Supergravity

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https://en.wikipedia.org/wiki/Supergravity 

 

Beyond the Standard Model
Simulated Large Hadron Collider CMS particle detector data depicting a Higgs boson produced by colliding protons decaying into hadron jets and electrons

In theoretical physics, supergravity (supergravity theory; SUGRA for short) is a modern field theory that combines the principles of supersymmetry and general relativity; this is in contrast to non-gravitational supersymmetric theories such as the Minimal Supersymmetric Standard Model. Supergravity is the gauge theory of local supersymmetry. Since the supersymmetry (SUSY) generators form together with the Poincaré algebra and superalgebra, called the super-Poincaré algebra, supersymmetry as a gauge theory makes gravity arise in a natural way.

 

In theoretical physics, supergravity (supergravity theory; SUGRA for short) is a modern field theory that combines the principles of supersymmetry and general relativity; this is in contrast to non-gravitational supersymmetric theories such as the Minimal Supersymmetric Standard Model. Supergravity is the gauge theory of local supersymmetry. Since the supersymmetry (SUSY) generators form together with the Poincaré algebra and superalgebra, called the super-Poincaré algebra, supersymmetry as a gauge theory makes gravity arise in a natural way.

Gravitons

Like all covariant approaches to quantum gravity, supergravity contains a spin-2 field whose quantum is the graviton. Supersymmetry requires the graviton field to have a superpartner. This field has spin 3/2 and its quantum is the gravitino. The number of gravitino fields is equal to the number of supersymmetries.

History

Gauge supersymmetry

The first theory of local supersymmetry was proposed by Dick Arnowitt and Pran Nath in 1975 and was called gauge supersymmetry.

Supergravity

The first model of 4-dimensional supergravity (without this denotation) was formulated by Dmitri Vasilievich Volkov and Vyacheslav A. Soroka in 1973, emphasizing the importance of spontaneous supersymmetry breaking for the possibility of a realistic model. The minimal version of 4-dimensional supergravity (with unbroken local supersymmetry) was constructed in detail in 1976 by Dan Freedman, Sergio Ferrara and Peter van Nieuwenhuizen. In 2019 the three were awarded a special Breakthrough Prize in Fundamental Physics for the discovery. The key issue of whether or not the spin 3/2 field is consistently coupled was resolved in the nearly simultaneous paper, by Deser and Zumino, which independently proposed the minimal 4-dimensional model. It was quickly generalized to many different theories in various numbers of dimensions and involving additional (N) supersymmetries. Supergravity theories with N>1 are usually referred to as extended supergravity (SUEGRA). Some supergravity theories were shown to be related to certain higher-dimensional supergravity theories via dimensional reduction (e.g. N=1, 11-dimensional supergravity is dimensionally reduced on T⁷ to 4-dimensional, ungauged, N = 8 supergravity). The resulting theories were sometimes referred to as Kaluza–Klein theories as Kaluza and Klein constructed in 1919 a 5-dimensional gravitational theory, that when dimensionally reduced on a circle, its 4-dimensional non-massive modes describe electromagnetism coupled to gravity.

mSUGRA

mSUGRA means minimal SUper GRAvity. The construction of a realistic model of particle interactions within the N = 1 supergravity framework where supersymmetry (SUSY) breaks by a super Higgs mechanism carried out by Ali Chamseddine, Richard Arnowitt and Pran Nath in 1982. Collectively now known as minimal supergravity Grand Unification Theories (mSUGRA GUT), gravity mediates the breaking of SUSY through the existence of a hidden sector. mSUGRA naturally generates the Soft SUSY breaking terms which are a consequence of the Super Higgs effect. Radiative breaking of electroweak symmetry through Renormalization Group Equations (RGEs) follows as an immediate consequence. Due to its predictive power, requiring only four input parameters and a sign to determine the low energy phenomenology from the scale of Grand Unification, it is widely investigated in particle physics.

See also: Gravity-Mediated Supersymmetry Breaking in the MSSM

11D: the maximal SUGRA

Main article: Eleven-dimensional supergravity

One of these supergravities, the 11-dimensional theory, generated considerable excitement as the first potential candidate for the theory of everything. This excitement was built on four pillars, two of which have now been largely discredited:

