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Beyond the Standard Model
Standard Model
Quantum gravity
String theory
Loop quantum gravity
Causal dynamical triangulation
Canonical general relativity
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Loop quantum gravity (LQG), also known as loop gravity and quantum geometry, is a proposed quantum theory of spacetime which attempts to reconcile the seemingly incompatible theories of quantum mechanics and general relativity. In Physics, the Standard Model of Particle physics is currently the best description of all experimental data The Standard Model of Particle physics is a theory that describes three of the four known Fundamental interactions together with the Elementary particles Quantum gravity is the field of Theoretical physics attempting to unify Quantum mechanics, which describes three of the fundamental forces of nature String theory is a still-developing scientific approach to Theoretical physics, whose original building blocks are one-dimensional extended objects called strings Causal dynamical triangulation (abbreviated as "CDT" invented by Renate Loll, Jan Ambjørn and Jerzy Jurkiewicz is an approach to Quantum In Physics, canonical quantum gravity is an attempt to quantize the canonical formulation of general relativity (or canonical gravity) In Theoretical physics, quantum geometry is the set of new mathematical concepts generalizing the concepts of Geometry whose understanding is necessary to describe SpaceTime is a patent-pending three dimensional graphical user interface that allows end users to search their content such as Google Google Images Yahoo! YouTube eBay Amazon and RSS Quantum mechanics is the study of mechanical systems whose dimensions are close to the Atomic scale such as Molecules Atoms Electrons General relativity or the general theory of relativity is the geometric theory of Gravitation published by Albert Einstein in 1916 It preserves many of the important features of general relativity, while at the same time employing quantization of both space and time at the Planck scale in the tradition of quantum mechanics. In Particle physics and Physical cosmology, the Planck scale is an Energy scale around 1 The technique of loop quantization was developed for the nonperturbative quantization of diffeomorphism-invariant gauge theory. In Mathematics, a diffeomorphism is an Isomorphism of Smooth manifolds It is an Invertible function that maps one Differentiable Gauge theory is a peculiar Quantum field theory where the Lagrangian is invariant under certain transformations Roughly, LQG tries to establish a quantum theory of gravity in which the very space, where all other physical phenomena occurs, becomes quantized. Quantum gravity is the field of Theoretical physics attempting to unify Quantum mechanics, which describes three of the fundamental forces of nature

LQG is one of a family of theories called canonical quantum gravity. In Physics, canonical quantum gravity is an attempt to quantize the canonical formulation of general relativity (or canonical gravity) A list of quantum gravity theories can be found on the quantum gravity page. Quantum gravity is the field of Theoretical physics attempting to unify Quantum mechanics, which describes three of the fundamental forces of nature The critics of this theory say that LQG is a theory of gravity and nothing more, though several LQG theorists are presently working to show that the theory can describe matter as well.

Contents

Loop quantum gravity in general, and its ambitions

Though not proven, it may be impossible to quantize gravity in 3+1 dimensions without creating matter and energy artifacts. SpaceTime is a patent-pending three dimensional graphical user interface that allows end users to search their content such as Google Google Images Yahoo! YouTube eBay Amazon and RSS Should LQG succeed as a quantum theory of gravity, the known matter fields will have to be incorporated into the theory a posteriori. Many of the approaches now being actively pursued (by Loll, Smolin, Bilson-Thompson, Freidel, Wise and others[1]) combine matter with geometry. Lee Smolin (born 1955 in New York City) is an American Theoretical physicist, a researcher at the Perimeter Institute for Theoretical Physics Several of these current efforts would be proven wrong if evidence were found of extra spatial dimensions.

The main claimed successes of loop quantum gravity are:

1. It is a nonperturbative quantization of 3-space geometry, with quantized area and volume operators. In Quantum mechanics, perturbation theory is a set of approximation schemes directly related to mathematical perturbation for describing a complicated quantum system In Physics, quantization is a procedure for constructing a Quantum field theory starting from a classical field theory. In physics an operator is a function acting on the space of Physical states As a resultof its application on a physical state another physical state is obtained

2. It includes a calculation of the entropy of black holes. In Thermodynamics (a branch of Physics) entropy, symbolized by S, is a measure of the unavailability of a system ’s Energy A black hole is a theoretical region of space in which the Gravitational field is so powerful that nothing not even Electromagnetic radiation (e

3. It replaces the Big Bang spacetime singularity with a Big Bounce. The Big Bang is the cosmological model of the Universe that is best supported by all lines of scientific evidence and Observation. A gravitational singularity (sometimes spacetime singularity) is approximately a place where quantities which are used to measure the Gravitational field become The Big Bounce is a theorized Scientific model related to the creation of the known Universe.

However, these claims are not generally accepted among the physics community. While many of the core results are rigorous mathematical physics, their physical interpretations remain speculative. Mathematical physics is the scientific discipline concerned with the interface of Mathematics and Physics. LQG may possibly be viable as a refinement of either gravity or geometry. It has not been shown that loop quantum gravity reproduces general relativity as a low energy limit. Thus for example, the entropy calculated in (2) is for a kind of hole which may, or may not, be a black hole.

