History of quantum field theory

From Wikipedia, the free encyclopedia

In particle physics, the history of quantum field theory starts with its creation by Paul Dirac, when he attempted to quantize the electromagnetic field in the late 1920s. Major advances in the theory were made in the 1950s, and led to the introduction of quantum electrodynamics (QED). QED was so successful and accurately predictive that efforts were made to apply the same basic concepts for the other forces of nature. By the late 1970s, these efforts successfully utilized gauge theory in the strong nuclear force and weak nuclear force, producing the modern standard model of particle physics.

Efforts to describe gravity using the same techniques have, to date, failed. The study of quantum field theory is still flourishing, as are applications of its methods to many physical problems. It remains one of the most vital areas of theoretical physics today, providing a common language to several different branches of physics.

Early developments

Quantum field theory originated in the 1920s from the problem of creating a quantum mechanical theory of the electromagnetic field. In particular, de Broglie in 1924 introduced the idea of a wave description of elementary systems in the following way: "we proceed in this work from the assumption of the existence of a certain periodic phenomenon of a yet to be determined character, which is to be attributed to each and every isolated energy parcel".^[1]

In 1925, Werner Heisenberg, Max Born, and Pascual Jordan constructed just such a theory by expressing the field's internal degrees of freedom as an infinite set of harmonic oscillators, and by then utilizing the canonical quantization procedure to these oscillators; their paper was published in 1926.^[2][3][4] This theory assumed that no electric charges or currents were present and today would be called a free field theory.

The first reasonably complete theory of quantum electrodynamics, which included both the electromagnetic field and electrically charged matter (specifically, electrons) as quantum mechanical objects, was created by Paul Dirac in 1927.^[5] This quantum field theory could be used to model important processes such as the emission of a photon by an electron dropping into a quantum state of lower energy, a process in which the number of particles changes—one atom in the initial state becomes an atom plus a photon in the final state. It is now understood that the ability to describe such processes is one of the most important features of quantum field theory.

The final crucial step was Enrico Fermi's theory of β-decay (1934).^[6][7] In it, fermion species nonconservation was shown to follow from second quantization: creation and annihilation of fermions came to the fore and quantum field theory was seen to describe particle decays. (Fermi's breakthrough was somewhat foreshadowed in the abstract studies of Soviet physicists, Viktor Ambartsumian and Dmitri Ivanenko, in particular the Ambarzumian–Ivanenko hypothesis of creation of massive particles (1930).^[8] The idea was that not only the quanta of the electromagnetic field, photons, but also other particles might emerge and disappear as a result of their interaction with other particles.)

Incorporating special relativity

It was evident from the beginning that a proper quantum treatment of the electromagnetic field had to somehow incorporate Einstein's relativity theory, which had grown out of the study of classical electromagnetism. This need to put together relativity and quantum mechanics was the second major motivation in the development of quantum field theory. Pascual Jordan and Wolfgang Pauli showed in 1928^[9][10] that quantum fields could be made to behave in the way predicted by special relativity during coordinate transformations (specifically, they showed that the field commutators were Lorentz invariant). A further boost for quantum field theory came with the discovery of the Dirac equation, which was originally formulated and interpreted as a single-particle equation analogous to the Schrödinger equation, but unlike the Schrödinger equation, the Dirac equation satisfies both the Lorentz invariance, that is, the requirements of special relativity, and the rules of quantum mechanics. The Dirac equation accommodated the spin-1/2 value of the electron and accounted for its magnetic moment as well as giving accurate predictions for the spectra of hydrogen.

The attempted interpretation of the Dirac equation as a single-particle equation could not be maintained long, however, and finally it was shown that several of its undesirable properties (such as negative-energy states) could be made sense of by reformulating and reinterpreting the Dirac equation as a true field equation, in this case for the quantized "Dirac field" or the "electron field", with the "negative-energy solutions" pointing to the existence of anti-particles. This work was performed first by Dirac himself with the invention of hole theory in 1930 and by Wendell Furry, Robert Oppenheimer, Vladimir Fock, and others. Schrödinger, during the same period that he discovered his famous equation in 1926,^[11] also independently found the relativistic generalization of it known as the Klein–Gordon equation but dismissed it since, without spin, it predicted impossible properties for the hydrogen spectrum. (See Oskar Klein and Walter Gordon.) All relativistic wave equations that describe spin-zero particles are said to be of the Klein–Gordon type.