• Werner Nahm showed 11 dimensions as the largest number of dimensions consistent with a single graviton, and more dimensions will show particles with spins greater than 2. However, if two of these dimensions are time-like, these problems are avoided in 12 dimensions. Itzhak Bars gives this emphasis.
• In 1981 Ed Witten showed 11 as the smallest number of dimensions big enough to contain the gauge groups of the Standard Model, namely SU(3) for the strong interactions and SU(2) times U(1) for the electroweak interactions. Many techniques exist to embed the standard model gauge group in supergravity in any number of dimensions like the obligatory gauge symmetry in type I and heterotic string theories, and obtained in type II string theory by compactification on certain Calabi–Yau manifolds. The D-branes engineer gauge symmetries too.
• In 1978 Eugène Cremmer, Bernard Julia and Joël Scherk (CJS) found the classical action for an 11-dimensional supergravity theory. This remains today the only known classical 11-dimensional theory with local supersymmetry and no fields of spin higher than two. Other 11-dimensional theories known and quantum-mechanically inequivalent reduce to the CJS theory when one imposes the classical equations of motion. However, in the mid-1980s Bernard de Wit and Hermann Nicolai found an alternate theory in D=11 Supergravity with Local SU(8) Invariance. While not manifestly Lorentz-invariant, it is in many ways superior, because it dimensionally-reduces to the 4-dimensional theory without recourse to the classical equations of motion.

• In 1980 Peter Freund and M. A. Rubin showed that compactification from 11 dimensions preserving all the SUSY generators could occur in two ways, leaving only 4 or 7 macroscopic dimensions, the others compact. The noncompact dimensions have to form an anti-de Sitter space. There are many possible compactifications, but the Freund-Rubin compactification's invariance under all of the supersymmetry transformations preserves the action.

Finally, the first two results each appeared to establish 11 dimensions, the third result appeared to specify the theory, and the last result explained why the observed universe appears to be four-dimensional.

Many of the details of the theory were fleshed out by Peter van Nieuwenhuizen, Sergio Ferrara and Daniel Z. Freedman.

The end of the SUGRA era

The initial excitement over 11-dimensional supergravity soon waned, as various failings were discovered, and attempts to repair the model failed as well. Problems included:

• The compact manifolds which were known at the time and which contained the standard model were not compatible with supersymmetry, and could not hold quarks or leptons. One suggestion was to replace the compact dimensions with the 7-sphere, with the symmetry group SO(8), or the squashed 7-sphere, with symmetry group SO(5) times SU(2).
• Until recently, the physical neutrinos seen in experiments were believed to be massless, and appeared to be left-handed, a phenomenon referred to as the chirality of the Standard Model. It was very difficult to construct a chiral fermion from a compactification — the compactified manifold needed to have singularities, but physics near singularities did not begin to be understood until the advent of orbifold conformal field theories in the late 1980s.
• Supergravity models generically result in an unrealistically large cosmological constant in four dimensions, and that constant is difficult to remove, and so require fine-tuning. This is still a problem today.
• Quantization of the theory led to quantum field theory gauge anomalies rendering the theory inconsistent. In the intervening years physicists have learned how to cancel these anomalies.

Some of these difficulties could be avoided by moving to a 10-dimensional theory involving superstrings. However, by moving to 10 dimensions one loses the sense of uniqueness of the 11-dimensional theory.

The core breakthrough for the 10-dimensional theory, known as the first superstring revolution, was a demonstration by Michael B. Green, John H. Schwarz and David Gross that there are only three supergravity models in 10 dimensions which have gauge symmetries and in which all of the gauge and gravitational anomalies cancel. These were theories built on the groups SO(32) and ${\displaystyle E_8\times E_8}$, the direct product of two copies of E₈. Today we know that, using D-branes for example, gauge symmetries can be introduced in other 10-dimensional theories as well.

The second superstring revolution

Initial excitement about the 10-dimensional theories, and the string theories that provide their quantum completion, died by the end of the 1980s. There were too many Calabi–Yaus to compactify on, many more than Yau had estimated, as he admitted in December 2005 at the 23rd International Solvay Conference in Physics. None quite gave the standard model, but it seemed as though one could get close with enough effort in many distinct ways. Plus no one understood the theory beyond the regime of applicability of string perturbation theory.

There was a comparatively quiet period at the beginning of the 1990s; however, several important tools were developed. For example, it became apparent that the various superstring theories were related by "string dualities", some of which relate weak string-coupling - perturbative - physics in one model with strong string-coupling - non-perturbative - in another.

Then the second superstring revolution occurred. Joseph Polchinski realized that obscure string theory objects, called D-branes, which he discovered six years earlier, equate to stringy versions of the p-branes known in supergravity theories. String theory perturbation didn't restrict these p-branes. Thanks to supersymmetry, p-branes in supergravity gained understanding well beyond the limits of string theory.

