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Comments about M-matrix and Connes tensor product
by Dr. Matti Pitkänen / September 6, 2009
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I have proposed that the identification of M-matrix as Connes tensor product defined by finite measurement resolution could lead to a universal definition of dynamics. This hypothesis is fascinating but -- mainly due to my poor understanding of HFFS of type II1 -- has remained just an interesting hypothesis.
In the following, I represent a formulation of this idea which is more precise than the earlier formulations and take the role of skeptic and reconsider also hyper-finite factors of type III1 appearing in quantum field theories.
I also consider the possibility that M-matrices could relate to a quantum variant of so-called 2-vector space formulated by John Baez and collaborators.
A. Finite measurement resolution and M-matrix
Consider first the formulation of finite measurement resolution in terms of inclusions of HFFs of type II1.
1. Finite measurement resolution means that M-matrix elements are defined by integrating (that is, taking a trace) over the degrees-of-freedom defined by the sub-factor N subset M in the sense that states created by N from a given state do not differ from it in given measurement resolution. As a consequence, N takes the role of complex numbers in the ordinary quantum theory.
2. This requires that the tensor product decomposition M= (M/N)⊗N exists in some sense. The factor space M/N could be seen as an analog of or even identical with a tensor product of quantum spinors with different quantum phases q=exp(i2π/n), n > 3 and would have a fractal quantum dimension fixed by the inclusion. Quantum spinors are finite-dimensional.
3. Consider M-matrix elements where r labels the states of M/N. Transition probabilities with a finite measurement resolution are defined by taking trace over N meaning summation over both n and n1. One obtains
. (The subscript '+' refers to Hermitian conjugation here.)
4. Suppose one multiplies the state by element n of N. Since one has a tensor product structure, the trace over N separates out so that one obtains a factor Tr(n+n) giving a multiplicative constant just as is obtained by multiplying the states appearing in the ordinary S-matrix by a constant. Note that the measurement resolution could be different for initial and final states. In this case, the smaller algebra would serve as measurement algebra.
5. The condition that N acts like complex numbers has not been used yet and the above picture makes sense even without this condition. For the ordinary S-matrix, the action of complex number on final state affects S-matrix in the same manner as the action of its conjugate on initial state. Generalizing, in positive energy ontology S-matrix elements would be anti-linear with respect to (say) initial quantum states as inner product like and the action of element n on the final state would be equal to the action of n+ on the initial state.
In Zero Energy Ontology, M-matrix is bilinear in positive and negative energy parts of the Zero Energy state. The condition that N acts like complex numbers means that the action of an element of N on the final state in M-matrix induces the same effect as the action of n (-rather than that of n+) on the initial state. This condition is rather powerful and expected to determine the M-matrix highly uniquely.
6. This picture does not require gauge invariance with respect to N. One can, however, also consider a stronger condition stating that the action of unitary elements of N act as gauge symmetries. In Zero Energy Ontology, this would mean that symmetries represent elements of infinite-dimensional orthogonal group.
7. Connes tensor product also makes sense in construction of many-particle states and gives rise to irreducible entanglement not visible in measurement resolution.
B. How unique is the Connes tensor product really?
I have used to state in a rather light-hearted manner that the Connes tensor product is highly unique. But what is the situation really? Is my belief just a folk wisdom that I have gathered somewhere?
Let us try to think ourselves. How unique the M-matrix could be.
1. The simplest solution to the conditions could be written as a formal tensor product M=M1⊗M2 where M1 is M-matrix in M/N and M2 projection operator PN in to N. The appearance of PNN would say that in N degrees-of-freedom entanglement coefficients are identical and basic condition would be satisfied since PN would commute with elements of N.
If one replaces PN with a more general operator, it must also commute with all elements of N. Which means that it must represent direct sum of HFFs of type II1 since the definition property of factors is that only unit matrix commutes with all operators of the factor.
2. This representation would suggest that M1 can be chosen completely freely as an analog of a complex square root of Hermitian density matrix. The skeptic would conclude that Connes tensor product does not say anything about the physically interesting part of M-matrix!
3. This formal representation very probably does not make sense as such since M1 is tensor product of quantum spinor spaces and tensor product property might reveal itself only as a factorization of trace into a product of traces over M/N and N parts of the tensor product-like structure.
This argument must be, of course, taken as childish babbling of a poor physicist knowing nothing about the delicacies of this branch of mathematics. It is quite possible that the delicacies of the Connes tensor product bring in uniqueness.
C. Is the resulting M-matrix realistic?
Are there hopes that the resulting M-matrix is realistic?
1. The resulting M-matrix defined in quantum Hilbert space has fractal quantum dimension. Since the unitary representations of quantum groups are finite-dimensional, one can expect that finite-dimensionality results. Therefore the skeptic would ask whether the use of mere HFFs of type II1 is enough and whether the outcome is just a topological S-matrix of braid theories. This would be disappointing.
One can, however, easily add factor of type I as a tensor product factor to give HFF of type II giving also the structure of ordinary wave mechanics.