Some alternative approaches to quantum gravity, such as spin foam models, are closely related to loop quantum gravity. In Physics, a spin foam is a topological structure made out of two-dimensional faces that represents one of the configurations that must be summed to obtain a Feynman's

The apparent incompatibility between quantum mechanics and general relativity

Main article: quantum gravity

In general relativity, the Einstein field equations assign a geometry (via a metric) to space-time. Quantum gravity is the field of Theoretical physics attempting to unify Quantum mechanics, which describes three of the fundamental forces of nature The Einstein field equations ( EFE) or Einstein's equations are a set of ten equations in Einstein 's theory of General relativity in which the In Mathematics, a metric space is a set where a notion of Distance (called a metric) between elements of the set is defined Before this, there is no physical notion of distance or time measurements. In this sense, general relativity is said to be background independent. An immediate conceptual issue that arises is that the usual framework of quantum mechanics, including quantum field theory, relies on a reference (background) space-time. In quantum field theory (QFT the forces between particles are mediated by other particles Therefore, one approach to finding a quantum theory of gravity is to understand how to do quantum mechanics without relying on such a background; this is the approach of the canonical quantization/loop quantum gravity/spin foam approaches.

Furthermore, in the framework of quantum field theory, and using the standard techniques of perturbative calculations, one finds that gravitation is non-renormalizable in contrast to the electroweak and strong interactions of the Standard Model of particle physics. In Quantum mechanics, perturbation theory is a set of approximation schemes directly related to mathematical perturbation for describing a complicated quantum system In Quantum field theory, the Statistical mechanics of fields and the theory of self-similar geometric structures renormalization refers to a collection This technical term implies that there are infinitely many free parameters in the theory and thus that it cannot be predictive.

Another interface between general relativity and quantum mechanics occurs in quantum field theory studied on curved (non-Minkowskian) backgrounds. In quantum field theory (QFT the forces between particles are mediated by other particles In Physics and Mathematics, Minkowski space (or Minkowski spacetime) is the mathematical setting in which Einstein's theory of Special relativity The vacuum, when it exists, is shown in general relativity to depend on the path of the observer through space-time (see Unruh effect). The Unruh effect, discovered in 1976 by Bill Unruh of the University of British Columbia, is the prediction that an accelerating observer will observe The Unruh effect can be described semi-classically in the case of a fixed background geometry on which propagate non-gravitational quantum mechanical particles (quanta). It's then natural to wonder about the inclusion of quantum gravitational effects, including interactions with the graviton (in analogy with the electron's interactions with the electromagnetic field of the nucleus within an atom). In Physics, the graviton is a hypothetical Elementary particle, a Boson to be exact that mediates the force of Gravity in the framework

History of LQG

Main article: history of loop quantum gravity

In 1986, Abhay Ashtekar reformulated Einstein's field equations of general relativity, using what have come to be known as Ashtekar variables, a particular flavor of Einstein-Cartan theory with a complex connection. General relativity is the theory of Gravitation published by Albert Einstein in 1915. Year 1986 ( MCMLXXXVI) was a Common year starting on Wednesday (link displays 1986 Gregorian calendar) Abhay Ashtekar (born July 5 1949) is an Indian physicist He is the Eberly Professor of Physics and the Director of the Institute for Gravitational Physics and In Theoretical physics, Ashtekar (new variables (named after Abhay Ashtekar who invented them represent an unusual way to rewrite the metric on the three-dimensional Einstein–Cartan theory in Theoretical physics extends General relativity to correctly handle Spin angular momentum. He was able to quantize gravity using gauge field theory. Gauge theory is a peculiar Quantum field theory where the Lagrangian is invariant under certain transformations In the Ashtekar formulation, the fundamental objects are a rule for parallel transport (technically, a connection) and a coordinate frame (called a vierbein) at each point. In Geometry, parallel transport is a way of transporting geometrical data along smooth curves in a Manifold. In Geometry, the notion of a connection (also connexion) makes precise the idea of transporting data along a curve or family of curves in a parallel and This page covers applications of the Cartan formalism. For the general concept see Cartan connection. Because the Ashtekar formulation was background-independent, it was possible to use Wilson loops as the basis for a nonperturbative quantization of gravity. In Gauge theory, a Wilson loop (named after Kenneth Wilson) is a gauge-invariant observable obtained from the Holonomy of the Gauge connection Explicit (spatial) diffeomorphism invariance of the vacuum state plays an essential role in the regularization of the Wilson loop states. In Mathematics, a diffeomorphism is an Isomorphism of Smooth manifolds It is an Invertible function that maps one Differentiable In Quantum field theory, the vacuum state (also called the vacuum) is the Quantum state with the lowest possible Energy.

Around 1990, Carlo Rovelli and Lee Smolin obtained an explicit basis of states of quantum geometry, which turned out to be labelled by Penrose's spin networks. Year 1990 ( MCMXC) was a Common year starting on Monday (link displays the 1990 Gregorian calendar) Carlo Rovelli is an Italian physicist and cosmologist who has worked in Italy the USA and France Lee Smolin (born 1955 in New York City) is an American Theoretical physicist, a researcher at the Perimeter Institute for Theoretical Physics 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 physics 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. In the mathematical field of Knot theory, a knot invariant is a quantity (in a broad sense defined for each knot which is the same for equivalent knots In the mathematical field of Knot theory, a knot polynomial is a Knot invariant in the form of a Polynomial whose coefficients encode some of

Being closely related to topological quantum field theory and group representation theory, LQG is mostly established at the level of rigour of mathematical physics. A topological quantum field theory (or topological field theory or TQFT) is a Quantum field theory which computes Topological invariants In the mathematical field of Representation theory, group representations describe abstract groups in terms of Linear transformations of Mathematical physics is the scientific discipline concerned with the interface of Mathematics and Physics. Physical interpretations are however more speculative.