Armed with this new nonperturbative tool, Edward Witten and many others could show all of the perturbative string theories as descriptions of different states in a single theory that Witten named M-theory. Furthermore, he argued that M-theory's long wavelength limit, i.e. when the quantum wavelength associated to objects in the theory appear much larger than the size of the 11th dimension, needs 11-dimensional supergravity descriptors that fell out of favor with the first superstring revolution 10 years earlier, accompanied by the 2- and 5-branes.

Therefore, supergravity comes full circle and uses a common framework in understanding features of string theories, M-theory, and their compactifications to lower spacetime dimensions.

Relation to superstrings

The term "low energy limits" labels some 10-dimensional supergravity theories. These arise as the massless, tree-level approximation of string theories. True effective field theories of string theories, rather than truncations, are rarely available. Due to string dualities, the conjectured 11-dimensional M-theory is required to have 11-dimensional supergravity as a "low energy limit". However, this doesn't necessarily mean that string theory/M-theory is the only possible UV completion of supergravity; supergravity research is useful independent of those relations.

4D N = 1 SUGRA

Main article: 4D N = 1 supergravity

Before we move on to SUGRA proper, let's recapitulate some important details about general relativity. We have a 4D differentiable manifold M with a Spin(3,1) principal bundle over it. This principal bundle represents the local Lorentz symmetry. In addition, we have a vector bundle T over the manifold with the fiber having four real dimensions and transforming as a vector under Spin(3,1). We have an invertible linear map from the tangent bundle TM to T. This map is the vierbein. The local Lorentz symmetry has a gauge connection associated with it, the spin connection.

The following discussion will be in superspace notation, as opposed to the component notation, which isn't manifestly covariant under SUSY. There are actually many different versions of SUGRA out there which are inequivalent in the sense that their actions and constraints upon the torsion tensor are different, but ultimately equivalent in that we can always perform a field redefinition of the supervierbeins and spin connection to get from one version to another.

In 4D N=1 SUGRA, we have a 4|4 real differentiable supermanifold M, i.e. we have 4 real bosonic dimensions and 4 real fermionic dimensions. As in the nonsupersymmetric case, we have a Spin(3,1) principal bundle over M. We have an R^4|4 vector bundle T over M. The fiber of T transforms under the local Lorentz group as follows; the four real bosonic dimensions transform as a vector and the four real fermionic dimensions transform as a Majorana spinor. This Majorana spinor can be reexpressed as a complex left-handed Weyl spinor and its complex conjugate right-handed Weyl spinor (they're not independent of each other). We also have a spin connection as before.

We will use the following conventions; the spatial (both bosonic and fermionic) indices will be indicated by M, N, ... . The bosonic spatial indices will be indicated by μ, ν, ..., the left-handed Weyl spatial indices by α, β,..., and the right-handed Weyl spatial indices by ${\displaystyle \dot{\alpha }}$, ${\displaystyle \dot{\beta }}$, ... . The indices for the fiber of T will follow a similar notation, except that they will be hatted like this: ${\displaystyle \hat{M},\hat{\alpha }}$. See van der Waerden notation for more details. ${\displaystyle M=(\mu ,\alpha ,\dot{\alpha })}$. The supervierbein is denoted by ${\displaystyle e_N^{\hat{M}}}$, and the spin connection by ${\displaystyle {\omega }_{\hat{M}\hat{N}P}}$. The inverse supervierbein is denoted by ${\displaystyle E_{\hat{M}}^N}$.

The supervierbein and spin connection are real in the sense that they satisfy the reality conditions

${\displaystyle e_N^{\hat{M}}(x,\overline{\theta },\theta {)}^{\ast }=e_{N^{\ast }}^{{\hat{M}}^{\ast }}(x,\theta ,\overline{\theta })}$ where ${\displaystyle {\mu }^{\ast }=\mu }$, ${\displaystyle {\alpha }^{\ast }=\dot{\alpha }}$, and ${\displaystyle {\dot{\alpha }}^{\ast }=\alpha }$ and ${\displaystyle \omega (x,\overline{\theta },\theta {)}^{\ast }=\omega (x,\theta ,\overline{\theta })}$.

The covariant derivative is defined as

${\displaystyle D_{\hat{M}}f=E_{\hat{M}}^N\left({\mathrm{\partial }}_{N}f+{\omega }_N[f]\right)}$.

The covariant exterior derivative as defined over supermanifolds needs to be super graded. This means that every time we interchange two fermionic indices, we pick up a +1 sign factor, instead of -1.

The presence or absence of R symmetries is optional, but if R-symmetry exists, the integrand over the full superspace has to have an R-charge of 0 and the integrand over chiral superspace has to have an R-charge of 2.

