unitary evolution of an undisturbed QM system in time and the non-unitary instantaneous "collapse" of the system during measurement.

There have been many attempts to resolve the measurement problem in QM, most of which tried to explain the non-unitary collapse of a system as a result of the effect of the environment on the unitary evolution of the system, i.e. the decoherence approach, though this approach has the fundamental flaw that however close one may be able to get to something that looks like non-unitary collapse using only unitary evolution, at the end any combination of unitary evolution can only give rise to unitary evolution, so the best the decoherence approach can do is to give us a FAPP (for all practical purposes) pseudo-explanation of the collapse of the wave function, but never a true explanation that goes the last step...

The discussion so far has been limited to NRQM. Needless to say, even more problems arise in dealing with the measurement problem in relativistic QM (see e.g. papers by Yakir Aharonov and David Albert.)

Ming writes: "The density matrix description of QM is entirely equivalent to the wave function description, as can easily be seen by the definition of the density matrix."

This is true if the density matrix is defined this way, but the paper shows a generalization of (pure) density matrices, whch are Hermitian, to non Hermitian states. These do not correspond to any wave function. In addition, even if they were equivalent, wave functions and density matrices treat time differently, which is the point of the discussion.

Ming's example of a system that evolves according to a unitary law between t1 and t2, and then undergoes a non unitary evolution at time t2 is not an uncommon way of describing collapse, but it is not entirely satisfactory in that it claims that sometimes evolution is unitary and sometimes it is not.

Regarding relativistic quantum mechanics and density matrices, the reader may find

"On the Role of Density Matrices in Bohmian Mechanics"

Detlef Duerr, Sheldon Goldstein, Roderich Tumulka, Nino Zanghi

quant-ph/0311127

Foundations of Physics 35(3): 449-467 (2005)

appropriate because it has a section on 2nd quantization.

See the literature for the large number of recent papers on relativistic Bohmian mechanics from the wave function approach and on density matrices in Bohmian mechanics.

Hello Carl,

I enjoyed your paper and especially your conclusion, "However, in this model observers do not interact per se, and consequently we may as well make the simplifying assumption that all observers have the same T. Then T becomes an attribute of the universe as a whole, and an explanation for that persistent human insistence

on free will and the uniqueness of the present."

Yes--my essay agrees "Time as an Emergent Phenomenon: Traveling Back to the Heroic Age of Physics:" http://fqxi.org/community/forum/topic/238

"All observers have the same T," you write, and indeed, dx4/dt = ic, from where time as we measure it on our watches emerges, is true for all observers.

The ever-moving T "is an attribute of the universe," providing us with "free will" and the "uniqueness of the present," which rests upon a deeper physical postulate -- the fourth dimension is expanding relative to the three spatial dimensions: dx4/dt = ic, from which Einstein's relativity is derived in my paper. Time, and all motion, emerge from the underlying physical reality of a fourth expanding dimension: dx4/dt = ic. Motion, nonlocality, time's arrows, entanglement, relativity, entropy, free will, and time itself all rest upon this deeper physical reality.

It is interesting to note that the velocity of all entities through space-time is uniquely c. To move at c relative to the fourth dimension means to be stationary in the three spatial dimensions. To move at c relative to the three spatial dimensions means to remain stationary in the fourth dimension, as does a photon. Ergo the fourth dimension is moving relative to the three spatial dimensions.

