Tuesday, May 25, 2010

About the Nature of Contemporary Mathematics

I found from Not Even Wrong a link to a very nice popular article The Nature of Contemporary Core Mathematics by topologist and topological quantum field theorist Frank Quinn. To my own surprise I had several times the impression of understanding (perhaps the aging brain of scientists learns to generate this experience when required;-)). Core mathematics differs from pure and applied mathematics in Quinn's classification in that it can but need not yet have direct applications.

I have been following the work of John Baez and others based on category theoretical inspirations and I admit my frustration because I am unable to follow the technicalities and fail to see the connection to quantum physics. Quinn represents interesting comments about this work which I could use as an excuse for giving up totally my attempts to understand what they are doing;-).

First of all Quinn notices the potential fatal consequences coming from the lack of a direct connection with physics. It is easy to agree: the lack of a concrete connection makes it possible for a brilliant mathematician to generate endlessly formal structures as generalizations of existing ones and to generate theorems as analogs of existing ones. This is not wrong as such but the lack of the evolutionary pressures leads to an inflation of these mathematical life forms and eventually the mass extinction is unavoidable since the academic metabolic resources are finite. I also think that something genuinely new and irreducible is required. Deformations of existing structures - what most of recent day theoretical physics is - are not enough.

The second point of Quinn is more technical. He claims that the decomposition structure of categories (composition of arrows) cannot hold true in the physically interesting 4-D case even in topological QFTs. What decomposition means here is the assumption that S-matrices - the important arrows now- are parametrized by the value of a continuous time parameter telling the duration of time evolution between initial and final states. This implies that one can decompose S-matrix to a product in many manners (arrow as product of arrays). One can do the same also for path integral (but only formally). Quinn interprets this as locality. One can indeed imagine of decomposing the S-matrix to a product of infinite number of S-matrices associated with infinitesimal values of time parameterm and to obtain representations as an exponent of interaction Hamiltonian. The relationship between what happens by two points separated by a wall can be reconstructed from what happens at the wall. This is how Quinn states it.

For a short time I was enthusiastic about cobordism category and S-matrices as as representation of this category and talked about TGD as almost topological QFT but soon realized that this does not lead to anywhere. My own objection against the decomposition is based on the same observation as Quinn's but formulated differently. The decomposition means that the counterpart of unitary time evolution in the sense of exponentiation of interaction Hamiltonian) is assumed. This is impossible in TGD framework as became clear during the first months of TGD when I learned that Hamiltonian formalism makes no sense in TGD because of the extreme non-linearity of any general coordinate invariant variational principle. Within about 7 years this led to the vision about physics as infinite-dimensional geometry of world of classical worlds (or configuration space as I used to say then) and the last five years I have been talking about zero energy ontology formulated in terms of causal diamonds (CDs) defined as intersections of future and past directed lightcones.

  1. The temporal distance between the tips of CD is quantized in powers of two (p-adic length scale hypothesis) and also the relative position of tips forms a discrete lattice like set in 3-D hyperboloid of future light-cone. Unions of CDs are allowed and they can also intersect and there are CDs within CDs so that one can speak about fractal hierarchy. Most of this is very relevant for the notions like radiative correction, coupling constant evolution, virtual particle, finite measurement resolution, second law, and even quantum cosmology (when one considers very large CDs). I feel that these fundamental quantum physical notions must be feeded explicitly to the mathematics at the fundamental level. I would be astonished if an approach starting from existing mathematical notions such as categories having motivations coming from the internal structure of mathematics of a particular era- in particular the idea to of transferring the results from one branch of mathematics to another one by using functors- could miraculously reproduce these notions.

  2. One can assign M-matrix ("complex" square root of density matrix decomposing to a product of positive square root of diagonal density matrix and unitary S-matrix) to any CD as a representation of zero energy state. The collection of M-matrices orthonormal as zero energy states defines unitary U-matrix. There is a fractal hierarchy of M-matrices (again a physical input) but the product property fails. One cannot arrange CDs to a sequences of CDs so that product decomposition fails and one cannot speak of cobordism category.

  3. Product property implying a representation of S-matrix as an exponential of interaction Hamiltonians is also inconsistent with number theoretical universality stating that M-matrices make sense in all number fields. In p-adic context the continuous unitary time evolution in non-sensical as follows from the fact that p-adic exponent function does not have the physical properties of real exponent function. To define the counterparts of the exponent of Hamiltonian one must introduce the counterparts of phases exp(iEt) as roots of unity appearing in algebraic extension of p-adic numbers. Discretization and number theoretical quantization are unvoidable. One has chronons and the classical Hamiltonian flow picture fails.

Quinn says many other things but just these points stimulated sufficient interest in me to write these comments. I recommend the article warmly for physicists willing to understand how mathematicians see the science.

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