Is the wavefunction ontological or epistemological?
Part 4: Building a Hilbert Space Intuition
Hilbert spaces are the natural arena of quantum mechanics
and their dimension can be either finite or infinite. As late Sidney Coleman
will joke, I intended to draw a 72 dimensional Hilbert space, but due to
budgetary constraints I can only draw the case of dimensionality three. However
this will turn out enough to understand the peculiarities of quantum mechanics.
As a reference for this post, I recommend this paper: http://arxiv.org/abs/quant-ph/0007041v1
by Diederik Aerts and collaborators and I will attempt to explain why the logical
operators “OR” and “NOT” are different in quantum mechanics.
As a physical theory subject to experimental verification,
quantum mechanics generates sets of propositions about a (quantum) physical
system, and those propositions are formally identified with projection
operators in a Hilbert space. Technically, those propositions generate an order
structure: a family of complete, separable, orthomodular, atomistic with the
covering property lattices (isn’t this a mouthful?). The reverse property stands as well, and Piron
was able to reconstruct quantum mechanics’ Hilbert space starting from those
properties. Diederik Aerts studied under Piron and his main research interest is
in composite systems: http://www.vub.ac.be/CLEA/aerts/
In general, a Hilbert space is a generalization of the usual
Euclidean space and its main property is the existence of an inner product. By definition,
a Hilbert space is defined over complex numbers, but it can be generalized over
any number system in which case it is called a Hilbert module. Since it is well
known that quantum mechanics can be expressed over reals, complex, or
quaternionic numbers, we can use the usual 3D space as an example of a three dimensional
Hilbert space (over real numbers) and try to understand the quantum mechanics
postulates in this case (the inner product would be the ordinary dot product). The
obvious advantage is that we can visualize right away what is going on.
In ordinary three dimensional Euclidean space, a point is
specified by three numbers: x,y,z, but in quantum mechanics, a point would
represent a particular quantum state. Statements (propositions) about the
quantum system correspond to projections on points, lines, and the entire
space. We can understand points, lines, and planes as quantum states
corresponding to either experimental preparation procedures, or to experimental
outcomes (collapsed subspaces).
So now let’s take a look at the picture below:
Suppose “State A”
it is a subspace of the original Hilbert space corresponding to a projection
P_a and suppose we can represent this as the vertical line in the picture. The projection
P_a would mean that measuring the observable “A” would yield the outcome “a” (“A”
could be the position, momentum, spin operators, etc).
The orthogonal complement of State A will be the horizontal plane called “Not A”, itself a Hilbert subspace corresponding to another
projection.
Let us say that the QM system is the green dot and measuring
“A” on it would yield the “a” value. Since the green dot is not on the “Not A” plane, this means that testing
for the P_{not a} projection would mean that the answer is false. Hence, if a is true, not a is false.
Now suppose the quantum state is prepared in the Hilbert
subspace Not A
The same game can be played in reverse and nothing out of
the ordinary happens.
But how about the case below?
Now the state is neither on “State A” nor on “Not A”.
Hence, we can state this: “the quantum
entity is in a state such that a is
not true, without not a being true”.
And this is not the case in Boolean logic. Quantum NOT has a nonclassical
behavior.
A similar game can be played with quantum OR:
In this case the green dot does not belong to either subspaces
A or B but it belongs to the closure of the union of the two subspaces (the
entire 3D space). Therefore: “a
∨ b
can be true without a ‘or’ b being true.” It is easy to see
that if the green dot belongs to either A
or B subspaces, it belongs to A∨B.
This means that if a ‘or’ b is true it follows that a
∨ b
is true.
Now what really happens in a Hilbert space is that in QM the ordinary Venn
diagrams which illustrate the boundary of finite sets are no longer confined on
a plane!!! In fact, the logic of quantum mechanics is the logic of projective
spaces and the Venn diagrams acquire an n-dimensional border.
