23 The Necessity of Gravitational Quantization

DE WITT called attention to the fact that there is an ambiguity in the choice of field variables in terms of which one may make an expansion about the Minkowski metric . For example, one might use either or where , . To be consistent in a self-energy calculation one should expand out to the second order, and the difference in choice leads to a difference in the trace of the non-gravitational (or matter) stress tensor. There are some fields, notably the electromagnetic field, for which this trace vanishes. In this case, you get the same result in the second order, no matter what you expand in terms of. It is a curious thing in the electromagnetic case (although, as Utiyama has pointed out, you do have the derivative coupling and therefore the divergence is of the second or third order), that if you include second order terms in the interaction, the old-fashioned self-energy will be exactly compensated. MISNER asked if the computation has also been done for the neutrino field. DE WITT replied that although it should be done, it has not been done for two reasons: (1) The mathematics of the spinor problem up to the second order is considerably more complicated and has not yet been fully worked out. (2) Interest has in the meantime shifted to the problem of tackling the complete nonlinear gravitational field.

WHEELER pointed out that the linear starting point is incompatible with any topology other than a Euclidean one; that if one has curved space or “wormholes,” you just can’t start expanding this way. BELINFANTE asked if anyone had found it possible to have hole theory in a curved space; i.e. can one make a covariant distinction between positive and negative energy states?

ROSENFELD said that Dirac, just after he had formulated hole theory and was faced with the objection that there is an infinite energy associated with these states, had attempted unsuccessfully to overcome this difficulty by introducing a closed universe.

MISNER said that one could get a quite good qualitative idea of what happens in curved space (the metric being externally impressed) by using the Feynman prescription. Since the action is still quadratic in the interesting field variables, there is no difficulty. In a spherical space there exist states of excitation which do not scatter each other; but as soon as the space has “bumps” in it photons in one state get scattered into other states. Hole theory is possible if the metric is static; however, a time dependent metric causes electrons to go to positive energy states.

BERGMANN pointed out that, on this account, one does not have to exclude hole theory, because if the electrons get excited, there is occasional pair production.

SALECKER introduced a thought experiment, involving a stream of particles falling on a diffraction grating. On account of the de Broglie relation for the waves associated with the stream, , one expects that particles of different mass will scatter differently if they fall from a given height. According to general relativity, one expects the same behavior for different masses with the same initial state of motion. Therefore, we arrive at a contradiction with the principle of equivalence.

FEYNMAN asked if the grating is here allowed to exert forces on the particles which are non-gravitational. DE WITT said that one needs rather a grating (made, for example, out of planets) which acts only through its gravitational field on the stream. FEYNMAN then said that he did not believe that the principle of equivalence denies the possibility of distinguishing between two different masses. Of course, the principle of equivalence would prevent one from distinguishing between masses by means of this particular experiment if only classical laws were operative. However, the introduction of Planck’s constant into the scheme of things introduces new possibilities, which are not necessarily in contradiction with the principle of equivalence. As far as this particular experiment is concerned, all that the principle of equivalence would say would be that if one performs the experiment in an elevator, he will obtain the same result as in a corresponding gravitational field. FEYNMAN also emphasized that the quantities and by themselves do not lead to a unit of mass, whereas such a unit exists if is included.

WHEELER pointed out that the principle of equivalence only denies the possibility of distinguishing between the gravitational and inertial masses of a single body, but definitely does not prevent one from distinguishing the masses of two different bodies, even when only gravitational forces are involved. For example, we know the relative sizes of the masses of the sun and the various planets solely from observation of their gravitational interactions. BERGMANN added that the principle of equivalence makes a statement about local conditions only. Therefore you can do one of two things: either (1) use a small diffraction grating that is not gravitational, or (2) use a diffraction grating made of planets. In this case, the conditions are certainly not local.

FEYNMAN characterized the point which Salecker had raised as an interesting point and a true point, but not necessarily a paradoxical one. If the falling particles are not allowed to react back on the grating, then according to the classical theory they will all follow the same paths. Whereas, in the quantum theory they will give rise to different diffraction patterns depending on their masses.

