Question

I'm begining to study Quantization of field with the second quantization formalism. I've studied phononic field, electromagnetic field in the vacuum and a generic relativistical scalar field.

I asked to me if is possible doing the same thing with the Gravitational Field Hamiltonian.

I've heard that we can do it only in the condition of linearized Gravity and we obtain a field with spin 2, but we can't do it in the general condition because we don't have quantization conditions.

What are these conditions? And how can we obtain only in linearized gravity the quantized field? And how it has got spin 2?

Answer 1

Spin 2 just means that the gravitational field is given by a metric field and general covariance, which is the nonlinear expression of a massless spin 2 representation of the Poincare group. The latter appears when linearizing around the Minkowski metric and dropping all interactions.

See the classical paper by S. Weinberg, Phys.Rev. 138 (1965), B988-B1002 and the entry ''Why do gravitons have spin 2?"" in Chapter B8: ''Quantum gravity'' of my theoretical physics FAQ at http://www.mat.univie.ac.at/~neum/physfaq/physics-faq.html

It is not true that only the linearized gravitational field can be quantized. Currently perhaps the best synopsis of canonical nonlinear quantum gravity is the following paper:

R. Brunetti, K. Fredenhagen, K. Rejzner, Quantum gravity from the point of view of locally covariant quantum field theory, 51 pages http://arxiv.org/abs/1306.1058 (Submitted on 5 Jun 2013)

Abstract: We construct perturbative quantum gravity in a generally covariant way. In particular our construction is background independent. It is based on the locally covariant approach to quantum field theory and the renormalized Batalin-Vilkovisky formalism. We do not touch the problem of nonrenormalizability and interpret the theory as an effective theory at large length scales.