It From Bit is Undecidable by Lawrence B Crowell

As a rule when somebody tells me that something is too abstract or is overly difficult I tend to translate that into, "This is to difficult for me, therefore it must be false." That is not exactly the most appropriate form of reasoning.

Cheers LC

Michel,

I reread your paper again this last Sunday. The desin d'enfant leads at the end to Mermin's pentagons. These are of course an aspect of the Kochen-Specker theorem. This is of course the main theorem on contextuality in QM. In my paper I discuss the quantum homotopies of associators at various dimensions, which are pentagonal systems. I copy this post on my essay blog page, so you can respond to this there as well.

I notice you have considerable interest in the G_2 group, which is the automorphism of the E8 group. The F_4 group is a centralizer in E8, whereby G_2 action keep it fixed; the elements of F_4 and G_2 commute.

The Kochen-Specker theorem is connected with the F_4 group, or the 24 cell. The 117 projectors with the original KS theorem in 3-dim Hilbert space is simplified by considering a four dimensional Hilbert space, or a system of 4 qubits. This involves only 18 projector operators. The space 24-cells is a system of root vectors for the F_4 group. Each root vector is paired with its negative to define a line through the origin in 4d space. These 24 lines are the 24 rays of Peres. The root vectors are

1 (2,0,0,0) 2 (0,2,0,0) 3 (0,0,2,0) 4 (0,0,0,2)

5 (1,1,1,1) 6 (1,1,-1,-1) 7 (1,-1,1,-1) 8 (1,-1,-1,1)

9 (-1,1,1,1) 10 (1,-1,1,1) 11 (1,1,-1,1) 12 (1,1,1,-1)

13 (1,1,0,0) 14 (1,-1,0,0) 15 (0,0,1,1) 16 (0,0,1,-1)

17 (0,1,0,1) 18 (0,1,0,-1) 19 (1,0,1,0) 20 (1,0,-1,0)

21 (1,0,0,-1) 22 (1,0,0,1) 23 (0,1,-1,0) 24 (0,1,1,0)

(I hope this table works out here) Consider these as 24 quantum states |ψ_i>, properly normalized, in a 4 dimensionl Hilbert Space e.g. it might be a system of two qubits. For each state we can define a projection operator

P_i = |ψ_i)(ψ_i| --- I have to use parentheses because carrot signs fail in this blog.

P_i are are Hermitian operators with three eigenvlaues of 0 and one of 1. They can be considered as observables and we could set up an experimental system where we prepare states and measure these observables to check that they comply with the rules of quantum mechanics. There are sets of 4 operators which commute because the 4 rays they are based on are mutually orthogonal. An example would be the four operators P_1, P_2, P_3, P4.

Quantum mechanics tells us if we measure these commuting observables in any order we will end up with a state which is a common eigenvector i.e. one of the first four rays. The values of the observables will always be given by 1,0,0,0 in some order. This can be checked experimentally. There exist 36 sets of 4 different rays that are mutually orthogonal, but we just need 9 of them as follows:

{P2, P4, P19, P20}

{P10, P11, P21, P24}

{P7, P8, P13, P15}

{P2, P3, P21, P22}

{P6, P8, P17, P19}

{P11, P12, P14, P15}

{P6, P7, P22, P24}

{P3, P4, P13, P14}

{P10, P12, P17, P20}

At this point you need to check two things, firstly that each of these sets of 4 observables are mutually commuting because the rays are othogonal, secondly that there are 18 observables each of which appears in exactly two sets.

Now assume there is some hidden variable theory which explains this system and which reproduces all the predictions of quantum mechanics. At any given moment the system is in a definite state, and values for each of the 18 operators are determined. The values must be 0 or 1. with the rules they are equal to 1 for exactly one observable in each of the 9 sets, the other three values in each set will be 0. Consequently, there must be nine values set to one overall. This leads to a contradiction, for each observable appears twice so which ever observables have the value of 1 there will always be an even number of ones in total, and 9 is not even.

