The Art Historian Physicist

Complex Analysis, Black Hole Thermodynamics

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  • An addendum to the core of a group

    Well, we all know Euclid´s “Elements” and weird problems such as prime counting functions (see Terence Tao´s blog for this). A simple addendum to Euclid´s ideas would be to say that instead of the normal axioms (0 dimension = point, 1 dimension = line etc…) a monomial or polynomial already encodes such properties.

    Instead of writing “Elements”, wouldn´t a “Polynomials” book be more accurate nowadays?

    I give three immediate examples: Einstein´s general relativity would be an harmonic oscillator of a “group” of polynomials (very witten-esque isn´t it!!!!).

    Jacob Lurie´s attempt of introducing path integrals (through his weird language) between algebra (weird since any first year calculus student knows algebra ≠ calculus) would simply be the least action of this “group”.

    Finally, I´m talking about “groups” here but what is a group? I think probably a type III von Neumann algebra coupled with some sort of flow (be it modular or not; see Tomita-Takesaki lecutres on Springer) would suffice to define these polynomials.

    Cheers

  • A reflection on the meromorphic properties of a black hole event horizon

    If the Universe could be observed as a generalized wavefunction $psi$, perpetually collapsing unto physical properties, what axioms would hold for a coherent picture of reality? A generalized Hilbert space, following von Neumann algebras? Not very pretty. Cauchy´s integral formula and theorems are prettier:

    Essentially, it says that if {\displaystyle f(z)} is holomorphic in a simply connected domain Ω, then for any simply closed contour {\displaystyle C} in Ω, that contour integral is zero.

    {\displaystyle \int _{C}f(z)\,dz=0.} (took this directly from Wikipedia: https://en.wikipedia.org/wiki/Cauchy%27s_integral_theorem). Now, Let U be an open subset of the complex plane C, and suppose the closed disk D defined as

    {\displaystyle D={\bigl \{}z:|z-z_{0}|\leq r{\bigr \}}}

    is completely contained in U. Let f : U → C be a holomorphic function, and let γ be the circle, oriented counterclockwise, forming the boundary of D. Then for every a in the interior of D,

    {\displaystyle f(a)={\frac {1}{2\pi i}}\oint _{\gamma }{\frac {f(z)}{z-a}}\,dz.\,}

    (https://en.wikipedia.org/wiki/Cauchy%27s_integral_formula). If f(z) = $psi$, the information paradox would literally like finding a needle on a haystack, only the needle would be a photon collapsing into the big haystack of an event horizon.

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