The lack of distributivity property (in QM we have orthomodular lattices and not Boolean algebras) can be explained geometrically by the fact that Venn diagrams for QM are n-dimensional objects no longer confined to a 2D plane.
In several quantum mechanics theorem (like Gleason’s theorem
http://en.wikipedia.org/wiki/Gleason's_theorem),
one basic assumption is that the Hilbert space dimensionality is higher than two.
The reason is that in two dimensional Hilbert spaces, Venn diagrams can be
confined in a plane and classical explanation of quantum effects can be easily
obtained.
Recalling the three circles in the prior Bell Theorem post,
it is clear now how the Bell
inequality is violated: the inequality was based on the classical physics
requirement that Venn diagrams must be represented on a 2D plane (which is not
the case in QM)
We can now also “visualize” the famous double slit experiment
(http://en.wikipedia.org/wiki/Double-slit_experiment
) question: which slit did the particle go through? This is a direct physical
illustration of the nonclassical OR behavior pictured above: one slit
corresponds to subspace A, the other to subspace B, and the quantum state is
located on neither one. Finding out “which way information” amounts to
projecting the state to either A or B and hence destroying the interference.
So QM is still strange, but at least now we can create a
mental picture of the Hilbert space geometry and we can visualize the meaning
of various famous QM results.
Next time I am going to use this technique on a recent
result called PBR which claims to show that
the wavefunction is not epistemological. Did PBR
actually succeed in proving this very strong claim? Yes and no. Please stay
tuned.
Florin,
ReplyDeleteI have been able to finally return to reading papers and your blog. BTW, the latest entry on your blog causes my machine to go into fits. I don’t know if this computer is trying to beat Bell theorem or not . I think some people trying to argue against Bell are invoking the fact that Bell is on a plane of simultaneity which is just a special frame in spacetime. Of course the KS theorem in 4-dim covers this with respect to contexuality (noncontextuality), but this is a different result. There is I think a bit of a hole in the q-foundation picture, though I don’t think this unravels Bell theorem.
I have been looking at you paper Unitary realization of wavefunction collapse, http://arxiv.org/pdf/1305.3594v1.pdf. This Grothendieck group realization is a K-theory approach. I think this is related in some way to the F4 for the KS theorem; which is one reason I sort of mentioned it above. The K4 is the stabilizer group for the G2, which in turn is the automorphism group of E8. I have this crazy idea that QM and GR are aspects of the same thing. Susskind is saying something similar with his EPR = ER. This equivalency occurs in the general SUGRA group of symmetries. I think what you are saying in effect is that there is a change in the wave function or collapse is an isomorphism between classes in the Gr-group G(R). I think this is related to G2 transformations which map F4 --- > F4 (F4 is preserved or a “constant of the motion”), but that a naïve interpretation of the data does not take this into account.
I’ll have more to say later as I digest this further.
Cheers LC
Lawrence,
DeleteI set the default script settings low at 20 and 400 to avoid trouble with slower browsers. What browser are you using? The script is running locally on your PC and is eating lots of memory for large input parameters to the point it will freeze your browser. Best to view it on Chrome.
On the EPR=ER idea, I think it is just junk (approaching it from the QM point of view). However, there is truth in unifying special relativity with QM because of the spin factor realization of Jordan algebras.
I have a crazy idea of having a different kind of QM inside a black hole, but I cannot either refute or prove it for now. The hypothesis is that next to the singularity QM changes from elliptic QM to hyperbolic QM but in this case there are no Hilbert spaces and the known math tools are ineffective because there are no metric, norm, or inner product spaces in the usual sense. The root cause is that the usual triangle inequality for complex numbers gets replaced by a reversed triangle inequality for split-complex numbers and any Cauchy sequence is no longer guaranteed to have a unique limit. I am reading functional analasys books to try to sort out the math.
