David Corfield
cohomology in physics

Idea

In that cohomology is an important part of the story of higher category theory as revolution, we should expect it to show itself in the development of current physics.

Cohomology plays a fundamental role in modern physics. (Zeidler, Quantum Field Theory, Volume 1, p. 14).

Fundamental physics is all controled by cohomology. (Schreiber) See his The use of nonabelian differential cohomology.

Stasheff, A Survey of Cohomological Physics.

The notions of homology and cohomology are deeply rooted in electrodynamics….the most simple approach to homology and cohomology groups is related to electrical circuits. This generalizes then to the Maxwell equations in electrodynamics by using differential forms and the de Rham cohomology. (Zeidler II, 227)

Cohomology is deeply routed (sic) in the following topics: Gauss’ surface theory, the Kirchhoff-Weyl theory of electrical networks, and Maxwell’s theory of electromagnetism. Cohomology lies at the heart of both modern differential topology and modern quantum field theory (the BRST approach).

Note: “Both the classical roots and the modern extensions will be thoroughly studies in Vol III on gauge theory and in Vol IV on quantum mathematics. (Zeidler II, 305)

The de Rham cohomology is a far-reaching generalization of Maxwell’s theory for the electromagnetic field. If the open set 𝒪\mathcal{O} is contractible to a point, then the existence of the electric (resp. magnetic) potantial is based on the local constraint curlE=0\mathbf{curl E} = 0 (resp. div B\mathbf{B} = 0) for the electric field E\mathbf{E} (resp. the magnetic field B\mathbf{B}). The number of global constraints for the existence of electric (resp. magnetic) potentials depends on the first Betti number β 1\beta_1 (resp. the second Betti number β 2\beta_2) of the open set 𝒪\mathcal{O}. The Betti number β 1\beta_1 (resp. β 2\beta_2) measures the number of essential 1-cycles (resp. 2-cycles) which are not boundaries. The Betti numbers are homotopical invariants, and hence the number of linearly independent constraints is also a homotopical invariant. (Zeidler III: 1039)

Zeidler IV will treat cohomology at length

(Note his idea of cohomology from energy levels of atom. See also Vol I 16.8).

Mass in classical mechanics

Mass “has a cohomological significance, it parametrizes the extensions of the Galileo group.” (Santiago Garcıa, hep-th/9306040). From discussion:

“Briefly: in classical mechanics, the Galilei group acts on the symplectic manifold of states of a free particle. But in quantum mechanics, we only have a projective representation of this group on the Hilbert space of states of the free particle. The cocycle is the particle’s mass.

Switching to a much more lowbrow way of talking: you can’t see the mass of a free classical particle by just watching its trajectory, since it goes along a straight line at constant velocity no matter what it’s mass is. But you can see the mass of a free quantum particle, because its wavefunction smears out faster if it’s lighter! So there’s some difference between classical and quantum mechanics. Ultimately this arises from the fact that the latter involves an extra constant, Planck’s constant.“ (John Baez)

“In slight disguise, one can see this cocycle also control already the classical free non-relativistic particle, in the sense that its action functional is of the form of a 1d WZW model with that cocycle being the ”WZW term“ that however comes down to be the ordinary free action.” (Urs Schreiber)

String theory and cohomology

H. Sati, Geometric and topological structures related to M-branes

“We consider the topological and geometric structures associated with cohomological and homological objects in M-theory.”

Minasian and Moore (1997) suggested that D-brane charges are classified by K-theory and not just by homology as was proposed first. (S. Fredenhagen, Basic Bundle Theory and K-Cohomology Invariants, p. 1)

Last revised on January 8, 2020 at 08:41:31. See the history of this page for a list of all contributions to it.