Herramientas de la metodología “DMAIC”

What we seek, I am therefore claiming, is a new theory, presumably non-linear and possibly non-deterministic, which: 1 is extremely accurately approximated, for reasonably small or simple systems, by a Schrödinger-evolving state vector; and

2 accords closely, for systems which are large or complicated, with reduction and with the standard replacement of probability amplitudes by the probabilities of actual alternative outcomes. (See Pearle14,15,16 for a very serious concrete proposal of this kind.)

In various other places17,18 I have put forward arguments to support my own viewpoint that the essential new and non-linear physical input should be general-relativistic gravity, together with certain considerations about the second law of thermodynamics and the apparently strongly time-asymmetric constraints on the structure of space-time singularities. I do not wish to repeat the details of my (rather round-about) argument here. My latest thinking on this question is being presented elsewhere19 _{and I refer the interested reader to that source for more details. In my opinion there are indeed strong}

grounds for believing that for a quantum gravity theory to have any chance of real success it must incorporate among its important features a precise and objective theory of state-vector reduction. (A different line of thought, also leading to the conclusion that state-vector reduction is a gravitational

phenomenon, had earlier been put forward by Károlyházy20; see Károlyházy, Frenkel and Lukács21 for an up-to-date account of these ideas; cf. also Komar22_.)

There are various reasons for thinking that the structure of quantum mechanics may well have to be modified in the context of general relativity. One of these concerns the choice of variables with respect to which the process of ‘quantization’ is to be conducted. In the standard non-relativistic (or special- relativistic) quantization procedure one has the canonical variables xa, pa of position and momentum which are made into non-commuting operators and

whose (canonical) Poisson bracket relations are replaced by (canonical) commutators. It would not be valid to apply this quantization procedure instead to some other variables , , obtained from xa and pa by an arbitrary canonical transformation, again replacing the canonical Poisson bracket relations

of and by canonical commutators. It might, for example, be the case that is related to xa _{by some arbitrary co-ordinate transformation, with}

canonical conjugate correspondingly chosen, and then the canonical quantization procedure will not in general apply to and What distinguishes the standard (xa_,p

a) from the more arbitrary (cf. Komar23) is its direct relation to the symmetry group of flat space-time, so that pa indeed

It will be noted that this selection principle for ‘preferred’ canonical variables will not work in a curved space-time. This is a much more immediate problem than the daunting difficulties involved in trying to quantize the gravitational field itself, since it arises already when gravity is simply taken to be an unquantized background field. Thus, even at this simplified level, the standard quantum-mechanical procedures run into difficulties. One does not, in fact, have any clear rules for a quantizing procedure, in the general case. When one is asked the question of whether the incorporation of space-time curvature into physics in any way affects the general framework and rule of procedure of quantum mechanics, one is forced to reply (if truthful) that one does not even know what the rules are, in a general curved-space setting!

It seems that in curved space, one must resort to some ‘patchwork’ formalism and, at best, much of the compelling elegance of quantum mechanics is removed. It has become fashionable to argue that, in view of the abounding difficulties that have been encountered in attempts to quantize general relativity, one should replace that theory by some other which might more willingly submit to being forced into the linear framework of standard quantum theory. Yet it is not often suggested that the quantum framework might itself be forced to yield. I would not dispute that some changes in classical general relativity must necessarily result if a successful union with quantum physics is to be achieved, but I would argue strongly that these must be accompanied by equally profound changes in the structure of quantum mechanics itself. The elegance and profundity of general relativity is no less than that of quantum theory. The successful bringing of the two together will never be achieved, in my view, if one insists on sacrificing the elegance and profundity of either one in order to preserve intact that of the other. What must be sought instead is a grand union of the two—some theory with a depth, beauty and character of its own (and which will be no doubt recognized by the strength of these qualities when it is found) and which includes both general relativity and standard quantum theory as two particular limiting cases.

This is all very well, the reader is no doubt thinking, but what relevance has the extremely weak phenomenon of even classical gravity—let alone the absurdly tiny and quite undetectable, ‘quantum corrections’ that one anticipates would result from a quantum gravity theory—to the commonplace

phenomenon of reduction? Whether one contends that reduction occurs early or late in the chain of alternatives—whether reduction has already physically taken place with the track in a cloud chamber or mark on a photographic plate, or whether it is delayed until it affects the state of a human brain—in either case only tiny amounts of energy are involved, by the standards of gravitation theory. Yet it is my contention that it is this link with gravity which

effects the apparent change in rules that takes place with reduction. When two states of differing energy distribution are linearly superposed, the slightly differing space-time geometries that these energy distributions produce (according to general relativity) must also be superposed. This is not clearcut matter since, as we have seen, the geometry is not just a ‘physical state’ but is something essentially involved in the very determination of the ‘procedure of quantization’ (cf. foregoing remarks concerning xa, pa and , ). It is hard to see how ‘quantization procedures’ are to be superposed.

