Fig. 5.5 A graphical representation of the two helicity states of the graviton. Given the momentum vector pμ that corresponds to propagation of the wave in a given direction, the transverse oscillations can describe a clock-wise or anti clock-wise circulation around the flight direction axis. These two possibilities correspond to the two available states of the graviton quantum particle

5.8 The Bottom-Up Approach, or Gravity à la Feynmann

There is no doubt that Richard Feynman (see Fig. 5.6) has been the most influential, creative and innovative among all the theoreticians living in the third quarter of the XX century. His very original ideas about perturbation theory and his graphical representation of quantum relativistic physical processes, not only led to the solution of Quantum Electrodynamics, but also provided a new paradigmatic way of conceiving Quantum Field Theory, that has become part of the forma mentis of all modern physicists. Similarly, his general conception of the path integral provided a new general framework to conceive all quantum theories. Feynman had also a special talent for teaching and his lecture courses at Caltech have become legendary. He used the most unusual analogies and stimulated student thought with unexpected arguments that led to deeper thinking and understanding. In the mid sixties he gave a course on General Relativity and, in order to introduce Einstein equations of Gravity, he resorted to a hypothetical Venusian civilization. The choice of Venus was not random. It was motivated by its terrible atmosphere. Venus is absolutely similar to the planet Earth as far as its dimension and its distance from the Sun are concerned, yet the average surface temperature is around 450° degrees centigrade and the pressure experienced by someone standing on its soil is about 92 atmospheres. Responsible for such a oven-like climate is the carbon dioxide (CO2) which traps almost all of the outgoing radiation: an extreme version of the Green-House effect. With such a temperature, all ancient oceans of Venus evaporated and water molecules, after flying to the high atmosphere, dissociated loosing their hydrogen atoms that flew away into the interplanetary space. Furthermore the intensive volcanic activity, ubiquitous on the surface of the planet, fills its diabolic atmosphere with dense clouds of sulfuric acid, which prevent Venusians from seeing the starry night and penetrating the depths of interplanetary and intergalactic space with their naked eye or optical instruments. Under such circumstances a hypothetical Venusian civilization quite unlikely had produced a Ptolemy, a Copernicus and a Newton and their scientists might not have discovered the Universal Law of gravitational attraction as an explanation of the observed motion of planets that they did not observe altogether. Yet,

Fig. 5.6 Richard Phillips Feynman (1918–1988) was born in Queens, New York, from a Jewish family, originated from Russia and Poland. Just as Albert Einstein, he was a late talker, namely he started uttering words only after his third year of life. Yet, by the age of fifteen he had already learned, by self-teaching, integral and differential calculus. His high school education was in New York and his first University education took place at the Massachusetts Institute of Technology. Then he was accepted by Princeton, where he obtained his Ph.D. degree in 1942, in times of war. His advisor was John Archibald Wheeler and his thesis contained the seeds of what was later to become the Feynman path integral approach to quantum mechanics and quantum field theory. Feynman was among the youngest physicists who took part in the Manhattan project, namely in the famous secret development of the atomic bomb, that happened in the secluded Laboratories of Los Alamos, New Mexico. In the early post-war years he conducted fundamental researches on Quantum Electrodynamics that lead him to share with Julian Schwinger and Sin-Itiro Tomonaga the 1965 Nobel Prize for the construction of that theory. Since the early sixties, up to his death in 1988, Feynman held the Richard Chace Tolman professorship in theoretical physics at the California Institute of Technology. Besides Quantum Electrodynamics and the path integral, Feynman gave very important contributions to the theory of superfluidity and to Quantum Chromodynamics, where he introduced the so called parton model. His personality was very open and brilliant. He had a special talent for teaching, where he resorted to unusual imaginative metaphors both to seduce and to penetrate the minds of his students. He liked very much joking and was typically very gentle, informal and unpretending with young people who sincerely wanted to discuss physics with him, while he could be at times very harsh and even rude with senior fellow scientists when they showed signs of excessive self-esteem

as Feynman advocated, intelligent beings, although by a different route, must have reached the same conclusions about the fundamental laws of nature.