Uncertainty, again

A subtle and careful analysis in 1933 by Niels Bohr and Léon Rosenfeld^[12] showed that there is a fundamental limitation on the ability to simultaneously measure the electric and magnetic field strengths that enter into the description of charges in interaction with radiation, imposed by the uncertainty principle, which must apply to all canonically conjugate quantities. This limitation is crucial for the successful formulation and interpretation of a quantum field theory of photons and electrons (quantum electrodynamics), and indeed, any perturbative quantum field theory. The analysis of Bohr and Rosenfeld explains fluctuations in the values of the electromagnetic field that differ from the classically "allowed" values distant from the sources of the field.

Their analysis was crucial to showing that the limitations and physical implications of the uncertainty principle apply to all dynamical systems, whether fields or material particles. Their analysis also convinced most physicists that any notion of returning to a fundamental description of nature based on classical field theory, such as what Einstein aimed at with his numerous and failed attempts at a classical unified field theory, was simply out of the question. Fields had to be quantized.

Second quantization

The third thread in the development of quantum field theory was the need to handle the statistics of many-particle systems consistently and with ease. In 1927, Pascual Jordan tried to extend the canonical quantization of fields to the many-body wave functions of identical particles^[13] using a formalism which is known as statistical transformation theory;^[14] this procedure is now sometimes called second quantization.^[15][16] In 1928, Jordan and Eugene Wigner found that the quantum field describing electrons, or other fermions, had to be expanded using anti-commuting creation and annihilation operators due to the Pauli exclusion principle (see Jordan–Wigner transformation). This thread of development was incorporated into many-body theory and strongly influenced condensed matter physics and nuclear physics.

The problem of infinities

Despite its early successes quantum field theory was plagued by several serious theoretical difficulties. Basic physical quantities, such as the self-energy of the electron, the energy shift of electron states due to the presence of the electromagnetic field, gave infinite, divergent contributions—a nonsensical result—when computed using the perturbative techniques available in the 1930s and most of the 1940s. The electron self-energy problem was already a serious issue in the classical electromagnetic field theory, where the attempt to attribute to the electron a finite size or extent (the classical electron-radius) led immediately to the question of what non-electromagnetic stresses would need to be invoked, which would presumably hold the electron together against the Coulomb repulsion of its finite-sized "parts". The situation was dire, and had certain features that reminded many of the "Rayleigh–Jeans catastrophe". What made the situation in the 1940s so desperate and gloomy, however, was the fact that the correct ingredients (the second-quantized Maxwell–Dirac field equations) for the theoretical description of interacting photons and electrons were well in place, and no major conceptual change was needed analogous to that which was necessitated by a finite and physically sensible account of the radiative behavior of hot objects, as provided by the Planck radiation law.

Renormalization procedures

This "divergence problem" was solved in the case of quantum electrodynamics through the procedure known as renormalization in 1947–49 by Hans Kramers,^[17] Hans Bethe,^[18] Julian Schwinger,^[19] Richard Feynman,^[20][21][22] and Shin'ichiro Tomonaga;^[23] the procedure was systematized by Freeman Dyson in 1949.^[24] Great progress was made after realizing that all infinities in quantum electrodynamics are related to two effects: the self-energy of the electron/positron, and vacuum polarization.

Renormalization requires paying very careful attention to just what is meant by, for example, the very concepts "charge" and "mass" as they occur in the pure, non-interacting field-equations. The "vacuum" is itself polarizable and, hence, populated by virtual particle (on shell and off shell) pairs, and, hence, is a seething and busy dynamical system in its own right. This was a critical step in identifying the source of "infinities" and "divergences". The "bare mass" and the "bare charge" of a particle, the values that appear in the free-field equations (non-interacting case), are abstractions that are simply not realized in experiment (in interaction). What we measure, and hence, what we must take account of with our equations, and what the solutions must account for, are the "renormalized mass" and the "renormalized charge" of a particle. That is to say, the "shifted" or "dressed" values these quantities must have when due systematic care is taken to include all deviations from their "bare values" is dictated by the very nature of quantum fields themselves.