Looking forward to your feedback.
best,
Florin
Florin,
ReplyDeleteI think you are being too hard on your self. With a black hole the size of the Hilbert space for states (harmonic oscillators) that make up the BH is by the Bekenstein bound on entropy S = k S_{bit} and
S_{bit} = A/2L_p^2 for L_p = sqrt{Għ/c^3}
given by dim(H) =~ e^S, where S is just B_{bit}. Entropy is really just #qubits/symbol string. However, these states are entangled with Hawking radiation which is emitted and this is another Hilbert space with dim(H) = e^S for S = A/4L_p^2. The total Hilbert space for the system is then dim(H) = e^{2S}. There is then a huge H-space of states that the system can evolve into, which leads to this issue currently waging on so called firewalls. What you indicate with the hyperbolic approach is indicative of the problem. This in the case of moduli space means that gauge connections do not converge in a Cauchy sequence.
However, this really is a matter of a massive redundancy. If you have a pure entangled system the information between a and b in regions A and B is
I(a,b) = S(a) + S(b) - S(a∪b)
The information is always positive and and vanishes only if the density matrix for a and b is a direct tensor product (non-entangled) so that S(a∪b) = S(a) + S(b). This means the mutual information between two regions decreases as their entanglement vanishes. The mutual information for operators in these two regions A and B for systems a and b is then
I(a,b) >= ( - )^2/()^2
As the regions A and B become decorrelated the expectation approaches zero. This overlap will decrease due to the formation of an event horizon. The expectation evaluates
O_aO_b = (1/2)[O_a, O_b] + (1/2){O_a, O_b}
where the commutator (same holds for anticommutator if fermions are relevant) is thought to be evaluated along geodesics which connect the two regions. Event horizons preclude such and the expectation vanishes over the scale connecting regions A and B. However, this is still evaluated in a small region or neighborhood which contains the horizon, where again this horizon separates regions A and B. All of that entanglement information is ultimately contained on the horizon, or a small neighborhood of the horizon. In the case of a black hole this would be a thickened shell. As such the huge Hilbert space with dim e^{2S} is an artifact of massive redundancy, and the physically relevant Hilbert space is still of dim = e^S.
The apparent massive loss of information is due to the fact that we have a isomorphism class of a huge number of state spaces that are formally equivalent. This is where the Grothendieck group enters into the picture. The state space is determined by a set of generators A in a group g, and there is a set of modules M such that for g/M the Hilbert space of dim = e^{2S} is reduced to the Hilbert space of dim = e^S. The set of transformations ψ --- >g^{-1}ψg is then over a set of equivalent or redundant states which can be reduced to a small Hilbert space with reduction which "mods-out" the modules M.
This moding out is essentially the formation of a type of macrostate, which we see is just a redundancy. This is I think an artifact of the automorphism of the gauge group, such as the Jordan matrix system --- an automorphism of the Fischer-Greiss monster group, or more selectively the g2 automorphism of E8. The existence of the horizon is a manifestation of a transition from elliptic properties to hyperbolic for by this process and the formation of horizons spacetime is being constructed from entanglements of fields with certain gauge symmetries. All of that hyperbolicity is ultimately a sort of illusion.
Cheers LC
The third equation did not come out right. I exchanged the carrot symbols for tex commands.
DeleteI(a,b) >= (\langle O_aO_b\rangle - \langle O_a\rangle\langle O_b\rangle)^2/(\langle O_a\langle\rangle O_b\rangle)^2
Cheers LC
Lawrence,
ReplyDeleteWith the BH stuff you are talking over my head and I am out of my league there. I don't really understand this new firewall stuff, and common sense dictates me that there is nothing there, especially for massive black holes of minuscule Hawking temperature.
The Grothendieck group is basically a way to introduce an inverse operation making the tensor product commutative monoid into a group. Since the sole characteristics of a Hilbert space is its dimensionality, for finite dimensional cases, the inverse operation corresponds to the collapse postulate.