You write, "In quantum mechanics, measurements are represented by operators. The state of a system is usually represented by a wave function which is operated on by the operators. This view of time is compatible with relativity in that each event is assigned a unique time coordinate; the wave function changes with time. The only diculty is the measurement or collapse processｻ this process must act outside of time asｬ in the language of special relativityｬ it modifes our representation of a single eventｬ for exampleｬ a particle experimentｬ converting our representation from a wave to a particleｮ｢ｼｯpｾｼpｾﾉndeedｭｭthe collapse of a wave function appears to be instantaneousｭｭ｢outside of timeｮ｢ ﾔhis is because the expanding fourth dimension distributes localityｬ so that a photonｧs spericallyｭsymmetric expanding probability wave yet represents a localityｬ no matter how far the photon propagatesｮ ﾈence the ﾅﾐﾒ paradoxｬ and ﾂellｧs ﾉnequalities and ﾁspect et alｧs experiments which demonstrated entanglement and actionｭatｭaｭdistanceｮ ﾑﾍ exhibits nonlocality throughoutｭｭin the double slit experimentｬ in the ﾅﾐﾒ paradoxｬ in the instantaneous collapse of the wave function in the photoelectric effectｬ in the very notion of a waveｬ and this nonlocality can be shown as a natural result arising from an inherently nonlocal fourth dimensionｬ which is nonlocal via its expansionｺ dxｴｯdt ｽicｬ from which ﾅinsteinｧs relativity is derived and entropy and timeｧs arrows are accounted forｮｼｯpｾｼpｾﾗellｬ ﾉ enjoyed your paper and will be studying up on itｬ as ﾉ think weｧre headed in the same general direction｡ｼｯpｾｼpｾﾔhanksｬｼｯpｾｼpｾﾅlliot

Thanks for reading my essay, Elliot.

There are some other people doing stuff in "Euclidean Relativity" (ER) that more or less fits in well with this essay and possibly your own more so. I'll write a comment over on your topic forum when I get some time and internet availability simultaneously.

I don't have a lot of hope for ER because prejudice against it is deeply ingrained. In fact, I don't have much hope for these essays on time. I just typed one up because I thought I shouldn't keep my opinions to myself forever on the subject. Instead, to change the foundations of physics I think we need to redo stuff that gives more concrete results.

Hopefully, Louise Riofrio will write an essay regarding her use of the equation R = c t, where t is the age of the universe and corresponds to the T used in the above essay.

Thanks Carl,

Actually my essay/Moving Dimensions Theory runs with Minkowski's/Einstein's relativity. I have looked into Euclidian Relativity only briefly.

Yes--we need concrete results, but after thirty years of String Theory and LQG, and probably fifty more, we should brave logic, reason, and *physics* with a faith in an apprehendable physical reality that is both simple and beautiful.

Sometimes science is not driven by an immediate novel experiment, but rather it is driven more by a need to provide a unique physical model underlying past experiments, thusly unifying once diverse realms in a physical model. I quote Einstein in my essay,

"Moving Dimensions Theory--which regards time as an emergent phenomena--was inspired in part by Einstein's words pertaining to the higher purpose of physical theories: "Before I enter upon a critique of mechanics as a foundation of physics, something of a broadly general

nature will first have to be said concerning the points of view according to which it is possible to criticize physical theories at all. The first point of view is obvious: The theory must not contradict empirical facts. . . The second point of view is not concerned with the relation to the material of observation but with the premises of the theory itself, with what may briefly but vaguely be characterized as the "naturalness" or "logical simplicity" of the premises (of the basic concepts

and of the relations between these which are taken as a basis). This point of view, an exact formulation of which meets with great difficulties, has played an important role in the selection and evaluation of theories since time immemorial.""

Time is a vast and mysterious phenomena that is so often glossed over. It almost seems that physicists have stopped asking fundamental physcial questions, content to count particles--I recall Wheeler refering to this obession as "ino-itis."

A great read is: "A World Without Time: The Forgotten Legacy of Godel and Einstein." On page 7 Yougrau writes, "Godel was quick to point out that if we can revisit the past, then it never really passed. But a time that fails to pass is no time at all. Einstein saw at once that if Godel was right, he had not merely domesticated time; he had killed it. . . In a word, if Einstein's relativity was real, time itself was merely ideal. The father of relativity was shocked. Though he praised Godel for his great contribution to the theory of relativity, he was fully aware that time, that elusive prey, had once agian slipped his net. . . But now the turly amazing took place: nothing."

How true--we just say, "well, relativity implies block time but time flows and moves, so let's just move on. well, entanglement and nonlocality don't have any physical explanation, so let's just move on and find more particles."

