Category theory is a unifying theory of mathematics that was initially developed in the second half of the 20th century. In this respect it is an alternative and complement to set theory. A key theme from the “categorical” point of view is that mathematics requires not only certain kinds of objects (Lie groups, Banach spaces, etc.) but also mappings between them that preserve their structure.

In particular, this clarifies exactly what it means for mathematical objects to be considered to be the same.

— Wikipedia on Unifying theories in mathematics

2010.01.11 Thursday ACHK

… more prescriptive. It frames a possible template for any mathematical theory: the theory should have nouns and verbs, i.e., objects, and morphisms, and there should be an explicit notion of composition related to the morphisms; the theory should, in brief, be packaged by a category. There is hardly any species of mathematical …

— When is one thing equal to some other thing?, Barry Mazur

2010.01.20 Wednesday ACHK

I should explain that “Yang-Mills” theories are nonlinear generalizations of electrodynamics which can describe forces such as the weak and strong interactions.

— How Far Are We from the Quantum Theory of Gravity?, R. P. Woodard

Yang–Mills theory is a gauge theory based on the SU(N) group. It was formulated by Yang and Mills in 1954 in an effort to extend the original concept of gauge theory for an Abelian group, as was quantum electrodynamics, to the case of a nonabelian group with the intention to get an explanation for strong interactions.

— Wikipedia on Yang–Mills theory

2010.01.19 Tuesday ACHK

The dynamics of photons in the double-slit experiment describes the relationship between classical electromagnetic waves and photons, the quantum counterpart of classical electromagnetic waves, in the context of the double-slit experiment. Superficially, it may look like the dynamics of a photon can be completely described by the classical Maxwell’s equations with only a reinterpretation of the classical field as a probability amplitude for the photon, however this notion is fraught with danger and ultimately leads to contradictions. One should not simply assume that the electromagnetic fields are a wave-function for the photon. For one thing, they are real and thus contain both positive and negative frequency components, which cannot be reconciled with the requirement for Schroedinger wavefunctions which are complex, positive frequency only. In addition, the electromagnetic fields are observable (e.g. with an oscilloscope) while Schroedinger wavefunctions are not observable, even in principle. Clearly, then, the fields are not wavefunctions, are physical, observable fields, rather than merely what you take the modulus-square of to obtain the probability of finding a photon somewhere. The existence of some “wavefunction of the photon” is not a fully settled issue.

— Wikipedia on Photon dynamics in the double-slit experiment

2010.01.17 Sunday ACHK