About QM and Hilbert spaces, I'll write more posts about very interesting stuff like rigged Hilbert spaces (the Hilbert space done right), Choi-Jamilkowski isomorphism and Riesz representation theorem which shows that there is an additional Hilbert space in the operator algebra.
Best,
Florin
Florin
ReplyDeleteThe firewall does not exist, but showing that it does not exist conclusively is a bit of a bugger. The problem centers around the fact that an entangled state does into swap into another without demolishing the first entanglement. The tensor product of two states with dim = N result in a space with dim = N^2. In an entanglement though the actual states which are occupied are of dim = 2N. A black hole is a system where there is an entanglement between the states on the horizon, as seen by a distant observer, and Hawking radiation states. The states on the event horizon are all the states which compose the BH. The problem is that an observer who enters a black hole witnesses these states in an entirely different way. These states are in the interior and they are on the singularity and are correlated with states this observer can witness up to the horizon. The two observers can hold an EPR pair. It is not possible for say Bob in the exterior to receive a copy of Alice's pair in the interior by jumping into the interior. In order for this to happen Alice must send her signal in a time interval t ~ 1/GM^2 after entering the horizon so the signal will "hug" near the horizon before plunging into the interior. In order to do this the clock neads to operate on an energy scale ~ GM^2, which is so large it would perturb the BH and make this process impossible. Hence the EPR in a BH can't lead to a cloning result. This means that quantum states interior to a BH are entangled with those exterior.
The problem is that we have a multiple form of entanglement. The exchange of one entanglement for another involves the demolition of the first. This is seen with the Hadmard operations in quantum teleporation which exchanges the EPR state with another. It is not possible to take two entangled states and further entangle them in this manner, which is the quantum monogamy result. The problem is this means that any entanglement between the interior and exterior states will over time result in the demolition between states on the horizon and Hawking radiation. When this happens it means a number of conditions are violated. These are field locality, the equivalence principle, or unitarity. If the equivalence principle is abandoned this is the firewall, where the horizon is over time converted into a singular condition.
Given this hypothesis, the firewall will occur or begin to significantly demolish entanglement fidelity in the exterior for an ordinary black hole until a a few percent of its lifetime. So for a stellar mass black hole the fire wall would not be a significant situation for at least 10^{65} years. Given that the observable universe is 1.4x10^{10} years old no black hole we observe will have a fire wall. The demolition of such entanglements between horizon states and Hawking radiation will be infinitesimal. For a quantum black hole with a short lifetime though it is possible to test whether this is the case. For a black hole that is a few Planck units of mass, say <~ 10^5 M_p, the entanglements between the interior and exterior states should be measurable (in principle) and with this whether or not a firewall emerges. A complementary experiment could be performed to measure whether the BH states are entangled with the product states of its decay.
continued
The fire wall means that the total Hilbert space e^S*e^S becomes potentially occupied by physical states, which is considerably larger than 2e^S. The state space is determined by a set of generators A in a group g, and there is a set of modules M such that for g/M the Hilbert space of dim = e^{2S} is reduced to the Hilbert space of dim ~ e^S. This operation I think would mean that the collapse of the wave function involves a massive redundancy. The isomorphisms under the set of modules would mean that fundamentally there is not quantum information lost in the collapse, or what appears to be a collapse. Black holes enter into this picture because Hawking radiation in the semiclassical limit is in mixed quantum states, and there has been long observed to be a similarity between the concealment of information in black holes, black hole thermodynamics and the problem with measurement in quantum mechanics.
ReplyDeleteCheers LC
Thanks Lawrence,
ReplyDeleteThe way I think about this situation is as follows: because of the singularity, maybe the no cloning theorem does not apply inside the BH (and no observer can actually notice that), and also the Hilbert space does reduce its dimensionality resulting in information erasure. Then on the BH outside, the information is always conserved and Hawking radiation is entangled with whatever falls in.