A chiral superfield X is a superfield which satisfies ${\displaystyle {\overline{D}}_{\hat{\dot{\alpha }}}X=0}$. In order for this constraint to be consistent, we require the integrability conditions that ${\displaystyle \left\{{\overline{D}}_{\hat{\dot{\alpha }}},{\overline{D}}_{\hat{\dot{\beta }}}\right\}=c_{\hat{\dot{\alpha }}\hat{\dot{\beta }}}^{\hat{\dot{\gamma }}}{\overline{D}}_{\hat{\dot{\gamma }}}}$ for some coefficients c.

Unlike nonSUSY GR, the torsion has to be nonzero, at least with respect to the fermionic directions. Already, even in flat superspace, ${\displaystyle D_{\hat{\alpha }}{e}_{\hat{\dot{\alpha }}}+{\overline{D}}_{\hat{\dot{\alpha }}}{e}_{\hat{\alpha }}\ne 0}$. In one version of SUGRA (but certainly not the only one), we have the following constraints upon the torsion tensor:

${\displaystyle T_{\widehat{\underset{\_}{\alpha }}\widehat{\underset{\_}{\beta }}}^{\widehat{\underset{\_}{\gamma }}}=0}$

${\displaystyle T_{\hat{\alpha }\hat{\beta }}^{\hat{\mu }}=0}$

${\displaystyle T_{\hat{\dot{\alpha }}\hat{\dot{\beta }}}^{\hat{\mu }}=0}$

${\displaystyle T_{\hat{\alpha }\hat{\dot{\beta }}}^{\hat{\mu }}=2i{\sigma }_{\hat{\alpha }\hat{\dot{\beta }}}^{\hat{\mu }}}$

${\displaystyle T_{\hat{\mu }\widehat{\underset{\_}{\alpha }}}^{\hat{\nu }}=0}$

${\displaystyle T_{\hat{\mu }\hat{\nu }}^{\hat{\rho }}=0}$

Here, ${\displaystyle \underset{\_}{\alpha }}$ is a shorthand notation to mean the index runs over either the left or right Weyl spinors.

The superdeterminant of the supervierbein, ${\displaystyle \left|e\right|}$, gives us the volume factor for M. Equivalently, we have the volume 4|4-superform${\displaystyle e^{\hat{\mu }=0}\wedge \cdots \wedge e^{\hat{\mu }=3}\wedge e^{\hat{\alpha }=1}\wedge e^{\hat{\alpha }=2}\wedge e^{\hat{\dot{\alpha }}=1}\wedge e^{\hat{\dot{\alpha }}=2}}$.

If we complexify the superdiffeomorphisms, there is a gauge where ${\displaystyle E_{\hat{\dot{\alpha }}}^{\mu }=0}$, ${\displaystyle E_{\hat{\dot{\alpha }}}^{\beta }=0}$ and ${\displaystyle E_{\hat{\dot{\alpha }}}^{\dot{\beta }}={\delta }_{\dot{\alpha }}^{\dot{\beta }}}$. The resulting chiral superspace has the coordinates x and Θ.

R is a scalar valued chiral superfield derivable from the supervielbeins and spin connection. If f is any superfield, ${\displaystyle \left({\overline{D}}^2-8R\right)f}$ is always a chiral superfield.

The action for a SUGRA theory with chiral superfields X, is given by

${\displaystyle S=\int d^{4}x{d}^2\mathrm{\Theta }2\mathcal{E}\left[\frac{3}{8}\left({\overline{D}}^2-8R\right)e^{-K(\overline{X},X)/3}+W(X)\right]+c.c.}$

where K is the Kähler potential and W is the superpotential, and ${\displaystyle \mathcal{E}}$ is the chiral volume factor.

Unlike the case for flat superspace, adding a constant to either the Kähler or superpotential is now physical. A constant shift to the Kähler potential changes the effective Planck constant, while a constant shift to the superpotential changes the effective cosmological constant. As the effective Planck constant now depends upon the value of the chiral superfield X, we need to rescale the supervierbeins (a field redefinition) to get a constant Planck constant. This is called the Einstein frame.

N = 8 supergravity in 4 dimensions

N = 8 supergravity is the most symmetric quantum field theory which involves gravity and a finite number of fields. It can be found from a dimensional reduction of 11D supergravity by making the size of 7 of the dimensions go to zero. It has 8 supersymmetries which is the most any gravitational theory can have since there are 8 half-steps between spin 2 and spin −2. (A graviton has the highest spin in this theory which is a spin 2 particle.) More supersymmetries would mean the particles would have superpartners with spins higher than 2. The only theories with spins higher than 2 which are consistent involve an infinite number of particles (such as string theory and higher-spin theories). Stephen Hawking in his A Brief History of Time speculated that this theory could be the Theory of Everything. However, in later years this was abandoned in favour of string theory. There has been renewed interest in the 21st century with the possibility that this theory may be finite.

