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**Physics, Astronomy & Cosmology / Re: What is the Hopf gauge?**

« **on:**07/05/2024 21:10:11 »

Ok, I have to include some actual stuff about what the big fuss is, why Penrose says much the same thing as, well a few people I suppose. The Hopf fibration does seem to be fundamental. Perhaps because the universe is just weird (except strips of paper are ordinary).

So from the Journal of Physics:

So from the Journal of Physics:

Quote

Hopf and the qubit?two-level quantum systems

The occurrence of the Hopf fibration in two-level quantum systems?systems that can be described using a two-dimensional complex Hilbert space, also nowadays known as qubits?was mentioned, e.g., by Penrose [36], and is described more extensively in [48]; so we can be brief and somewhat schematic about it here.

We take the two-dimensional Hilbert space just to be

with its standard Hermitian inner product ≔

. The states of the system are given by density matrices, i.e.,

Hopf and mechanics?the harmonic oscillator

It may surprise some that the simple two-dimensional isotropic classical harmonic oscillator provides an example of the occurrence of the Hopf fibration in physics. But indeed, its phase flow, restricted to a non-trivial energy shell, Hopf-fibers the latter! To see this easily, we scale everything such that the mass and the angular frequency of the oscillator become 1; the Hamiltonian then reads

whence the equations of motion

Employing the usual

Hopf and General Relativity?Taub-NUT space

Around 1950, G?del and Taub independently started the investigation of spatially homogeneous cosmological models in General Relativity. The ones constructed by them are space-times

having line elements of the form where the ωk are essentially left-invariant 1-forms on a three-dimensional Lie group and the product of forms in ds2 is symmetric tensor multiplication as usual. More precisely, , where

is some interval for τ, to be determined during the process of

Hopf and twistors?Robinson congruences

The concept of twistor can, according to its inventor, Penrose, be introduced in a number of ways, and all conformally invariant field laws in flat space-time can be reformulated in terms of twistors, generating new ways of looking at such laws; this has already been very fruitful in the past in many cases.

Clearly, this is not the place to give any of the different ways of approaching or using twistors. We rather pick out one way of representing a twistor space-time-geometrically and highlight

Hopf and Wignerism?helicity representations

In all examples discussed so far, the bundle, or total, space

was of immediate importance and thus in the forefront of the formalism; the base space

, on the other hand, did figure in the qubit case and is used in technical treatments of NUT space that work with the space of Killing trajectories. In the two examples to follow, it will be the other way round: it is the base space that is in the foreground, while the bundle space appears as derived, e.g., via a patching construction. The

Hopf and magnetic monopoles

This example of our collection is probably the first one that caught the attention of a larger number of physicists. As announced, the Hopf fibration comes here from a bottom-up construction.

In his 1931 article ?Quantised Singularities in the Electromagnetic Field? (magnetic monopoles), Dirac [19] starts out looking for a generalization of wave mechanics in which the wave function does not have a definite value for its phase difference between two points. In the 1970s, it was realized [50] that

Hopf and the Dirac equation9

The two-level quantum systems of Section 2 may be considered, in particular, as describing the spin states of a non-relativistic spin-

particle?i.e., one ignores its spatial degrees of freedom. If the latter is not done, one must write spinorial wave functions, subject to appropriate wave equations. In the relativistic regime, the relevant wave equation is the Dirac equation. We shall point out here that the Hopf fibration has significance also in this context. Most directly this comes about

Acknowledgements

I am indebted at least to the following persons for directly or indirectly educating me on matters included in this article: R. Beig, I. Bengtsson, P. Nurowski, R. Penrose, H. Rumpf, W. Simon, and A. Trautman. In particular, important information came from participants of a meeting in honor of E.T. Newman held in June 2001 at the Banach Center, Warsaw; I thank the organizers of that meeting for inviting me there to present a preliminary version of this article, for hospitality, and

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