Hilbert action

Hilbert action
Name	Hilbert action
Field	Theoretical physics, Differential geometry
Introduced	1915
Introduced by	David Hilbert
Related	Einstein–Hilbert action, Hilbert space, Riemann curvature

Hilbert action The Hilbert action is a variational functional introduced in the early twentieth century that plays a central role in the mathematical formulation of classical field theories and geometric analysis. It connects the work of David Hilbert, Bernhard Riemann, Élie Cartan, Felix Klein, and Albert Einstein through a synthesis of differential geometry, tensor calculus, and variational methods developed during the relativity revolution and subsequent mathematical physics research at institutions such as the University of Göttingen, Prussian Academy of Sciences, and University of Berlin.

History and origin

The origin of the Hilbert action traces to contributions by David Hilbert, Albert Einstein, Marcel Grossmann, Hermann Minkowski, and Bernhard Riemann amid exchanges at the Kaiser Wilhelm Institute for Mathematics and correspondences preserved in archives at the Royal Society and Max Planck Society. Influences include classical works by Carl Friedrich Gauss, Adrien-Marie Legendre, Joseph-Louis Lagrange, William Rowan Hamilton, and later formalization by Élie Cartan and Hermann Weyl during lectures at the ETH Zurich and Princeton University. Debates involving Emmy Noether, Felix Klein, Max Born, and contemporaries over variational principles and conservation laws shaped its acceptance in the curricula of University of Göttingen, University of Leipzig, and University of Cambridge.

Definition and mathematical formulation

Formally built from the scalar curvature and the metric tensor, the Hilbert action uses constructs from Riemannian geometry, Ricci curvature, Levi-Civita connection, tensor calculus, and the calculus of variations developed by Joseph-Louis Lagrange, Pierre-Simon Laplace, Adrien-Marie Legendre, and William Rowan Hamilton. The integrand involves the volume element associated with the metric and objects studied by Bernhard Riemann, Gregorio Ricci-Curbastro, Tullio Levi-Civita, and Élie Cartan as used in expositions by J. L. Synge and A. Schild. Works appearing in proceedings of the Prussian Academy of Sciences, Proceedings of the Royal Society, and monographs from Cambridge University Press and Springer contain detailed derivations and coordinate expressions using techniques from Finsler geometry and methods championed by André Weil.

Relation to Einstein–Hilbert action and general relativity

The Hilbert action is closely related to the Einstein–Hilbert action introduced in the same historical period and used in formulations by Albert Einstein, David Hilbert, Marcel Grossmann, Arthur Eddington, and Willem de Sitter for general relativity. Its adoption influenced research programs at Princeton University, Cambridge University, California Institute of Technology, and institutes such as the Institute for Advanced Study where figures like John Archibald Wheeler, Robert Oppenheimer, and Subrahmanyan Chandrasekhar lectured on gravitational theory. Debates involving Emmy Noether and Felix Klein clarified the role of diffeomorphism invariance discussed in seminars at Imperial College London and University of Chicago.

Variational principle and field equations

Applying the variational principle to the Hilbert action yields field equations through methods pioneered by Joseph-Louis Lagrange, William Rowan Hamilton, Sofia Kovalevskaya, and formalized by Emmy Noether and Felix Klein. The stationarity condition under metric variations leads to equations analogous to those derived by Albert Einstein and cast in tensorial form by Gregorio Ricci-Curbastro and Tullio Levi-Civita. Techniques from functional analysis advanced at Harvard University, Massachusetts Institute of Technology, and University of Paris are used to handle the Euler–Lagrange equations and boundary-value problems central to work by André Lichnerowicz and Yakov Frenkel.

Symmetries, conservation laws, and boundary terms

Symmetries of the Hilbert action under diffeomorphisms link to conservation statements articulated by Emmy Noether and debated by Felix Klein, with implications for conserved currents discussed by Arthur Eddington, Richard Feynman, and John Bell. Treatment of boundary terms in variational procedures invoked methods from George David Birkhoff, James Clerk Maxwell, and modern regularization approaches developed at CERN, Stanford Linear Accelerator Center, and research groups at Perimeter Institute. The role of Gibbons–Hawking–York type boundary contributions and counterterms has been examined in contexts involving Stephen Hawking, Gary Gibbons, and James York.

Applications and extensions

Extensions of the Hilbert action underpin research in quantum gravity, string theory, Kaluza–Klein theory, and modified gravity models pursued at CERN, Institute for Advanced Study, California Institute of Technology, Kavli Institute for Theoretical Physics, and universities such as University of Cambridge and Harvard University. Applications include cosmological models used by Edwin Hubble, Georges Lemaître, Alan Guth, Andrei Linde, and compact-object studies involving Subrahmanyan Chandrasekhar, Kip Thorne, and Roger Penrose. Generalizations incorporate higher-curvature terms studied by David Lovelock, B. Zwiebach, Juan Maldacena, and phenomenology explored at SLAC National Accelerator Laboratory.

Mathematical properties and regularization methods

Analysis of the Hilbert action draws on spectral theory from work by John von Neumann, David Hilbert, Israel Gelfand, and Marcel Riesz, and on renormalization methods developed by Kenneth Wilson, Gerard 't Hooft, and Murray Gell-Mann. Regularization and functional determinants are treated using techniques from Atiyah–Singer index theory, heat-kernel methods by H. P. McKean, and zeta-function regularization employed in studies by Stephen Hawking and Julian Schwinger. Mathematical rigor for boundary-value formulations has been advanced by André Lichnerowicz, Richard Palais, and researchers affiliated with Institut des Hautes Études Scientifiques and Courant Institute.

Category:Variational principles Category:Mathematical physics