Higher-dimensional SUGRA

Main article: Higher-dimensional supergravity

Higher-dimensional SUGRA is the higher-dimensional, supersymmetric generalization of general relativity. Supergravity can be formulated in any number of dimensions up to eleven. Higher-dimensional SUGRA focuses upon supergravity in greater than four dimensions.

The number of supercharges in a spinor depends on the dimension and the signature of spacetime. The supercharges occur in spinors. Thus the limit on the number of supercharges cannot be satisfied in a spacetime of arbitrary dimension. Some theoretical examples in which this is satisfied are:

• 12-dimensional two-time theory
• 11-dimensional maximal supergravity
• 10-dimensional supergravity theories
  • Type IIA supergravity: N = (1, 1)
  • Type IIB supergravity: N = (2, 0)
  • Type I supergravity: N = (1, 0)
• 9d supergravity theories
  • Maximal 9d supergravity from 10d
  • T-duality
  • N = 1 Gauged supergravity

The supergravity theories that have attracted the most interest contain no spins higher than two. This means, in particular, that they do not contain any fields that transform as symmetric tensors of rank higher than two under Lorentz transformations. The consistency of interacting higher spin field theories is, however, presently a field of very active interest.

History of classical field theory

From Wikipedia, the free encyclopedia
https://en.wikipedia.org/wiki/History_of_classical_field_theory

Iron filings used to show the magnetic field lines of a bar magnet.

In the history of physics, the concept of fields had its origins in the 18th century in a mathematical formulation of Newton's law of universal gravitation, but it was seen as deficient as it implied action at a distance. In 1852, Michael Faraday treated the magnetic field as a physical object, reasoning about lines of force. James Clerk Maxwell used Faraday's conceptualisation to help formulate his unification of electricity and magnetism in his field theory of electromagnetism.

With Albert Einstein's special relativity and the Michelson–Morley experiment, it became clear that electromagnetic waves could travel in a vacuum without the need of a medium or luminiferous aether. Einstein also developed general relativity, in which spacetime was treated as a field and its curvature was the origin of the gravitational interactions, putting an end to action at a distance.

In quantum field theory, fields become the fundamental objects of study, and particles are excitations of these fields. To differentiate from quantum field theory, previously developed field theories were called classical field theories.

Early mechanical explanations of forces

Magnetism

René Descartes drawing of a magnetic effluvia from 1644. It shows the magnetic effluvia of the Earth (D) attracting several round lodestones (I, K, L, M, N) and illustrates his theory of magnetism.

The first record of explanations of how magnets works comes from ancient Greece. Thinkers like Thales of Miletus, Aristotle and Diogenes Laertius considered that magnets were animated and should have a soul in order to move. Empedocles tried to provide a mechanical explanation of why magnets could influence each other by introducing the concept of "effluences" emanated by magnets. According to book Quaestiones by Alexander of Aphrodisias from about 200 AD, this was Empedocles view:

On the reason why the lodestone attracts iron. Empedocles says that the iron is attracted to the stone by the effluences which issue from both, and because the pores of the stone are commensurate with the effluences from the iron. The effluences from the stone stir and disperse the air lying upon and obstructing the pores of the iron and when this is removed the iron is drawn on by a concerted outflow. As the effluences from the iron travel towards the pores of the stone, because they are commensurate with them and fit into them the iron itself follows and moves together with them.

Democritus had a similar view as Empedocles but added that the effluences created a void. Metals and rocks could also contain void in order to be less or more attracted to magnets.

This idea survived up to the Scientific Revolution. In 1664, René Descartes produced his theory of magnetism, in which the flow of effluences or effluvia rarified the air, creating differences in air pressure. According to Descartes, these effluvia circulated inside and around the magnet in closed loops.

Gravitation

Further information: Mechanical explanations of gravitation

In ancient times, Greek thinkers like Posidonius (1 BC) noticed a relation between the tides and the position of the Moon in the sky. He considered that light from the Moon had an influence on the tides.