SALECKER then raised again the question why the gravitational field needs to be quantized at all. In his opinion, charged quantized particles already serve as sources for a Coulomb field which is not quantized. (Editor’s Note: Salecker did not make completely clear what he meant by this. If he meant that some forces could be represented by actions-at-a-distance, then, although he was misunderstood, he was right. For the corresponding field can then be eliminated from the theory and hence remain unquantized. He may have meant to imply that one should try to build up a completely action-at-a-distance theory of gravitation, modified by the relativistic necessities of using both advanced and retarded interactions and imbedded in an “absorber theory of radiation” to preserve causality. In this case, gravitation per se could remain unquantized. However, these questions were not discussed until later in the session.)

BELINFANTE insisted that the Coulomb field is quantized through the -field. He then repeated DeWitt’s argument that it is not logical to allow an “expectation value” to serve as the source of the gravitational field. There are two quantities which are involved in the description of any quantized physical system. One of them gives information about the general dynamical behavior of the system, and is represented by a certain operator (or operators). The other gives information about our knowledge of the system; it is the state vector. Only by combining the two can one make predictions. One should remember, however, that the state vector can undergo a sudden change if one makes an experiment on the system. The laws of nature therefore unfold continuously only as long as the observer does not bring extra knowledge of his own into the picture. This dual aspect applies to the stress tensor as well as to everything else. The stress tensor is an operator which satisfies certain differential equations, and therefore changes continuously. It has, however, an expectation value which can execute wild jumps depending on our knowledge of the number and behavior of mass particles in a certain vicinity - if this expectation value were used as the source of the gravitational field then the gravitational field itself - at least the static part of it - would execute similar wild jumps. One can avoid this subjective behavior on the part of the gravitational field only by letting it too become a continuously changing operator, that is, by quantizing it. These conclusions apply at least to the static part of the gravitational field, and it is hard to see how the situation can be much different for the transverse part of the field, which describes gravitational radiation.

FEYNMAN then made a series of comments of which the following is a somewhat condensed but approximately verbatim transcript:

“I’d like to repeat just exactly what Belinfante said with an example - because it seems clear to me that we’re in trouble if we believe in quantum mechanics but don’t quantize gravitational theory. Suppose we have an object with spin which goes through a Stern-Gerlach experiment. Say it has spin 1/2, so it comes to one of two counters.

Connect the counters by means of rods, etc., to an indicator which is either up when the object arrives at counter 1, or down when the object arrives at counter 2. Suppose the indicator is a little ball, 1 cm in diameter.”

“Now, how do we analyze this experiment according to quantum mechanics? We have an amplitude that the ball is up, and an amplitude that the ball is down. That is, we have an amplitude (from a wave function) that the spin of an electron in the first part of the equipment is either up or down. And if we imagine that the ball can be analyzed through the interconnections up to this dimension ( cm) by the quantum mechanics, then before we make an observation we still have to give an amplitude that the ball is up and an amplitude that the ball is down. Now, since the ball is big enough to produce a real gravitational field (we know there’s a field there, since Coulomb measured it with a 1 cm ball) we could use that gravitational field to move another ball, and amplify that, and use the connections to the second ball as the measuring equipment. We would then have to analyze through the channel provided by the gravitational field itself via the quantum mechanical amplitudes.”

“Therefore, there must be an amplitude for the gravitational field, provided that the amplification necessary to reach a mass which can produce a gravitational field big enough to serve as a link in the chain does not destroy the possibility of keeping quantum mechanics all the way. There is a bare possibility (which I shouldn’t mention!) that quantum mechanics fails and becomes classical again when the amplification gets far enough, because of some minimum amplification which you can get across such a chain. But aside from that possibility, if you believe in quantum mechanics up to any level then you have to believe in gravitational quantization in order to describe this experiment.”

