_{1}

^{*}

Viewing gravitational energy-momentum <i>P<sub>G</sub><sup style="margin-left:-7px;"> μ </sup></i> as equal by observation, but different in essence from inertial energy-momentum <i>P<sub>I</sub><sup style="margin-left:-7px;"> μ </sup></i> naturally leads to the gauge theory of volume-preserving diffeomorphisms of a four-dimensional inner space. To analyse scattering in this theory, the gauge field is coupled to two Dirac fields with different masses. Based on a generalized LSZ reduction formula the S-matrix element for scattering of two Dirac particles in the gravitational limit and the corresponding scattering cross-section are calculated to leading order in perturbation theory. Taking the non-relativistic limit for one of the initial particles in the rest frame of the other the Rutherford-like cross-section of a non-relativistic particle scattering off an infinitely heavy scatterer calculated quantum mechanically in Newtonian gravity is recovered. This provides a non-trivial test of the gauge field theory of volume-preserving diffeomorphisms as a quantum theory of gravity.

Imagine a world in which physicists would be forced from the outset to think about gravitation in terms and in the language of relativistic quantum field theory―a language consisting of terms such as state vectors in Fock spaces, causal quantum fields, operators, probability amplitudes, observables, propagators, conserved quantities such as the electric charge, energy-momentum and the like. Within that framework they might try to answer questions such as “Given a certain number of incoming particles described by free state vectors with given inertial energy-momenta

To answer such questions imagine those physicists following a similar reasoning as in quantum electrodynamics to construct a theory yielding the S-matrix for quantum gravity. Hence they would start with asymptotic states and fields describing matter as in the case of QED implementing microcausality and space-time symmetries from the outset. And as in the case of QED they would look for a conserved quantity related to a global gauge symmetry which could generate the gravitational interaction through gauging the symmetry locally―in the case of QED it is the electric charge related to a global

To identify a conserved quantity which our physicists could relate to a gauge field transmitting the gravitational interaction at the quantum field level they would have to go back to the very outset of what is known about gravity. Ultimately this is the observed equality of inertial and gravitational mass

Now a) the observed equality of inertial and gravitational mass of an on-shell physical object in its rest frame together with b) the conservation of the inertial energy-momentum

assuming that the gravitational energy-momentum

To explore this route our physicists might postulate that both

Our physicists would finally assure the observed equality of inertial and gravitational energy-momentum for on-shell observable physical objects in this approach by taking the gravitational limit, i.e. equating both types of momenta. They would also note―being forced from the outset to think about gravitation in terms and in the language of relativistic quantum field theory―that the language of classical physics with its reference to spacetime trajectories of particles makes no sense in the context of constructing such a theory―as would the principle of equivalence. Only in the limit of

In fact, we have worked out the above line of thinking on the basis of which we have defined the classical and quantum gauge field theories of the group of volume-preserving diffeomorphisms [

This being the case we can now directly analyze physical situations for which we can compare predictions both within the framework of the theory presented as well as within the standard framework of Newtonian gravity dealt with quantum-mechanically such as the gravitational scattering of two particles with different masses.

Hence in this paper, we calculate the scattering cross-section of two Dirac particles with different masses and compare it in an appropriate limit with the cross-section of a non-relativistic particle scattering off an infinitely heavy scatterer calculated quantum mechanically in Newtonian gravity―and determine the numerical value of the coupling constant of our theory in the process.

In this section we calculate the scattering amplitude of two Dirac particles with different masses in quantum gravity to lowest order in perturbation theory in natural units

Our starting point is the action for two Dirac fields

Above, x and X denote spacetime and inner space coordinates [

denotes the gravitational field strength and

the covariant derivative [

We want to calculate the scattering amplitude of two Dirac particles with incoming and outgoing inertial equal to gravitational energy-momenta

and

tial and final states.

In quantum gravity S-matrix elements are related by generalized LSZ reduction formulae [

Above

Next we have to calculate the time-ordered product of the four interacting field operators in Equation (5) which is obtained from the generating functional

where

Above

where

and gauge field propagator as in Equation (103) in [

for the fields

introduced in [

The part of

Note the arrows on the derivatives w.r.t. inner coordinates indicating the directions in which they act.

Evaluating the functional derivatives in Equations (6) and (7), setting the source terms equal to zero and discarding disconnected and higher order contributions we obtain the vacuum expectation value for the time-or- dered product of the four field operators to leading order

Inserting this expression in Equation (5) and performing the truncation we find the scattering amplitude to be

Performing the remaining integrations the amplitude finally becomes

Note that before taking the gravitational limit the amplitude is scale-invariant under

Trying to take the limits above we are left with an expression of the type

with

This is the regularization we employed in [

Noting that in the limit above

vanishes we see that the inner longitudinal part of the gauge field propagator Equation (11) does not contribute to the amplitude.