In the 9th century, Abu Ma'shar al-Balkhi (Latinized as Albumasar) wrote his book on The Great Introduction to the Science of Astrology (Kitāb al-madkhal al-kabīr) recorded the correlation between the tides and the Moon, noticing that there were two tides in a day. As there is no moonlight when the Moon is the opposite side of Earth, he proposed that the Moon had some intrinsic virtue that attracted the water. The Sun would have some of that virtue but less than the moon. This book was translated to Latin and was a reference for European medieval scholars. One writer that rejected this astrological reading was Robert Grosseteste who wrote On the Ebb and Flow of the Sea (Latin: Questio de fluxu et refluxu maris), written around 1227, in which he insisted that light from the Moon rarefied the air producing the tides. He explained the tides when the Moon is below the horizon as reflections from the celestial sphere. Two theories coexisted, the idea of light influencing the tides and Albumasar' virtue. Roger Bacon supported the idea of Grosseteste, Albertus Magnus supported a mix of both, and others like Jean Buridan hesitated between the two.

In 17th century, Johannes Kepler who came up with the Kepler's laws of planetary motion, proposed the idea that the Sun emitted some sort of invisible "species" that traveled instantaneously and acted more strongly depending on the distance, size and density of the planet. Kepler considered that if the Sun rotated, it would create a whirlpool of species that drags all planets to orbit around it. The idea of the rotation of the Sun was confirmed by Galileo Galilei, but the frequency did not match Kepler's calculations. To explain the tides, Kepler considered that the species would behave similar to magnetic forces.

Descartes rejected Kepler's theory and also constructed also a mechanical explanation of gravitation based on the ideas vortices, considering space continuous. Descartes pushed the Aristotelian idea of the plenum, considering that there was no void and the entirety of space was filled with corpuscles.

Newtonian gravitation

Field lines between Earth and the Moon. In Isaac Newton's classical gravitation, mass is the source of an attractive gravitational field.

Newtonian mechanics

Before Newton, only a few mechanical explanations of gravity existed.

In 1687, Newton's Principia in 1687 provided a framework with which to investigate the motion and forces. He introduced mathematical definition of gravitational force with his law of universal gravitation, in which the gravitational force between two bodies is directed along the line separating the bodies and its magnitude is proportional to the product of their masses, divided by the square of their distance apart.

While Newton explanation of gravity was very successful in astronomy, it did not explain how it could act at a distance and instantaneously. Newton, considered action at a distance to be:

so great an Absurdity that I believe no Man who has in philosophical Matters a competent Faculty of thinking can ever fall into it.

— Isaac Newton, Letters to Bentley, 1692/3

Gottfried Wilhelm Leibniz complained that Newtonian mechanics violated the metaphysics of continuity according to natura non facit saltus, in which every cause and effect should be connected to one another. Roger Joseph Boscovich rejected Leibniz take considering that bodies would have discontinuous changes in density at the boundaries and that if came into contact their velocities would change discontinuously.

British empiricist like John Locke, George Locke and David Hume regarded Newton's second law of motion as sufficient, as it establishes a causal relation between force and acceleration.

Beginning of aether theories

Main article: Aether theories

To solve the issue of action at a distance, aether theories were developed. The aether was considered as a yet undetected medium and responsible agent for conducting the force. In a letter to Robert Boyle in 1679 Newton proposed an "aethereal substance" to explain gravity. Later in his work Opticks of 1717 he considered the aether to be made of impenetrable corpuscules. Newtonian aether was very dilute and elastic. Immanuel Kant considered Newton's aether inconsistent as requiring additional forces between corpuscles. Leibniz on the other hand considered a continuous medium.

Eulerian fluid dynamics

Velocity field lines around a ball in a fluid. The arrows indicate the velocity of the fluid.

An important development of field theories appeared with Leonhard Euler who expanded Newtonian mechanics in his work Mechanica of 1736. Euler work expanded on how to deal with rotations of rigid bodies, elasticity and fluid mechanics. To describe fluids he considered a flow velocity function (today called velocity field) defined at every point in space. However, this function was for a long time considered significantly different from that of the forces of gravitation as it was only defined inside a medium and thus was considered a real quantity. Modern science historian Mary Hesse attributed the origin of field theory to Euler flow velocity field.

Euler also introduced between 1755 and 1759 the Lagrangian and Eulerian specifications for the flow that would be important to detach motion of particles from field properties.

Potential theory

Joseph-Louis Lagrange is often cited for introducing the concept of a potential in 1777, and independently by Adrien-Marie Legendre (1784–1794) and Pierre-Simon Laplace (1782–1799).Lagrange noticed that he could introduce a gravitational potential to derive the gravitational force. This function was called a potential function by mathematician George Green 1828 and by Carl Friedrich Gauss in 1840 just as "potential".

Forces of electricity and magnetism

Charles-Augustin de Coulomb showed in 1785 that the repulsive force between two electrically charged spheres obeys the same (up to a sign) force law as Newton's law of universal gravitation. In 1823, Siméon Denis Poisson introduced the Poisson's equation, explaining the electric forces in terms of an electric potential. The same year André-Marie Ampère showed that the force between infinitesimal lengths of current-carrying wires similarly obeys an inverse-square law such that the force is directed along the line of separation between the wire elements. These law suffered from the same problem of action-at-a-distance.