Yougrau continues, "Although in the immediate aftermath of Godel's discoveries a few physicists bestirred themselves to refute him and, when this failed, tried to generalize and explore his results, this brief flurry of interest soon died down. . . A conspiracy of silence descended on the Einstein-Godel freindship and its scientfic consequences." --A World Without Time

Try bringing up Godel's universe and the paradoxical, unresolved implications regarding time, and you will likely get strange, if not snarky, looks in a modern physics department, as foundational quations are deemed impolite, while inscrutable math with lesser ambitions is smiled upon. But somehow, someday, we must be liberated from a block universe and given a *physical* explantation of free will, time, and its arrows that corresponds to our empirical reality. I look across this coffee shop, and I see quantum mechanics and general relativity existing in perfect harmony, and I see time progressing in a unique, constant direction. I cannot travel back to earlier this evening--I cannot go back a microsecond, and thus time is something far beyond a fourth dimension. And the cool thing is Einstein and Minkowski never stated that time is the fourth dimensions, but rather they wrote x4=ict, implying that dx4/dt = ic, and the fourth dimension is expanding relative to the three spatial dimensions, distributing locality for QM, and fathering time and its arrows in QM, stat-mech, and relativity.

I think a lot can come of these essays. :)

The author has submitted an article somewhat related to this, "Density Matrices and the Weak Quantum Numbers", to Foundations of Physics. The paper gives a derivation of the weak hypercharge and weak isospin quantum numbers of the left and right handed elementary fermions from a few simple assumptions about their density matrix representations. A copy of the submitted paper is attached.Attachment #1: WeakQNs.pdf

In fact both 'Quantum mechanics' and 'Einstein's Relativity' are combinations of the Empiricism method tools (new algebraic Geometry born in XVIIth century in Europe).

One can even say that 'Quantum mechanics' is preceeding Einstein's ideology although Einstein seems to speak about high spheres and high speeds and Planck about small particles.

Einstein's theory is just a reflex as Quantum mechanics is.

The 'collapse process' as you say is still in Descartes algebraic Geometry (and string theory too). This collapse has exactly the same cause than the Higgs Boson paralogism and than the dualisme of particle splitted in two.

F. Le Rouge,

I agree with you completely on classifying QM and relativity, both GR and SR, as due to empiricism. String theory gets away from it, but since string theory is completely compatible with both relativity and QM, it cannot avoid the rot built into its foundations.

Mathematically, the essence of the problem is that both QM and relativity depend on symmetries to describe the world. The symmetries are obtained from experimental observations. Thus the empiricism. Neither theory is an attempt at explaining the world, they're both very elaborate curve fitting procedures, not theories in the Descartes sense.

The reason I wrote the paper linked above, "WeakQNs" was to give an alternative explanation to the weak quantum numbers of the elementary fermions. Rather than just saying "this is what experimenters tell us", I feel that theorists should look for an underlying explanation.

For me, symmetry is an attribute that allows one to solve a differential equation. The differential equation is the fundamental object, not the symmetries it possesses. To guess the nature of the universe we must guess the differential equations, not their symmetries.

That symmetries have done us well so far is not proof that the universe is constructed of symmetries "all the way down". To me, it's just proof that differential equations have symmetries, a fact that any mathematician knows without the need for experimental evidence.

Newton's gravitation is a great example of this. The differential equation is very simple, F = ma = gmM/r^2 (uh, if I remember correctly). The symmetries are somewhat more complicated; i.e the conserved quantities observed by Kepler. This is the natural order of things, the fundamental object is simple, its symmetries are more complicated. With modern physics, the increasing complexity of the observed symmetries is a sign that the next big steps will be in the other direction, in guessing the fundamental differential equations.

Carl Brannen,

You quoted a paper by Duerr, Goldstein, et al.

If I recall correctly, they wrote "The Emperor's New Swindle" and faced distrust from those who consider entanglement proved. Doesn't this issue have serious practical consequences?

As far as I know, quantum computing does still not work as promised, and there are reasons for me to question an application of single electron counting published in PRL in 1997.

