Goldstone theorem
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| Goldstone theorem | |
|---|---|
| Name | Goldstone theorem |
| Field | Theoretical physics |
| Introduced | 1961 |
| Discovered by | Jeffrey Goldstone |
| Related | Spontaneous symmetry breaking, Nambu–Goldstone boson, Higgs mechanism, Anderson mechanism |
Goldstone theorem
The Goldstone theorem is a result in theoretical physics that guarantees gapless excitations when continuous symmetries are spontaneously broken in relativistic quantum field theories. It links work by Jeffrey Goldstone with concepts developed by Yoichiro Nambu, Jona-Lasinio model, Peter Higgs, Philip Anderson, and influenced later developments involving the Higgs boson, Anderson mechanism, and aspects of condensed matter physics and particle physics phenomenology. The theorem underpins reasoning used in research at institutions such as CERN, Fermilab, SLAC National Accelerator Laboratory, and Institute for Advanced Study.
Statement of the theorem
In relativistic quantum field theory, suppose a continuous global symmetry associated with a Lie group such as SU(2), U(1), SO(3), or SU(3) is realized in the Lagrangian but not in the vacuum state. Then the Goldstone theorem asserts the existence of one massless scalar particle per broken generator, often called a Nambu–Goldstone boson, appearing in spectra studied at places like Brookhaven National Laboratory and DESY. The statement is formalized using conserved currents tied to Noether's theorem originally associated with Emmy Noether and applied in contexts explored by Murray Gell-Mann and Richard Feynman. In systems described by relativistic invariance a one-to-one correspondence between broken generators and gapless modes holds; modifications arise in nonrelativistic settings studied by Philip W. Anderson and others.
Physical significance and applications
Goldstone modes explain low-energy behavior in contexts ranging from spontaneous magnetization in materials probed at Bell Labs and Los Alamos National Laboratory to long-wavelength phonons and magnons measured in experiments at MIT and University of Cambridge. In particle physics, Goldstone bosons inform chiral dynamics of pions within the chiral symmetry framework developed by Steven Weinberg and Murray Gell-Mann and underlie models such as the linear sigma model and chiral perturbation theory. The theorem's implications shaped searches for the Higgs boson at Large Hadron Collider experiments like ATLAS and CMS, and influenced theoretical tools used at Perimeter Institute and Kavli Institute for Theoretical Physics. In condensed matter, applications include superfluidity studies linked to Lev Landau, superconductivity insights tied to John Bardeen, Leon Cooper, and Robert Schrieffer, and topological phases investigated at Institute for Quantum Information and Matter.
Proofs and formal derivations
Rigorous proofs use canonical commutation relations, current algebra techniques developed by Murray Gell-Mann and Shmuel Weinberg, and path integral formalisms refined by Richard Feynman and Julian Schwinger. Early derivations employed operator methods akin to those by Jeffrey Goldstone and later generalizations invoked Ward–Takahashi identities attributed to John Clive Ward and Yasutaka Takahashi. Nonperturbative approaches utilize effective action methods popularized by Roman Jackiw and Joseph Polchinski, while axiomatic frameworks draw on insights from Gerard 't Hooft and Alexander Polyakov. Lattice regularizations used in numerical proofs are implemented in programs at CERN and Brookhaven influenced by work of Kenneth Wilson.
Examples and models
Classic examples include the linear sigma model developed in analogy to ideas by Yoichiro Nambu and Gell-Mann, where spontaneous breaking of SU(2)_L × SU(2)_R produces massless pions in chiral limits explored by Steven Weinberg. The XY model and Heisenberg model in statistical mechanics, studied by researchers at Princeton University and Harvard University, exhibit Goldstone modes manifesting as spin waves investigated by Philip W. Anderson and Lev Landau. Superfluid helium systems studied by Pierre-Gilles de Gennes and Lev Landau display phonon modes reflecting broken U(1) symmetry, while superconductors described by the BCS theory of Bardeen, Cooper, and Schrieffer show the Anderson mechanism that alters naive Goldstone counting. Nonlinear sigma models used by Curtis Callan, Harvey Coleman, and Sidney Coleman illustrate how target-space geometry affects mode structure.
Extensions and related results
Beyond the original relativistic result, generalized counting rules for Nambu–Goldstone modes were established in work by Hidetsugu Watanabe, Tomotaka Brauner, and Yukawa Institute collaborators, relevant to cold atom experiments at JILA and NIST. The Higgs mechanism formulated by Peter Higgs, François Englert, and Robert Brout shows how gauge symmetry converts would-be Goldstone bosons into massive gauge bosons observed in electroweak experiments associated with CERN and Fermilab. Anomalies in current conservation studied by Adler and Bell modify naive conclusions, while topological solitons from work by Alexander Polyakov and Gerard 't Hooft interplay with broken symmetries in ways that produce localized excitations rather than gapless spectra. Modern developments connect Goldstone physics to topological order studied at Microsoft Station Q and Institute for Quantum Information.
Historical development and attribution
The conceptual roots lie in spontaneous symmetry breaking introduced in condensed matter contexts by Lev Landau and formalized in particle physics by Yoichiro Nambu and collaborators. Jeffrey Goldstone's 1961 papers provided a clear field-theoretic statement and derivation that stimulated subsequent contributions from Jeffrey Goldstone, Yoichiro Nambu, Yoichiro Nambu and Giovanni Jona-Lasinio, Peter Higgs, François Englert, and Robert Brout. The interplay between condensed matter insights by Philip Anderson and particle-theory formulations by Steven Weinberg and Gerard 't Hooft shaped the pathway to the electroweak theory developed by Sheldon Glashow, Abdus Salam, and Steven Weinberg. Recognition of these ideas influenced prize awards such as the Nobel Prize in Physics conferred to individuals for related discoveries performed at institutions including Imperial College London and Cambridge University.