The ingredients of loop quantum gravity

Loop quantization

At the core of loop quantum gravity is a framework for nonperturbative quantization of diffeomorphism-invariant gauge theories, which one might call loop quantization. In Mathematics, a diffeomorphism is an Isomorphism of Smooth manifolds It is an Invertible function that maps one Differentiable Gauge theory is a peculiar Quantum field theory where the Lagrangian is invariant under certain transformations While originally developed in order to quantize vacuum general relativity in 3+1 dimensions, the formalism can accommodate arbitrary spacetime dimensionalities, fermions,[2] an arbitrary gauge group (or even quantum group), and supersymmetry,[3] and results in a quantization of the kinematics of the corresponding diffeomorphism-invariant gauge theory. Gauge theory is a peculiar Quantum field theory where the Lagrangian is invariant under certain transformations In Mathematics and Theoretical physics, quantum groups are certain Noncommutative algebras that first appeared in the theory of Quantum integrable systems Much work remains to be done on the dynamics, the classical limit and the correspondence principle, all of which are necessary in one way or another to make contact with experiment.

In a nutshell, loop quantization is the result of applying C*-algebraic quantization to a non-canonical algebra of gauge-invariant classical observables. C*-algebras (pronounced "C-star" are an important area of research in Functional analysis, a branch of Mathematics. In Physics, the canonical commutation relation is the relation between Canonical conjugate quantities (quantities which are related by definition such that one is Non-canonical means that the basic observables quantized are not generalized coordinates and their conjugate momenta. Instead, the algebra generated by spin network observables (built from holonomies) and field strength fluxes is used.

Loop quantization techniques are particularly successful in dealing with topological quantum field theories, where they give rise to state-sum/spin-foam models such as the Turaev-Viro model of 2+1 dimensional general relativity. A much studied topological quantum field theory is the so-called BF theory in 3+1 dimensions. Since classical general relativity can be formulated as a BF theory with constraints, scientists hope that a consistent quantization of gravity may arise from the perturbation theory of BF spin-foam models.

This discrete structure may require modifications of quantum mechanics, and a line of research called polymer quantum mechanics has been pursued.

Lorentz invariance

Main article: Lorentz covariance

LQG is a quantization of a classical Lagrangian field theory which is equivalent to the usual Einstein-Cartan theory in that it leads to the same equations of motion describing general relativity with torsion. In standard Physics, Lorentz covariance is a key property of Spacetime that follows from the Special theory of relativity, where it applies globally Einstein–Cartan theory in Theoretical physics extends General relativity to correctly handle Spin angular momentum. General relativity or the general theory of relativity is the geometric theory of Gravitation published by Albert Einstein in 1916 The term torsion may refer the following In geometry Torsion of curves Torsion tensor in differential geometry As such, it can be argued that LQG respects local Lorentz invariance. Global Lorentz invariance is broken in LQG just as in general relativity. General relativity or the general theory of relativity is the geometric theory of Gravitation published by Albert Einstein in 1916 A positive cosmological constant can be realized in LQG by replacing the Lorentz group with the corresponding quantum group. In Physical cosmology, the cosmological constant (usually denoted by the Greek capital letter Lambda: Λ was proposed by Albert Einstein as a modification In Physics (and mathematics the Lorentz group is the group of all Lorentz transformations of Minkowski spacetime, the classical setting In Mathematics and Theoretical physics, quantum groups are certain Noncommutative algebras that first appeared in the theory of Quantum integrable systems

Diffeomorphism invariance and background independence

General covariance is the invariance of physical laws under arbitrary coordinate transformations. In Theoretical physics, general covariance (also known as Diffeomorphism covariance or general invariance) is the Invariance of the This condition is most noteworthy in the context of general relativity where it has some profound implications, as Einstein discovered. The argument involves only the very basics of GR, as we will see below. More details and discussions can be found in Rovelli's book or the papers by Rovelli and Gaul[4] and by Smolin. [5]

It begins with a mathematical observation. Here is written the differential equation for the simple harmonic oscillator twice

Eq(1) \frac{d^2 f(x)}{dx^2} + f(x) = 0
Eq(2) \frac{d^2 g(y)}{dy^2} + g(y) = 0

except in Eq(1) the independent variable is x and in Eq(2) the independent variable is y. Once we find out that a solution to Eq(1) is f(x) = cosx, we immediately know that g(y) = cosy solves Eq(2). This observation combined with general covariance has profound implications for GR.

Assume pure gravity first. Say we have two coordinate systems, x-coordinates and y-coordinates. General covariance demands the equations of motion have the same form in both coordinate systems, that is, we have exactly the same differential equation to solve in both coordinate systems, except in one the independent variable is x and in the other the independent variable is y. In Theoretical physics, general covariance (also known as Diffeomorphism covariance or general invariance) is the Invariance of the Once we find a metric function gab(x) that solves the EQM in the x-coordinates we immediately know (by exactly the same reasoning as above!) that the same function written as a function of y solves the EOM in the y-coordinates. As both metric functions have the same functional form but belong to different coordinate systems, they impose different spacetime geometries. Thus we have generated a second distinct solution! Now comes the problem. Say the two coordinate systems coincide at first, but at some point after t = 0 we allow them to differ. We then have two solutions, they both have the same initial conditions yet they impose different spacetime geometries. The conclusion is that GR does not determine the proper-time between spacetime points! The argument I have given (or rather a refinement of it) is what's known as Einstein's hole argument. It is straightforward to include matter - we have a larger set of differential equations but they still have the same form in all coordinates systems, so the same argument applies and again we obtain two solutions with the same initial conditions which impose different spacetime geometries. It is very important to note that we could not have generated these extra distinct solutions if spacetime were fixed and non-dynamical, and so the resolution to the hole argument, background independence, only comes about when we allow spacetime to be dynamical. Before we can go on to understand this resolution we need to better understand these extra solutions. We can interpret these solutions as follows. For simplicity we first assume there is no matter. Define a metric function \tilde{g}_{ab} whose value at P is given by the value of gab at P0, i. e.