“You will note that I use gravity as part of the link in a system on which I have not yet made an observation. The only way to avoid quantization of gravity is to suppose that if the amplification gets big enough then interference effects can in principle no longer play a role beyond a certain point in the chain, and you are not allowed to use quantum mechanics on such a large scale. But I would say that this is the only ‘out’ if you don’t want to quantize gravity.”

BONDI: “What is the difference between this and people playing dice, so that the ball goes one way or the other according to whether they throw a six or not?”

FEYNMAN: “A very great difference. Because I don’t really have to measure whether the particle is here or there. I can do something else: I can put an inverse Stern-Gerlach experiment on and bring the beams back together again. And if I do it with great precision, then I arrive at a situation which is not derivable simply from the information that there is a 50 percent probability of being here and a 50 percent probability of being there. In other word , the situation at this stage is not 50-50 that the die is up or down, but there is an amplitude that it is up and an amplitude that it is down - a complex amplitude - and as long as it is still possible to put those amplitudes together for interference you have to keep quantum mechanics in the picture.”

“It may turn out, since we’ve never done an experiment at this level, that it’s not possible - that by the time you amplify the thing to a level where the gravitational field can have an influence, it’s already so big that you can’t reverse it - that there is something the matter with our quantum mechanics when we have too much action in the system, or too much mass - or something. But that is the only way I can see which would keep you from the necessity of quantizing the gravitational field. It’s a way that I don’t want to propose. But if you’re arguing legally as to how the situation stands...”

WITTEN: “What prevents this from becoming a practical experiment?”

FEYNMAN: “Well, it’s a question of what goes on at the level where the ball flips one way or the other. In the amplifying apparatus there’s already an uncertainty - loss of electrons in the amplifier, noise, etc. - so that by this stage the information is completely determined. Then it’s a die argument.”

“You might argue this way: Somewhere in your apparatus this idea of amplitude has been lost. You don’t need it any more, so you drop it. The wave packet would be reduced (or something). Even though you don’t know where it’s reduced, it’s reduced. And then you can’t do an experiment which distinguishes interfering alternatives from just plain odds (like with dice).”

“There’s certainly nothing to prevent this experiment from being carried out at the level at which I make the thing go ’clink-clank,’ because we do it every day: We sit there and we wait for a count in the chamber - and then we publish, in the Physical Review, the information that we’ve obtained one pi meson - And then it’s printed (bang!) on the printing presses - stacked and sent down to some back room - and it moves the gravitational field!”

“There’s no question that if you have allowed that much amplification you have reduced the wave packet. On the other hand it may be that we can think of an experiment - it may be worthwhile, as a matter of fact, to try to design an experiment where you can invert such an enormous amplification.”

BERGMANN: “In other words, if it is established that nobody reads the Physical Review, then there is a definite 50 percent uncertainty...”

FEYNMAN: “Well, some of the copies get lost. And if some of the copies get lost, we have to deal with probabilities again.”

ROSENFELD: “I do not see that you can conclude from your argument that you must quantize the gravitational field. Because in this example at any rate, the quantum distinction here has been produced by other forces than gravitational forces.”

FEYNMAN: “Well, suppose I could get the whole thing to work so that there would be some kind of interference pattern. In order to describe it I would want to talk about the interaction between one ball and the other. I could talk about this as a direct interaction like . (This is related to the discussion of whether electrostatics is quantized or not.) However, if you permit me to describe gravity as a field then I must in the analysis introduce the idea that the field has this value with a certain amplitude, or that value with a certain amplitude. This is a typical quantum representation of a field. It can’t be represented by a classical quantity. You can’t say what the field is. You can only say that it has a certain amplitude to be this and a certain amplitude to be that, and the amplitudes may even interfere again... possibly. That is, if interference is still possible at such a level.”

ROSENFELD: “But what interferes has nothing to do with gravitation.”