Gauge invariance

The first approach that bore fruit is known as the "interaction representation" (see the article Interaction picture), a Lorentz-covariant and gauge-invariant generalization of time-dependent perturbation theory used in ordinary quantum mechanics, and developed by Tomonaga and Schwinger, generalizing earlier efforts of Dirac, Fock and Podolsky. Tomonaga and Schwinger invented a relativistically covariant scheme for representing field commutators and field operators intermediate between the two main representations of a quantum system, the Schrödinger and the Heisenberg representations. Within this scheme, field commutators at separated points can be evaluated in terms of "bare" field creation and annihilation operators. This allows for keeping track of the time-evolution of both the "bare" and "renormalized", or perturbed, values of the Hamiltonian and expresses everything in terms of the coupled, gauge invariant "bare" field-equations. Schwinger gave the most elegant formulation of this approach. The next and most famous development is due to Feynman, who, with his brilliant rules for assigning a "graph"/"diagram" to the terms in the scattering matrix (see S-matrix and Feynman diagrams). These directly corresponded (through the Schwinger-Dyson equation) to the measurable physical processes (cross sections, probability amplitudes, decay widths and lifetimes of excited states) one needs to be able to calculate. This revolutionized how quantum field theory calculations are carried-out in practice.

Two classic text-books from the 1960s, James D. Bjorken, Sidney David Drell, Relativistic Quantum Mechanics (1964) and J. J. Sakurai, Advanced Quantum Mechanics (1967), thoroughly developed the Feynman graph expansion techniques using physically intuitive and practical methods following from the correspondence principle, without worrying about the technicalities involved in deriving the Feynman rules from the superstructure of quantum field theory itself. Although both Feynman's heuristic and pictorial style of dealing with the infinities, as well as the formal methods of Tomonaga and Schwinger, worked extremely well, and gave spectacularly accurate answers, the true analytical nature of the question of "renormalizability", that is, whether ANY theory formulated as a "quantum field theory" would give finite answers, was not worked-out until much later, when the urgency of trying to formulate finite theories for the strong and electro-weak (and gravitational interactions) demanded its solution.

Renormalization in the case of QED was largely fortuitous due to the smallness of the coupling constant, the fact that the coupling has no dimensions involving mass, the so-called fine structure constant, and also the zero-mass of the gauge boson involved, the photon, rendered the small-distance/high-energy behavior of QED manageable. Also, electromagnetic processes are very "clean" in the sense that they are not badly suppressed/damped and/or hidden by the other gauge interactions. By 1965 James D. Bjorken and Sidney David Drell observed: "Quantum electrodynamics (QED) has achieved a status of peaceful coexistence with its divergences ...".^[25]

The unification of the electromagnetic force with the weak force encountered initial difficulties due to the lack of accelerator energies high enough to reveal processes beyond the Fermi interaction range. Additionally, a satisfactory theoretical understanding of hadron substructure had to be developed, culminating in the quark model.

Non-abelian gauge theory

Thanks to the somewhat brute-force, ad hoc and heuristic early methods of Feynman, and the abstract methods of Tomonaga and Schwinger, elegantly synthesized by Freeman Dyson, from the period of early renormalization, the modern theory of quantum electrodynamics (QED) has established itself. It is still the most accurate physical theory known, the prototype of a successful quantum field theory. Quantum electrodynamics is the most famous example of what is known as an Abelian gauge theory. It relies on the symmetry group U(1) and has one massless gauge field, the U(1) gauge symmetry, dictating the form of the interactions involving the electromagnetic field, with the photon being the gauge boson.

Beginning in the 1950s with the work of Yang and Mills, following the previous lead of Weyl and Pauli, deep explorations illuminated the types of symmetries and invariances any field theory must satisfy. QED, and indeed, all field theories, were generalized to a class of quantum field theories known as gauge theories. That symmetries dictate, limit and necessitate the form of interaction between particles is the essence of the "gauge theory revolution". Yang and Mills formulated the first explicit example of a non-abelian gauge theory, Yang–Mills theory, with an attempted explanation of the strong interactions in mind. The strong interactions were then (incorrectly) understood in the mid-1950s, to be mediated by the pi-mesons, the particles predicted by Hideki Yukawa in 1935,^[26] based on his profound reflections concerning the reciprocal connection between the mass of any force-mediating particle and the range of the force it mediates. This was allowed by the uncertainty principle. In the absence of dynamical information, Murray Gell-Mann pioneered the extraction of physical predictions from sheer non-abelian symmetry considerations, and introduced non-abelian Lie groups to current algebra and so the gauge theories that came to supersede it.

