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| Nambu–Goto action | |
|---|---|
| Name | Nambu–Goto action |
| Field | Theoretical Physics |
| Introduced | 1970 |
| Introduced by | Yoichiro Nambu; Tetsuo Goto |
Nambu–Goto action
The Nambu–Goto action is a classical action functional for relativistic strings introduced in 1970 by Yoichiro Nambu and Tetsuo Goto that describes the dynamics of one-dimensional objects propagating in spacetime. It encodes the principle that a string sweeps out a minimal-area worldsheet in a background spacetime and serves as a cornerstone in the development of String theory, influencing work at institutions such as the CERN and Princeton University and informing research by figures including Edward Witten, Michael Green, John Schwarz, Leonard Susskind, and David Gross.
The Nambu–Goto action models a classical relativistic string by assigning an action proportional to the area of the worldsheet embedded in a target spacetime used in formulations at places like Harvard University, Massachusetts Institute of Technology, California Institute of Technology, Stanford University, and University of Cambridge. Early proponents such as Yoichiro Nambu and Tetsuo Goto framed the action in analogy with the Dirac action for membranes and in the same era that theorists like Gerard 't Hooft, Alexander Polyakov, Stephen Hawking, Subrahmanyan Chandrasekhar, and Roger Penrose advanced adjacent fields. The action connects to classical results by Joseph Louis Lagrange and Adrien-Marie Legendre in variational calculus and resonates with quantum developments by Paul Dirac, Richard Feynman, Murray Gell-Mann, and Stanislaw Ulam.
The action is derived by extremizing the invariant area of a two-dimensional surface traced by a string through a D-dimensional Minkowski or curved background spacetime, a formulation shared in contexts investigated at Yale University, Columbia University, University of Chicago, University of California, Berkeley, and University of Oxford. Starting from embedding coordinates X^μ(σ^a) with μ labeling target-space indices associated with groups studied at Los Alamos National Laboratory and σ^a parameterizing the worldsheet similar to parameterizations used by Carl Friedrich Gauss and Bernhard Riemann, the action equals the string tension times the integral of the square root of the determinant of the induced metric, echoing extremal surface problems traced back to Joseph-Louis Lagrange and modernized by scholars linked to Max Planck Institute for Physics and SLAC National Accelerator Laboratory. Variational treatment yields the minimal-surface equations analogous to geodesic equations used by Elie Cartan and Tullio Levi-Civita in differential geometry.
The action exhibits local reparameterization invariance of the worldsheet and global Poincaré invariance of the target spacetime, symmetry structures central to research programs at Princeton Plasma Physics Laboratory and Institute for Advanced Study. Worldsheet diffeomorphism symmetry and Weyl invariance in closely related formulations underpin constrained Hamiltonian analyses developed by Paul Dirac and extended in canonical approaches by Richard Arnowitt, Stanley Deser, and Charles Misner. The presence of first-class constraints generates Virasoro-like algebraic structures familiar from studies by Miguel Virasoro collaborators, influencing canonical quantization methods pursued at Rutgers University and University of California, Santa Barbara. Gauge-fixing choices such as static gauge or light-cone gauge echo techniques applied by Julian Schwinger and Gerard 't Hooft in gauge theories.
Quantization of the action proceeds via canonical quantization, path-integral quantization, and covariant operator methods, techniques refined by researchers at Imperial College London, University of Tokyo, Max Planck Institute for Gravitational Physics, and Kavli Institute for Theoretical Physics. Canonical light-cone quantization yields a physical spectrum consistent with pioneering results obtained by Peter Goddard and Charles Thorn, while covariant quantization requires handling of the conformal anomaly and critical dimensions discovered in analyses by Miguel Virasoro, John Schwarz, Michael Green, and David Olive. BRST quantization methods, influenced by work at CERN and DESY, provide a systematic framework for dealing with gauge redundancies, with ghosts introduced in line with procedures developed by C. Becchi, A. Rouet, and R. Stora.
The Polyakov action, introduced by Alexander Polyakov and further studied by scholars at Landau Institute for Theoretical Physics and Yerevan Physics Institute, is classically equivalent to the Nambu–Goto action upon elimination of an auxiliary worldsheet metric, a correspondence exploited in conformal field theory work by Belavin, Polyakov, and Zamolodchikov. The Polyakov formulation makes Weyl invariance manifest, facilitating perturbative quantization and renormalization approaches pioneered by Kenneth Wilson, Gerard 't Hooft, and Alexander Migdal at laboratories and universities worldwide. The equivalence links to treatments of anomalies and modular invariance examined in the context of research by Eugene Wigner, Emmy Noether, Niels Bohr, and Werner Heisenberg.
Beyond fundamental String theory research at Institute of Theoretical Physics, Chinese Academy of Sciences and Perimeter Institute, the action informs models of cosmic strings investigated by Alexander Vilenkin and Tom Kibble and analogues in condensed-matter systems explored at Bell Laboratories and IBM Research. It underpins calculations of string scattering amplitudes building on techniques by Gabriele Veneziano, Leonard Susskind, and Miguel Virasoro and influences dualities studied by Juan Maldacena and Shiraz Minwalla. Phenomenological implications connect to studies at Fermilab, Brookhaven National Laboratory, and observational programs involving LIGO and Planck (spacecraft) where effective string-like descriptions occasionally arise.
Mathematically, the action sits at the intersection of minimal surface theory developed by Jesse Douglas, T. Rado, and Marston Morse and of two-dimensional conformal field theory cultivated by P. Ginsparg and Daniel Friedan. Generalizations include higher-dimensional brane actions inspired by Paul Townsend and Joseph Polchinski, sigma-model couplings to background fields paralleling work at Institute for Advanced Study, and supersymmetric extensions associated with Edward Witten and Nathan Seiberg. Twistor-inspired and integrability-based approaches connect to efforts by Roger Penrose and Ludwig Faddeev, while geometric quantization viewpoints tie into research at École Normale Supérieure and Collège de France.