Loop quantum gravity (LQG) is a theory of
quantum gravity
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 vi ...
, which aims to merge
quantum mechanics
Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...
and
general relativity
General relativity, also known as the general theory of relativity and 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 ...
, incorporating matter of the
Standard Model
The Standard Model of particle physics is the theory describing three of the four known fundamental forces ( electromagnetic, weak and strong interactions - excluding gravity) in the universe and classifying all known elementary particles. It ...
into the framework established for the pure quantum gravity case. It is an attempt to develop a quantum theory of gravity based directly on Einstein's geometric formulation rather than the treatment of gravity as a 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 network
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 ...
s. The evolution of a spin network, or
spin foam, has a scale above 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.
The areas of research, which involves about 30 research groups worldwide, share the basic physical assumptions and the mathematical description of quantum space. Research has evolved in two directions: the more traditional canonical loop quantum gravity, and the newer covariant loop quantum gravity, called
spin foam theory. The most well-developed theory that has been advanced as a direct result of loop quantum gravity is called
loop quantum cosmology
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.
The distinguishing feature of LQC is the prominent role play ...
(LQC). LQC advances the study of the early universe, incorporating the concept of the
Big Bang
The Big Bang event is a physical theory that describes how the universe expanded from an initial state of high density and temperature. Various cosmological models of the Big Bang explain the evolution of the observable universe from the ...
into the broader theory of the
Big Bounce
The Big Bounce is a hypothesized cosmological model for the origin of the known universe. It was originally suggested as a phase of the ''cyclic model'' or ''oscillatory universe'' interpretation of the Big Bang, where the first cosmological even ...
, which envisions the Big Bang as the beginning of a
period of expansion that follows a period of contraction, which one could talk of as the ''
Big Crunch
The Big Crunch is a hypothetical scenario for the ultimate fate of the universe, in which the expansion of the universe eventually reverses and the universe recollapses, ultimately causing the cosmic scale factor to reach zero, an event potential ...
.''
History
In 1986,
Abhay Ashtekar
Abhay Vasant Ashtekar (born 5 July 1949) is an Indian theoretical physicist. He is the Eberly Professor of Physics and the Director of the Institute for Gravitational Physics and Geometry at Pennsylvania State University. As the creator of Ash ...
reformulated Einstein's general relativity in a language closer to that of the rest of fundamental physics, specifically
Yang–Mills theory
In mathematical physics, Yang–Mills theory is a gauge theory based on a special unitary group SU(''N''), or more generally any compact, reductive Lie algebra. Yang–Mills theory seeks to describe the behavior of elementary particles using t ...
. Shortly after,
Ted Jacobson and
Lee Smolin
Lee Smolin (; born June 6, 1955) is an American theoretical physicist, a faculty member at the Perimeter Institute for Theoretical Physics, an adjunct professor of physics at the University of Waterloo and a member of the graduate faculty of the ...
realized that the formal equation of quantum gravity, called the
Wheeler–DeWitt equation
The Wheeler–DeWitt equation for theoretical physics and applied mathematics, is a field equation attributed to John Archibald Wheeler and Bryce DeWitt. The equation attempts to mathematically combine the ideas of quantum mechanics and gene ...
, admitted solutions labelled by loops when rewritten in the new
Ashtekar variables
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 q_ (x) on the spatial slice and the metric's conjugate momentum K^ (x), which is relate ...
.
Carlo Rovelli
Carlo Rovelli (born May 3, 1956) is an Italian theoretical physicist and writer who has worked in Italy, the United States and, since 2000, in France. He is also currently a Distinguished Visiting Research Chair at the Perimeter Institute, and ...
and Smolin defined a
nonperturbative and background-independent quantum theory of gravity in terms of these loop solutions.
Jorge Pullin and
Jerzy Lewandowski understood that the intersections of the loops are essential for the consistency of the theory, and the theory should be formulated in terms of intersecting loops, or
graphs.