Eq(3) \tilde{g}_{ab}(P) = g_{ab}(P_0).

(see figure 1(a)). Now consider a coordinate system which assigns to P the same coordinate values that P0 has in the x-coordinates (see figure 1(b)). We then have

Eq(4) \tilde{g}_{ab} (y_0=u_0,y_1=u_1, y_2=u_2, y_3=u_3) = g_{ab} (x_0=u_0,x_1=u_1, x_2=u_2 , x_3=u_3),

where u0,u1,u2,u3 are the coordinate values of P0 in the x-coordinate system.

Figure 1
Figure 1

When we allow the coordinate values to range over all permissible values, Eq(4) is precisely the condition that the two metric functions have the same functional form! We see that the new solution is generated by dragging the original metric function over the spacetime manifold while keeping the coordinate lines "attached", see Fig 1. It is important to realise that we are not performing a coordinate transformation here, this is what's known as an active diffeomorphism (coordinate transformations are called passive diffeomorphisms). It should be easy to see that when we have matter present, simultaneously performing an active diffeomorphism on the gravitational and matter fields generates the new distinct solution.

The resolution to the hole argument (mainly taken from Rovelli's book) is as follows. As GR does not determine the distance between spacetime points, how the gravitational and matter fields are located over spacetime, and so the values they take at spacetime points, can have no physical meaning. What GR does determine, however, are the mutual relations that exist between the gravitational field and the matter fields (i. e. the value the gravitational field takes where the matter field takes such and such value). From these mutual relations we can form a notion of matter being located with respect to the gravitational field and vice-versa, (see Rovelli's for exposition). What Einstein discovered was that physical entities are located with respect to one another only and not with respect to the spacetime manifold. This is what background independence is! And that is the context for Einstein's remark "beyond my wildest expectations".

Since the Hole Argument is a direct consequence of the general covariance of GR, this led Einstein to state:

"That this requirement of general covariance, which takes away from space and time the last remnant of physical objectivity, is a natural one, . . . "[6]

LQG preserves this symmetry under active diffeomorphisms by requiring that the physical states remain invariant under the generators of active diffeomorphisms. The interpretation of this condition is well understood for purely spatial active diffeomorphisms. However, the understanding of active diffeomorphisms involving time (the Hamiltonian constraint) is more subtle because it is related to dynamics and the so-called problem of time in general relativity. In Loop quantum gravity, dynamics such as time-evolutions of fields are controlled by the Hamiltonian constraint. A generally accepted calculational framework to account for this constraint is yet to be found.

The term "active diffeomorphism" has been used, instead of just "diffeomorphism", to emphasize that this is not a case of simple coordinate transformations. It is active diffeomorphisms which are the gauge transformations of GR and they should not be confused with the freedom of choosing coordinates on the space-time M. Invariance under coordinate transformations is not a special feature of GR as all physical theories are invariant under coordinate transformations. (Indeed, the mathematical definition of a diffeomorphism is a transformation which relates manifolds with equivalent topological and differentiable structure, but not necessarily equivalent metrics. For example, a diffeomorphism can turn a doughnut into a tea cup. )

Whether or not Lorentz invariance is broken in the low-energy limit of LQG, the theory is formally background independent. Background independence is a condition in theoretical physics especially in Quantum gravity (QG that requires the defining equations of a theory to be independent of the actual The equations of LQG are not embedded in, or presuppose, space and time, except for its invariant topology. Instead, they are expected to give rise to space and time at distances which are large compared to the Planck length. The Planck length, denoted by \scriptstyle\ell_P \, is the unit of Length approximately 1 At present, it remains unproven that LQG's description of spacetime at the Planckian scale has the right continuum limit, described by general relativity with possible quantum corrections.

LQG and the big bang singularity

Main article: loop quantum cosmology

Abhay Ashtekar and Martin Bojowald have released papers stating that according to loop quantum gravity, the singularity of the Big Bang is avoided. Loop quantum cosmology (LQC is a finite symmetry reduced model of Loop quantum gravity. Abhay Ashtekar (born July 5 1949) is an Indian physicist He is the Eberly Professor of Physics and the Director of the Institute for Gravitational Physics and Martin Bojowald is a German -born physicist who now works at the Institute for Gravitational and the Cosmos of the Pennsylvania State University USA In Mathematics, a singularity is in general a point at which a given mathematical object is not defined or a point of an exceptional set where it fails to be The Big Bang is the cosmological model of the Universe that is best supported by all lines of scientific evidence and Observation. What the researchers found was a prior collapsing universe. Since gravity becomes repulsive near Planck density according to their simulations, this resulted in a "Big Bounce" and the birth of our current universe. The Planck density is the unit of Density, denoted by ρ P in the system of Natural units known as Planck units. The Big Bounce is a theorized Scientific model related to the creation of the known Universe. [7] These topics are an active research in loop quantum cosmology. Loop quantum cosmology (LQC is a finite symmetry reduced model of Loop quantum gravity. However these results involve a truncation of the theory and so do not properly apply to loop quantum gravity itself.