FEYNMAN: “That’s true ... when you finish the whole experiment and analyze the results. But, if we analyze the experiment in time by the propagation of an amplitude - saying there is a certain amplitude to be here, and then a certain amplitude that the waves propagate through there, and so on - when we come across this link - if you’ll permit me to represent it by a gravitational field - I must, at this stage in time, be able to say that the situation is represented now not by a particle here, not by a result over there, but by a certain amplitude for the field to be this way and a certain amplitude to be that way. And if I have an amplitude for a field, that’s what I would define as a quantized field.”

BONDI: “There is a little difficulty here (getting onto one of my old hobby horses again!) if I rightly understand this, which I’m not sure that I do: The linkage must not contain any irreversible elements. Now, if my gravitational link radiates, l’ve had it!”

FEYNMAN: “Yes, you’ve had it! Right. So, as you do the experiment you look for such a possibility by noting a decrease of energy of the system. You only take those cases in which the link doesn’t radiate. The same problem is involved in an electrostatic link, and is not a relevant difficulty.”

BONDI: “Oh yes, because in the electrostatic case I can put a conducting sphere around it ...”

FEYNMAN: “It doesn’t make any difference if it radiates. If every once in a while the particle which is involved is deflected irreversibly in some way, you just remove those cases from your experiment. The occurrence could be observed by some method outside.”

BERGMANN: “Presumably the cross section for gravitational radiation is extremely...”

FEYNMAN: “And furthermore, we can estimate what the odds are that it will not happen.”

BONDI: “I’m just trying to be difficult.”

GOLD: “But that need not mean that there is some profound thing wrong with your quantum theory. It can mean merely that when you go into the details of how to make an op...”

FEYNMAN: “There would be a new principle! It would be fundamental! The principle would be: - roughly: Any piece of equipment able to amplify by such and such a factor grams or whatever it is) necessarily must be of such a nature that it is irreversible. It might be true! But at least it would be fundamental because it would be a new principle. There are two possibilities. Either this principle - this missing principle - is right, or you can amplify to any level and still maintain interference, in which case it’s absolutely imperative that the gravitational field be quantized ... I believe! or there’s another possibility which I haven’t thought of.”

BUCKINGHAM: “The second possibility lands you back in the same difficulty again. If you could amplify to any factor, you could reduce to a negligible proportion an additional signal to take an observation on, say, those balls.”

FEYNMAN: “No!”

BUCKINGHAM: “Because you only need one light quantum.”

FEYNMAN: “No!”

BUCKINGHAM: “If you could amplify up to any factor this becomes negligible.”

FEYNMAN: “It depends! ... You see (pointing to a blank space on the blackboard) this statement that I have written here is not written very precisely as a matter of fact if you look at it you probably can’t even see the words. I haven’t thought out how to say it properly. It isn’t simply a matter of amplifying to any factor. It’s too crude - I’m trying to feel my way. We know that in any piece of apparatus that has ever been built it would be a phenomenally difficult thing to arrange the experiment so as to be reversible. But is it impossible? There’s nothing in quantum mechanics which says that you can’t get interference with a mass of gram - or one gram.”

BUCKINGHAM: “Oh, yes. What I’m saying, though, is that the laws have to be such that the effect of one light quantum is sufficient to determine which side the ball is on, and would be enough to disturb the whole experiment.”

FEYNMAN: “Certainly! That’s always true. That’s just as true no matter what the mass is.”

ANDERSON: “Suppose a neutral elementary particle really has a gravitational field associated with it which you could actually use in the causal link. The thing that bothers you is that you may be getting something that is too small to produce a gravitational field.”

FEYNMAN: “It’s a question of design. I made an assumption in this analysis that if I make the mass too small the fields are so weak I can’t get the experiment to operate. That might be wrong too. It may be that if you analyze it close enough, you’ll see that I can make it go through a gravitational link without all that amplification - in which case there’s no question. At the moment all I can say is that we’d better quantize the gravitational field, or else find a new principle.”

SALECKER: “If you assume that gravitation arises as a sort of statistical phenomenon over a large number of elementary particles, then you also cannot perform this experiment.”

