LQG for the Bewildered

Abstract

We present a pedagogical introduction to the notions underlying the connection formulation of General Relativity -Loop Quantum Gravity (LQG) -with an emphasis on the physical aspects of the framework. We begin by reviewing General Relativity and Quantum Field Theory, to emphasise the similarities between them which establish a foundation upon which to build a theory of quantum gravity. We then explain, in a concise and clear manner, the steps leading from the Einstein-Hilbert action for gravity to the construction of the quantum states of geometry, known as spin-networks, which provide the basis for the kinematical Hilbert space of quantum general relativity. Along the way we introduce the various associated concepts of tetrads, spin-connection and holonomies which are a pre-requisite for understanding the LQG formalism. Having provided a minimal introduction to the LQG framework, we discuss its applications to the problems of black hole entropy and of quantum cosmology. A list of the most common criticisms of LQG is presented, which are then tackled one by one in order to convince the reader of the physical viability of the theory.An extensive set of appendices provide accessible introductions to several key notions such as the Peter-Weyl theorem, duality of differential forms and Regge calculus, among others. The presentation is aimed at graduate students and researchers who have some familiarity with the tools of quantum mechanics and field theory and/or General Relativity, but are intimidated by the seeming technical prowess required to browse through the existing LQG literature. Our hope is to make the formalism appear a little less bewildering to the un-initiated and to help lower the barrier for entry into the field.

Quantum Error Correction in Loop Quantum Gravity

“…With each edge we can associate a triangle of area A j i . The four triangles associated with the given vertex, will then close up 3 to form a tetrahedron with which we can associate an operator whose eigenvalues can be interpreted as the volume of the resulting "quantum 2 there are many introductory reviews on LQG, at various levels of technical sophistication ranging from advanced [20,21,22] to intermediate [23,24,25,26] to (relatively speaking) elementary [27,28,29] 3 The four angular momenta must satisfy the closure condition 4 i=1 j i = 0. This is analogous to the requirement that a classical tetrahedron satisfies tetrahedron".…”