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Chapter 1
Complex geometry of nuclei and atoms∗
M. F. Atiyah
School of Mathematics, University of Edinburgh,
James Clerk Maxwell Building,
Peter Guthrie Tait Road, Edinburgh EH9 3FD, UK [email protected] N. S. Manton
Department of Applied Mathematics and Theoretical Physics,
University of Cambridge,
Wilberforce Road, Cambridge CB3 0WA, UK [email protected] We propose a new geometrical model of matter, in which neutral atoms are modelled by compact, complex algebraic surfaces. Proton and neutron numbers are determined by a surface’s Chern numbers. Equivalently, they are determined by combinations of the Hodge numbers, or the Betti numbers. Geometrical constraints on algebraic surfaces allow just a finite range of neutron numbers for a given proton number. This range encompasses the known isotopes.
Keywords: Atoms; nuclei; algebraic surfaces; 4-manifolds.
PACS numbers: 02.40.Tt, 02.40.Re, 21.60.–n
1.Introduction
It is an attractive idea to interpret matter geometrically, and to identify conserved attributes of matter with topological properties of the geometry. Kelvin made the pioneering suggestion to model atoms as knotted vortices in an ideal fluid.1 Each atom type would correspond to a distinct knot, and the conservation of atoms in physical and chemical processes (as understood in the 19th century) would follow from the inability of knots to change their topology. Kelvin’s model has not survived because atoms are now known to be structured and divisible, with a nucleus formed of protons and neutrons bound together, surrounded by electrons. At high energies, these constituents can be separated. It requires of order 1 eV to remove an electron from an atom, but a few MeV to remove a proton or neutron from a nucleus.
Atomic and nuclear physics has progressed, mainly by treating protons, neutrons and electrons as point particles, interacting through electromagnetic and strong nuclear forces.2 Quantum mechanics is an essential ingredient, and leads to a discrete spectrum of energy levels, both for the electrons and nuclear particles. The nucleons (protons and neutrons) are themselves built from three pointlike quarks, but little understanding of nuclear structure and binding has so far emerged from quantum chromodynamics (QCD), the theory of quarks. These point particle models are conceptually not very satisfactory, because a point is clearly an unphysical idealisation, a singularity of matter and charge density. An infinite charge density causes difficulties both in classical electrodynamics3 and in quantum field theories of the electron. Smoother structures carrying the discrete information of proton, neutron and electron number would be preferable.
In this paper, we propose a geometrical model of neutral atoms where both the proton number P and neutron number N are topological and none of the constituent particles are pointlike. In a neutral atom the electron number is also P, because the electron’s electric charge is exactly the opposite of the proton’s charge. For given P, atoms (or their nuclei) with different N are known as different isotopes.
A more recent idea than Kelvin’s is that of Skyrme, who proposed a nonlinear field theory of bosonic pion fields in 3 + 1 dimensions with a single topological invariant, which Skyrme identified with baryon number.
4,5 Baryon number (also called atomic mass number) is the sum of the proton and neutron numbers,
B =
P +
N. Skyrme’s baryons are solitons in the field theory, so they are smooth, topologically stable field configurations. Skyrme’s model was designed to model atomic nuclei, but electrons can be added to produce a model of a complete atom. Protons and neutrons can be distinguished in the Skyrme model, but only after the internal rotational degrees of freedom are quantised.
6 This leads to a quantised “isospin,” with the proton having isospin up
I3 =
and the neutron having isospin down
I3 = −
, where
I3 the third component of isospin. The model is consistent with the well-known Gell-Mann–Nishijima relation
7where
Q is the electric charge of a nucleus (in units of the proton charge) and
B is the baryon number.
Q is integral, because
I3 is integer-valued (half-integer-valued) when
B is even (odd).
Q equals the proton number
P of the nucleus and also the electron number of a neutral atom. The neutron number is
N =
B −
I3. The Skyrme model has had considerable success providing models for nuclei.
8–11 Despite the pion fields being bosonic, the quantised Skyrmions have half-integer spin if
B is odd.
12 But a feature of the model is that proton number and neutron number are not separately topological, and electrons have to be added on.
The Skyrme model has a relation to 4-dimensional fields that provides some motivation for the ideas discussed in this paper. A Skyrmion can be well approximated by a projection of a 4-dimensional Yang–Mills field. More precisely, one can take an SU(2) Yang–Mills instanton and calculate its holonomy along all lines in the (Euclidean) time direction.13 The result is a Skyrme field in 3-dimensional space, whose baryon number B equals the instanton number.
So a quasi-geometrical structure in 4-dimensional space (a Yang–Mills instanton in flat R4) can be closely related to nuclear structure, but still there is just one topological charge. A next step, first described in Ref. 14, was to propose an identification of smooth, curved 4-manifolds with the fundamental particles in atoms — the proton, neutron and electron. Suitable examples of manifolds were suggested. These manifolds were not all compact, and the particles they modelled were not all electrically neutral. One of the more compelling examples was Taub-NUT space as a model for the electron. By studying the Dirac operator on the Taub-NUT background, it was shown how the spin of the electron can arise in this context.15 There has also been an investigation of multi-electron systems modelled by multi-Taub-NUT space.16,17 However, there are some technical difficulties with the models of the proton and neutron, and no way has yet been found to geometrically combine protons and neutrons into more complicated nuclei surrounded by electrons. Nor is it clear in this context what exactly should be the topological invariants representing proton and neutron number.
A variant of these ideas is a model for the simplest atom, the neutral hydrogen atom, with one proton and one electron. This appears to be well modelled by CP2, the complex projective plane. The fundamental topological property of CP2 is that it has a generating 2-cycle with self-intersection 1. The second Betti number is b2 = 1, which splits into b2+ = 1 and b2− = 0. A complex line in CP2 represents this cycle, and in the projective plane, two lines always intersect in one point. A copy of this cycle together with its normal neighbourhood can be interpreted as the proton part of the atom, whereas the neighbourhood of a point dual to this is interpreted as the electron. The neighbourhood of a point is just a 4-ball, with a 3-sphere boundary, but this is the same as in the Taub-NUT model of the electron, which ...
