Deepak Vaid

LQG for the Bewildered

We present a pedagogical introduction to the notions underlying the connec- tion 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 foun- dation 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 min- imal 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 dierential 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

Expanding on Classical GR

We now return to the discussion of general relativity. Equipped with the preceding discussions of both the quantisation of field theories, and the geometrical interpretations of gauge transformations, it is time to set about formulating what will eventually become a theory of dynamical spacetime obeying rules adapted from quantum field theory. But before we get there we must cast classical GR into a form amenable to quantisation.

Kinematical Hilbert Space

The allowed loop states that spacetime is composed of can take several forms. They may consist of simple closed loops. These loops may be linked through each other.

Quantum Field Theory

Quantum Field Theory should be familiar to most (if not all) modern physicists, however we feel it is worth mentioning the basic details here, in order to emphasize the similarities between QFT and GR, and hence illustrate how GR can be written as a gauge theory. In short, we will see that a local change of phase of the wavefunction is equivalent to the position-dependent change of basis we considered in the case of GR. Just as the partial derivative of a vector gave (via the product rule) a derivative term corresponding to the change in basis, we will see that a derivative term arises corresponding to the change in phase of the quantum field. This introduces a connection and a covariant derivative defined in terms of the connection.

First Steps to a Theory of Quantum Gravity

As discussed in the previous section, we wish to attempt to canonically quantise GR, which means turning the Hamiltonian, diffeomorphism and Gauss constraints into operators and replacing Poisson brackets with commutation relations.