Luminiferous aether

In 1800, Thomas Young proved the wave nature of light using the double-slit experiment. This discovery led him in 1802 to consider the existence of luminiferous aether in which light traveled. Augustin-Jean Fresnel considered it to be an elastic medium. The motion of this aether were described mathematically by scientist like Claude-Louis Navier (in 1821) and Augustin-Louis Cauchy (in 1828) as discrete medium. About 1840, George Stokes and Lord Kelvin extended the formalism to describe a continuous aether using the idea of a potential theory. This development was important as it allowed to describe any deformable medium in terms of continuous functions.

Introduction of fields

Main article: History of electromagnetic theory

Faraday's lines of force

Main article: Line of force

Magnetic field lines of a magnetic dipole.

Michael Faraday developed the concept of lines of force to describe electric and magnetic phenomena. In 1831, he writes

By magnetic curves, I mean the lines of magnetic forces, however modified by the juxtaposition of poles, which would be depicted by iron filings; or those to which a very small magnetic needle would form a tangent."

He provided a definition in 1845,

But before I proceed to them, I will define the meaning I connect with certain terms which I shall have occasion to use: thus, by line of magnetic force, or magnetic line of force, or magnetic curve, I mean that exercise of magnetic force which is exerted in the lines usually called magnetic curves, and which equally exist as passing from or to magnetic poles, or forming concentric circles round an electric current. By line of electric force, I mean the force exerted in the lines joining two bodies, acting on each other according to the principles of static electric induction, which may also be either in curved or straight lines.

In his work, he also coined the term "magnetic field" in this sense in 1845, which he later used frequently. He provided a clear definition in 1850, stating

I will now endeavour to consider what the influence is which paramagnetic and diamagnetic bodies, viewed as conductors, exert upon the lines of force in a magnetic field. Any portion of space traversed by lines of magnetic power, may be taken as such a field, and there is probably no space without them. The condition of the field may vary in intensity of power. from place to place, either along the lines or across them; but it will be better to assume for the present consideration a field of equal force throughout, and I have formerly described how this may, for a certain limited space, be produced.

Faraday did not conceive of this field as a mere mathematical construct for calculating the forces between particles—having only rudimentary mathematical training, he had no use for abstracting reality to make quantitative predictions. Instead he conjectured that there was force filling the space where electromagnetic fields were generated and reasoned qualitatively about these forces with lines of force:

Important to the definition of these lines is that they represent a determinate and unchanging amount of force. Though, therefore, their forms, as they exist between two or more centers or sources of power, may vary greatly, and also the space through which they may be traced, yet the sum of power contained in any one section of a given portion of the lines is exactly equal to the sum of power in any other section of the same lines, however altered in form or however convergent or divergent they may be at the second place.

However, Faraday never used the term "electric field" explicitly. Nevertheless, Faraday's insights into the behavior of magnetic fields would prove invaluable for the development of electromagnetism and field theory.

Kelvin's definition

In 1845, Lord Kelvin formalized the mathematical similarities between the fields of electromagnetic phenomena and Joseph Fourier work on heat; and in 1847 between electric conduction and elasticity. These similarities led Lord Kelvin to propose a formal definition of magnetic field in 1851:

Any space at every point of which there is a finite magnetic force is called ‘a field of magnetic force’ or (magnetic being understood) simply ‘a field of force,’ or sometimes ‘a magnetic field’.

— Lord Kelvin, On the theory of magnetic induction in crystalline and non-crystalline substances,

Kelvin also introduced the concept of a magnetic vector potential.

Maxwell's electromagnetic field

Electric and magnetic fields of an electromagnetic wave along an axis. In vacuum these two fields are orthogonal and propagate at the speed of light as predicted by Maxwell.

In 1864, James Clerk Maxwell published "A Dynamical Theory of the Electromagnetic Field" in which he compiled all known equations of electricity and magnetism. Maxwell's equations led to an electromagnetic wave equation with waves that propagated in vacuum at the speed of light. He describes his research as

(3) The theory I propose may therefore be called a theory of the Electromagnetic Field, because it has to do with the space in the neighbourhood of the electric and magnetic bodies, and it may be called a Dynamical Theory, because it assumes that in that space there is matter in motion, by which the observed electromagnetic phenomena are produced

(4) The electromagnetic field is that part of space which contains and surrounds bodies in electric or magnetic conditions

In A Treatise on Electricity and Magnetism of 1873, he writes "the electric field is the portion of space in the neighbourhood of electrified bodies, considered with reference to electric phenomena." And for magnetic fields

lt appears therefore that in the space surrounding a wire transmitting an electric current a magnet is acted on by forces dependent on the position of the wire and on the strength of the current. The space in which these forces act may therefore be considered as a magnetic field, and we may study it in the same way as we have already studied the field in the neighbourhood of ordinary magnets, by tracing the course of the lines of magnetic force, and measuring the intensity of the force at every point.