2. The physical picture is that the orbits of fermions and anti-fermions at light-like 3-surfaces defined braids and the topological M-matrix can be assigned with the legs of generalized Feynman diagrams. The remaining configuration space degrees-of-freedom could correspond to factors of type I.
3. In finite measurement resolution, the space-time surface decomposes to strings connecting the braid strands at different light-like 3-surfaces. This stringy interaction should guarantee that the theory does not reduce to a topological QFT. This interaction is between braids so that something new is needed. Does this something new allow a description in terms of finite measurement resolution too or is factor of type I enough? Or could the almost TQFT property mean that N does not act anymore like complex numbers so that Connes tensor product is deformed?
4. One can also wonder what is the role of the dynamics in the parameters labeling the copies of HFF. One interpretation is in terms of zero modes for the Kähler metric of World of Classical Worlds and therefore as non-quantum fluctuating degrees-of-freedom. One can also consider interpretation of superposition over these parameters in terms of thermodynamical degrees-of-freedom.
D. Should one replace HFFs of type II1 with HFFs of type III1?
These skeptical arguments encourage a serious reconsideration of an alternative approach for achieving uniqueness by extending the HFF of type II1 with HFF of type III1 (note: I have already considered this generalization below). The dream would be that any M-matrix with a finite measurement resolution is obtained from a universal M-matrix with infinite measurement resolution existing in some sense by tracing over HFF N of type III1 and multiplying by the projector to N.
The Tomita-Takesaki theorem raises the hope that this universal M-matrix indeed exists and is unique apart from the inner automorphisms of HFF of type III1 and complex parameter t whose real and imaginary parts have interpretation as Time and Temperature.
1. In quantum field theories in Minkowski space, hyper-finite factors of type III1 appear. It would not be surprising that they would appear also in Quantum-TGD. These factors can be expressed as so called crossed products of hyper-finite factors of type hyper-finite II∞ factor and real numbers whereas the latter correspond to a tensor product of hyper-finite II1 factor and I∞ factor (for the basic wisdom about von Neumann algebras, see this).
2. One can assign to any M-matrix in factor M an M-matrix associated with any quantum space M/N by tracing over N. This could also make sense for HFFs of type III1. In this case, the trace Tr(n*n) over N would be always infinite so that multiplication by operator of n of N would be analogous to a multiplication of the moduli square of S-matrix elements with an infinite scale factor. Maybe this relates to infinite multiplicative renormalization factors appearing in quantum field theories.
3. The Tomita-Takesaki theorem assumes a von Neuman algebra acting on complex Hilbert space and the existence of a cyclic state Ω of Hilbert space (i.e., vacuum state in Physics). Furthermore, the existence of a faithful state ω(x) = (xΩ,Ω) satisfying by definition ω(x*x) > 0 is assumed. Faithful state is analogous to a thermodynamical ensemble. For any anti-linear operator S, one has the polar decomposition S=JΔ½ where Δ=S*S>0 is positive self-adjoint operator and J is anti-unitary involution with maps the algebra acting in Hilbert space to the algebra commutating with it by x→JxJ.
ω→σωt = AdΔit is canonical "evolution" associated with ω and maps von Neumann algebra to itself. This automorphism is unique apart of inner automorphism and thus characterizes the von Neumann algebra itself. So-called KMS condition holds true: ω(yx) =ω(σtω(x)y), t→i in the sense of analytic continuation. This condition is easy to understand by checking it for matrix algebras.
4. The facts that the automorphism defined by Δ characterizes the algebra itself and the automorphism is non-trivial for factors of type III led Connes and Rovelli to propose that thermodynamics represents a deeper level of physics than Hamiltonian (unique apart from inner automorphism) defining it and one could define Lorentz invariance thermodynamical time using thermodynamics for von Neumann algebras of type III.
5. In Zero Energy Ontology where M-matrix corresponds to a generalized thermodynamical state, a variant of this idea suggests itself. The automorphism for hyper-finite factor of type III1 would define M-matrix depending on complex parameter t apart from inner automorphism of the factor. The real part of the parameter t would correspond to the Lorentz invariant temporal distance between the tips of CD quantized in powers of 2. Imaginary part of t would correspond to the inverse temperature.
E. The notion of 2-vector space and entanglement with finite measurement resolution
John Baez and collaborators are playing with very formal looking formal structures by replacing vectors with vector spaces. Direct sum and tensor product serve as basic arithmetic operations for vector spaces and one can define category of n-tuples of vectors spaces with morphisms defined by linear maps between vectors spaces of the tuple.
n-tuples also allow elementwise product and sum. They obtain results which make them happy. For instance, the category of linear representations of give group forms 2-vector spaces since direct sums and tensor products of representations as well as n-tuples make sense. The 2-vector space however looks more or less trivial from the point of Physics.