Venusians were very good particle theorists and besides atomic physics they already mastered classical and quantum field theory and knew that all interactions are mediated by the exchange of quanta of an appropriate field. The history of their science did not include a Newton but the Venusian analogues of Dirac, Pauli and Emmy Noether had lived and produced their results. Moreover every Venusian child was able to drop things and, while sitting on his chair, experience their falling down.

Hence Venusian scientists knew that there existed a mysterious force of attraction they named gravitation. Yet what sort of field was responsible for such an interaction? The reader should be aware that the Venusian scientists had a perfect knowledge of special relativity and of the equivalence of mass and energy E = mc2, which they were daily experiencing with their thermonuclear reactors and particle accelerators. In order to investigate the mysterious gravitational phenomenon, Venusians had constructed sophisticated analogues of the Cavendish experiment and they knew that masses attract each other with a force that decreases with the inverse squared distance. So sitting at their working desk, the Venusian physicists could summarize the knowledge they had accumulated on gravitation.

(a)Gravitation is a long range interaction since its force decreases with a r12 law. Hence the quantum of the mediator field must be a massless particle traveling at the speed of light.

(b)Mass seems to be the conserved charge and the source of such an interaction.

Yet there were two fundamental obstacles in order to proceed further, namely:

(c)mass is equivalent to energy because of special relativity, so energy rather than mass should be the true source that couples to the sought for gravitational field,

(d)as far as the Venusians knew, mass is associated with no generator of any symmetry and it was difficult to imagine how to relate it with Noether’s theorem.

The solution to these two problems was the same and came from the following observation. Energy E and not mass m should be the source of gravitation. Yet differently from mass, energy is not a scalar invariant, rather it is the fourth component of a quadri-vector, to be specific the momentum vector P μ. Hence the charges of gravitation have to be assumed to be the momenta P μ and these, contrary to mass, are associated with the generators of a bona fide symmetry of classical and quantum theories. It is the symmetry of space-time translations in Minkowski space. Here was the clue. The relevant canonical conserved Noether current was the current of quadri-momentum, which in Venusian physics was known under the name of stress-energy tensor. Given any Poincaré invariant field theory, via Noether theorem (1.7.3)–(1.7.4) the Venusians could calculate its canonical stress-energy tensor T λν which, in most cases turned out to be symmetric after lowering of its upper index with the Lorentz invariant metric η:

Some Venusian scientists had observed that there exists exceptional systems where the canonical Noetherian stress-energy tensor turns out to be non-symmetric, yet other Venusian scientists had shown that, without spoiling its two fundamental properties of being conserved and associated with the translation generators, the canonical stress-energy tensor could always be improved (essentially by the addition of some surface terms to the Lagrangian) and made symmetric in all cases. So the Venusians concluded that the mediator of gravitational interactions, that couples to the stress-energy tensor, must be a two-index symmetric tensor field and they named

it hμν = hνμ. Venusians also knew that since it couples to a conserved current, in some way or another, the gravitational field hμν should be endowed with some kind of local symmetries, whose number was predetermined by the number of conserved charges, namely by the number of components of the momentum vector P μ, i.e. four in 4-dimensional Minkowski space. If gravity existed in higher m-dimensional Minkowski space-times, of which some foolish Venusian scientists had started to dream, then the number of gauge transformations should be precisely m.

Hence, following the Feynman-Venusian approach let us then consider the most general form of a Lorentz invariant linear equation for a symmetric tensor field hμν (x) that is quadratic in partial derivatives, describes a massless particle and has the stress-energy tensor as a source. Such an equation is necessarily of the following form:

where α, β, γ , δ are some numerical coefficients to be determined. In the electromagnetic case the conservation of the electric current and the gauge invariance of the field equation were shown to be just two equivalent statements. This is essentially true also in the spin two-case, yet there are some additional subtleties which is worth mentioning. To this effect let us compare the relations imposed on the coefficients of (5.8.2), by the two conditions:

(a)Conservation of the stress-energy tensor: ∂μTμν = 0.

(b)Invariance of the right hand side of the equation under the gauge transformations (5.7.5).