The 1960s and 1970s saw the formulation of a gauge theory now known as the Standard Model of particle physics, which systematically describes the elementary particles and the interactions between them. The strong interactions are described by quantum chromodynamics (QCD), based on "color" SU(3). The weak interactions require the additional feature of spontaneous symmetry breaking, elucidated by Yoichiro Nambu and the adjunct Higgs mechanism, considered next.

Electroweak unification

The electroweak interaction part of the standard model was formulated by Sheldon Glashow, Abdus Salam, and John Clive Ward in 1959^[27][28] with their discovery of the SU(2)xU(1) group structure of the theory. In 1967, Steven Weinberg brilliantly invoked the Higgs mechanism for the generation of the W and Z masses^[29] (the intermediate vector bosons responsible for the weak interactions and neutral-currents) and keeping the mass of the photon zero. The Goldstone and Higgs idea for generating mass in gauge theories was sparked in the late 1950s and early 1960s when a number of theoreticians (including Yoichiro Nambu, Steven Weinberg, Jeffrey Goldstone, François Englert, Robert Brout, G. S. Guralnik, C. R. Hagen, Tom Kibble and Philip Warren Anderson) noticed a possibly useful analogy to the (spontaneous) breaking of the U(1) symmetry of electromagnetism in the formation of the BCS ground-state of a superconductor. The gauge boson involved in this situation, the photon, behaves as though it has acquired a finite mass.

There is a further possibility that the physical vacuum (ground-state) does not respect the symmetries implied by the "unbroken" electroweak Lagrangian from which one arrives at the field equations. The electroweak theory of Weinberg and Salam was shown to be renormalizable (finite) and hence consistent by Gerardus 't Hooft and Martinus Veltman. The Glashow–Weinberg–Salam theory (GWS theory) is a triumph and, in certain applications, gives an accuracy on a par with quantum electrodynamics.

Quantum chromodynamics

In the case of the strong interactions, progress concerning their short-distance/high-energy behavior was much slower and more frustrating. For strong interactions with the electro-weak fields, there were difficult issues regarding the strength of coupling, the mass generation of the force carriers as well as their non-linear, self interactions. Although there has been theoretical progress toward a grand unified quantum field theory incorporating the electro-magnetic force, the weak force and the strong force, empirical verification is still pending. Superunification, incorporating the gravitational force, is still very speculative, and is under intensive investigation by many of the best minds in contemporary theoretical physics. Gravitation is a tensor field description of a spin-2 gauge-boson, the "graviton", and is further discussed in the articles on general relativity and quantum gravity.

Quantum gravity

From the point of view of the techniques of (four-dimensional) quantum field theory, and as the numerous efforts to formulate a consistent quantum gravity theory attests, gravitational quantization has been the reigning champion for bad behavior.^[30]

There are technical problems underlain by the fact that the gravitational coupling constant has dimensions involving inverse powers of mass, and, as a simple consequence, it is plagued by perturbatively badly behaved non-linear self-interactions. Gravity is itself a source of gravity, analogously to gauge theories (whose couplings, are, by contrast, dimensionless) leading to uncontrollable divergences at increasing orders of perturbation theory.

Moreover, gravity couples to all energy equally strongly, as per the equivalence principle, so this makes the notion of ever really "switching-off", "cutting-off" or separating, the gravitational interaction from other interactions ambiguous, since, with gravitation, we are dealing with the very structure of space-time itself.