Sure, it is a speculation, but here is the line of argument I will investigate. The Hilbert space is not a single space, but a sandwich of 3 spaces (http://en.wikipedia.org/wiki/Rigged_Hilbert_space) and the outside space which corresponds to distributions may be shrunk by the singularity reducing the original Hilbert space dimensionality near the singularity also (how can you have test functions when space-time itself vanishes at the singularity?). If true, information is not conserved inside the BH and if this is the case, the no clonning theorem does not apply either.
I have to do the hard math after I understand better the spectral theory to see if this speculation pans out.
Florin,
ReplyDeleteIt is plausible that a rigged Hilbert space could be involved with the sort of dual properties, or entanglements between inner and outer spaces.
The physical problem I can maybe see with this idea of a different QM that diminishes information inside a BH is when the BH becomes quantum mechanical. For a BH with a small number of Planck units of mass the scale of horizon fluctuations approaches the classical scale of the horizon radius. For this scale of BH state interior and exterior may be entangled in such as way that the exterior observe has some uncertainty as to whether they are measuring states in or out of the BH. This would tend to imply that the destruction of information interior to the BH results in the destruction of information outside.
The uncertainty with the event horizon goes back to the Bohr-Einsten debate. Einstein argued that a box on a scale could be used to beat the uncertainty principle. Assume there is a box with a hole in one wall covered by a shutter which may be opened and closed by a clock mechanism inside the box. Radiation in the box would add to the weight of the box. The box would be weighed and then at a given moment the clock would open the shutter allowing a single photon of radiation to escape. The box could then be re-weighed, the difference between the two weights telling us the amount of energy that escaped using the formula E = mc^2. Under the uncertainty principle it is not possible to obtain an exact measurement of the energy of the released photon and the time at which it was released. Einstein’s experiment was designed to show such exact measurements were possible, the clock measuring the time of release of the energy and the weighing of the box disclosing the amount of energy involved. The gravity field for a change in potential ΔΦ = gΔh is such there is an uncertainty in the height of the box on the scale and that ΔhΔp = ħ/2 for Δp ~ mc. This defeated Einstein’s attempt to work around the uncertainty principle.
The box could then be a black hole and the quantum tunneling of quanta out of the black hole. The Schwarzschild metric element g_{tt} = 1 – 2GM/rc^2 = 1 – 2Φ is defined for a mass contained in a radius R_s = 2GM/c^2. The uncertainty in the mass of the black hole is then a fluctuation in the metric element
Δg_{tt} = 2GΔM/rc^2 = ΔR_s/r
The gravity potential fluctuates with an uncertainty ΔΦΔt = ħ/2, and for r = R_s we have that
ΔRΔt = Għ/c^2,
which gives a space-time complementary fluctuation of the radius (horizon radius) and time. This is the radius uncertainty as seen by an exterior observer associated with an uncertainty in time. The term Għ/c^2 is a space-time measure of the area of the event horizon Planck units.
Maybe the issue has to do with the fact that black holes generally have angular momentum or a BPS charge. A black hole with
r_{+/-} = m +/- sqrt{m^2 – Q^2}
where there is an interior region beyond the inner horizon r_-. The extremal condition with m = Q is the lightlike situation, where the normal black hole is with m > Q. However, for the quantum black hole the mass may fluctuate so that a near extremal black hole may in the Regge trajectory fluctuate to the extremal condition. In particle physics this is an off-shell condition. The extremal condition for black hole qubits is they form a nilpotent ring. The timelike condition defines qubits with a more general structure. If the inner horizon and outer horizon have complementary quantum states, or states that form entanglement pairs then the extremal condition is an interesting case. There the two horizons merge and form a type of naked singularity that does not transmit causal information to the outside world.
I do however think the Grothendieck group plays a role in determining how the states occupied in the dim(H) ~ e^{2S} state space is actually 2e^S. This might even connect to the restriction imposed by the extremal condition.
