PH YSI CAL REVIEW 130, NUMBER VOLUME 1 Plasmons, Gauge Invariance, and Mass p. W. ANDERsoN BdI TelePhoee Laboratories, MNrray IIN, (Received 8 November Department of Physics Mathematics and Astronomy ¹mJersey 1962) Schwinger has pointed out that the Yang-Mills vector boson implied by associating a g transformation with a conservation law (of baryonic charge, for instance) does not neces mass, if a certain criterion on the vacuum fluctuations of the generalized current is satisfied the theory of plasma oscillations is a simple nonrelativistic example exhibiting all of the fea ger's idea. It is also shown that Schwinger's criterion that the vector field m&0 implies spectrum before including the Yang-Mills interaction contains m=0, but that the examp ductivity illustrates that the physical spectrum need not. Some comments on the relationship ideas and the zero-mass difhculty in theories with broken symmetries are given. ECKXTLY, H-boson connections: from Bell Labs to Geneva Schwinger' has given an argument that associating a gauge strongly suggesting transformation with a local conservation law does not necessarily require the existence of a zero-mass vector boson. For instance, it had previously seemed impossible to describe the conservation of baryons in such a manner because of the absence of a zero-mass boson and of the accompanying forces. ' The long-range problem of the mass of the bosons represents the major stumbling block in Sakurai's attempt to treat the dynamics of strongly interacting particles in terms of the Yang-Mills gauge fields which seem to be required to accompany the known conserved currents of baryon number and hypercharge. ' (We use the term "YangMills" in Sakurai's sense, to denote any generalized gauge field accompanying a local conservation law. ) The purpose of this article is to point out that the familiar plasmon theory of the free-electron gas excalifornia theory Institute Technology emplifies Schwinger's in aof straightforward very PMA Salon, June 2013 manner. In the plasma, transverse electromagnetic waves do not propagate below the "plasma frequency, which is usually thought of as the frequency of longwavelength longitudinal oscillation of the electron gas. April 16 2014, maria spiropulu " is equivalent to the mass, w electrons leading to diverg fluctuations resembles the stro of Schwinger's theory. In s low-frequency photons, gauge conservation are clearly satisfi In fact, one can draw a di dielectric constant treatment Schwinger's argument. Schwin commutation relations for th one sum rule for the vacuum those for the matter field give value for the Auctuations of m is the source for A and the tw equations, the two sum rules a unless there is a contribution homogeneous, weakly interacti the 6eld equations. If, howe large enough, there can be n the massless solutions cannot The usual theory of the pla electromagnetic field quantumvacuum Quctuations; yet the
H-boson history march 25 2014,BU Physics Colloquium, maria spiropulu
Baird Physics Research Conference Philip Anderson Princeton University Thursday, February 27, 2014 4:00 PM Caltech Feynman Lecture Hall 201 East Bridge Refreshments served in 114 E. Bridge at 3:45 p.m. The Discovery of the Anderson-Higgs Mechanism Landau introduced the idea of the ground state of a condensed matter system as a vacuum and of the elementary excitations as quasiparticles moving in this vacuum. He and Tisza noted that spontaneous orderings such as magnetism could be thought of as spontaneous symmetry-breaking of this vacuum, and based theories of phase transitions on this idea. In studying antiferromagnetism, I realized that symmetrybreaking could also have dynamical consequences, and suggested for condensed matter what later came to be known as Goldstone s theorem. With the appearance of the BCS theory of superconductivity and Nambu s transcription of it into a theory of the interaction mass of hadrons, interest grew in broken symmetry in the real vacuum. At the same time, there was concern about gauge invariance of BCS; and papers by Nambu, Bogoliubov and Shirkov, and PWA in 1958 addressed that issue by studying the collective excitation spectrum; but only PWA correctly included the electromagnetic interaction and found an empty energy gap: Goldstone s theorem fails! When I learned in 1962 that Goldstone s theorem was an obstacle to serious theories, I tried as best I could to explain the physics in quasirelativistic terms, hence my 1963 paper. It can be seen as successfully predicting heavy gauge bosons, and PWA 1958 even contains a brief remark on a Higgs; Littlewood and Varma claim to have found such an object in the 90 s. At least two of the three Higgs groups were quite familiar with either the 63 or 58 paper. As for me, I was too aware of the zero-point energy problem and busy doing other things.
Examples of the zeroth theorem of the history of science J. D. Jackson a Department of Physics, University of California, Berkeley and Lawrence Berkeley National Laboratory, Berkeley, California 94720 Received 11 October 2007; accepted 10 March 2008 Discoveries, rules, or insights that are named after someone (often) did not originate with that person Avogadro s number was first determined by Loschmidt Olber s paradox was first mentioned by Kepler The Dirac delta function was invented by Heaviside Lorentz gauge was invented by Lorenz etc.
Nambu (1960) The importance of Spontaneous Symmetry Breaking Physical system Ferromagnets Crystals Superconductors Broken symmetry Rotational invariance with respect to spin Translational and rotational invariance modulo discrete values Local gauge invariance particle number Apply condensed matter ideas to particle physics Now the quantum vacuum is the medium
Goldstone s Sombrero The Higgs field effective potential describes the energetics of turning on the field to a certain (complex) value Even before including quantum corrections, the scalar field selfinteractions may energetically favor a nonzero vev Because of the symmetry there are degenerate vacua And it is usually claimed that spontaneous symmetry breaking is obvious V( ϕ) NOT SO Re ϕ Im ϕ Figure 2. The Mexican-hat potential energy density considered by
Goldstone s Sombrero V( ϕ) Re ϕ Im ϕ Figure 2. The Mexican-hat potential energy density considered by e.g in the double well quantum mechanics problem, there is a degeneracy associated with a Z 2 symmetry but the ground state is a superposition that preserves the symmetry!
V( ϕ) Re ϕ Figure 2. The Mexican-hat potential energy density considered by The key difference is that in quantum field theory it is much more difficult to transition from one one degenerate ground state to another The quantum vacuum is like a many-body system in this sense (complexity) Phillip Anderson emphasized in his 1972 article More is Different that spontaneous symmetry breaking is a property of large/complex systems. Im ϕ
Im ϕ V( ϕ) Goldstone s Sombrero Re ϕ Figure 2. The Mexican-hat potential energy density considered by Even though it is difficult to transition from one degenerate vacuum to another, there are single particle excitations corresponding to locally deforming along the trough These are the massless Goldstone bosons that Higgs and co. were so worried about Physical system Broken symmetry Goldstone modes Antiferromagnets Rotational invariance spin waves Crystals Translational and rotational acoustic phonons BCS Superconductors U(1) phase symmetry???