Eckard Blumschein

Do you have any ideas about the scale of the time T in the rho(x,t,x',t',T)? Is it T G' or Ag = g' so that g^{-1}Ag = g^{-1}g'. This can define exterior products as well so that the right hand side is a density matrix rho_{gg'} = |g>

My post was transmitted in a garbled form, so I am trying again

Do you have any ideas about the scale of the time T in the rho(x,t,x',t',T)? Is it T G' or Ag = g' so that g^{-1}Ag = g^{-1}g'. This can define exterior products as well so that the right hand side is a density matrix rho_{gg'} = |g>

Evidently this does not like carot signs! So bra-kets I replace with ( and )

Do you have any ideas about the scale of the time T in the rho(x,t,x',t',T)? Is it T must less than 1/freq, for freq pertaining to the system.

I have been working on a system of noncommutative geometry which includes associators. For an associator A and g, g' in the quantum groups G and G' an associator acts on these so that A:G rightarrow G' or Ag = g' so that g^{-1}Ag = g^{-1}g'. This can define exterior products as well so that the right hand side is a density matrix rho_{gg'} = |g)(g'|, writen symbolically here. If we coarse grain over the associator the value of an observable O is then

(O) = tr(O rho_{gg'}.

For O = unit this gives tr(rho_{gg'}) = e^{&E/kT}, for &E the energy functional (error) induced by tracing over the associators. Clearly then &E = kT ln(tr(rho_{gg'}) = &s, which is the entropy or information loss due to the coarse graining.

This leads to the Bogoliubov algebra for quantum fields in curved spacetime, which is due to the nonEuclidean nature of time. Associative quantum mechanics is nonunitary, but if it is due to an error correction code then q-bits are preserved, at least on a fine grained scale. Then the appearance of entropy (information loss or "burial") and its identification with time is a large scale emergence.

I have a piece here #371 on an aspect of this physics with AdS spacetimes and the scaling of quantum fields.

Cheers,

Lawrence B. Crowell

Lawrence,

As it turns out, I'm not a fan of using symmetry groups to define the foundations of particle physics. I think it's been rather well picked over and is mathematically naive; yes differential equations have symmetries but symmetries are never foundational in mathematics. In my view, the situation we've ended up in is from people making a series of lucky guesses about symmetries but to get further, we have to make lucky guesses about differential equations (that have the observed symmetries).

Adding T changes the geometry of spacetime and geometries imply symmetries. When you add an extra time dimension T, you end up having to modify Dirac's gamma matrices. That is, there are normally four gamma matrices because there are 3+1 space time dimensions. When you go to 3+1+1, you naturally end up with 5 gamma matrices. String theorists do similar things.

This all suggests that we should look at the density matrix states in the Clifford algebra C(4,1). This is done in a paper I wrote a few years ago called "The Geometry of Fermions" (which string you can google to get a copy). The theme of the paper is "count the hidden dimensions using Clifford algebra and density matrices". The density matrix states correspond to the primitive idempotents of the Clifford algebra.

That paper was written from a slightly different point of view (the extra time dimension has to do with proper time, more or less), but as far as counting dimensions it works out the same. Your paper uses imaginary time; what I'm doing in that paper is related. It's also related to the "Euclidean relativity" work by various people. There are complications associated with the choice of which parameters you assume contribute to the geometry but it's too complicated to discuss here.

A short description of the particle content of the density matrices of a Clifford algebra is that you get N particles for a theory that needs NxN complex matrices in its representation. N is 2 to some integer power k. The N particles have properties that can be described as +-1 for k different parameters. For the Dirac gamma matrices k=2 and the two properties can be chosen to be particle / antiparticle, and spin up / spin down. This is just Clifford algebra and density matrices, no need to assume any symmetries or anything else; it all follows directly from density matrices and geometry.

All this implies a preon structure for the elementary particles which is further explored in various other papers I've done. Most recently, Marni Sheppeard has been helping me with the CKM and MNS matrices.

After this contest ends (probably with my essay winning nothing), I'll type the essay into a more complete submission to Foundations of Physics maybe.

Hello Carl,

The 120-cell defines the icosians which are the D_8, which with the 128 half-spinor part gives the CL(16), which embeds E_8. The icosians are a system of quaternions, gamma matrix valued elements. So in part I would agree with you. This defines an error correction code [4,2,2], which is an elementary (if you call E_8 elementary) quantum error correction code. The physical point is that q-bits which are "processed" by an instanton of the gravity field, such as a black hole, pass through the information channel completely preserved, but encrypted in a form which is difficult to cypher.