Cheers LC
Lawrence,
ReplyDeleteGood point, I did not consider small BHs.
I don't think the Grothendieck group has anything to do with what you say because for field theory, the cardinality is infinite and the Grothendieck group breaks down (e.g. alef - 7 = alef). For field theory there are other problems and tools available and in some cases there is no Hilbert space (for example due to IR divergences) and there are ambiguities in defining a vacua and selecting one particular representation from an infinite number of them. So field theory is a different beast.
What do you make of my paper attempting to derive QM?
Best,
Florin
Right at the moment I don’t have a lot of time to go into great detail. If I can I might later tonight or the next couple of days. You are absolutely correct that spacetimes involve many vacua. I think gravitation is really a frame bundle system which preserves a vacuum in any local inertial reference frame. I think in fact there is a general principle which involves any frame bundle. The observer at a distance though who has EPR pairs with states on an infalling frame finds there is a decline in the fidelity of this entanglement due to the fact the vacua for the two previously entangled states is changing.
ReplyDeleteThe supergravity vacua is governed by the SO(9), or SO(8,1), group for supergravity. This is given in the short exact sequence
F4: 0 --- > B_4 --- > spin_{52\36} --- > OP^2 --- > 0
The 36 of B_4 connects to the OP^2 or the projective Fano plane in 16 dim. F_4 is the system which illustrates the Kochen-Specker theorem. The E8 group in effect holds lots of F_4’s. The G_2 group is the automorphism of the E_8 group and transformation of G_2 leave F_4 invariant. F_4 is a stabilizer with G_2, so during G_2 transformations it is conserved, or physically it is a constant of the motion. This is similar to a gauge system, but if a theory lacks the appropriate gauge covariant operator things appear to go wrong. All of these vacua are ultimately equivalent in some moduli system. The question is whether the moduli space on the field or gauge level (the transformations of states etc) can result in this Grothendieck construction with states.
Which paper do you try to derive QM? I have read your pages on Bell theorem. It looks as if you want a Randi for challenges such as Joy Christian’s. BTW, he still occasionally appears on the FQXi page with comments.
LC
Lawrence,
ReplyDeleteThe paper is http://arxiv.org/abs/1303.3935 The core idea is the invariance of the laws of nature under tensor composition. From this one can obtain a categorical derivation of QM. I did not write the paper in the language of category theory, but funny enough, 11 days after I uploaded my result, Anton Kapustin from Caltech uploaded a very similar paper written in category theory formalism (and we had the same original inspiration)
The derivation is not complete and I need to derive the C* algebra condition ||T^* T|| = ||T||^2 to nail it for good. I think I am making good progress on this...
On Joy, some time ago, using sock-puppetry he attacked a book by Scott Aaronson on Amazon. Then recently he claimed he obtained a "local realistic" computer simulation of his model and bragged on Amazon's review site for Scott's book. Because I took part in a discussion there I got a notification from Amazon and I discovered that Joy made strong claims at FQXi too. Since I had this Java script done for quite some time, I used it to challenge his bogus claims at FQXi. And the timing was also right on my blog posts where I was talking about Bell theorem anyway.
Florin,
ReplyDeleteI read your paper “Quantum mechanics from invariance laws,” with some attention to details. I found no particular problem with it as far as the mathematics. I agree with your conclusion that quantum mechanics is complex. QM that is quaternion is a “double copy” of sorts, such as the Dirac equation and the relationship between spinors and vectors. The Dirac equation is though really a pair of spinor equations, where the block form of the Dirac matrices according to Pauli matrices indicates this reducible representation. The two spinor valued equations are the Weyl equations.
We can then say that quantum mechanics, even with the Dirac equation, is entirely complex. Where then are quaternions? Quaternions exist as the generators of quantum states. These are field operators which generate unitary transformations of quantum states. This is the nature of equation 3 with the parallel translation by a vector field. The rest of this machinery follows. What you have done is to illustrate how the GNS construction emerges from the properties of the generators of the states. The associator is computed but with the conditions you have on the α and τ the associator is zero for QM states or products thereof.
