![]() ![]() 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). It requires the existence of a positive cosmological constant, which is consistent with observed acceleration in the expansion of the Universe. The finiteness of these amplitudes was proven in 2011. It was completed in 2008, leading to the definition of a family of transition amplitudes, which in the classical limit can be shown to be related to a family of truncations of general relativity. 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. 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. This result defines an explicit basis of states of quantum geometry, which turned out to be labelled by Roger Penrose's spin networks, which are graphs labelled by spins. In 1994, Rovelli and Smolin showed that the quantum operators of the theory associated to area and volume have a discrete spectrum. 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. Carlo Rovelli and Smolin defined a nonperturbative and background-independent quantum theory of gravity in terms of these loop solutions. Shortly after, Ted Jacobson and Lee Smolin realized that the formal equation of quantum gravity, called the Wheeler–DeWitt equation, admitted solutions labelled by loops when rewritten in the new Ashtekar variables. In 1986, Abhay Ashtekar reformulated Einstein's general relativity in a language closer to that of the rest of fundamental physics, specifically Yang–Mills theory. Main article: History of loop quantum gravity LQC advances the study of the early universe, incorporating the concept of the Big Bang into the broader theory of the Big Bounce, which envisions the Big Bang as the beginning of a period of expansion that follows a period of contraction, which has been described as the Big Crunch. The most well-developed theory that has been advanced as a direct result of loop quantum gravity is called loop quantum cosmology (LQC). 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 areas of research, which involve about 30 research groups worldwide, share the basic physical assumptions and the mathematical description of quantum space. Consequently, not just matter, but space itself, prefers an atomic structure. 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. These networks of loops are called spin networks. 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. 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 mysterious mechanism (force). Donaldson–Thomas invariants have close connections to Gromov–Witten invariants of algebraic three-folds and the theory of stable pairs due to Rahul Pandharipande and Thomas.ĭonaldson–Thomas theory is physically motivated by certain BPS states that occur in string and gauge theory pg 5.Loop quantum gravity ( LQG) is a theory of quantum gravity, which aims to reconcile quantum mechanics and general relativity, incorporating matter of the Standard Model into the framework established for the intrinsic quantum gravity case. The invariants were introduced by Simon Donaldson and Richard Thomas ( 1998). The Donaldson–Thomas invariant is a holomorphic analogue of the Casson invariant. Given a compact moduli space of sheaves on a Calabi–Yau threefold, its Donaldson–Thomas invariant is the virtual number of its points, i.e., the integral of the cohomology class 1 against the virtual fundamental class. In mathematics, specifically algebraic geometry, Donaldson–Thomas theory is the theory of Donaldson–Thomas invariants.
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