Hamilton, Chandrasekhar, and Dublin: Regularized Fermion Degeneracy in High Dimensions

Dublin News
01 Sep 2022, 10 GMT+10

https://www.dublinnews.com/newsr/15627

Dr Jonathan Kenigson, FRSA

The Irish mathematical tradition is steeped in an adoration of historicity. However, this tradition is also superbly compatible with the demands of modern mathematical practice. Trinity College, Dublin, is the most influential Irish educational institution and is a prime alternative to Oxbridge in rigor and faculty achievements in mathematics. Schrödinger developed a monumental corpus of results in Wave Mechanics. Salmon and Walton contributed fundamentally to Optics and the kinetic theory of particles. Hamilton is synonymous with energy methods in physics. He is also responsible for the development of Noncommutative Field Theory particularly Quaternionic and n-Onionic extensions to Complex Numbers. In the current submission, I will briefly explore a paradigm made possible by Hamilton’s contributions in Field Theory.

Consider a degenerate star with global radial symmetry situated in an ambient space of Euclidean dimension n. Under the physical condition that n must be three or greater, one considers the Degeneracy Pressure exerted by Fermions within the sphere. If the gravitational force exceeds the Degeneracy Pressure of the Fermions, the star collapses into a yet more degenerate object a Black Hole. Chandrasekhar (1935) established the critical pressure necessary for a star comprised of Fermionic matter to collapse into a Black Hole. The proof relies heavily upon the quantum mechanics proposed by Schrödinger's Wavefunctions (1926). The Chandrasekhar case applies in n=3 dimensions. Higher dimensions were more elusive until the general n-space case was established by Sami Al-Jaber in 2008. The proof relies upon identical quantum-mechanical properties and reduces to Chandrasekhar’s limit when the Fermions are Electrons, and the ambient dimension is precisely 3. In a recent New York Weekly article, I briefly discuss the Renormalization of Al-Jaber’s paradigm and related applications in Economics (Kenigson 2022).

Consider the following more general problem: Let K be a spherical, stationary gravitating body of fixed mass M comprised of Fermions having an identical but dimension-dependent Degeneracy Pressure. If each dimension n is assigned to the ambient space in which K as situated (informally, “parking” K in an n-dimensional space) via a suitably regular probability distribution function (PDF), it is possible to derive the mean Degeneracy Pressure summing over all dimensions. One must, however, prove that this expression converges to a finite value. A consistent Zeta Renormalization Paradigm exists. In other words, one can “force-balance” each term in the expansion of the expected value to correspond to a mirror term. When this process is completed “ad-infinitum,” the resulting quantity can be proved to converge to a real value in the limit. This process of Hawking renormalization converges quite slowly but predictably.

The resulting limit makes an infinite quantity finite and calculable at the expense of renormalizing the finite terms to other more complicated ones using the theory of Integral Transforms. I attempt the process via a “Diagonalization” argument like that of Cantor, exploiting the rapid convergence of Geometric Series. The resulting Zeta Renormalization converges absolutely to a real number. Dublin provided the Schrödinger paradigm that defines Wavefunctions. But what of Hamilton’s contributions? I propose that these also bear on all problems concerning radially symmetric bodies because the Quaternion Ring encodes the symmetries of a sphere. In higher dimensions as in Al-Jaber’s case one may generalize to n-Onionic Rings (Mebius 2005). Degeneracy Pressure in dimension n should be invariant under nonsingular n-Onionic transforms in dimension n. This would demonstrate the physicality of the underlying renormalization model.

Works Cited.

Al-Jaber, S. M. (2008). “Degenerate electron gas and star stabilization in D dimensions”. Nuovo. Cimento., 123, 17.

Chandrasekhar, S. (1935). “The Highly Collapsed Configurations of a Stellar Mass (second paper)”. Monthly Notices of the Royal Astronomical Society. 95 (3): 207225.

Hawking, S. W. (1977). “Zeta function regularization of path integrals in curved spacetime”. In Euclidean quantum gravity (pp. 114–129).

Kenigson, J. (2022). “How Alive is the Quadrivium: Dr. Jonathan Kenigson on Quantum Advances in the Markets”. New York Weekly (Online). https://nyweekly.com/news/how-alive-is-the-quadrivium-dr-jonathan-kenigson-on-quantum-advances-in-the-markets/

Mebius, J. (2005). “A matrix-based proof of the quaternion representation theorem for four-dimensional rotations”. arXiv:math/0501249

Schrödinger, E. (1926). “Quantisierung als Eigenwertproblem”. Annalen der Physik. 384 (4): 273376.