Topology of the Quantum Dimension – Topology and Quantum Nodes

In U2-5 we classified standing waves in a usual two-dimensional sphere by the number l of nodal lines. Consider a circular line on the equator of the sphere, always for the state m=l. Each of the “l” nodal lines crosses this circular line on the equator twice. Therefore, on this equatorial line, 2 l nodal points arise, that is, always an even number.

All other states can be reached by applying the “d minus” operator. In the 360° world, this is the full spectrum of standing waves on a sphere in three dimensions. If we scale the distance to h bar, the quantum states arise, for example those from the orbitals as shown in U2-6.

What happens, however, if we go from the 360° world to the 720° world, that is, to the three-dimensional sphere in four dimensions? Now, spin states arise, which, as we have seen in U2-9, are actually nothing but usual standing waves in the 720°-world. Only due to the mapping into the 360°-world by winding the amplitude twice onto a circle in 360°, a double-valued amplitude arises with a single nodal point.

In 360°, the spin states thus fill the gaps: these states have an odd number of nodes in the 360°-world. The difference between quantum states with an even or odd number of nodes is huge: The so-called “bosons” are the states that are well known to us from the usual two-dimensional sphere, with an even number of nodes. At low temperatures, bosons can condense into the lowest quantum state; there is no ban on multiple occupations of the same state.

The so-called “fermions” have an odd number of nodes. Due to the double-valuedness, which arises upon projecting four dimensions onto three, each state observed in three dimensions can only be occupied at the most once – that is the so-called “Pauli exclusion principle”.

The Pauli exclusion principle is the basis for the stability of matter, as discussed in U2-4.

The deeper cause for the stability of the matter thus lies in the projection of the quantum states from the 720°-quantum dimension into the 360° world, and the associated double-valuedness of the spin, which we will examine more closely in the next slide.