# Quantization of Gravitational Field: Quantization conditions

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?

• Take a look at arxiv.org/abs/gr-qc/9405057 and some of the Refs. therein. This should be a good starting point, which may allow you to answer the questions yourself. – Daniel Grumiller Feb 20 '11 at 15:10
• Good question, looking forward to answers. But your last question is trivial if you know what scalar and vector field is. Namely, they are fields transforming as spin 0 and spin 1 representations of the Lorentz group. These numbers actually correspond to number of tensor indices of the field, so it shouldn't be surprising that metric as a rank two tensor is associated with spin 2 particles. There is some more math relating to group theory involved (and also some subtleties because of the masslessness) but this is the basic reason. – Marek Feb 20 '11 at 15:16
• @Marek , sorry I haven't (yet) studied group theory. I can understand what are you saying because the scalar field has not elicity and polarizations, and a vectorial field has 3 components and I can connect to them 3 polarization and the field has elicity. but the electromagnetic field is a tensor too $F^{\mu\nu}$, we quantize the vectorial field potential $A^\mu$ don't we? And for a tensorial field like gravity what are elicities? – Boy Simone Feb 20 '11 at 15:26
• it's a little bit more complicated than that. EM field has only 2 physical polarizations (the transverse ones) because it is massless. Massive vector field would also have a longitudinal polarization (for a total of 3) but again one polarization is unphysical. As for tensor fields there are more options. It depends on what kind of tensor we are talking about (e.g. whether it is symmetric, traceless, and so on) and again also whether it is massive. But in general when something is massless it has has only two helicities (the left- and right-handed ones). – Marek Feb 20 '11 at 15:33
• I've bean using EM based quantum formula(Braket, Expectation and Operators) for some time. I still feel uncomfortable using it, as compared with other forms of mathematical methods. I think if I got a chance to see its application with other forces (other than EM), I might get a better understanding of how it operates. Even if the final formulas show an unattainable solution, I like this question, and always appreciate a good demonstration. – Luke Burgess Dec 10 '13 at 20:10

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.

• Good paper, still trying to pece together some of the formulas. I sappose I could try and answer this myself, would be my first real attempt at a quantum question on the physics stack. inspirehep.net/record/1237092/files/arXiv%3A1306.1058.pdf – Luke Burgess Dec 11 '13 at 16:31
• -1; i) completely inaccessible link for a person who is beginning to study second quantization. ii) obscure and imprecise explanation of the reason for graviton being a spin-2 particle. – user24155 Dec 12 '13 at 6:41
• @ArashArabi: Criticising is easy. Be constructive, and post a clearer and more precise explanation of the reason for the graviton being a spin-2 particle! – Arnold Neumaier Dec 13 '13 at 9:30
• Oh Arnold, it is good to see you back. – Jia Yiyang Dec 13 '13 at 13:09
• @ArnoldNeumaier I will try Arnold. Also, I did not mean to say your answer was not helpful; it was just not helpful enough for a beginner. The edit made it certainly better though, in my opinion. – user24155 Dec 13 '13 at 23:50