At this stage I would agree that working out explicit irreps of groups and the like is probably secondary. What I am more interested in is a "proof of existence," or maybe more like a demostration of applicability. What particular irrep the E_8 takes is at this time secondary, where of course this gets into the Leech lattice and irreps are practically impossible to find.

To be honest I think that to find appropriate irreps of these large groups with huge numbers of irreps, such as the E_8 with 60,779,787 some sort of quantum computation might be required. The quantum computer would then find the minimal energy or configuration for a wave function set over all possible irreps. I will need some time to think about his, but some quantum computation over the root space of Kazhdan-Lusztig-Vogan polynomials, thought here as eigen-numbers might result in a heirarchy of irreps, where some extremization principle might give the irrep appropriate for physics.

The quantum groups I outline are defined on quaternions (noncommutative quaternions) and the overlap is for different quaternions under different representations or bases. So this connects in some way with your ideas about using Clifford algebra.

I suppose I will also rewrite my paper as well and try to publish it. I will pretty clearly not win. I entered late anyway, for I thought the deadline was Jan 1, and then was informed it was Dec 1 around Thanksgiving. So I wrote it up in about three days.

Cheers,

Lawrence B. Crowell

Lawrence, interesting about the error correcting codes. My cohort, Marni Sheppeard, talks about this a lot. It comes up in the context of "mutually unbiased bases" which is how she got a postdoc at Oxford's quantum information group that starts in January. That is, she's the only person on the planet who knows MUBs and category theory. The MUBs come from my application of them to the Koide mass formulas. Eventually I'll get around to publishing this, but I'm hoping to get more stuff complete first.

Right now I'm messing around with Gullstrand-Painleve coordinates and writing the gravitational force of a black hole as a series in powers of the radial distance. (GP coordinates do this exactly with a finite number of terms and so are kind of interesting.)

On the subject of E8, it arises naturally from density matrices in a way that can be described in a few paragraphs.

Density matrices are operators and so they can act on states. When you make a bound state by combining a number of particles which are individually represented by density matrices you are making an operation on the general set of density matrices M that is a mapping. That is, "M" means all possible density matrices of all possible symmetries.

There is a peculiarly natural way to describe a bound state built from density matrices and that is to assemble them into a matrix form. In doing this, you have to make a minor generalization of density matrices to non Hermitian density matrices. These sorts of density matrices represent states where the outgoing and incoming state differ. They can be represented by products of the usual Hermitian density matrices.

Anyway, E8 is the only group whose algebra has symmetry given by the group. That is, E8's algebra has symmetry E8. Now when you consider a matrix of density matrices, what you are describing can also be considered as a symmetry operation on the density matrices themselves.

This is better described in a series of posts on the subject that can be located by googling for

density+matrix+E8+bound+state

and clicking around on references. It's about a half dozen posts in total.

The reason it takes so much is that it requires rethinking the concept of "quantum state" quite a bit. But the mathematics is very simple and straightforward. And to help comprehension, I put in a lot of examples. The thing I don't like about it is that the E8 symmetry comes from the assumption that the density matrix represents the bound state exactly. Of course this is not at all exact and one gets a broken E8 instead. It seems to me that it's better to try and understand the exact bound state instead of a bad approximation.

Sorry it took some time to get back. I looked at some of the references on density matrices. The identification with projectors or idempotents is interesting. Yet I am not sure how they can be identified with primitive idempotents.

I am not sure how this connects with associators and MUBs. Yet I think that systems of quaternions in an octonionic system define groups g and that g^{-1}Ag = g^{-1}g' for A an associator map between these groups. This defines for exterior products of elements (states) g in a quantum group density matrices, which are defined across in commensurate quantum groups for associators. I looked at MUBs about a year ago, so I can't say with any certainty how this connects with MUBs. This does have connections with Hadamard matrices however, which as I recall are utilized in MUBs.