This is valuable for what I started with the FQXi essay that I wrote for this contest. Towards the end I argue that the axiom which needs to be changed for quantum gravity is the assumption of quantum field locality. This is different from quantum locality as a causality condition on the field amplitudes on a spatial surface of simultaneity. Things go a bit differently if there are event horizons present. At the end of the essay and a bit more in the supplemental section I indicate how an S-matrix is a time (or space) ordered set of fields (generators of q-states) and how the presence of an event horizon gives a nonzero associator.
Again I will try to break this out in greater detail in the next couple of days. You can read my essay at http://fqxi.org/community/forum/topic/1625. This is interesting what you have done. I have to confess that I have been for quite some time interested in the prospect for some sort of nonassociativity in quantum field theory or quantum gravity. The reason is this leads into Mathieu group structure as quantum error correction codes. This I think should be the coup de grace on the problem of information conservation in the universe, in particular with black holes, horizons and quantum gravity/cosmology.
Your take on Joy C’s recent activities is interesting. I have seen in recent days that he appears on the FQXi blog roll for recent entries. I have not gone to the trouble of actually reading his stuff. It is all rubbish I am sure. I think FQXi dropped any future sponsorship of his research on quantum foundations, so he is really reduced to being a sort of blog-meister or guru with his little blog following. To be honest what people seem to not appreciate is that what is really mysterious is how it is the classical or macroscopic world exists! How this gets built up from the quantum substratum of nature is to me a fascinating open question. The quantum world by itself is actually rather simple, but how there is this classical reality, or what we perceive as reality, that gets built up from QM is to me mind boggling.
Cheers LC
Florin,
ReplyDeleteI am reposting this, since it has not appeared. Maybe something went wrong with my posting yesterday afternoon.
I read your paper “Quantum mechanics from invariance laws,” with some attention to details. I found no particular problem with it as far as the mathematics. I agree with your conclusion that quantum mechanics is complex. QM that is quaternion is a “double copy” of sorts, such as the Dirac equation and the relationship between spinors and vectors. The Dirac equation is though really a pair of spinor equations, where the block form of the Dirac matrices according to Pauli matrices indicates this reducible representation. The two spinor valued equations are the Weyl equations.
We can then say that quantum mechanics, even with the Dirac equation, is entirely complex. Where then are quaternions? Quaternions exist as the generators of quantum states. These are field operators which generate unitary transformations of quantum states. This is the nature of equation 3 with the parallel translation by a vector field. The rest of this machinery follows. What you have done is to illustrate how the GNS construction emerges from the properties of the generators of the states. The associator is computed but with the conditions you have on the α and τ the associator is zero for QM states or products thereof.
This is valuable for what I started with the FQXi essay that I wrote for this contest. Towards the end I argue that the axiom which needs to be changed for quantum gravity is the assumption of quantum field locality. This is different from quantum locality as a causality condition on the field amplitudes on a spatial surface of simultaneity. Things go a bit differently if there are event horizons present. At the end of the essay and a bit more in the supplemental section I indicate how an S-matrix is a time (or space) ordered set of fields (generators of q-states) and how the presence of an event horizon gives a nonzero associator.
Again I will try to break this out in greater detail in the next couple of days. You can read my essay at http://fqxi.org/community/forum/topic/1625. This is interesting what you have done. I have to confess that I have been for quite some time interested in the prospect for some sort of nonassociativity in quantum field theory or quantum gravity. The reason is this leads into Mathieu group structure as quantum error correction codes. This I think should be the coup de grace on the problem of information conservation in the universe, in particular with black holes, horizons and quantum gravity/cosmology.
