Is Loop quantum gravity canonical quantization of Ashtekar's new variablesThis year I’m hoping to integrate ideas about “quantization” into the course more than in the past, starting off with the mathematics behind what physicists often call “canonical quantization”. This topic is worked out very explicitly and in great detail in this book-
Spring Course
Posted on January 10, 2023 by woit
Demystifier said:Loop quantum gravity is not exactly a quantization of Ashtekar variables. Instead, Ashtekar variables are an important step towards construction of variables that are quantized in loop quantum gravity. As far as I know, there is no direct quantization of Ashtekar variables. And finally, those variables are no longer "new".
is quantization of Ashtekar variables possible and would be a
quantum gravity
Canonical quantization is a procedure used in quantum field theory to quantize classical fields. In the context of Ashtekar's new variables, this involves promoting the classical variables, which reformulate general relativity, to quantum operators and imposing commutation relations between them. This approach aims to provide a quantum description of gravity.
Ashtekar's new variables are a reformulation of general relativity using a set of complex variables that simplify the equations of the theory. These variables include a self-dual connection and a densitized triad, which replace the traditional metric and its conjugate momentum. This reformulation is particularly useful in the context of loop quantum gravity.
Canonical quantization is important for Ashtekar's new variables because it provides a systematic way to transition from classical to quantum gravity. By quantizing these variables, one can attempt to develop a consistent theory of quantum gravity, addressing the problem of how to unify general relativity with quantum mechanics.
Several challenges arise in the canonical quantization of Ashtekar's new variables, including the implementation of the reality conditions to ensure the variables correspond to real physical quantities, the proper treatment of the constraints inherent in general relativity, and the definition of a suitable Hilbert space for the quantum states. Additionally, finding solutions to the quantum constraints, such as the Hamiltonian constraint, remains a significant challenge.
The canonical quantization of Ashtekar's new variables is a foundational aspect of loop quantum gravity (LQG). In LQG, the quantized Ashtekar variables lead to a discrete structure of space at the quantum level, represented by spin networks. This approach provides insights into the quantum geometry of spacetime and offers potential resolutions to singularities, such as those found in black holes and the Big Bang.