There is another element to what I am thinking which are projective varieties and Goppa codes. These are codes on elliptic curves and varieties. Since they work for projective varieties these seem to have connections with null congruences (light cones, Robonson congruences, horizons etc). What I have been attempting so far with little success is to find connections between Goppa codes and some elliptic curve conditions with norms of cyclotomic rings of quaternions.

I think there is some sort of connection between Golay codes, which really work on Euclidean lattices, and Goppa codes which have properties similar to Lorentzian systems with Zariski point-set topological moduli.

Cheers,

Lawrence B. Crowell

The notion of primitive idempotents isn't that common in the literature. You have to be a bit of a conosewer to run into it. I'm sure there's references in the literature as it's kind of obvious, if you spend enough time playing with the things. My website, www.densitymatrix.com has a connection to Frank Porter's (Cal Tech) class notes on quantum mechanics. If you click on that link, you will see a link labeled "Physics 125c Course Notes Density Matrix Formalism" or similar. Page 11 of those class notes will verify what I've said about "Hermitian primitive idempotents" being a characterization of pure density matrices. Non Hermitian density matrices are even more rarely discussed than the primitive idempotents and pure density matrices. I seem to recall seeing some reference at a conference on non Hermiticity.

Re connection between pure density matrices and MUBs. The powerful thing about MUBs is that they define a set of quantum states that interact. The usual way one must define a set of mathematical objects (spinors) to represent the fermions, and then define a set of stuff that allows them to interact. It is in defining these TWO things that it is inevitable that you have to choose a symmetry. With states chosen from a complete MUB set, the interactions are already built into them.

The interactions between MUB basis states are trivial in that they all have the same amplitude, but the phases are not completely trivial and this gives them a structure which can be analyzed. As an example of the non triviality of the phases of pure density matrices chosen from the simplest (Pauli algebra) MUB, consider the following product of pure density matrices:

(1+x)/2 (1+y)/2 (1+z)/2 (1+x)/2

where "x", "y", and "z" stand for the three Pauli spin matrices. You will get a complex multiple k of the pure density matrix (1+x)/2. The complex phase that gets picked up in that product is a Berry-Pancharatnam or quantum phase, not one of the arbitrary complex phases that confuse spinor calculations. The magnitude of k is 1/8 because there are three transitions between quantum states with transition probabilities of 1/2. The phase of k is -pi/4 or +pi/4, I forget.

And what does the above product represent physically in quantum mechanics? Because these are operators, and because they are written in geometric language i.e. x, y, z, they have more meaning than the equivalent spinor calculation which is:

( +x | +y ) ( +y | +z ) ( +z | +x ). I hope that parsed.

The meaning of the product "(1+x)/2 (1+y)/2 (1+z)/2 (1+x)/2" is:

(a) It is a product of projection operator for a quantum state; this is a history of a particle. See the interpretation of quantum mechanics known as "Consistent Histories" for more.

(b) It represents a sequence of measurements of a quantum state. See Schwinger's "Measurement Algebra" for more on this. His papers are linked at another one of my websites, www.measurementalgebra.com.

(c) It represents a sequence of four Stern-Gerlach filters set up so as to allow only spins oriented in the x, z, y, and finally x directions to pass. That is, this is a representation of a sequence of four polarizing filters. The phase is the phase that a particle passing all four filters would possess. From a field theory point of view, each Stern-Gerlach filter is a source of gauge bosons that interact with a beam of fermions. The point here is that with a Stern-Gerlach filter the physicist is not thinking of the effect of the beam on the filters, but only the reverse effect, how does the filter act upon the beam.

(d) The individual terms like (1+x)/2 represents the field that is present in the Stern-Gerlach field. That is, a Stern-Gerlach filter oriented in the +x direction has a magnetic field that is inhomogeneous in the +x direction. This is a hint that there is something going on here that has something to do with gauge bosons and the geometry of space-time.

(e) Finally, most important to me, and getting back to the MUB theory, the product represents a spin-1/2 fermion that emits three gauge bosons (in the quantum information limit where we just keep track of 1 quantum bit for the particle and don't keep track of what happens to the object that absorbed the gauge boson). The first boson emitted is one that converts a (1+x)/2 fermion to a (1+z)/2. The second converts a (1+z)/2 to a (1+y)/2, etc. As such, each of these products, i.e. "(1+x)/2 (1+z)/2", represents a gauge boson in the quantum information limit. You can think of these sort of like the six off diagonal gluons; so a red/green gluon converts a green to a red.