Contemporary framework of renormalization

Parallel breakthroughs in the understanding of phase transitions in condensed matter physics led to novel insights based on the renormalization group. They involved the work of Leo Kadanoff (1966)^[31] and Kenneth Geddes Wilson & Michael Fisher (1972)^[32]—extending the work of Ernst Stueckelberg–André Petermann (1953)^[33] and Murray Gell-Mann–Francis Low (1954)^[34]—which led to the seminal reformulation of quantum field theory by Kenneth Geddes Wilson in 1975.^[35]

This reformulation provided insights into the evolution of effective field theories with scale, which classified all field theories, renormalizable or not (cf. subsequent section). The remarkable conclusion is that, in general, most observables are "irrelevant", i.e., the macroscopic physics is dominated by only a few observables in most systems.

During the same period, Kadanoff (1969)^[36] introduced an operator algebra formalism for the two-dimensional Ising model, a widely studied mathematical model of ferromagnetism in statistical physics. This development suggested that quantum field theory describes its scaling limit. Later, there developed the idea that a finite number of generating operators could represent all the correlation functions of the Ising model.

The renormalization group spans a set of ideas and methods to monitor changes of the behavior of the theory with scale, providing a deep physical understanding which sparked what has been called the "grand synthesis" of theoretical physics, uniting the quantum field theoretical techniques used in particle physics and condensed matter physics into a single powerful theoretical framework.

The gauge field theory of the strong interactions, quantum chromodynamics, QCD, relies crucially on this renormalization group for its distinguishing characteristic features, asymptotic freedom and color confinement.

Supergravity

From Wikipedia, the free encyclopedia

In theoretical physics, supergravity (supergravity theory; SUGRA for short) is a modern field theory that combines the principles of supersymmetry and general relativity where supersymmetry obeys locality, contrasting non-gravitational supersymmetric theories such as the Minimal Supersymmetric Standard Model. Since the generators of supersymmetry (SUSY) use the Poincaré group to form a super-Poincaré algebra, gravity follows naturally from local supersymmetry.^[1]

Gravitons

Like any field theory of gravity, a supergravity theory 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^[2] and was called gauge supersymmetry.

Supergravity

The minimal version of four-dimensional Supergravity was discovered in 1976 by Dan Freedman, Sergio Ferrara and Peter van Nieuwenhuizen,^[3] and 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 four-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 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, its interest is a widely investigated model of particle physics

11D: the maximal SUGRA

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^[4] 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^{[citation needed]} gives this emphasis.
• In 1981 Ed Witten showed^[5] 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.^{[citation needed]} 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^[6] 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^{[citation needed]}. 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^{[permanent dead link]}. While not manifestly Lorentz-invariant, 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.^[7] 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.^[8]

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.^[9]

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, equates 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, need 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;^{[citation needed]} supergravity research is useful independent of those relations.

4D N = 1 SUGRA

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

${\displaystyle e_{N}^{\hat {M}}(x,{\overline {\theta }},\theta )^{*}=e_{N^{*}}^{{\hat {M}}^{*}}(x,\theta ,{\overline {\theta }})}$ 

where ${\displaystyle \mu ^{*}=\mu }$, ${\displaystyle \alpha ^{*}={\dot {\alpha }}}$, and ${\displaystyle {\dot {\alpha }}^{*}=\alpha }$ 

and

${\displaystyle \omega (x,{\overline {\theta }},\theta )^{*}=\omega (x,\theta ,{\overline {\theta }})}$.

The covariant derivative is defined as

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

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, ${\displaystyle D_{\hat {\alpha }}e_{\hat {\dot {\alpha }}}+{\overline {D}}_{\hat {\dot {\alpha }}}e_{\hat {\alpha }}\neq 0}$. In one version of SUGRA (but certainly not the only one), we have the following constraints upon the torsion tensor:

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

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

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

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

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

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

Here, ${\displaystyle {\underline {\alpha }}}$ 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, ${\displaystyle \left({\bar {D}}^{2}-8R\right)f}$ is always a chiral superfield.

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

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

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

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 SUGRA
• 10-dimensional SUGRA theories
  • Type IIA SUGRA: N = (1, 1)
  • IIA SUGRA from 11d SUGRA
  • Type IIB SUGRA: N = (2, 0)
  • Type I gauged SUGRA: N = (1, 0)
• 9d SUGRA theories
  • Maximal 9d SUGRA from 10d
  • T-duality
  • N = 1 Gauged SUGRA

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.