LQG and particle physics

Main article: preon

There have been recent claims that loop quantum gravity may be able to reproduce features resembling the Standard Model. In Particle physics, preons are postulated "point-like" particles conceived to be subcomponents of Quarks and Leptons The word was coined The Standard Model of Particle physics is a theory that describes three of the four known Fundamental interactions together with the Elementary particles So far only the first generation of fermions (leptons and quarks) with correct parity properties have been modelled by Sundance Bilson-Thompson using preons constituted of braids of spacetime as the building blocks. In Particle physics, fermions are particles which obey Fermi-Dirac statistics; they are named after Enrico Fermi. Leptons are a family of fundamental Subatomic particles comprising the Electron, the Muon, and the Tauon (or tau particle as well as their In Physics, a quark (kwɔrk kwɑːk or kwɑːrk is a type of Subatomic particle. Dr Sundance O Bilson-Thompson is an Australian theoretical particle physicist. In Particle physics, preons are postulated "point-like" particles conceived to be subcomponents of Quarks and Leptons The word was coined [8] However, there is no derivation of the Lagrangian that would describe the interactions of such particles, nor is it possible to show that such particles are fermions, nor that the gauge groups or interactions of the Standard Model are realised. The Lagrangian, L of a Dynamical system is a function that summarizes the dynamics of the system Utilization of quantum computing concepts made it possible to demonstrate that the particles are able to survive quantum fluctuations. A quantum computer is a device for Computation that makes direct use of distinctively Quantum mechanical Phenomena, such as superposition In Quantum physics, a quantum fluctuation is the temporary change in the amount of energy in a point in space arising from Werner Heisenberg 's Uncertainty principle [9] Other recent results suggest that LQG's framework may allow for the derivation of certain spin-1 bosons such as the photon, and gluon, and possibly the spin-2 graviton. In Particle physics, bosons are particles which obey Bose-Einstein statistics; they are named after Satyendra Nath Bose and Albert Einstein In Physics, the photon is the Elementary particle responsible for electromagnetic phenomena Gluons ( Glue and the suffix -on) are Elementary particles that cause Quarks to interact and are indirectly responsible for the In Physics, the graviton is a hypothetical Elementary particle, a Boson to be exact that mediates the force of Gravity in the framework This line of research follows the reductionist paradigm of finding a building block of elementary particles.

Independent of the above discussion, there exist various proposals on how to incorporate fermions (matter) within LQG's framework. A single framework that can account for the standard model and gravity is known in physics as a theory of everything. The Standard Model of Particle physics is a theory that describes three of the four known Fundamental interactions together with the Elementary particles A theory of everything ( TOE) is a putative Theory of Theoretical physics that fully explains and links together all known physical phenomena

It currently appears that nothing forbids coupling anomalous - i. e. quantum mechanically inconsistent - chiral fermions to LQG [citation needed].

LQG and the Graviton

There have been recent results in LQG using the spinfoam formalism by Carlo Rovelli, Eugenio Bianchi, Leonardo Modesto, and Simone Speziale[10][11]that LQG does give rise to gravitons, and allows gravitons to interact as expected, reproducing Newton's law of gravity.

The Kodama state

Main article: Immirzi parameter

In 1988, Hideo Kodama wrote down the equations of the Kodama state, but as it described a positive (de Sitter universe) spacetime, which was believed to be inconsistent with observation, it was largely ignored. The Immirzi parameter (also known as the Barbero-Immirzi parameter) is a numerical Coefficient appearing in Loop quantum gravity, a nonperturbative theory A de Sitter universe is a solution to Einstein 's field equations of General Relativity which is named after Willem de Sitter.

Lee Smolin's paper, "Quantum gravity with a positive cosmological constant"[12] suggests that the Kodama state is a ground state which has a good semiclassical limit which reproduces the dynamics of general relativity with a positive (de Sitter) cosmological constant, 4 dimensions, and gravitons, and is an exact solution to ordinary constraints on background independent quantum gravity, providing evidence that loop quantum gravity is indeed a quantum gravity with the correct semiclassical description. Lee Smolin (born 1955 in New York City) is an American Theoretical physicist, a researcher at the Perimeter Institute for Theoretical Physics General relativity or the general theory of relativity is the geometric theory of Gravitation published by Albert Einstein in 1916 Edward Witten published a paper titled "A Note On The Chern-Simons And Kodama Wavefunctions" in response to Lee Smolin's, arguing that the Kodama state is unphysical, due to an analogy to a state in Chern-Simons theory wavefunction resulting in negative energies,[13] and citing Smolin's paper. Edward Witten (born August 26, 1951) is an American Theoretical physicist and Professor at the Institute for Advanced Study The Chern-Simons theory is a 3-dimensional Topological quantum field theory of Schwarz type, developed by Shiing-Shen Chern and James Harris Simons Recently, Andrew Randono has published two papers that cite Witten's paper,[14][15] and address these objections, by generalizing the Kodama state, with the conclusion that the Immirzi parameter, when generalized with a real value, fixed by matching with black hole entropy, describes parity violation in quantum gravity, and is CPT invariant, and is normalizable, and chiral, consistent with known observations of both gravity and quantum field theory. The Immirzi parameter (also known as the Barbero-Immirzi parameter) is a numerical Coefficient appearing in Loop quantum gravity, a nonperturbative theory In Physics, black hole thermodynamics is the area of study that seeks to reconcile the Laws of thermodynamics with the existence of Black hole Event CPT symmetry is a fundamental symmetry of Physical laws under transformations that involve the inversions of charge, parity and In Quantum mechanics, Wave functions which describe real particles must be normalisable: the probability of the particle to occupy any place must A phenomenon is said to be chiral if it is not identical to its Mirror image (see Chirality) Randono claims that Witten's conclusions rest on the immirizi parameter taking on an imaginary number, which simplifies the equation. Geometric interpretation Geometrically imaginary numbers are found on the vertical axis of the complex number plane The physical inner product may resemble the MacDowell-Mansouri action formulation of gravity. The MacDowell-Mansouri action is a mathematical object (an action) that is used to derive Einstein's field equations of General relativity.

