History of loop quantum gravity

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The history of loop quantum gravity spans more than three decades of intense research.

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History

Classical theories of gravitation

General relativity is the theory of gravitation published by Albert Einstein in 1915. According to it, the force of gravity is a manifestation of the local geometry of spacetime. Mathematically, the theory is modelled after Bernhard Riemann's metric geometry, but the Lorentz group of spacetime symmetries (an essential ingredient of Einstein's own theory of special relativity) replaces the group of rotational symmetries of space. (Later, loop quantum gravity inherited this geometric interpretation of gravity, and posits that a quantum theory of gravity is fundamentally a quantum theory of spacetime.)

In the 1920s, the French mathematician Élie Cartan formulated Einstein's theory in the language of bundles and connections, [1] a generalization of Riemannian geometry to which Cartan made important contributions. The so-called Einstein–Cartan theory of gravity not only reformulated but also generalized general relativity, and allowed spacetimes with torsion as well as curvature. In Cartan's geometry of bundles, the concept of parallel transport is more fundamental than that of distance, the centerpiece of Riemannian geometry. A similar conceptual shift occurs between the invariant interval of Einstein's general relativity and the parallel transport of Einstein–Cartan theory.

Spin networks

In 1971, physicist Roger Penrose explored the idea of space arising from a quantum combinatorial structure. [2] [3] His investigations resulted in the development of spin networks. Because this was a quantum theory of the rotational group and not the Lorentz group, Penrose went on to develop twistors. [4]

Loop quantum gravity

In 1982, Amitabha Sen tried to formulate a Hamiltonian formulation of general relativity based on spinorial variables, where these variables are the left and right spinorial component equivalents of Einstein–Cartan connection of general relativity. [5] Particularly, Sen discovered a new way to write down the two constraints of the ADM Hamiltonian formulation of general relativity in terms of these spinorial connections. In his form, the constraints are simply conditions that the spinorial Weyl curvature is trace free and symmetric. He also discovered the presence of new constraints which he suggested to be interpreted as the equivalent of Gauss constraint of Yang–Mills field theories. But Sen's work fell short of giving a full clear systematic theory and particularly failed to clearly discuss the conjugate momenta to the spinorial variables, its physical interpretation, and its relation to the metric (in his work he indicated this as some lambda variable).

In 1986–87, physicist Abhay Ashtekar completed the project which Amitabha Sen began. He clearly identified the fundamental conjugate variables of spinorial gravity: The configuration variable is as a spinoral connection (a rule for parallel transport; technically, a connection) and the conjugate momentum variable is a coordinate frame (called a vierbein) at each point. [6] [7] So these variable became what we know as Ashtekar variables, a particular flavor of Einstein–Cartan theory with a complex connection. General relativity theory expressed in this way, made possible to pursue quantization of it using well-known techniques from quantum gauge field theory.

The quantization of gravity in the Ashtekar formulation was based on Wilson loops, a technique developed by Kenneth G. Wilson in 1974 [8] to study the strong-interaction regime of quantum chromodynamics (QCD). It is interesting in this connection that Wilson loops were known to be ill-behaved in the case of standard quantum field theory on (flat) Minkowski space, and so did not provide a nonperturbative quantization of QCD. However, because the Ashtekar formulation was background-independent, it was possible to use Wilson loops as the basis for nonperturbative quantization of gravity.

Due to efforts by Sen and Ashtekar, a setting in which the Wheeler–DeWitt equation was written in terms of a well-defined Hamiltonian operator on a well-defined Hilbert space was obtained. This led to the construction of the first known exact solution, the so-called Chern–Simons form or Kodama state. The physical interpretation of this state remains obscure.

In 1988–90, Carlo Rovelli and Lee Smolin obtained an explicit basis of states of quantum geometry, which turned out to be labeled by Penrose's spin networks. [9] [10] In this context, spin networks arose as a generalization of Wilson loops necessary to deal with mutually intersecting loops. Mathematically, spin networks are related to group representation theory and can be used to construct knot invariants such as the Jones polynomial. Loop quantum gravity (LQG) thus became related to topological quantum field theory and group representation theory.

In 1994, Rovelli and Smolin showed that the quantum operators of the theory associated to area and volume have a discrete spectrum. [11] Work on the semi-classical limit, the continuum limit, and dynamics was intense after this, but progress was slower.

On the semi-classical limit front, the goal is to obtain and study analogues of the harmonic oscillator coherent states (candidates are known as weave states).

