Spin Foams

Abstract

Ever since special relativity, space and time have become seamlessly merged into a single entity, and space-time symmetries, such as Lorentz invariance, have played a key role in our fundamental understanding of nature. Quantum mechanics, however, did not originally conform to this new way of thinking. The original formulation of quantum mechanics, called 'canonical', involves wavefunctions, operators, Hamiltonians, and time evolution in a way that treats time very differently from space. This situation was improved by Feynman, who formulated quantum mechanics in terms of probabilities calculated by summing over amplitudes associated to classical histories -the path integral formulation of quantum mechanics. As histories are naturally space-time objects in which space and time can be viewed 'on equal footing', the path integral formulation allowed, for the first time, space-time symmetries to be manifest in a general quantum theory.The key insight of Einstein's theory of gravity, general relativity, is that gravity is spacetime geometry. Space-time geometry, the one 'background structure' -i.e., non-dynamical space-time structure -remaining after special relativity, was discovered to be dynamical and to describe the gravitational field, revealing nature to be 'background independent.' Background independence can equivalently be expressed in terms of a profound enlargement of the basic spacetime symmetry group of physics: invariance under Lorentz transformations and translations is replaced by invariance under the much larger group of space-time diffeomorphisms.We have already seen in the chapter by Sahlmann on gravity, geometry and the quantum, a canonical quantization of Einstein's gravity, and hence of geometry, in which geometric operators are derived with discrete eigenvalues [1, 2, 3]. Instead of space being a smooth continuum, we see that it comes in discrete quanta -minimal 'chunks of space.' Furthermore, as discussed in the chapter by Agullo and Corichi, when applied to cosmology, this quantum theory of gravity leads to a new understanding of the Big Bang in which usually problematic infinities are resolved, and one can actually ask what happened before the Big Bang. In spite of these successes, because it is a canonical theory, it has as a drawback that space-time symmetries, in particular space-time diffeomorphism symmetry, are not manifest. Equivalently, the preferred separation between space and time prevents full background independence from being manifest.One can ask: Is there a way to construct a path-integral formulation of quantum gravity, in which the most radical discovery of general relativity, background independence, or equivalently, space-time diffeomorphism invariance, can be fully manifest, which nevertheless retains the successes of the canonical theory? This is the question leading to the spin foam program. In answering it one must understand more carefully the relationship between the canonical and path integral formulations of quantum mechanics, and in particular how these apply to g...

Spin-foam model for gravity coupled to massless scalar field

“…Our motivation to study the coupling of spin foams with massless scalar field comes from several directions. The first direction is the spin-foam theory [1][2][3][4][5][6][7][8]. The research in the theory has been focused on studying gravitational field and there are interesting proposals for such models [9][10][11][12][13].…”