As there is no infrared problem for

with the invariant matrix element

It contains the information about the underlying dynamics of the theory and is completely symmetric under the interchange of the two particles, i.e.

We note the similarity of

which changes the dynamics completely. Note that in the rest frame of the particle with mass M the coupling strenght reduces to

In this section we calculate the cross-section for the scattering of two Dirac particles with different masses in quantum gravity to lowest order in perturbation theory.

We start with the usual Lorentz-invariant expression for the cross-section with two incoming and two outgoing Dirac fermions [

As we are interested in the unpolarized cross-section we first average over initial and sum over final states

Proceeding with the calculation of

Inserting these and performing the Lorentz sums in

To further extract the physics of the two-particle scattering process we choose as coordinate system the one in which the particle with mass

Energy conservation for the chosen coordinates relates the energy

Performing the phase space integrals over

which after a little algebra can be expressed in terms of

The first and last lines above are exactly the same as in the case of scattering of two Dirac particles with different masses in quantum electrodynamics [

tional interaction strength replacing the square of the fine structure constant

We next evaluate both the limits of a heavy scatterer

If the energy E of the incoming particle of mass m is much smaller than the mass M of the scatterer Equation (26) yields up to higher orders in

In addition we have from Equation (26) in this limit

where

Setting

we find the analogue to the Mott scattering cross-section [

recalling that

Equation (32) reduces in the non-relativistic limit

obtained from a quantum mechanical (and incidentially a classical) treatment of the scattering of a particle of mass m off an infinitely heavy scatterer M in Newtonian gravity [

Note that the value of

We again stress the fact that the scattering of two Dirac particles with different masses in the limit of a heavy scatterer and a non-relativistic incoming particle is physically equivalent to gravitational Rutherford scattering― and hence provides a non-trivial comparison and test for our claim that the theory presented in [

We finally note that Equation (32) does not depend on the specific properties of the incoming particle, but just on the kinematical factor

If on the other hand the energy E is much larger than the mass m of the incoming particle we have

and Equation (26) yields

where

A little algebra yields the scattering cross-section in this limit in terms of

Note that for a heavy scatterer

We finally note that Equation (37) does depend on the specific properties of the incoming particle, i.e. its mass m, as does the general formula Equation (28) for the scattering cross-section―an expression that the principle of equivalence seems not to generally hold in a quantum context.

In this paper, we have calculated the gravitational scattering cross-section of two Dirac particles of different masses to leading order in perturbation theory within quantum gravity described by the gauge field theory of volume-preserving diffeomorphisms. We have demonstrated that this cross-section in the limit of one very heavy particle and the other non-relativistic becomes equal to the Rutherford-like cross-section for a non-relati- vistic particle scattering off a Newton potential. This has allowed us to determine the value of the coupling constant appearing in the gauge field theory of volume-preserving diffeomorphisms.

This result is much less trivial than the analogous one in QED because in that case the theory describing electrodynamics at the classical and the quantum level is the same. In the case of gravity all: the invariance groups and the gauge fields, the Lagrangians, the spaces on which they are defined, the coupling mechanisms describing gravitation at the classical and the quantum level are very different―and yet the same result emerges for one of the few quantities which can be calculated in both approaches.

If indeed the gauge field theory of volume-preserving diffeomorphisms consistently describes gravity at the quantum level then this is due to the fact that the framework of relativistic quantum fields and renormalizable gauge field theories offers a very economical way to consistently implement what we know from experiment about elementary particles and their processes. It offers the necessary classical and quantum degrees of freedom to describe all: observable states labelled by a complete set of quantum numbers, causality at the micro-level, the conservation of energy, momentum and angular momentum as well as the conservation of various types of “charges”―in our case the conservation of gravitational energy-momentum and its dynamical implementation through gauging the group of volume-preserving diffeomorphisms of an inner space.

ChristianWiesendanger, (2015) Scattering Cross-Sections in Quantum Gravity—The Case of Matter-Matter Scattering. Journal of Modern Physics,06,273-282. doi: 10.4236/jmp.2015.63032

Generally,

Specifically,

by

Working in Minkowskian gauges [

Lorentz group

The same lower and upper indices are summed unless indicated otherwise.

All further conventions related e.g. to spinors, phase space integrals etc. are standard and taken from [