Dirac–Born–Infeld action

Dirac–Born–Infeld action
Name	Dirac–Born–Infeld action
Field	Theoretical physics
Introduced	1934
Inventor	Paul Dirac, Max Born, Leopold Infeld
Related	String theory, D-brane, Born–Infeld electrodynamics, Nambu–Goto action

Dirac–Born–Infeld action The Dirac–Born–Infeld action is a nonlinear action principle originally formulated to regularize singularities in classical electrodynamics and later repurposed as an effective action for extended objects in string theory, M-theory, and related frameworks. It appears in analyses of D-brane dynamics, tachyon condensation, and low-energy limits of superstring theory compactifications, providing a geometric and gauge-invariant description of worldvolume dynamics. The action has connections to historical work by Paul Dirac, Max Born, and Leopold Infeld and modern developments involving Edward Witten, Joseph Polchinski, and others.

Introduction

The Dirac–Born–Infeld action arose from efforts by Paul Dirac and contemporaries to address self-energy problems in the electron model and was formalized by Max Born and Leopold Infeld as a nonlinear modification of Maxwell's theory. It reemerged in the context of string theory when the effective worldvolume action of D-branes was identified with the DBI functional, linking to constructs from Nambu–Goto action and proposals by T. Banks, M. R. Douglas, and N. Seiberg. Researchers including André Neveu, John H. Schwarz, Michael Green, Edward Witten, Joseph Polchinski, Ashoke Sen, and Cumrun Vafa elaborated its role in modern high-energy theory and supersymmetry.

Historical development

The origin traces to work by Paul Dirac on the electron model and to an influential paper by Max Born and Leopold Infeld seeking finite self-energy for the electron, predating developments in quantum electrodynamics by Richard Feynman and Sin-Itiro Tomonaga. Rediscovery in string theory occurred through studies by Joseph Polchinski on D-branes and by Edward Witten on effective actions, with important contributions from Ashoke Sen on tachyon dynamics and Juan Maldacena in the context of the AdS/CFT correspondence. Subsequent work by Nathan Seiberg, Leonard Susskind, Michael Douglas, Paul Townsend, and Andrei Losev extended the formalism to brane intersections, noncommutative geometry inspired by Alain Connes, and duality symmetries explored by Peter West and Chris Hull.

Mathematical formulation

The DBI action for a p-dimensional brane in a target spacetime with metric g_{μν} and gauge field F_{ab} is constructed from the determinant of the induced metric plus gauge contributions, paralleling the structure of the Nambu–Goto action and borrowing techniques from differential geometry used by S. Kobayashi and K. Nomizu. In supersymmetric settings one supplements with a Wess–Zumino term as developed in works by Bruno Zumino and Jerome Polchinski. The determinant structure links to variational principles studied by David Hilbert and Emmy Noether, while gauge invariances relate to analyses by Hermann Weyl and Élie Cartan. For inclusion of fermions, constructions rely on spinor methods advanced by Élie Cartan, Hermann Weyl, and Paul Dirac, and superspace techniques championed by Salam and John Strathdee.

Physical interpretations and applications

Physically, the DBI action encodes nonlinear electromagnetic responses analogous to those sought by Max Born and Leopold Infeld and models brane tension and coupling to background fields as in analyses by Joseph Polchinski and Edward Witten. It is used to study soliton solutions and BPS states in the tradition of Stanley Mandelstam and Murray Gell-Mann, and to analyze holographic probes in Juan Maldacena’s AdS/CFT correspondence with applications exploited by Ofer Aharony, Steven Gubser, Igor Klebanov, and Alexander Polyakov. Phenomenological uses include models of inflation inspired by Andrei Linde and Alan Guth, dark energy constructions considered by Sean Carroll, and condensed matter analogues investigated by Philip Anderson and Subir Sachdev.

Examples in string theory and brane dynamics

Specific realizations appear for D0-brane quantum mechanics studied by Tom Banks and W. Fischler, for D1-brane systems connected to Polyakov action analyses by Alexander Polyakov, and for higher Dp-brane configurations elaborated by Joseph Polchinski and Paul Townsend. Intersections and bound states are treated using techniques developed by Andrew Strominger, Curtis Callan, and Gary Horowitz, and tachyon condensation scenarios analyzed by Ashoke Sen employ DBI-like effective actions. Duality relations invoking T-duality and S-duality studied by Ed Witten and Cumrun Vafa further situate DBI terms within the web of string dualities.

Quantization and perturbative expansions

Quantization approaches range from semiclassical expansions around classical DBI solutions used by Gerard ’t Hooft and Alexander Belavin to effective field theory treatments relying on renormalization methods developed by Kenneth Wilson and John Zinn-Justin. Perturbative expansions in derivatives produce higher-derivative corrections calculated in frameworks by Michael Green and John Schwarz, while nonperturbative effects are investigated via instanton techniques pioneered by Gregory Moore and Edward Witten. Noncommutative deformations connected to Seiberg–Witten map work by Nathan Seiberg and Edward Witten alter quantization and radiative corrections.

Extensions and generalizations

Generalizations include supersymmetric DBI actions constructed by Peter West and Paul Howe, non-Abelian DBI proposals examined by Andrei Tseytlin and Alexander Tseytlin, and curved background extensions related to General relativity work by Albert Einstein and geometric analyses by Bernhard Riemann. Relations to Born–Infeld electrodynamics in classical contexts recall the original Max Born program, while modern links to noncommutative geometry by Alain Connes and to topological string theory by Hirosi Ooguri provide fertile ground for future research. Recent advances tie DBI structures to investigations by Nathan Seiberg, Cumrun Vafa, Edward Witten, Joseph Polchinski, Ashoke Sen, Juan Maldacena, Leonard Susskind, Andrew Strominger, Alexander Polyakov, Michael Green, and John Schwarz.

Category:Theoretical physics