Hamiltonian dynamics

LQG was initially formulated as a quantization of the Hamiltonian ADM formalism, according to which the Einstein equations are a collection of constraints (Gauss, Diffeomorphism and Hamiltonian). The kinematics are encoded in the Gauss and Diffeomorphism constraints, whose solution is the space spanned by the spin network basis. The problem is to define the Hamiltonian constraint as a self-adjoint operator on the kinematical state space. The most promising work[ according to whom? ] in this direction is Thomas Thiemann's Phoenix Project. [12]

Covariant dynamics

Much of the recent[ as of? ] work in LQG has been done in the covariant formulation of the theory, called "spin foam theory." The present version of the covariant dynamics is due to the convergent work of different groups, but it is commonly named after a paper by Jonathan Engle, Roberto Pereira and Carlo Rovelli in 2007–08. [13] Heuristically, it would be expected that evolution between spin network states might be described by discrete combinatorial operations on the spin networks, which would then trace a two-dimensional skeleton of spacetime. This approach is related to state-sum models of statistical mechanics and topological quantum field theory such as the Turaeev–Viro model of 3D quantum gravity, and also to the Regge calculus approach to calculate the Feynman path integral of general relativity by discretizing spacetime.

See also

Related Research Articles

<span class="mw-page-title-main">General relativity</span> Theory of gravitation as curved spacetime

General relativity, also known as the general theory of relativity, and as Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics. General relativity generalizes special relativity and refines Newton's law of universal gravitation, providing a unified description of gravity as a geometric property of space and time, or four-dimensional spacetime. In particular, the curvature of spacetime is directly related to the energy and momentum of whatever present matter and radiation. The relation is specified by the Einstein field equations, a system of second-order partial differential equations.

<span class="mw-page-title-main">Quantum gravity</span> Description of gravity using discrete values

Quantum gravity (QG) is a field of theoretical physics that seeks to describe gravity according to the principles of quantum mechanics. It deals with environments in which neither gravitational nor quantum effects can be ignored, such as in the vicinity of black holes or similar compact astrophysical objects, as well as in the early stages of the universe moments after the Big Bang.

<span class="mw-page-title-main">Loop quantum gravity</span> Theory of quantum gravity, merging quantum mechanics and general relativity

Loop quantum gravity (LQG) is a theory of quantum gravity that incorporates matter of the Standard Model into the framework established for the intrinsic quantum gravity case. It is an attempt to develop a quantum theory of gravity based directly on Albert Einstein's geometric formulation rather than the treatment of gravity as a mysterious mechanism (force). As a theory, LQG postulates that the structure of space and time is composed of finite loops woven into an extremely fine fabric or network. These networks of loops are called spin networks. The evolution of a spin network, or spin foam, has a scale on the order of a Planck length, approximately 10−35 meters, and smaller scales are meaningless. Consequently, not just matter, but space itself, prefers an atomic structure.

<span class="mw-page-title-main">Spin network</span> Diagram used to represent quantum field theory calculations

In physics, a spin network is a type of diagram which can be used to represent states and interactions between particles and fields in quantum mechanics. From a mathematical perspective, the diagrams are a concise way to represent multilinear functions and functions between representations of matrix groups. The diagrammatic notation can thus greatly simplify calculations.

Abhay Vasant Ashtekar is an Indian theoretical physicist who created Ashtekar variables and is one of the founders of loop quantum gravity and its subfield loop quantum cosmology. Ashtekar has also written a number of descriptions of loop quantum gravity that are accessible to non-physicists. He is an Evan Pugh Professor Emeritus of Physics and former Director of the Institute for Gravitational Physics and Geometry and Center for Fundamental Theory at Pennsylvania State University.

In theoretical physics, the Einstein–Cartan theory, also known as the Einstein–Cartan–Sciama–Kibble theory, is a classical theory of gravitation, one of several alternatives to general relativity. The theory was first proposed by Élie Cartan in 1922.

The Immirzi parameter is a numerical coefficient appearing in loop quantum gravity (LQG), a nonperturbative theory of quantum gravity. The Immirzi parameter measures the size of the quantum of area in Planck units. As a result, its value is currently fixed by matching the semiclassical black hole entropy, as calculated by Stephen Hawking, and the counting of microstates in loop quantum gravity.

In the ADM formulation of general relativity, spacetime is split into spatial slices and a time axis. The basic variables are taken to be the induced metric on the spatial slice and the metric's conjugate momentum , which is related to the extrinsic curvature and is a measure of how the induced metric evolves in time. These are the metric canonical coordinates.

In theoretical physics, geometrodynamics is an attempt to describe spacetime and associated phenomena completely in terms of geometry. Technically, its goal is to unify the fundamental forces and reformulate general relativity as a configuration space of three-metrics, modulo three-dimensional diffeomorphisms. The origin of this idea can be found in an English mathematician William Kingdon Clifford's works. This theory was enthusiastically promoted by John Wheeler in the 1960s, and work on it continues in the 21st century.

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<span class="mw-page-title-main">Canonical quantum gravity</span> A formulation of general relativity

In physics, canonical quantum gravity is an attempt to quantize the canonical formulation of general relativity. It is a Hamiltonian formulation of Einstein's general theory of relativity. The basic theory was outlined by Bryce DeWitt in a seminal 1967 paper, and based on earlier work by Peter G. Bergmann using the so-called canonical quantization techniques for constrained Hamiltonian systems invented by Paul Dirac. Dirac's approach allows the quantization of systems that include gauge symmetries using Hamiltonian techniques in a fixed gauge choice. Newer approaches based in part on the work of DeWitt and Dirac include the Hartle–Hawking state, Regge calculus, the Wheeler–DeWitt equation and loop quantum gravity.

