Arcadian Functor

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Marni D. Sheppeard

Saturday, November 03, 2007

Connes-onical Time

In his latest post, Alain Connes comments on the canonical nature of time evolution for noncommutative spaces. In M Theory, the analogue should be a whole heirarchy of Planck constants. For example, the Weyl relations of the quantum torus depend on the parameter $\hbar$. These spaces are studied in a very nice paper from 1993 by Alan Weinstein, where $\hbar$ is associated to time evolution for a particle on an ordinary torus. The particle is initially concentrated at a point, for $\hbar = 0$, but quickly becomes non-localised.

Louise Riofrio also has many fascinating posts on an emergent thermodynamic Time associated to Planck's number. In M theory we may view this as an approximation to a 3-Time picture, which brings to mind the twistor triality of Sparling, or the 2-Time theory of Itzhak Bars. It seems that wherever we look, the canonical Arrow raises its head.


Blogger Matti Pitkanen said...

There is a nice discussion about the canonical time evolution for factors of type III, which could teach a lot about what is involved to a motivated layman (such as me).

Type III time evolution is unique apart from the time parameter defining the duration of evolution and inner automorphism, the presence of which could be interpreted in terms of universal gauge invariance.

One can imagine two kinds of almost unique dynamics both having kind of universal gauge invariances.

1. For factors of type III canonical time evolution with inner automorphisms as gauge symmetries. I have tried to understand whether this kind of dynamics could emerge in TGD as a kind of fusion of HFF II_1 dynamics: the differences in the realizations of these factors are rather delicate: same infinite braid system seems to allow both IIs and IIIs. What is problematic with factors of type III is that the trace of unit is infinite: this could lead to difficulties and the divergences of QFT could relate very closely to this.

2. For HFFs of type II_1 the M-matrix defined by Connes tensor product and with measurement resolution defined by inclusion N subset M. This codes to the statement that Hermitian elements of N are symmetries of M-matrix and thus define gauge invariance like symmetry but due to measurement resolution. The old flavor SU(n) of hadron physics, almost forgotten now, might be a physical example of this kind of symmetry.

These symmetries are indeed very much like local gauge symmetries acting on a finite number of tensor factors in the tensor product representation of HFF. In the representation as a group algebra of S_infty or braid algebra B_infty these transformations act on finite number of braid strands only. Gauge symmetries in discrete real line: this is the interpretation.

The counterparts of global gauge transformations -outer automorphisms - correspond to transformations acting on the closure that is on infinite number of braid strands/tensor factors. A typical example is representation of S_n as diagonal group with elements gxgxg.... with g element of S_n.

The physical beauty here is that the periodicity of the imbedding of S_n (or of any finite group G, or compact gauge group actually) allows to represent the infinite system situation by a finite braid at space-time level: no need to have infinite number of replicas of same basic situation.

Inner-outer<---->local-global: this translation should reduce considerably the frustrations of poor physicist trying to understand what these mathematicians are saying.

There are also other questions (in TGD context). What about HFF of type II_infty with trace having all possible values. Could bosonic and fermionic degrees of freedom combine to form this kind of factor as a tensor product. I feel that this is not a good option and superconformal symmetry would suggest direct sum of II_1:s in both degrees of freedom.

Most naturally one would have a direct infinite direct sum of HFFs of II_1 with M-matrix in each summand in both F and B degrees of freedom. This would bring naturally p-adic thermodynamics (each value of conformal weight defining one summand) and other kinds of thermodynamics. But I could be wrong;-)!

The main message would be that one could start from a hierarchy of universal M-matrices with different measurement resolutions and work out their physical representations. Just the opposite for the usual approach. One should show many things. If one believes in TGD one should show that the universal gauge invariance provides a representation of various superconformal symmetries of light-like 3-surfaces; that number theoretic braids emerge naturally; etc... Of course, also other representations of this universal dynamics could exist looking totally different (dualities).

November 03, 2007 7:30 PM  
Blogger Matti Pitkanen said...

A comment about spectrum of hbar, which Kea assigns to M theory.

In my own approach the spectrum of hbars emerges naturally from the generalization of the imbedding space. There are close connections to Jones inclusions but when I try to express this connection with two sentences I hear only strange humming in my head;-). Perhaps, a correlate for quantum entanglement to my adviser which I am unable to reduce;-).

In any case, the hierarchy of Planck constant is an essential element of quantum TGD proper now and realizes the notion quantum criticality. For instance, the understanding of Higgs expectation led to the identification of number theoretic braids and to a direct construction of the space-time sheet associated with given light-like 3-surfaces and this is a real victory.

Time evolutions (M-matrix) involve Planck constant but I cannot see how these time evolutions alone could imply the spectrum of Planck constants. I wish I would understand this better.

November 03, 2007 8:54 PM  
Blogger Matti Pitkanen said...

Still a comment about the posting of Alain Connes. Connes mentions 3-D foliations V which give rise to type III factors. Foliation property requires a slicing of V one-form v to which slices are orthogonal. v satisfies thus integrability conditions. If so called Godbillon-Vey invariant is non-vanishing, factor of type III is obtained using Schrodinger amplitudes for which the flow lines of foliation define the time evolution.

In TGD light-like 3-surfaces are natural candidates for V and it is interesting to concretize the situation in this context. The one-form v defined by the induced Kahler gauge potential A defining also a braiding is a unique identification for v. The foliation property requires that v multiplied by suitable scalar is gradient. This gives dA= w\wedge A, A=-dpsi/psi =-dlog(psi). Something proportional to log(psi) can be taken as a third coordinate varying along flow lines of A: the braiding flow defines a continuous sequence of maps of partonic 2-surface to itself.

Physically this means the possibility of a super-conducting phase with order parameter, essentially phase depending only on psi only. This would describe supra current flowing along flow lines of A. Integrability in general is not true and one cannot assign Schrodinger time evolution with the flow lines of v (except possibly trivial time evolution for which energy vanishes!). One could not speak about supra flow along lines of A since the flow would be mixing.

Connes tells that a factor of type III results if Godbillon-Vey invariant defined as the integral of dw\wedge w over 3-manifold is non-vanishing. The operators of the algebra in question are transversal operators acting on Schrodinger amplitudes in each slice. Essentially Schrodinger equation in 3-D space-time would be in question with factor of type III resulting from the exotic choice of the time coordinate defining the slicing.

In TGD Schroedinger amplitudes are replaced by second quantized induced spinor fields. Hence one does not face the problem whether it makes sense to speak about Schroedinger time evolution along the time-lines of foliation or not. Also the fact that "time evolution" for the modified Dirac operator corresponds to single position dependent generalized eigenvalue identified as Higgs expectation same for all transversal modes (essentially z^n labelled by conformal weight) is crucial since it saves from the problems caused by the possible non-existence of Schroedinger evolution.

November 03, 2007 9:04 PM  
Blogger L. Riofrio said...

Thanbks again for many interesting links to researchers like Connes. The most interesting science seems to be happening outside the mainstream.

November 04, 2007 4:28 AM  

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