A superconductor of any kind is nothing more or less than a material in which a particular symmetry of the laws of nature, electromagnetic gauge invariance, is spontaneously broken. ... These rotations act on a two-dimensional vector, whose two components are the real and imaginary parts of the electron field, the quantum mechanical operator that in quantum field theories of matter destroys electrons. The rotation angle of the broken symmetry group can vary with location in the superconductor, and then the symmetry transformations also affect the electromagnetic potentials ... The symmetry breaking in a superconductor leaves unbroken a rotation by 180°, which simply changes the sign of the electron field. In consequence of this spontaneous symmetry breaking, products of any even number of electron fields have non-vanishing expectation values in a superconductor, though a single electron field does not. All of the dramatic exact properties of superconductors – zero electrical resistance, the expelling of magnetic fields from superconductors known as the Meissner effect, the quantization of magnetic flux through a thick superconducting ring, and the Josephson formula for the frequency of the AC current at a junction between two superconductors with different voltages – follow from the assumption that electromagnetic gauge invariance is broken in this way, with no need to inquire into the mechanism by which the symmetry is broken.

The fundamental qualitative difference between the superconducting and normal ground state wave function is produced when the large degeneracy of the single particle electron levels in the normal state is removed. If we visualize the Hamiltonian matrix which results from an attractive two-body interaction in the basis of normal metal configurations, we find in this enormous matrix, sub-matrices in which all single-particle states except for one pair of electrons remain unchanged. These two electrons can scatter via the electron-electron interaction to all states of the same total momentum. We may envisage the pair wending its way (so to speak) over all states unoccupied by other electrons.

This ability to liquefy at such low temperatures makes helium a superconductor of electricity. For example, liquid helium is used as a coolant in magnetic resonance imaging, a breakthrough technology that generates high-resolution images of the human body, avoiding the need for radiation.

Ideas and techniques known in quantum electrodynamics have been applied to the Bardeen-Cooper-Schrieffer theory of superconductivity. In an approximation which corresponds to a generalization of the Hartree-Fock fields, one can write down an integral equation defining the self-energy of an electron in an electron gas with phonon and Coulomb interaction. The form of the equation implies the existence of a particular solution which does not follow from perturbation theory, and which leads to the energy gap equation and the quasi-particle picture analogous to Bogolyubov's.

The long and imposing list of physicists (among them Bohr, Heisenberg and Feynman) who had tried or were trying their hand at superconductivity should have given me pause. Even Einstein, in 1922 — before the quantum theory of metals was in place — had attempted to construct a theory of superconductivity. Fortunately, I was unaware of these many unsuccessful attempts. So when John invited me to join him (he, somehow, neglected to mention these previous efforts), I decided to take the plunge.

As was emphasized by Feynman ... among others, the physics of superfluidity lies not in the presence of gapless excitations, but in the paucity of gapless excitations. (After all, the Fermi liquid has a continuum of gapless modes.) There are too few modes that the superfluid can lose energy and momentum to.

Consider a pair of electrons which interact above a quiescent Fermi sphere with an interaction of the kind that might be expected due to the phonon and the screened Coulomb fields. If there is a net attraction between the electrons, it turns out that they can form a bound state, though their total energy is larger than zero. The properties of a noninteracting system of such bound pairs are very suggestive of those which could produce a superconducting state.

In the last few years, an interesting new subject, the study of topological quantum fluids, has emerged. Examples of topological quantum fluids include the Hall fluid, the chiral spin fluid, and the anyon superfluid.

The Higgs boson is an essential part of the analogy to the Meissner effect in superconductivity that leads us to an excellent understanding of the masses of the electroweak gauge bosons W^± and Z⁰ as consequences of electroweak symmetry breaking.

He's like a storybook spirit, a little djinn or something, except instead of air or water his element is imagination.

[Hallucinogenic drugs] now formed a circle, one could almost say a magic circle: the starting point had been the synthesis of lysergic acid amides, among them the naturally occurring ergot alkaloid ergobasin. This led to the synthesis of lysergic acid diethylamide, LSD. The hallucinogenic properties of LSD were the reason why the hallucinogenic magic mushroom teonanacatl found its way into my laboratory. THe work with teonanacatl, from which psilocybin and psilocin were isolated, proceeded to the investigation of another Mexican magic drug, ololiuhqui, in which hallucinogenic principles in the form of lysergic acid amides were again encountered, including ergobasin — with which the magic circle closed.

Ordinarily he was insane, but he had lucid moments when he was merely stupid.

When I arrived in Leipzig, Heisenberg was working on the theory of ferromagnetism. It was known the magnetism of such substances as iron was due to the "spin" of the electrons inside the substance. Each electron spins like a little top, and in the iron there is a "molecular field", a force that tends to align the spin of each electron with that of its neighbors. But the nature of this field was unknown. It could not be a magnetic effect because magnetic forces are much too weak to account for the observed behaviour. Heisenberg saw that the answer lay in the Pauli exclusion principle, which says that no two electrons can be in exactly the same state. Thus two electrons with the same spin orientation keep out of each other's way; while this repulsion may increase their energy of motion, it diminishes their mutual repulsion, and can therefore lead to a decrease in total energy, making the parallel alignment of the electron spins energetically favourable. He had encountered this mechanism in the theory of atomic spectra and concluded that it was also responsible for ferromagnetism.

Perhaps a lunatic was simply a minority of one.