The situation could become more interesting in quantum measurement theory with finite measurement resolution described in terms of inclusions of hyperfinite factors of type II1. The reason is that Connes tensor product replaces ordinary tensor product and brings in interactions via irreducible entanglement as a representation of finite measurement resolution. The category in question could give Connes tensor products of quantum state spaces and describing interactions. For instance, one could multiply M-matrices via Connes tensor product to obtain category of M-matrices having also the structure of 2-operator algebra.
1. The included algebra represents measurement resolution and this means that the infinite-D sub-Hilbert spaces obtained by the action of this algebra replace the rays. Sub-factor takes the role of complex numbers in generalized QM so that one would obtain non-commutative quantum mechanics.
For instance, quantum entanglement for two systems of this kind would not be between rays but between infinite-D subspaces corresponding to sub-factors. One could build a generalization of QM by replacing rays with sub-spaces. It seems that quantum group concept more-or-less does this. The states in representations of quantum groups could be seen as infinite-dimensional Hilbert spaces.
2. One could speak about both operator algebras and corresponding state spaces modulo finite measurement resolution as quantum operator algebras and quantum state spaces with fractal dimension defined as factor space like entities obtained from HFF by dividing with the action of included HFF. Possible values of the fractal dimension are fixed completely for Jones inclusions. Maybe these quantum state spaces could define the notions of quantum 2-Hilbert space and 2-operator algebra via direct sum and tensor production operations. Fractal dimensions would make the situation interesting both mathematically and physically.
Suppose one takes the fractal factor spaces as the basic structures and keeps the information about inclusion.
1. Direct sums for quantum vectors spaces would be just ordinary direct sums with HFF containing included algebras replaced with direct sum of included HFFs.
2. The tensor products for quantum state spaces and quantum operator algebras are not anymore trivial. The condition that measurement algebras act effectively like complex numbers would require Connes tensor product involving irreducible entanglement between elements belonging to the two HFFs. This would have direct physical relevance since this entanglement cannot be reduced in state function reduction.
3. The sequences of super-conformal symmetry breakings identifiable in terms of inclusions of super-conformal algebras and corresponding HFFs could have a natural description using the 2-Hilbert spaces and quantum 2-operator algebras.
For background, see the chapter "Construction of Quantum Theory: M-matrix" of Towards M-matrix.
My defense statement
Following von Neumann known from his factors of type I, II, and III, one could classify mathematicians to those of type I and type II. I hope that mathematicians do not feel insulted by this kind of classifications;-).
Mathematicians of type I believe in the uniqueness of those mathematical structures which are really God-given. Classical number fields and groups would be the basic example about God-given-ness. Also Physics would be fixed uniquely by the condition of the mathematical existence of the theory and by its maximal richness.
Mathematicians of type II prefer genericity. Everything that can be imagined is possible. In Physics, this would mean that space-time dimension four just happens to be 4 in this particular corner of the Multiverse and symmetries of physics (if real rather than figments of our imagination) just happen to be just those of Standard Model in this little pocket of Multiverse.
String theorists began as mathematicians of type I and were fascinated by the idea that they had the 'Theory of Everything' allowing to calculate proton mass to arbitrary number digits within decade-or-two. As they suffered a phase transition to Mbrane-theorists, they woke up as mathematicians of type II and realized how wonderful it is after all that M-theory predicts nothing. Lubos is one of few exceptions. Personally, I am also still of type I and I can only represent excuses for my conservatism. I hope that the jury bothers to listen for humanitarian reasons if not for anything else.
1. The Kähler geometry of the Worlds of Classical Worlds is unique already for loop spaces. In 3-D context, there are excellent hopes that even the imbedding and therefore symmetries of physics can be fixed from the existence condition. The basic outcome is that infinite-dimensional symmetries are not figments of our imagination. They are absolutely essential for the infinite-dimensional geometric existence. Extended super-conformal invariance indeed fixes space-time dimension correctly and implies M4×S decomposition for imbedding space.
2. The isometries of this particular infinite-D space which has the special property that it exists and also of corresponding finite-dimensional imbedding space must have very special meaning. Number theoretic one is the natural guess. Classical number fields and their complexifixations are natural here.
3. The choice of S remains to be fixed from number theoretic considerations. The existence of M8-M4×CP2 duality ("number theoretical compactification") -- with M8 is interpreted as hyper-octonionic plane of complexified octonions with Minkowskian metric -- means the possibility to map space-time surfaces identified as hyper-quaternionic surfaces of M8 to M4xCP2. This map preservers induced metric and Kahler form. Standard Model symmetries are the outcome.
M8-M4×CP2 duality implies also the core element of gauge invariance and stringy picture.
1. Only if the hyper-quaternionic plane assignable to a point of space-time surface contains preferred M2 (hypercomplex plane) of M8 having interpretation as plane of non-physical polarizations, it can be labeled by a point of CP2 so that the map of X4 in M8 to X4 in M4×CP2 exists naturally and maps M4 point to M4 point and hyper-quaternionic plane to CP2 point. This is the physical essence of gauge conditions. It seems to be possible to assume that the choices of the preferred polarization plane is local.