Spinfoam

Main article: spin foam

Spinfoam may be regarded as the path integral formulation of LQG. In Physics, a spin foam is a topological structure made out of two-dimensional faces that represents one of the configurations that must be summed to obtain a Feynman's This article is about a formulation of quantum mechanics For integrals along a path also known as line or contour integrals see Line integral. There have been results to show that in 2+1 dimensions, LQG and spin foam are exactly equivalent, [16], but spin foam models based on the Barrett-Crane model are known to be inequivalent to 3+1 Hamiltonian formulation. The Barrett-Crane model is a model in Quantum gravity which was defined using the Plebanski action. Rovelli has provided results with a modified spin foam, one that incorporates an immirizi parameter, is equivalent in spinfoam in 3+1 dimensions. [17].

non commutative geometry and loop gravity

Main article: Noncommutative standard model

Non commutative geometry and loop quantum gravity may be linked, according to recent papers published by Johannes Aastrup, Jesper M. The noncommutative standard model is an extension of the Standard Model to include a modified form of General relativity, developed by Alain Connes Grimstrup, Ryszard Nest. [18], as well as Abhay Ashtekar [19]. Abhay Ashtekar (born July 5 1949) is an Indian physicist He is the Eberly Professor of Physics and the Director of the Institute for Gravitational Physics and Should this research program succeed, Alaine Conne's Noncommutative geometry would give gravity and the standard model, with LQG providing techniques to quantize the gravity. LQG on a noncommutative space may have the standard model, hence making it a candidate theory of everything.

LQG and analogues to condensed matter physics

Condensed matter physics is the field of physics that deals with the macroscopic physical properties of matter. Condensed matter physics is the field of Physics that deals with the macroscopic physical properties of Matter. In particular, it is concerned with the "condensed" phases that appear whenever the number of constituents in a system is extremely large and the interactions between the constituents are strong.

Loop quantum gravity offers a candidate spacetime atom, the spin network, that may strongly interact with one another quantum mechanically. SpaceTime is a patent-pending three dimensional graphical user interface that allows end users to search their content such as Google Google Images Yahoo! YouTube eBay Amazon and RSS 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 physics There maybe 10^99 spin network units in a volume contained by a hydrogen atom, a number far in excess to the 10^28 atoms present in a molar amount of a substance. There have been two research programs, "quantum graphity" and group field theory" that attempt to apply principles and mechanisms that condensed matter physics uses when they apply it to matter. There are two outstanding issues in loop quantum gravity -- incorporating the standard model within its framework, or even better, deriving the standard model, and explaining how a continuous spacetime emerges from a discrete model. The Standard Model of Particle physics is a theory that describes three of the four known Fundamental interactions together with the Elementary particles The two may be interrelated, and perhaps condensed matter physics mechanisms may play an analogous role. Condensed matter physics is the field of Physics that deals with the macroscopic physical properties of Matter. There has been an increasing interest and research programs in the loop quantum gravity community to appropriate results from condensed matter physics analogues. Various forms of topological order may give rise to some features of the standard model through string nets, and group field theory may model continuous spacetime from discrete atoms as a Bose Einstein condensate. In Physics, topological order is a new kind of order (a newkind of organization of particles in a Quantum state that is beyond theLandau symmetry-breaking description A Bose–Einstein condensate (BEC is a State of matter of Bosons confined in an external Potential and cooled to Temperatures very near to Interestingly, both group field theory and quantum graphity make use of bosonic systems, rather than fermionic. In Particle physics, bosons are particles which obey Bose-Einstein statistics; they are named after Satyendra Nath Bose and Albert Einstein In Particle physics, fermions are particles which obey Fermi-Dirac statistics; they are named after Enrico Fermi.

LQG and string nets

Main article: string-net

MIT's Xiao-Gang Wen and Harvard University's Michael Levin are two condensed matter physics researchers who have attempted to model elementary particles such as electrons and photons as resulting from a discrete lattice structure of spacetime in analogy to phonons in solid state physics. In Condensed matter physics, a string-net is an extended object whose collective behavior has been proposed as a physical explanation for Topological order by Condensed matter physics is the field of Physics that deals with the macroscopic physical properties of Matter. In Physics, a phonon is a quantized mode of vibration occurring in a rigid crystal lattice, such as the Atomic lattice of a Solid Solid-state physics, the largest branch of Condensed matter physics, is the study of rigid Matter, or Solids The bulk of solid-state physics theory and They attempt to model elementary particles as emergent properties of a string-net condensation based on a mechanism called topological order in condensed matter physics, and LQG's spin networks have the properties necessary to reproduce the standard model as the result of the collective behavior of a group of spin networks. In Condensed matter physics, a string-net is an extended object whose collective behavior has been proposed as a physical explanation for Topological order by In Physics, topological order is a new kind of order (a newkind of organization of particles in a Quantum state that is beyond theLandau symmetry-breaking description The Standard Model of Particle physics is a theory that describes three of the four known Fundamental interactions together with the Elementary particles [20][21] This approach differs from the preon approach, in that Wen and Levin see particles as an emergent property of quantum spacetime, rather than built up of smaller substructures as is the case with Bilson-Thompson's preon theory. For other uses see Emergence (disambiguation, Emergent, and Emergency. In Particle physics, preons are postulated "point-like" particles conceived to be subcomponents of Quarks and Leptons The word was coined

