Category theory has some potential for physics. Quantum logic I am less sanguine about. It has always struck me as a way of expressing something we understand in set theoretic language. It has always struck me as a formalistic study that brings little additional content.
Philip Goyal demonstrated how the above summation over intermediate points is equivalent in a discrete setting to the concatenation of measurements, where the summation is over all possible outcomes [1]. The complex amplitudes are products of complex numbers, and in a discrete setting this is a multiplication rule which requires complex numbers. The result is that quantum mechanics is reduced to a simple system of associative and commutative mathematics of complex numbers with no reference to classical mechanics, or any notion of space or spacetime. However, the one requirement is that the points intermediate to the initial and final points be intermediate in time.
The Goyal logic is a summation of Stern-Gerlach experiments. The intermediate point corresponds to some intermediate measurement between the source of particles and the final Stern-Gerlach (SG) apparatus. If the outcome of the intermediate SG apparatus is ignored, or no measurement is performed of their outcomes, the split beams recombine as a discrete summation. So the intermediate SG apparatus represents a sum over elementary quantum events. This summation in the complex algebra corresponding to this logic recovers the quantum superposition.
These summations over SG experiments occur in a sequence. There is no ambiguity in the ordering of these events. Further, the process appears well defined in a discrete setting. The Goyal approach for discrete quantum mechanics, even if the number of elements is enormous, but not infinite, indicates some sort of quantization of time, and a discrete spacetime.
This has elements of Zariski topology. Consider the affine space An as the n-dimensional space over a closed field F. The topology is constructed from closed sets defined by the polynomial set S ∈ F by
V(S) = {x ∈ An|f(x) = 0; ∀f ∈ S}
For two polynomials in the set S we have the following rules:
V(p1)∪V(p2) = V(p1×p2), V(p1)∩V(p2) = V(p1 + p2)
which serve as the representation map between the logic of outcomes and the algebra of quantum operations demonstrated by Goyal. This closed set topology defines the Zariski topology on the affine set
An. So a connection to quantum mechanics exists within this system with respect to Zariski topology. This is the topology of {\’E}tale and Grothendieck, or topos theory. An overview of topos theory in physics is in [2] by Isham.
[1] P. Goyal, K. H. Knuth, J. Skilling, "Origin of Complex Quantum Amplitudes and Feynman's Rules," {\it Phys. Rev.} {\bf A 81}, 022109 (2010) http://arxiv.org/abs/0907.0909
[2] C. J. Isham, "Topos Methods in the Foundations of Physics," {\it"Deep Beauty,} ed. Hans Halvorson, Cambridge University Press (2010) http://arxiv.org/abs/1004.3564
This post imported from StackExchange Physics at 2014-04-05 03:01 (UCT), posted by SE-user Lawrence B. Crowell