To add another ingredient into this mix I reference , which illustrates how the Kochen-Specker result is an aspect of the 24-cell. The 24-cell has a number of representations. The full representation is the F_4 group with 1154 Hurwitz quaternions. The other is the B_4, which is the 16 cell Plus an 8-cell, and the other is D_4 which is three 8-cells. The more general automorphism is then F_4. The quotient between the 52 dimensional F_4 and the 36 dimensional so(9) ~ B_4 defines the short exact sequence

F_4/B_4:1 --> spin(9) --> F_{52\16} --> {\cal O}P^2 --> 1,

where F_{52\16} means F_4 restricted to 36 dimensions, which are the kernel of the map to the 16 dimensional Moufang or Cayley plane OP^2. The occurrence of 36 and 9 is no accident, and this is equivalent to the structure used to prove the KS theorem.

F_4 is the isometry group of the projective plane over the octonions. There are extensions to this where the bi-ocotonions CxO have the isometry group E_6, HxO has E_7 and OxO has E_8. This forms the basis of the "magic square." F_4 plays a prominent role in the bi-octonions, which is J^3(O) or the Jordan algebra as the automorphism which preserves the determinant of the Jordan matrix

The exceptional group G_2 is the automorphism on O, or equivalently that F_4xG_2 defines a centralizer on E_8. The fibration G_2 --> S^7 is completed with SO(8), where the three O's satisfy the triality condition in SO(8). The G_2 fixes a vector basis in S^7 according to the triality condition on vectors V \in J^3(O) and spinors θ in O, t:Vxθ_1xθ_2 --> R. The triality group is spin(8) and a subgroup spin(7) will fix a vector in V and a spinor in θ_1. To fix a vector in spin(7) the transitive action of spin(7) on the 7-sphere with spin(7)/G_2 = S^7 with dimensions

dim(G_2) = dim(spin(7)) - dim(S^7) = 21 - 7 = 14.

The G_2 group in a sense fixes a frame on the octonions, and has features similar to a gauge group. The double covering so(O) ~= so(8) and the inclusion g_2 \subset spin(8) determines the homomorphism g_2 hook--> spin(8) --> so(O). The 1-1 inclusion of g_2 in so(O) maps a 14 dimensional group into a 28 dimensional group. This construction is remarkably similar to the moduli space construction of Duff et al. .

Cheers LC

Michel,

I don't have as much time this morning to expand on this, so I will just make this rather brief for now. I will try to expand on this later today or tomorrow.

The three-qubit entanglement corresponds to a BPS black hole. The four qubit entanglement is the case of an extremal black hole. I think there is an underlying relationship between functions of the form (ψ|ψ) = F(ψψψ), an elliptic curve with the cubic form corresponding to the 3-qubit, and the "bounding" Jacobian curve that defines a quartic for G(ψψψψ). This I think is some sort of cohomology.

The G2 I think defines a frame bundle on the E8 which defines the F4 condition for 18 rays in the spacetime version of Kochen-Specker.

As I said I should have more time later to discuss this in greater depth.

Cheers LC

This is a sort of code. The J^3(O) Freudenthal matrix of 3 octonions or E8s embeds the Leech lattice which is a Steiner system S(5, 8, 24) for error correction.

LC

Torsten,

I finally got a little bit of time to write more on what I had mused about a couple of weeks ago. This all seems to center in a way around a type of cobordism with respect to these replacements of handles or Casson handles. The replacement of a circle with a knot suggests a type of theory that involves Hopf links. The trefoil for instance is by the Jones polynomial such that a left - right trefoil equals a Hopf link.

The manifold constructed from the knot K is

M_k = ((M^3\D^2xS^1)xS^1)∪_T^3 ((S^3\(D^2xK))xS^1).