Maxwell had to settle for the idea of a luminiferous aether. He wrote

We have therefore some reason to believe, from the phenomena of light and heat, that there is an aethereal medium filling space and permeating bodies, capable of being set in motion and of transmitting that motion from one part to another, and of communicating that motion to gross matter so as to heat it and affect it in various ways.

Maxwell was conflicted on the idea on the nature of the fields, he considered the aether to a mechanical medium in order to carry energy. In 1868 Carl Neumann discussed the idea of the electromagnetic field being an independent energy field.

In 1887, Heinrich Hertz published his experimental evidence of the existence of electromagnetic waves.

Relativistic field theory

Special relativity

Main article: History of special relativity

The 1887 Michelson–Morley experiment attempted to prove that electromagnetic radiation were oscillations of a luminiferous aether; however, the result was negative, indicating that the electromagnetic field could exist and travel in vacuum. To explain this phenomenon, Albert Einstein developed his theory of special relativity (1905) which resolved the conflicts between classical mechanics and electromagnetism. Einstein introduced the Lorentz transformation for electromagnetic fields between reference frames.

Space-time as a field

Main article: History of general relativity

Einstein developed the Einstein field equations of general relativity in 1915, consistent with special relativity and that could explain gravitation in terms of a field theory of spacetime. This removed the need of a gravitational aether.

In 1918, Emmy Noether publishes her theorem on the relations between symmetries and conservation laws. Noether's theorem was adapted to general relativity as well as to non-relativistic field theories.

The geometric aspect of space-time can be used to study Newtonian gravitational field. This was developed by Élie Cartan in 1923 and leads to a classically covariant formulation known as geometrized Newtonian gravitation.

Unification attempts

Main article: Classical unified field theories

Attempts to create a unified field theory based on classical physics are classical unified field theories. During the years between the two World Wars, the idea of unification of gravity with electromagnetism was actively pursued by several mathematicians and physicists like Einstein, Theodor Kaluza, Hermann Weyl, Arthur Eddington, Gustav Mie and Ernst Reichenbacher.

Early attempts to create such theory were based on incorporation of electromagnetic fields into the geometry of general relativity. In 1918, the case for the first geometrization of the electromagnetic field was proposed in 1918 by Weyl. In this work Weyl coins the term gauge theory. Weyl, in an attempt to generalize the geometrical ideas of general relativity to include electromagnetism, conjectured that Eichinvarianz or invariance under the change of scale (or "gauge") might also be a local symmetry of general relativity.

In 1919, the idea of a five-dimensional approach was suggested by Kaluza. From that, a theory called Kaluza–Klein theory was developed. It attempts to unify gravitation and electromagnetism, in a five-dimensional space-time. There are several ways of extending the representational framework for a unified field theory which have been considered by Einstein and other researchers. These extensions in general are based in two options. The first option is based in relaxing the conditions imposed on the original formulation, and the second is based in introducing other mathematical objects into the theory. An example of the first option is relaxing the restrictions to four-dimensional space-time by considering higher-dimensional representations. That is used in Kaluza–Klein theory. For the second, the most prominent example arises from the concept of the affine connection that was introduced into general relativity mainly through the work of Tullio Levi-Civita and Weyl.

Further development of quantum field theory changed the focus of searching for unified field theory from classical to quantum description. Because of that, many theoretical physicists gave up looking for a classical unified field theory. Quantum field theory would include unification of two other fundamental interactions of nature, the strong and weak nuclear force which act on the subatomic level.

Quantum fields

Main article: History of quantum field theory

Fields become the fundamental object of study in quantum field theory. Mathematically, quantum fields are formalized as operator-valued distributions. Although there is no direct method of measuring the fields themselves, the framework asserts that all particles are "excitations" of these fields. For example: whereas Maxwell's theory of classical electromagnetism describes light as a self-propagating wave in the electromagnetic field, in quantum electrodynamics light is the massless gauge boson particle called the photon. Furthermore, the number of particles in an isolated system need not be conserved; an example of a process for which this is the case is bremsstrahlung. More detailed understanding of the framework is obtained by studying the Lagrangian density of a field theory which encodes the information of its allowed particle interactions.