Loop quantum cosmology (LQC) is a finite, symmetry-reduced model of loop quantum gravity (LQG) that predicts a "quantum bridge" between contracting and expanding cosmological branches.

A quantum field theory of general relativity provides operators that measure the geometry of spacetime. The volume operator of a region is defined as the operator that yields the expectation value of a volume measurement of the region , given a state of quantum General Relativity. I.e. is the expectation value for the volume of . Loop Quantum Gravity, for example, provides volume operators, area operators and length operators for regions, surfaces and path respectively.

In the field of theoretical physics, the Holst action is an equivalent formulation of the Palatini action for General Relativity (GR) in terms of vierbeins by adding a part of a topological term (Nieh-Yan) which does not alter the classical equations of motion as long as there is no torsion,

In the ADM formulation of general relativity one splits spacetime into spatial slices and time, the basic variables are taken to be the induced metric, , on the spatial slice, and its conjugate momentum variable related to the extrinsic curvature, ,. These are the metric canonical coordinates.

In theoretical physics, the problem of time is a conceptual conflict between general relativity and quantum mechanics in that quantum mechanics regards the flow of time as universal and absolute, whereas general relativity regards the flow of time as malleable and relative. This problem raises the question of what time really is in a physical sense and whether it is truly a real, distinct phenomenon. It also involves the related question of why time seems to flow in a single direction, despite the fact that no known physical laws at the microscopic level seem to require a single direction.

<span class="mw-page-title-main">Joseph Kouneiher</span> French mathematical physicist

Joseph Kouneiher is a French mathematical physicist. He is a professor of mathematical physics and engineering sciences at Nice SA University, France. He works primarily on the foundations of science, and his work in the domains of quantum field theory, quantum gravity, string theory and conformal field theory is widely cited and is well known. He holds three PHDs in mathematical physics and Epistemology and history of sciences.

References

  1. ↑ Élie Cartan. "Sur une généralisation de la notion de courbure de Riemann et les espaces à torsion." C. R. Acad. Sci. (Paris) 174, 593–595 (1922); Élie Cartan. "Sur les variétés à connexion affine et la théorie de la relativité généralisée." Part I: Ann. Éc. Norm.40, 325–412 (1923) and ibid. 41, 1–25 (1924); Part II: ibid. 42, 17–88 (1925).
  2. ↑ Penrose, Roger (1971). "Applications of negative dimensional tensors". Combinatorial Mathematics and its Applications. Academic Press. ISBN   0-12-743350-3.
  3. ↑ Penrose, Roger (1971). "Angular momentum: an approach to combinatorial space-time". In Bastin, Ted (ed.). Quantum Theory and Beyond. Cambridge University Press. ISBN   0-521-07956-X.
  4. ↑ Penrose, Roger (1987). "On the Origins of Twistor Theory". In Rindler, Wolfgang; Trautman, Andrzej (eds.). Gravitation and Geometry, a Volume in Honour of Ivor Robinson. Bibliopolis. ISBN   88-7088-142-3.
  5. ↑ Amitabha Sen, "Gravity as a spin system," Phys. Lett.B119:89–91, December 1982.
  6. ↑ Abhay Ashtekar, "New variables for classical and quantum gravity," Phys. Rev. Lett., 57, 2244-2247, 1986.
  7. ↑ Abhay Ashtekar, "New Hamiltonian formulation of general relativity," Phys. Rev.D36, 1587-1602, 1987.
  8. ↑ Wilson, K. (1974). "Confinement of quarks". Physical Review D . 10 (8): 2445. Bibcode:1974PhRvD..10.2445W. doi:10.1103/PhysRevD.10.2445.
  9. ↑ Carlo Rovelli and Lee Smolin, "Knot theory and quantum gravity," Phys. Rev. Lett., 61 (1988) 1155.
  10. ↑ Carlo Rovelli and Lee Smolin, "Loop space representation of quantum general relativity," Nuclear PhysicsB331 (1990) 80-152.
  11. ↑ Carlo Rovelli, Lee Smolin, "Discreteness of area and volume in quantum gravity" (1994): arXiv:gr-qc/9411005.
  12. ↑ Thiemann, T (2006). "The Phoenix Project: Master constraint programme for loop quantum gravity". Classical and Quantum Gravity. 23 (7): 2211–2247. arXiv: gr-qc/0305080 . Bibcode:2006CQGra..23.2211T. doi:10.1088/0264-9381/23/7/002. S2CID   16304158.
  13. ↑ Jonathan Engle, Roberto Pereira, Carlo Rovelli, "Flipped spinfoam vertex and loop gravity". Nucl. Phys.B798 (2008). 251–290. arXiv:0708.1236.

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