I would contend, therefore, that when two geometries involved in a linear superposition become too different from one another—in some yet-to-be- determined precise sense—then linear superposition fails to hold, and some effective non-linear instability sets in, resulting in one or the other geometry winning out, the result being reduction. However, the criterion for deciding when such an instability becomes operative cannot depend solely on the energies or masses involved. The masses involved in cloud chamber droplets, or relevant collections of silver iodide molecules in a photograph, or configurations in a brain, would appear to be very tiny—apparently tinier than the Planck-Wheeler mass of 10−5 _{g which seems to characterize the scale}

of quantum gravity—which, in turn, might perhaps be smaller than the total electron mass involved in a coherent quantum state for a large

superconducting coil. Some care will be needed in deciding upon the precise measure of mass-energy distribution difference that is needed to trigger off the instabilities of reduction.

My earlier arguments17,18 _{have emphasized the time-asymmetric role of entropy in reduction and the gravitational origin (via time-asymmetry in space-}

time singularity structure) of the second law of thermodynamics. For an overall consistency of the physics involved, it is argued that gravity ought also to be critically relevant to the quantum-mechanical reduction process. But it is gravity in its role of providing a gravitational entropy that enters. In the early stages of the universe (at the big bang) the entropy content of the matter was high. The second law arises because, at the singularity, something

constrained the entropy of gravity to be extremely low, whereas the potential for entropy content of (conformally) curved space-time was enormously high.

As the universe evolves, this unused potential of gravitational entropy is gradually taken up (clumping of gas into stars, etc.) and is directly or indirectly taken advantage of by systems requiring a low entropy reservoir (e.g. plants making use of degradable photons from the sun, which is a hot spot in the sky by virtue of its gravitational clumping). The ultimate high value of entropy for a gravitationally clumped object is achieved by the black-hole state. This entropy can be given in precise terms by the Bekenstein-Hawking formula (cf. Wald24) and we find that for an ordinary black hole, resulting from

the collapse of, say, a ten solar-mass star, the entropy per baryon is some twenty orders of magnitude larger than even the seemingly huge value of 108 _or

so (taking units with Boltzmann’s constant as unity) for the thermal black-body radiation left over from the hot big bang.

Even when this maximum entropy value is not achieved, the gravitational entropy can be quite sizable, though as yet no precise determination of this entropy has been suggested. In a rough way, we can say that this entropy is intended to estimate (the logarithm of) the number of quantum states that go to make up a given classical geometry. Such an entropy measure would have to be very much a non-local expression. But this is not surprising in general relativity, where even energy must be given by a non-local expression (cf. Penrose and Rindler25).

The idea is that, for ‘reduction’ to take place, we must be in a situation where, such as in the localization of a photon on a photographic plate, the lowering of entropy that this ‘reduction’ would seem to entail must be at least compensated by a corresponding increase in the gravitational entropy, this gravitational entropy being higher for clumped localized energy distributions than for those which are spread out more uniformly. (This cannot be just the ‘quantum-mechanical entropy’, which remains at zero so long as one uses a pure-state description, but is some more ‘commonsense’, though somewhat ill-defined, entropy concept which increases as more and more degrees of freedom become involved. I am grateful to P.Pearle for illuminating discussions concerning this. Quantum reduction must be a non-local process, and the hope is that this non-locality can be matched with the non-locality involved in the gravitational entropy concept.

It will be clear to the reader that there is much speculation and lack of precision in this picture. (For more clarification, see Penrose19.) But my purpose here is not to spell out in detail how the reduction procedure might work (which I cannot do, not having a proper theory) but merely to attempt to

persuade the reader of the plausibility of there being some new physical process going on which has a perfectly objective character, even though we do not understand how this process works in detail. I would certainly anticipate, however, that the process is likely to defy any meaningful local description in ordinary space-time terms. But any speculation involved in the ‘details’ of this process will, I suppose, pale to insignificance by comparison with the speculative aspects of what I have to suggest in the next section—where I turn to what will (at first) seem to be a totally different topic!