In 1994, Rovelli and Smolin showed that the quantum
operators
Operator may refer to:
Mathematics
* A symbol indicating a mathematical operation
* Logical operator or logical connective in mathematical logic
* Operator (mathematics), mapping that acts on elements of a space to produce elements of another sp ...
of the theory associated to area and volume have a discrete spectrum. That is, geometry is quantized. This result defines an explicit basis of states of quantum geometry, which turned out to be labelled by
Roger Penrose
Sir Roger Penrose (born 8 August 1931) is an English mathematician, mathematical physicist, philosopher of science and Nobel Laureate in Physics. He is Emeritus Rouse Ball Professor of Mathematics in the University of Oxford, an emeritus f ...
's
spin network
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 ...
s, which are
graphs labelled by
spins
The spins (as in having "the spins")Diane Marie Leiva. ''The Florida State University College of Education''Women's Voices on College Drinking: The First-Year College Experience"/ref> is an adverse reaction of intoxication that causes a state of v ...
.
The canonical version of the dynamics was established by Thomas Thiemann, who defined an anomaly-free Hamiltonian operator and showed the existence of a mathematically consistent background-independent theory. The covariant, or "
spin foam", version of the dynamics was developed jointly over several decades by research groups in France, Canada, UK, Poland, and Germany. It was completed in 2008, leading to the definition of a family of transition amplitudes, which in the
classical limit
The classical limit or correspondence limit is the ability of a physical theory to approximate or "recover" classical mechanics when considered over special values of its parameters. The classical limit is used with physical theories that predict n ...
can be shown to be related to a family of truncations of general relativity. The finiteness of these amplitudes was proven in 2011. It requires the existence of a positive
cosmological constant
In cosmology, the cosmological constant (usually denoted by the Greek capital letter lambda: ), alternatively called Einstein's cosmological constant,
is the constant coefficient of a term that Albert Einstein temporarily added to his field eq ...
, which is consistent with observed
acceleration in the expansion of the Universe.
Background independence
LQG is formally
background independent. Meaning the equations of LQG are not embedded in, or dependent on, space and time (except for its invariant topology). Instead, they are expected to give rise to space and time at distances which are 10 times the
Planck length. The issue of background independence in LQG still has some unresolved subtleties. For example, some derivations require a fixed choice of the
topology
In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
, while any consistent quantum theory of gravity should include topology change as a dynamical process.
Space-time as a "container" over which physics takes place has no objective physical meaning and instead the gravitational interaction is represented as just one of the fields forming the world. This is known as the relationalist interpretation of space-time. In LQG this aspect of general relativity is taken seriously and this symmetry is preserved by requiring that the physical states remain invariant under the generators of diffeomorphisms. The interpretation of this condition is well understood for purely spatial diffeomorphisms. However, the understanding of diffeomorphisms involving time (the
Hamiltonian constraint
The Hamiltonian constraint arises from any theory that admits a Hamiltonian formulation and is reparametrisation-invariant. The Hamiltonian constraint of general relativity is an important non-trivial example.
In the context of general relativ ...
) is more subtle because it is related to
dynamics and the so-called "
slave nature of time" in general relativity. A generally accepted calculational framework to account for this constraint has yet to be found. A plausible candidate for the quantum Hamiltonian constraint is the operator introduced by Thiemann.
Constraints and their Poisson bracket algebra
Dirac observables
The constraints define a constraint surface in the original phase space. The gauge motions of the constraints apply to all phase space but have the feature that they leave the constraint surface where it is, and thus the orbit of a point in the hypersurface under gauge transformations will be an orbit entirely within it.
Dirac observables are defined as phase space functions,
, that Poisson commute with all the constraints when the constraint equations are imposed,
:
that is, they are quantities defined on the constraint surface that are invariant under the gauge transformations of the theory.
Then, solving only the constraint
and determining the Dirac observables with respect to it leads us back to the
Arnowitt–Deser–Misner (ADM) phase space with constraints
. The dynamics of general relativity is generated by the constraints, it can be shown that six Einstein equations describing time evolution (really a gauge transformation) can be obtained by calculating the Poisson brackets of the three-metric and its conjugate momentum with a linear combination of the spatial diffeomorphism and Hamiltonian constraint. The vanishing of the constraints, giving the physical phase space, are the four other Einstein equations.