The project to embed topological order in loop quantum gravity is called quantum graphity, and Tomasz Konopka, Fotini Markopoulou and Simone Severini have argued that loop quantum gravity's spin networks are equivalent to Wen and Levin's string net condensation, and give rise directly to U(1) gauge charge and electrons. In Physics, topological order is a new kind of order (a newkind of organization of particles in a Quantum state that is beyond theLandau symmetry-breaking description Dr Fotini G Markopoulou-Kalamara is a Greek Theoretical physicist interested in foundational Mathematics and Quantum mechanics. [22] In the quantum graphity model, points in spacetime are represented by nodes on a graph connected by links that can be on or off. This indicates whether or not the two points are directly connected as if they are next to each other in spacetime. When they are on the links have additional state variables which are used to define the random dynamics of the graph under the influence of quantum fluctuations and temperature. At high temperature the graph is in Phase I where all the points are randomly connected to each other and no concept of spacetime as we know it exists. As the temperature drops and the graph cools, it is conjectured to undergo a phase transition to a Phase II where spacetime forms. It will then look like a spacetime manifold on large scales with only near-neighbour points being connected in the graph. The hypothesis of quantum graphity is that this geometrogenesis models the condensation of spacetime in the big bang.

LQG and group field theory

Einstein's theory of spacetime, general relativity, describes spacetime as a continuum, whereas loop quantum gravity describes the quanta of spacetime as discrete. There is some analogy to fluids, which can often be approximated as continuous, even if they are actually made of discrete atoms. Daniele Oriti has proposed group field theory as an approach to derive a classical, continuous spacetime from a discrete atomic manner in analogy to mechanisms found in condensed matter physics, and suggests that spin networks can undergo a condensation that is analogous to a Bose Einstein condensate[23] as an example of a transition from discrete to continuous. Condensed matter physics is the field of Physics that deals with the macroscopic physical properties of Matter. A Bose–Einstein condensate (BEC is a State of matter of Bosons confined in an external Potential and cooled to Temperatures very near to

Problems

While there has been a recent proposal relating to observation of naked singularities,[24] and doubly special relativity, as a part of a program called loop quantum cosmology, as of now there is no experimental observation for which loop quantum gravity makes a prediction not made by the Standard Model or general relativity. Doubly-special relativity (DSR— also called deformed special relativity or by some extra-special relativity — is a modified theory of Special relativity Loop quantum cosmology (LQC is a finite symmetry reduced model of Loop quantum gravity. This problem plagues all current theories of quantum gravity (except those that have been proven wrong).

Making predictions from the theory of LQG has been extremely difficult computationally, also a recurring problem with modern theories in physics.

Another problem is that a crucial free parameter in the theory known as the Immirzi parameter can only be computed by demanding agreement with Bekenstein and Hawking's calculation of the black hole entropy. The Immirzi parameter (also known as the Barbero-Immirzi parameter) is a numerical Coefficient appearing in Loop quantum gravity, a nonperturbative theory Jacob David Bekenstein (born May 1, 1947) is a Physicist who has contributed to the foundation of Black hole thermodynamics and to other aspects Stephen William Hawking CH, CBE, FRS, FRSA (born 8 January 1942 is a British theoretical physicist. Loop quantum gravity predicts that the entropy of a black hole is proportional to the area of the event horizon, but does not obtain the Bekenstein-Hawking formula S = A/4 unless the Immirzi parameter is chosen to give this value. A prediction directly from theory would be preferable.

Presently, no semiclassical limit recovering general relativity has been shown to exist.