The gluon analogy can be taken literally. With the Pauli MUBs, there are a total of six states (1+x)/2, (1+y)/2, (1+z)/2, (1-x)/2, (1-y)/2, and (1-z)/2. These are just enough to represent red, green, blue, anti-red, anti-green, and anti-blue, respectively. Instead of representing the color states as orthogonal states with a gauge boson that interacts between them, I'm representing the color states as states that are not orthogonal; the non orthogonality automatically defines the interaction.

This use of MUBs gives a derivation of Koide's mass formulas for the hadrons and is the subject of my the paper it appears I will release first. I've finished off the mesons this morning and have just started classifying the baryons. Tommaso Dorigo has kindly undertaken to waive the submission fee at Phys Math Central.

This (e) interpretation is the one that is necessary to use to see how E8 falls out of the density matrix formalism. Calculations where we ignore the gauge bosons are not that uncommon in QM. An example is the bound state of hydrogen and Schroedinger's equation. We think of the electron as emitting photons without thinking (at first order) of how those photons are absorbed by the nucleus. This simplifies the problem to one where we watch only one of the participants in the dance. You can do the same thing with a meson, whatever is emitted by one quark has to be absorbed by the antiquark. (Of course some gluons are reabsorbed by the quark but we just sum over those and look at the quark only in terms of what it exchanges with the antiquark.)

It is in ignoring what happens to the gauge bosons that you end up with E8 from density matrix formalism. The matrices you work with have fermions down the diagonal and the gauge boson interactions off diagonal. This is simpler than it probably sounds. It's just consistency relations; there has to be just as much stuff becoming red as there is stuff that used to be red becoming something else. The E8 is only approximate because the assumption that you can ignore the gauge bosons is only approximate.

From what I can recall, the Hadamard matrices come up when you try and find complete sets of MUBs. So far, I understand that as a purely mathematical endeavor and don't have an interpretation of it. I would guess that if I was hanging around more with Marni Sheppeard she'd eventually get me to see it differently.

Part of my problem with associators is that I don't see how to give them a physical interpretation. Getting back to (c) above, the product is interpreted physically as a sequence of polarization filters. For this, clearly associativity applies and the associator is trivial.

While the octonions are not associative, my understanding of them is that they are almost associative in the sense that one only picks up a negative sign when you fool around with the parentheses on a product of basis vectors. That smells to me like something that will go away when one converts from a state vector / spinor representation to a density matrix representation. So in that sense, my feelings about octonions is better when they are used for state vector representations than density matrix as I can see a way of eventually rescuing them with a direct physical interpretation.

I will comment more later on the MUB issue. What you indicate is a standard aspect of associator rules for basis elements. For e_i e_j and e_k associator tables usually have e_i(e_je_k) = -(e_ie_j)e_k and the sum of these defines some other element. The quantum group G has elements g = exp(ia*e) (a is small) and the associator A = exp(ia*e') is such that g^{-1)Ag is an associated product. The Baker-Campbell-Hausdorff expansion gives commutators of the elements plus associators. The associators are between lattices which tesselate the manifold, where the lattice is E_8 and the group is E_8. The miracle of E_8 is that the symmetry of the root space is that of the group.

The physical idea is that this is a nonunitary transformation of quantum elements which still preserves quantum bits. A density matrix of such states when course grained gives thermal distributions seen in Hawking radiation.

I will get back to MUBs later. It has been a while since I have looked at that topic.

Cheers,

Lawrence B. Crowell

Lawrence,

I've got a new paper out. Well, I'm basically asking friends to review it before I submit it to Phys Math Central. It's about the masses of hadron excitations. Kind of like Regge trajectories, but about radial excitations instead of angular momentum. Anyway, it's also about MUBs and has an introduction to them in the background section. Right now, it's here:

http://www.brannenworks.com/koidehadrons.pdf

After this I'm writing a joint paper with Marni Sheppeard that uses the same methods to do the quark and lepton mixing angles.