FEYNMAN: “Yes, it depends what the origin is. One should think about designing an experiment which uses a gravitational link and at the same time shows quantum interference - what dimensions are involved, etc. Or if you suppose that every experiment of this kind is impossible to do, you must try to state what the general principle is, by trying a few examples. But you have to state it right, and that will take some thinking.”

DE WITT then remarked that there is still another type of experiment which might some day help to decide the question of whether or not the gravitational field is quantized - namely, producing (or finding in cosmic rays) particles having energies of Bev and observing their interactions. At the level of such structures as “wormholes” these are the energies in which one is interested.

ROSENFELD then gave an amusing historical survey in which he presented some of the ideas which Faraday and Maxwell had on gravitation. Faraday attempted to measure the ability of a moving gravitational field to induce electric current by moving a coil of wire or a heavy mass up and down. Although he detected no effect, he was convinced that such an effect must nevertheless exist. It struck Faraday that an important difference between the gravitational and electric fields was that, apparently, gravitational energy was not absorbed by matter. It also puzzled Maxwell that the gravitational lines of force in the vicinity of two interacting masses exhibited a behavior which did not seem to permit the introduction of stresses in the ether to explain the attraction of the masses; the lines of force have the masses for their sources, but do not end on them: they exert a push, and not a pull, on the masses. It seems that they have to end on some far-away mass distribution, which they can so to speak take as support to push the masses nearer to each other and this gives rise to their apparent “attraction.”

ROSENFELD then raised the question of the existence of gravitational waves. Formally, one can get a spherical wave which depends on the third time derivative of the quadrupole distribution of masses. This solution has been obtained, in the linear program, by expanding around the Minkowski metric. However, Rosenfeld does not know if this wave has a definite physical meaning because the energy transported in this way is not described by a tensor but by an expression which has meaning only with reference to the space chosen for background. This uncertainty about the existence of waves has long prevented people like Gupta, for instance, from quantizing them. “It seems to me that the question of the existence and absorption of waves is crucial for the question whether there is any meaning in quantizing gravitation. In electrodynamics the whole idea of quantization comes from the radiation field, and the only thing we know for sure how to quantize is the pure radiation field.

“Now the arguments that Feynman has just raised about the necessity of quantizing gravitation, if one has to avoid contradiction with the uncertainty principle, did not convince me completely - though I must confess I cannot refute them. In his famous device of the two holes, the idea is that the motion of the mass will show through which hole the electron has passed. I do not know if it is as easy as that, because if you put the condition that the exchange of momentum between the electron and this mass does not disturb the interference pattern, it means that the deviation of the electron due to the momentum it has received must lead to a deviation which is less than the distance between the fringes. This distance between the fringes is . Now if you give an extra momentum to the electron,this would produce a displacement and this must be smaller than . This means that . But this means that the uncertainty in positionof the mass must be larger than and we don’t know whether the mass is at this hole or that hole.”

FEYNMAN: “Uh, yes ... I know. I might have to use gravitons to scatter the particle, and then make some kind of assumption that I can measure the gravity wave, no matter how weak it is. Now if I consider only gravito-statics, I still have a problem. I still have a quantum theory of gravity. Although it is said that there is no quantum theory of electrostatics, there is really, I think. The writing of in the Schrödinger equation removes electro-statics from a field theory and makes it into the quantization of a rather simple field. I still think, in a certain sense, that when you represent it as a field, and not as a solution which is the result of a field, that we still have to quantize it in order to get it to work.”

GOLD: “It could be that inaccuracies are always introduced because no experiment can be, finally, gravitational only. It is possible that the existing quantum theory will already always make sure that nothing can be measured without sufficient accuracy.”

FEYNMAN: “If I write down a term in the equation for the universe and if I have only particles in the system with no question of field variable, is this a quantum theory of gravitation or is it a classical theory? That’s a question of definition. In other words, when this term is included has the gravitation field been quantized?”