Quantization of the constraints – the equations of quantum general relativity
Pre-history and Ashtekar new variables
Many of the technical problems in canonical quantum gravity revolve around the constraints. Canonical general relativity was originally formulated in terms of metric variables, but there seemed to be insurmountable mathematical difficulties in promoting the constraints to
quantum operator
In physics, an operator is a function over a space of physical states onto another space of physical states. The simplest example of the utility of operators is the study of symmetry (which makes the concept of a group useful in this context). Beca ...
s because of their highly non-linear dependence on the canonical variables. The equations were much simplified with the introduction of Ashtekar's new variables. Ashtekar variables describe canonical general relativity in terms of a new pair of canonical variables closer to those of gauge theories. The first step consists of using densitized triads
(a triad
is simply three orthogonal vector fields labeled by
and the densitized triad is defined by
) to encode information about the spatial metric,
:
(where
is the flat space metric, and the above equation expresses that
, when written in terms of the basis
, is locally flat). (Formulating general relativity with triads instead of metrics was not new.) The densitized triads are not unique, and in fact one can perform a local in space
rotation
Rotation, or spin, is the circular movement of an object around a '' central axis''. A two-dimensional rotating object has only one possible central axis and can rotate in either a clockwise or counterclockwise direction. A three-dimensional ...
with respect to the internal indices
. The canonically conjugate variable is related to the extrinsic curvature by
. But problems similar to using the metric formulation arise when one tries to quantize the theory. Ashtekar's new insight was to introduce a new configuration variable,
that behaves as a complex
connection where
is related to the so-called
spin connection
In differential geometry and mathematical physics, a spin connection is a connection on a spinor bundle. It is induced, in a canonical manner, from the affine connection. It can also be regarded as the gauge field generated by local Lorentz tr ...
via
. Here
is called the chiral spin connection. It defines a covariant derivative
. It turns out that
is the conjugate momentum of
, and together these form Ashtekar's new variables.
The expressions for the constraints in Ashtekar variables; Gauss's theorem, the spatial diffeomorphism constraint and the (densitized) Hamiltonian constraint then read:
:
:
:
respectively, where
is the field strength tensor of the connection
and where
is referred to as the vector constraint. The above-mentioned local in space rotational invariance is the original of the
gauge invariance here expressed by Gauss's theorem. Note that these constraints are polynomial in the fundamental variables, unlike the constraints in the metric formulation. This dramatic simplification seemed to open up the way to quantizing the constraints. (See the article
Self-dual Palatini action for a derivation of Ashtekar's formalism).
With Ashtekar's new variables, given the configuration variable
, it is natural to consider wavefunctions
. This is the connection representation. It is analogous to ordinary quantum mechanics with configuration variable
and wavefunctions
. The configuration variable gets promoted to a quantum operator via:
:
(analogous to
) and the triads are (functional) derivatives,
:
(analogous to
). In passing over to the quantum theory the constraints become operators on a kinematic Hilbert space (the unconstrained
Yang–Mills Hilbert space). Note that different ordering of the
's and
's when replacing the
's with derivatives give rise to different operators – the choice made is called the factor ordering and should be chosen via physical reasoning. Formally they read
:
:
:
There are still problems in properly defining all these equations and solving them. For example, the Hamiltonian constraint Ashtekar worked with was the densitized version instead of the original Hamiltonian, that is, he worked with
. There were serious difficulties in promoting this quantity to a quantum operator. Moreover, although Ashtekar variables had the virtue of simplifying the Hamiltonian, they are complex. When one quantizes the theory, it is difficult to ensure that one recovers real general relativity as opposed to complex general relativity.
Quantum constraints as the equations of quantum general relativity
The classical result of the Poisson bracket of the smeared Gauss' law
with the connections is
:
The quantum Gauss' law reads
:
If one smears the quantum Gauss' law and study its action on the quantum state one finds that the action of the constraint on the quantum state is equivalent to shifting the argument of
by an infinitesimal (in the sense of the parameter
small) gauge transformation,
:
and the last identity comes from the fact that the constraint annihilates the state. So the constraint, as a quantum operator, is imposing the same symmetry that its vanishing imposed classically: it is telling us that the functions