Getting a handle on Symmetries is an always work in progress for me, so as to keep abreast of the science and the theoretic involved.
Given string theory’s high ambition to account for all nature’s forces and particles, given the number of string theorists working to achieve that ambition, and given the general public’s interest in string theory, two stories in seven years might seem low. But is it? (See above link)
So traveling back in time, one can move forward.
Spontaneous symmetry breaking
Introduced into particle physics by Nambu in 1960, spontaneous symmetry breaking was to become a pillar of the field’s standard model, which since its completion in the mid-1970s has survived every experimental challenge. When a physical state does not exhibit all the symmetries of the dynamical laws that govern it, the violated symmetries are said to be spontaneously broken.
The idea had been around for a long time in classical mechanics, fluid dynamics, and condensed-matter physics. An oft-cited example is ferromagnetism. Its underlying laws of atomic physics are absolutely invariant under rotation. Nonetheless, below a critical temperature the atomic spins spontaneously line up in some arbitrary direction to create a state that is not rotationally symmetric. Similarly, the cylindrical symmetry of a state in which a pencil is perfectly poised on its tip is spontaneously broken when the pencil inevitably falls over. But such examples give little hint of the subtlety and power of the notion once Nambu began exploiting it in quantum field theory.
It began with a paper Nambu wrote in 1959 about gauge invariance in superconductivity.1 The paper exhibits his virtuosity in two disparate specialties—quantum field theory and condensed-matter theory. He became conversant with both as a graduate student at the University of Tokyo after he was mustered out of the army in 1945. Eventually he began working with the group around Sin-itiro Tomonaga, one of the creators of modern quantum electrodynamics (QED). Tomonaga was actually based at another university in Tokyo. But the University of Tokyo was strong in condensed-matter physics. So Nambu started out working on the Ising model of ferromagnetism.
After two years at the Institute for Advanced Study in Princeton, Nambu came to the University of Chicago in 1954, just before the untimely death of Enrico Fermi. When John Bardeen, Leon Cooper, and Robert Schrieffer published their theory of superconductivity in 1957, Nambu and others noted that the BCS superconducting ground state lacked the gauge invariance of the underlying electromagnetic theory. In classical electrodynamics, gauge invariance refers to the freedom one has in choosing the vector and scalar potentials. In QED that freedom is linked to the freedom to change the phase of the electron wavefunction arbitrarily from point to point in space. Did the gauge-symmetry violation mean that the BCS theory was simply wrong? Or perhaps superconductivity was a manifestation of some yet unknown force beyond electromagnetism and atomic physics.
Having heard Schrieffer give a talk about the new theory in 1957 without mentioning gauge invariance, Nambu spent the next two years thinking about its role in the theory. He recast the BCS theory into the perturbative quantum-field-theoretic formalism with which Richard Feynman had solved—independently of Tomonaga—the problem of the intractable infinities in QED. From that reformulation, Nambu concluded that the superconducting ground state results from the spontaneous breaking of the underlying gauge symmetry. He showed that all the characteristic manifestations of superconductivity—including the expulsion of magnetic flux and the energy gap that assures lossless current flow—follow simply from that spontaneous symmetry breaking.
See:Physics Nobel Prize to Nambu, Kobayashi, and Maskawa for theories of symmetry breaking by Bertram Schwarzschild Physics Today and references cited in article below.
- 1. Y. Nambu, Phys. Rev. 117, 648 (1960) [SPIN].
- 2. Y. Nambu, Phys. Rev. Lett. 4, 380 (1960) [SPIN].
- 3. Y. Nambu, G. Jona-Lasinio, Phys. Rev. 122, 345 (1961) [SPIN]; 124, 246 (1961) [SPIN].
- 4. P. W. Anderson, Phys. Rev. 130, 439 (1963) [SPIN].
- 5. F. Englert, R. Brout, Phys. Rev. Lett. 13, 321 (1964) [SPIN]; P. W. Higgs, Phys. Rev. Lett. 13, 508 (1964) [SPIN]; G. S. Guralelnik, C. R. Hagen, T. W. B. Kibble, Phys. Rev. Lett. 13, 585 (1964) [SPIN].
- 6. S. Weinberg, Phys. Rev. Lett. 19, 1264 (1967) [SPIN].
- 7. M. Kobayashi, T. Maskawa, Prog. Theor. Phys. 49, 652 (1973) .
- 8. S. L. Glashow, J. Iliopoulos, L. Maiani, Phys. Rev. D 2, 1285 (1970) [SPIN].