See also

References

  1. ^ See List of loop quantum gravity researchers
  2. ^ Baez, John, Krasnov,Kirill : Quantization of Diffeomorphism-Invariant Theories with Fermions http://front.math.ucdavis.edu/9703.3112
  3. ^ Ling,Yi; Smolin, Lee : Supersymmetric Spin Networks and Quantum Supergravity http://www.arxiv.org/abs/hep-th/9904016
  4. ^ Gaul, Marcus; Rovelli, Carlo (2000). This is a list of researchers in the Physics field of Loop quantum gravity. Loop quantum cosmology (LQC is a finite symmetry reduced model of Loop quantum gravity. In Mathematics, Heyting algebras are special Partially ordered sets that constitute a generalization of Boolean algebras named after Arend Heyting In Mathematics, category theory deals in an abstract way with mathematical Structures and relationships between them it abstracts from sets Noncommutative geometry, or NCG, is a branch of Mathematics concerned with the possible spatial interpretations of Algebraic structures for which the In Mathematics, a topos (plural "topoi" or "toposes" is a type of category that behaves like the category of sheaves of sets C*-algebras (pronounced "C-star" are an important area of research in Functional analysis, a branch of Mathematics. In General relativity, Regge calculus is a formalism for producing simplicial approximations of spacetimes which are solutions to the Einstein field equation Doubly-special relativity (DSR— also called deformed special relativity or by some extra-special relativity — is a modified theory of Special relativity In physics invariance mechanics, in its simplest form is the rewriting of the laws of Quantum field theory in terms of invariant quantities only This is a list of researchers in the Physics field of Loop quantum gravity. "Loop Quantum Gravity and the Meaning of Diffeomorphism Invariance" (subscription required). Lect. Notes Phys. 541: 277–324.  
  5. ^ Smolin, Lee. "The case for background independence" (subscription required). hep-th/0507235.  
  6. ^ Einstein, Albert; H. Albert Einstein ( German: ˈalbɐt ˈaɪ̯nʃtaɪ̯n; English: ˈælbɝt ˈaɪnstaɪn (14 March 1879 – 18 April 1955 was a German -born theoretical A. Lorentz, H. Weyl, and H. Minkowski (1916). The Principle of Relativity, 117. A principle of relativity is a criterion for judging physical theories, stating that they are inadequate if they do not prescribe the exact same laws of physics in  
  7. ^ "Researchers Look Beyond the Birth of the Universe", Eberly College of Science, 12 May 2006.  
  8. ^ Bilson-Thompson, Sundance O. ; Markopoulou,Fotini; Smolin, Lee. "Quantum gravity and the standard model".  
  9. ^ Castelvecchi, Davide; Valerie Jamieson (August 12 2006). "You are made of space-time". New Scientist (2564).  
  10. ^ Bianchi, Eugenio; Leonardo Modesto, Carlo Rovelli, Simone Speziale (10 Apr 2006). "Graviton propagator in loop quantum gravity". Class. Quant. Grav. 23: 6989–7028. doi:10.1088/0264-9381/23/23/024. A digital object identifier ( DOI) is a permanent identifier given to an Electronic document. arXiv:gr-qc/0604044. The arXiv ( pronounced " Archive " as if the "X" were the Greek letter Chi, χ is an Archive for electronic  
  11. ^ Rovelli, Carlo (23 Dec 2005). "Graviton propagator from background-independent quantum gravity". Phys. Rev. Lett. 97: 151301. arXiv:gr-qc/0508124. The arXiv ( pronounced " Archive " as if the "X" were the Greek letter Chi, χ is an Archive for electronic  
  12. ^ Smolin, Lee (9 Sep 2002). "Quantum gravity with a positive cosmological constant". arXiv:hep-th/0209079. The arXiv ( pronounced " Archive " as if the "X" were the Greek letter Chi, χ is an Archive for electronic  
  13. ^ "{{{title}}}" . 0306083.  | title=A Note On The Chern-Simons And Kodama Wavefunctions| first= Edward| last= Witten| month=19 Jun| year= 2003}}
  14. ^ Randono, Andrew (14 Nov 2006). "Generalizing the Kodama State I: Construction". arXiv:gr-qc/0611073. The arXiv ( pronounced " Archive " as if the "X" were the Greek letter Chi, χ is an Archive for electronic  
  15. ^ Randono, Andrew (14 Nov 2006). "Generalizing the Kodama State II: Properties and Physical Interpretation". arXiv:gr-qc/0611074. The arXiv ( pronounced " Archive " as if the "X" were the Greek letter Chi, χ is an Archive for electronic  
  16. ^ arXiv:gr-qc/0402112 "We present a rigorous regularization of Rovellis's generalized projection operator in canonical 2+1 gravity. This work establishes a clear-cut connection between loop quantum gravity and the spin foam approach in this simplified setting"
  17. ^ http://cift.fuw.edu.pl/users/kostecki/zakopane08/rovelli.pdf
  18. ^ http://arxiv.org/abs/0802.1783
  19. ^ http://arxiv.org/abs/0802.2527
  20. ^ Levin, Michael; Wen, Xiao-Gang (23 Sep 2005) "Photons and electrons as emergent phenomena" http://arxiv.org/abs/cond-mat/0407140 page 8 "loop quantum gravity appears to be a string net condensation. . . "
  21. ^ Konopka, Tomasz; Markopoulou, Fotini; Smolin, Lee (17 Nov 2006) "Quantum Graphity" http://arxiv.org/abs/hep-th/0611197 page 3: "we argue, but do not prove, that loop quantum gravity's spin networks can reproduce Wen's and Levin's string net condensation"
  22. ^ Konopka, Tomasz; Markopoulou, Fotini; Severini, Simone (6 Jan 2008) "Quantum Graphity: a model of emergent locality" http://arxiv.org/abs/0801.0861
  23. ^ Oriti, Daniele (17 Oct 2007). "Group field theory as the microscopic description of the quantum spacetime fluid: a new perspective on the continuum in quantum gravity". arXiv:0710.3276. The arXiv ( pronounced " Archive " as if the "X" were the Greek letter Chi, χ is an Archive for electronic  
  24. ^ 404 error. Institute of Physics. Retrieved on 2006-08-19. Year 2006 ( MMVI) was a Common year starting on Sunday of the Gregorian calendar. Events 43 BC - Octavian, later known as Augustus compels the Roman Senate to elect him Consul.

Bibliography

External links


Dictionary

loop quantum gravity

-noun

  1. (physics) A proposed quantum theory of spacetime which attempts to reconcile the seemingly incompatible theories of quantum mechanics and general relativity
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