This is easily done. Let us begin with condition (a). If we take the ∂μ derivative of both sides of (5.8.2), the left-hand side vanishes, consistently with the assumed vanishing of the right-hand side, if:

β = 1

δ = 1 (5.8.3)

α + γ = 0

Consider then condition (b), namely let us suppose that the right-hand side of (5.8.2) is invariant under the transformation (5.7.5). This is true if:

β = 1

α = 1 (5.8.4)

γ + δ = 0

The two systems of (5.8.3) and (5.8.4) admit the common solution:

which exactly corresponds to the linearization of Einstein equation displayed in (5.7.9). Hence the conservation of the stress-energy tensor and the gauge transformation (5.8.4) do not exactly imply each other, yet this is not a real problem, since

we have so far disregarded the existence of a field redefinition which changes the coefficients of (5.8.2) preserving the stress-energy conservation. Consider the following transformation:

equation identical in form to the original one but with new coefficients, related to the old ones in the following way:

˜=

ββ

˜ =

δ δ

(5.8.7)

α˜ = α(1 + 4u) − 2u γ˜ = γ (1 + 4u) + 2u

It is immediately evident from (5.8.7) that if the stress-energy conservation condition (5.8.3) was satisfied by the old coefficients, so it is by the new ones with a tilda. Hence we actually have a one-parameter family of field equations consistent with the conservation of the stress-energy tensor and they are related to each other by the field transformation (5.8.6). In this family, the linearization of Einstein equation, namely (5.7.9), is gauge-invariant under the transformation (5.7.5) which is the infinitesimal form of diffeomorphisms. The other members of the same family have also a gauge symmetry with the same number of arbitrary functions, the only difference being the following slight modification of the transformation laws:

where we have denoted ∂ · ξ = ∂μξν . Hence the principle that conservation of the source is equivalent to the existence of a local gauge-symmetry for the interactionmessenger field is respected. We just have the freedom of choosing the form of the local gauge transformations displayed in (5.8.8) and the form associated with the linearization of Einstein equations just corresponds to the simplest choice u = 0. Yet we have already seen that the choice u = 12 is quite convenient for the maximal simplification of the equation, after gauge-fixing.

So, coming back to Feynman’s story, the good Venusian particle physicists arrived at (5.7.9) and could also easily derive the quadratic Lagrangian from which such an equation follows by means of variational calculus. It is as follows:

where, just as before, h denotes the η-trace of the hμν -tensor. The Lagrangian (5.8.9) is not only Lorentz but also Poincaré invariant and, as such, the Venusians

could easily calculate the stress-energy tensor carried by the gravitational field hμν , which is just the Noether current associated with translational symmetries. They arrived at the following result:

T λμ(h) = −∂λhρσ ∂μhρσ − ∂λhμσ ∂σ h − ∂λh∂ρ hμρ − 2∂λhμσ ∂ρ hρσ

The reader may note that this canonical Noether tensor, after lowering the upper index with the flat metric is not automatically symmetric. Namely it falls among those exceptions alluded above. Also on Earth the fact that the Noether conserved current associated with translations is not automatically symmetric in its two indices is a topic that generated a large literature over the years [1]. Various proposals were put forward how to symmetrize the canonical stress-energy tensor in such a way that it is still conserved (divergenceless) and still defines the correct charges, namely the momentum vector P μ. The most popular and most frequently adopted of these symmetrization procedures is due to Belinfante and Rosenfeld and dates back to the years 1939–1940. We will not dwell on this issue. In this context what is of interest to us and to the Venusians is the following point. We have a symmetrized energy momentum tensor:

which means that also the gravitational field hμν carries energy and momentum. Why this energy and momentum should not be on the same footing as the energy and momentum carried by the other matter fields? Hence why should we not modify the gravity field equation as it follows:

hμν + ∂μ∂ν hρσ ηρσ − ∂μ∂ρ hρν − ∂ν ∂ρ hρμ

− ημν hρσ ηρσ + ημν ∂ρ ∂σ hρσ = κTμν + κTμν(grav)(h) (5.8.12)

Obviously we should and the good Venusians did it. The news is that the gravitational equation is no longer a linear one! We can bring the term κTμν(grav)(h) on the

left hand-side of the equation and consider the resulting non-linear differential expression the truly correct form of the propagation equation for the hμν field. With some effort we can also reconstruct the Lagrangian from which such a non-linear equation (including quadratic terms) derives. It will contain both quadratic and cubic terms in hμν and have the following structure:

From L (h) via Noether theorem we can calculate once again the canonical stressenergy tensor which now will involve both quadratic and cubic terms. If we repeat