ROSENFELD: “The classical theory of the gravitation field would give you this term.”

FEYNMAN: “No, it would not, because it cannot produce a Schrödinger equation; it gives you a force between two points, and then you interpret that energy as a Hamiltonian for the particles.”

ROSENFELD: “That is only if you impose upon me the obligation to start from a Hamiltonian.”

FEYNMAN: “Yes - So if we consider Newtonian gravitation alone we would not have to argue whether to quantize or not (whatever you call this process of including a term in the Hamiltonian); but if we consider the dynamic aspect and if we do not consider this process as quantization, then my experiment does not prove the necessity for quantization.”

BONDI: “Does this mean, then, if there were no gravitational waves, you would not feel that this experiment would prove the need for quantization?”

FEYNMAN: “It depends on your definition, as I tried to explain, of quantization. If I can use my other experiment - which is just exactly the same, but is a little clearer in a certain respect, because the mass moves back and forth - the question is: How do we analyze the situation? Now listen, we can analyze as we go along and cut the thing in the middle if we want to, and say that this produces a field and the field acts on the other one. That’s one way of representing it. If we do it that way, then we have to have an amplitude for the field being here and an amplitude for the field being there. The gravitational field has to be quantized. Incidentally, this uses only gravito-statics. But there’s another way of representing the same thing, and that is there is an action at a distance between the two particles: then we do not have to analyze it in the intermediate range as a function of time. I am sorry I did stop at these subtleties when I made up this example.”

BONDI: “That is very much clearer now. It does seem to me that this vexed question of the existence of gravitation waves does become more important for this reason.”

FEYNMAN: “Yes. There’s a delay in this equipment. If there’s real delay in this equipment, the information is stored in the field and can’t be extracted by looking at the particles of the equipment; and if a quantum theory of the gravitational field is not necessary, there is another possible theory: the action at a distance theory of gravity. That is another way out. It may not be necessary to have a gravitational field at all.”

ROSENFELD: “Well, the last point about which I worry very much is this: It is difficult for me to imagine a quantized metric unless, of course, this quantization of the metric is related to the deep-seated limitations of the definitions of space and time in very small domains corresponding to internal structures of particles. That is one prospect we may consider. The whole trouble, of course, which raises all these doubts, is that we have too few experiments to decide things one way or the other.”

BELINFANTE then raised the question of the validity of Gupta’s successive approximations. These approximations (though possibly valid in a different scheme) cannot be made using Papapetrou’s formula in the way proposed by Gupta, for the following reason: This method mixes the various orders of approximation and thus leads to nonsense if fixed up (by adding adjustable stresses after each step) in such a way as to avoid the other contradictions to which the method leads when used uncorrected - the reason being that intermediate approximations lead to sources for the next approximation which do not satisfy conservation laws and thus contradict the differential equations of Papapetrou. (The trouble may be due to Papapetrou’s use of a gravitational energy density depending on second derivatives of the field.)

The question of the absorption and production of gravitational waves was raised again. FEYNMAN discussed a device which would absorb gravitational energy, provided one assumes the existence of gravitational radiation (but as he pointed out, “My instincts are that if you can feel it, you can make it.”). For this purpose one can use a result already presented at an earlier session that the displacement of a particle in the path of a gravitational wave satisfies the differential equation

A particle situated initially near a long light rod, oriented parallel to the propagation direction, could be made to scrape against the rod by the transverse-transverse wave amplitudes.

FEYNMAN then presented the result of a calculation of the energy radiated by a double star system in a circular orbit:

where the masses of the stars are and , and is the magnitude of their relative velocity. The effect is extremely small, the “lifetime” of the earth going around the sun being of the order years.

WHEELER said that, from the point of view of expanding the field about the Minkowski metric, the behavior of the Schwarzschild solution might be described by saying that the field away from the singularity itself acts partly as a source of gravitation. In this connection, one might conceive of a “gravitational” geon - an object in which the gravitational waves traveling around in circles provide the energy to hold the system together.