Your take on Joy C’s recent activities is interesting. I have seen in recent days that he appears on the FQXi blog roll for recent entries. I have not gone to the trouble of actually reading his stuff. It is all rubbish I am sure. I think FQXi dropped any future sponsorship of his research on quantum foundations, so he is really reduced to being a sort of blog-meister or guru with his little blog following. To be honest what people seem to not appreciate is that what is really mysterious is how it is the classical or macroscopic world exists! How this gets built up from the quantum substratum of nature is to me a fascinating open question. The quantum world by itself is actually rather simple, but how there is this classical reality, or what we perceive as reality, that gets built up from QM is to me mind boggling.
Cheers LC
PS
ReplyDeleteI just looked at the FQXi blog page “Disproofs of disproofs …,” where JC recently posted something. It does look as if the debate has ramped up there, where you have taken his claims to task. For myself I prefer not to get mired in that again. In my opinion the highest statement of intellectual honesty is contained in the words “I am wrong,” or just as well, “We are wrong.” JC seems unable to do that. I and a friend of mine are considering doing a mod on Minecraft to construct a quantum world. If that works then we will further modify it to look at position dependent communication between logic gates to simulate an event horizon. In this way a working and somewhat visual-virtual world that emulates quantum mechanics and maybe quantum black holes and gravity might serve to facilitate work, and education
Lawrence,
ReplyDeleteI did ponder on quaternionic QM for quite some time (and I read Adler's monograph). From the Jordan algebra side, quaternions correspond to a projective space and in his classic "Octonions" paper Baez explains well how this comes about.
However, this is one of the red herrings in understanding QM. Quaternions represent 2 symmetries: invariance of the state to a phase rotation (and from this one gets the usual complex numbers and a sqrt (-1) = i) and an invariance to time reversal for the case of bosons (I don't recall precisely if is bosons or fermions, I have to look it up). From time invariance one gets the other sqrt (-1) = j and from ij one gets k=ij and in the end quaternions.
Physically there are no experiments which will reveal a phase or a violation of time reversal symmetry. Also since the tensor product of 2 bosons is not a boson, there is no tensor composition in quaternionic QM.
A similar game is played on real QM which corresponds to fermions, and there are some subtleties there regarding computing the expectation values.
On Joy, he got a programmer to model some of his equations as a program and claims the got the correct correlation. Not so fast I say, and I will not let this unchallenged. The physics community already "excommunicated" him and no one is taking him seriously, but he enjoys some residual support from former friends and this is why fqxi did not cut him lose completely.
Lawrence,
Deletehere is the reference for quaternionic QM: "Geometry of Quantum States" by Ingemar Bengtsson and Karol Zyczkowski, page 137.
Let N be the number of QM system and Theta the time reversal operator. By Wigner theorem, Theta can be unitary on anti-unitary. We demand = and Thera^2 = +/- 1.
If N is the dimesion of the QM state:
N odd , bosons -> Theta^2 = +1
N even , fermions -> Theta^2 = -1.
The observables commute with Theta: [O, Theta] = 0 making this a superselection rule. The observable restriction (the superselection rule) restricts the state space to be the base manifold of the fibre bundle whose bundle space is CP^{2k+1} and whose fibres are CP^1 This results in quaternionic QM. Hence quaternionic QM is complex QM with a superselection rule.
The superselection rule is applied to decompose the QM into irreducible representations. In that sense it means that quaternion QM is really a sort of double copy. An interesting form of this is with twistor theory that expresses a quaternion a + ib + jc + kd in a matrix
Delete|a + bi c + di|
|-c + di a – bi|
where the i, j, k in the quaternion are Pauli matrices in this matrix form with i = sqrt{-1}. This gives an ambispinor theory, which is again a sort of doubling.
There is a Grothendieck form of the Riemann-Roch theorem. I think this is an interesting thing to study. I don’t understand it yet in completeness. This does segue into your idea about the structure of QM resulting from the product of states.
Cheers LC