On Thursday, July 2, 2020 at 6:48:08 AM UTC-5, Philip Thrift wrote:
>
> https://arxiv.org/abs/2007.00418
>
> [Submitted on 1 Jul 2020]
> *Forcing as a computational process*
>
> Joel David Hamkins
> <https://arxiv.org/search/math?searchtype=author&query=Hamkins%2C+J+D>,
> Russell
> Miller <https://arxiv.org/search/math?searchtype=author&query=Miller%2C+R>
> , Kameryn J. Williams
> <https://arxiv.org/search/math?searchtype=author&query=Williams%2C+K+J>
>
> We investigate how set-theoretic forcing can be seen as a computational
> process on the models of set theory. Given an oracle for information about
> a model of set theory ⟨M,∈M⟩, we explain senses in which one may
> compute M-generic filters G⊆ℙ∈M and the corresponding forcing
> extensions M[G]. Specifically, from the atomic diagram one may compute G,
> from the Δ0-diagram one may compute M[G] and its Δ0-diagram, and from the
> elementary diagram one may compute the elementary diagram of M[G]. We also
> examine the information necessary to make the process functorial, and
> conclude that in the general case, no such computational process will be
> functorial. For any such process, it will always be possible to have
> different isomorphic presentations of a model of set theory M that lead to
> different non-isomorphic forcing extensions M[G]. Indeed, there is no Borel
> function providing generic filters that is functorial in this sense.
>
> @philipthrift
>
Forcing is a bizarre idea, where one introduces some new category or
objects that extends a model. In some ways we do this with the extension of
mathematics from the reals to complex numbers. The forcing of one system
into another has some connections to the Jordan-Banach algebra as well. The
replacement of a Poisson bracket structure {X, Y} → i/ħ[X, Y], is given by
the replacement of generators that are Jordan with product rule A∘B = ½(AB
+ BA) to a Lie algebraic ad_A(B) = [A, B] product. For a general quantum
system with a Lindblad type equation
Idρ/dt = [H, ρ] + ½([Aρ, A^†] + [A, ρA^†]),
such that the Hamiltonian is an element of the Lie group L, usually as a
weight, and the operators A and A^† are operators so that AρA^†, AA^†ρ,
and ρAA^† are operations on the density matrix that are Jordan. Depending
upon how one looks at this we can think of the quantum mechanical theory on
Fubini-Study metric on ℂP^n, say a metric based action or Lagrangian ℒ =
g_{ab}y^ay^b to ℒ = g_{ab}y^ay^b + U(x_1, x_2, … ) with a general
Euler-Lagrange equation
F_i = ∂/∂x_i(∂ℒ/∂y_a)y^a - ∂ℒ/∂x_i.
This is component version in Finsler geometry of a differential 1-form ω =
πdq – dη. It should be apparent this is similar to the first law of
thermodynamics. This is a “practical lesson” in what is meant by forcing.
The imaginary valued system is extended to components that are further
multiplied by i = √(-1). This is a sort of Wick rotation version of the
forcing from reals to complex numbers.
The Jordan-Banach algebra though is just a start. We have the skew symmetry
that corresponds to the conjugate on I = √(-1). This Lie algebraic system
is in contrast to the Jordan product and algebra. We also have in physics
fundamental properties due to bosonic and fermionic statistics. In
supersymmetry there are the Grassmann numbers that enter in to intertwine
between these. Further, on a 2-dimensional space the interchange of two
bosons or fermions is ambiguous. This is because a clockwise and
counterclockwise rotational exchange are topologically distinct. In 3
dimensions of space the paths can be exchanged, but no in two. This means a
sign change is possible in either boson or fermion cases. The stretched
horizon of a black hole is a 2-dimensional surface and an anionic system
for quantum fields. I think this is a foundation for how there are these
dichotomies in physics, such as quantum mechanics and macroscopic physics
and different statistics. Instead of physics being founded on very large
algebraic systems I think that Wigner had something right with the idea of
small groups. The big symmetry systems come about from symmetry breaking or
the removal of a degeneracy.
The fibration and Noether’s theorem in differential forms is λ = p_idq_i –
Hdt, where the 1-form has a horizonal p_idq_i and vertical Hdt principal
bundles. This is a Finsler geometric way of seeing Lagrangian dynamics. The
differential dλ = dp_i∧dq_i – dH∧dt, vanishes on the contact manifold. In a
strict Finsler context the horizontal and vertical bundles are annulled
separately. The q_i is the generator of p_i and dp_i = 0 means momentum is
invariant under translation and similarly energy is conserved under time
translations. If there is a Lagrange multiplier that connects the vertical
and horizontal parts of the bundle then we have
dp_i∧dq_i – dH∧dt = (dp_i/dt)dt∧dq_i – (∂H/dq_i)dq_i∧dt = [(dp_i/dt) +
(∂H/dq_i)]dq_i∧dt
which gives us the dynamical equation (dp_i/dt) = -∂H/dq_i.
Hamkins et al paper is dense in set theoretic constructions. These things I
have some introductory acquaintance with. I am also more interested in the
practical transduction of such ideas. I will try to see what I can gather
from this paper. Forcing does have some algorithmic connection and the
above connects 1/kT ↔ it/ħ with macroscopic system with a temperature, or
gravity in the case of spacetime physics, to time in the case of quantum
mechanics.
LC
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