Quantization of Newton’s Gravity

In this work we will use a recently developed non relativistic (NR) quantization methodology that successfully overcomes troubles with infinities that plague non-renormalizable quantum field theories (QFTs). The ensuing methodology is here applied to Newton’s gravitation potential. We employ here the concomitant mathematical apparatus to formulate the NR QFT discussed in the well known classical text-book by Fetter and Walecka. We emphasize the fact that we speak of non relativistic QFT. This is so because we appeal to Newton’s gravitational potential, while in a relativistic QFT one does not employ potentials. Our main protagonist is the notion of propagator. This notion is of the essence in non relativistic quantum field theory (NR-QFT). Indeed, propagators are indispensable tools for both nuclear physics and condensed matter theory, among other disciplines. In the present work we deal with propagators for both fermions and bosons.


Preliminaries
In this work we will use a recently developed non relativistic quantization methodology that successfully overcomes all troubles of non-renormalizable QFT [1]. The essential result of such procedures is that we can dispense with renormalization and counter-terms. The reader can consult the recent references [1] The above claims are validated because infinities in Feynman diagrams, that arise in the convolution of quantum propagators (QP), disappear if one 1) represents QP by ultra-hyperfinctions (a generalization of Schwartz' distributions) and follows this technique with an appropriate Laurent expansion. The facts 1) and 2) above are clearly explained, with all kind of details, in [1] [2] [3] [4] [5]. Accordingly, no more mathematical aspects of the procedure need to be given in this paper.
The techniques of [1] [2] [3] [4] [5] are here applied to Newton's gravitation potential. We strongly emphasize the fact that, since we will be inserting a gravitational potential into a Schrödinger Equation (SE), the ensuing discussion is per force non-relativistic, and as such is the character of SE.

Organizing Our Material
In Section 2 we revisit Newton's gravity. Section 3 is devoted to an explicit display of results belonging to [6], concerning non relativistic quantum field theory (NR-QFT). In Section 4 we apply the results of Sections 2 and 3 so as to obtain the N-QFT of Newton's gravity. We discuss, as examples, the calculation of the self-energy for fermions and of the dressed propagator for both, bosons and fermions, to first order in perturbation theory. Some conclusions are drawn in Section 5.

Newton's Gravity
As stated above, 1 r − is viewed here as ( ) ( ) Remember also that We need now the Fourier transform of Integrating over φ one finds Finally, dealing with the variable r we arrive at As an example, consider now the anti transform of 2 4 k − π and verify that it is

Self Energies
The energy that a particle gains as the result of environment-modifications that it itself generates is called a self-energy Σ . This quantity denotes the contribution to the particle's effective mass due to interactions particle-surrounding medium (SM). Consider the particular (and common) condensed matter scenario: electrons moving in a material. Σ represents there the potential felt by a given electron due to the SM's interactions with it. Given that electrons repel each other, a moving electron does polarize the electrons in its vicinity, This, in turn, changes the potential of the moving electron fields. Such effects necessarily involve self-energy.

Fermion Dressed Propagators
The dressed propagator is defined to be the two-point function to all orders of the perturbation expansion. It changes the bare mass to the physical mass. We Θ is the Heaviside's step function. We appeal now to the very well known rela- Thus, the pertinent expression in momentum space reads where k = k and 2 2 k k m ω = We already stated above that PV signifies ''principal value of a function''. The system's interaction's Hamiltonian is defined by a two-body F V potential such that ( ) ( ) ( ) ( ) where 1 is the unity matrix. The dressed propagator here verifies ˆˆ, so that the dressed propagator becomes diagonal. Then, ( the self-energy. We can pass now to its perturbative expansion at first order  (3.11) Consequently (up to first order),

Bosons' Dressed Propagators from FW's Book
For free bosons FW introduce the propagator in momentum space as One has then where the primed part refers to the noncondensate ( 0

Fermions
We wish to calculate ( ) and then ( ) Starting here, a father lengthy manipulation leads to ( ) ∫ k k (4.5) so that the self energy reads Accordingly, one writes for the dressed propagator noting that We recall at this stage that, in Ref. [9], it was been proved that Then, using the result (4.12) If V → ∞ , F k → ∞ , n finite, we find and thus

B V r
We calculate now the dressed propagator for For this result, we have used the relation of [7] concerning the regularization of integrals that depend upon a power of x. Thus, for the dressed propagator we find, up to first order We must proceed from here as we did for the fermion case to obtain ( ) ( ) and we obtain for the dressed propagator the relation
• We emphasize that our methodology successfully tackles all renormalization issues. We made full use ultra-hyperfunctions' theory, in particular the results reported in [2]. • With such tools we have been able to construct a non-relativistic quantum field theory (NR QFT) of Newton's gravitation (NG).
• This was done for pairs of fermions or bosons that interact between themselves via NG.
• Our manipulations were based on the results of the classical book [6].
• As special examples, we have obtained the dressed propagators for both types of particles, up to first order in perturbation theory, and also the fermions' self-energy.
• The examples indicate that we have indeed constructed, both for fermions and bosons, a viable non-relativistic quantum field theory of gravitation.
• Remark that we were here concerned only with Newton's gravitation.