On the left the R^1 in M^4 = M^3xR is replaced by S^1, and we can think of the S^1 as a periodic cycle with a real number line as a covering. Think of a wheel rolling on the real number line, or a spiral covering of a circle. In this setting the crux of the matter involves replacing a circle S^1 with a knot K. Physically this avoids topologies with circular time or closed timelike loops such as the Godel universe. The S^1 to the right of each expression is the embedding "time cycle" and the three manifolds of interest are (M^3\D^2xS^1) and S^3\(D^2xK). In a thin sandwich, a narrow section of spacetime separated by two spatial surfaces, we may think of the bottom spatial surface or bread slice as (M^3\D^2xS^1) and the second one as S^3\(D^2xK). We might further be so bold as to say the bottom surface is a left handed trefoil and there is a superposition of two surfaces, one with a right handed trefoil and the other with two S^1s in a link. There is then a type of cobordism between the bottom slice of bread and the top, which in this case might be a map from (M^3\D^2xS^1) ∪_T^3 S^3\(D^2xLT), for LT = left refoil to (M^3\D^2xS^1★S^1)∪_T^3 S^3\(D^2xRT). There the star means linking.

This is a theory of topology change in spacetime, or of some underlying topological change in topology which still maintains an "overall smooth" structure. This is then a type of topological quantum field theory (TQFT). A TQFT just means a theory that is a quantum field theory up to homotopy. This is a way of looking at fields (eg the knots as Wilson loops of fields) according to the underlying space they exist on. This approach amounts to cutting up the space into pieces, examining the fields there and then looking at the entire ensemble (pieces up back). This then has an underlying locality to it this way. However, the connection between knot polynomials and quantum groups indicates there is also something nonlocal as well.

This conjecture means that TQFT assigns data to all possible geometric element to a space, from a 0-dim point to the full manifold in an n-dim cobordism. For a space of n-dimensions there is a functor F

F:bord_n^f --- > A

For A an algebra. The algebra is the generator of the group G = quantum group. Physically the algebra corresponds to the connection coefficients A which form the Wilson loops ∮A•dx = ∫∫∇•Ada (to express this according to basic physics). This is a sort of Grothendieck topos or category system, which relates a knot group with a cobordism. I conjecture that a complete understanding of this system is a TQFT.

I will write in greater detail later on this, for I have sketched out some of this. Physically (or philosophically if you will) the description of spacetime this way is I think equivalent to a description of TQFT in general. In fact one result of the AdS/CFT correspondence is that a 4-spacetime as the boundary of an AdS_5 is equivalent to 10-dim supergravity. The exotic structure of 4-dim manifolds may then be a manifestation of 10-dim supergravity.

I copied this on my essay blog site, so if you respond to this there I get an email alert.

Cheers LC

My mention of consciousness at the end, in connection with top-down physical or causal theories, was a bit conjectural. In a way I put that in there because I know a lot of people want to hear about consciousness and physics. Call it a bit of self-promotion.

To be honest I don't know what role consciousness has with physcs or the universe. A lot of people think it is a quantum process. I don't know about that honestly. I think it could be argued that consciousness is the ultimate classical or macroscopic non-quantum system. Consciousness at least generates an epiphenonenon of wave function collapse.

Cheers LC

This mathematics is involved with kissing numbers. The big sourcebook on this is Conway and SLoane, "Sphere Packing, Lattices and Codes." These spheres in the 4-dim case are connected to Planck units of volume. The packing system is the 24-cell or equivalently the F4 group.

These systems are error correction codes. In the case of sporadic groups the quantum error correction is meromorphic which preserves quantum information. In the simple case there is no pole this recovers unitarity. In the study of this it is important to keep the connection to Jacobi theta functions. This also connects up with the Ramanujan Mock theta function and the partition function for the integers. This partition function is related to the density of states of a bosonic string as well as the thermal partition function of a black hole.

There are deep relationships involved with this. My essay here is an attempt to lay down some physical arguments for this.

Cheers LC