BORN'S RECIPROCITY IN THE CONFORMAL DOMAIN
Institute of Theoretical Physics, University of Wroclaw,
pl. Maksa Borna 9, PL-50-204 Wroclaw, Poland
in Z. Oziewicz et al. (eds.) Spinors, Twistors,
Clifford Algebras and Quantum Deformations, 129-140.
Kluwer Academic Publishers 1993
Max Born's reciprocity principle is revisited and complex four dimensional Kahler manifold D4=SU(2,2)/S(U(2)xU(2)) is proposed as a replacement for space-time on micro scale. It is suggested that the geodesic distance in D4 plays a role of a quark binding super-Hamiltonian.
Some 55 years ago, in the Scottish city of Edinburgh, Max Born wrote 'A suggestion for unifying quantum theory and relativity'[Born, 1938], the paper that introduced his 'principle of reciprocity'. He started there with these words:
'There seems to be a general conviction that the difficulties of our present theory of ultimate particles and nuclear phenomena (the infinite values of the self energy, the zero energy and other quantities) are connected with the problem of merging quantum theory and relativity into a consistent unit. Eddington's book, "Relativity of the Protoll and the Electron", is an expression of this tendency; but his attempt to link the properties of the smallest particles to those of the whole universe contradicts strongly my physical intuition. Therefore I have considered the question whether there may exist (other possibilities of unifying quantum theory and the principle of general invariance, which seems to me the essential thing, as gravitation by its order of magnitude is a molar effect and applies only to masses in bulk, not to the ultimate particles. I present here an idea which seems to be attractive by its simplicity and may lead to a satisfactory theory. '
Born then went on to introduce the principle of reciprocity - a primary symmetry between coordinates and momenta. He explained that
'The word reciprocity is chosen because it is already generally used in the lattice theory of crystals where tim motion of the particle is described in the p~space with help of the reciprocal lattice.' A year later, in a paper "Reciprocity and the Number 137. Part 1", [Born, 1939] he makes an attempt to derive from his new principle the numerical value of the fine structure constant. 1 The most recent and clear exposition of the principle of reciprocity appears in his paper 'Reciprocity Theory of Elementary Particles ', published in 1949 in honor of 70th birthday of Albert Einstein [Born, 1949]. The following extensive quotation from the Introduction to this paper brings us closer to Born's original motivations.
'The theory of elementary particles which I propose in the following pages is based on the current concepts of quantum mechanics and differs widely from the ideas which Einstein himself has developed in regard to this problem.(...) Relativity postulates that all laws of nature are invariant with respect to such linear transformations of space time xk : (x,t) for which the quadratic form R = xkxk = t2 -- x2 is irtvariant (...). The underlying physical assumption is that the 4-dimensional distance r: R½ has an absolute significance and can be measured. This is natural and plausible assumption as long as one has to do with macroscopic dimensions when measuring rods and clocks can be applied. But is it still plausible in the domain of atomic phenomena? (...) I think that the assumptions of the observability of the 4-dimensional distance of two events inside atomic dimensions is an extrapolation which can only be justified by its consequences; and I am inclined to interpret the difficulties which quantum mechanics encounters in describing elementary particles and their interactions as indicating the failure of this assmnption.
The well-known limits of observability set by Heisenberg's uncertainty rules have little to do with this question; they refer to the measurements and momenta of a particle by an instrument which defines a macroscopic frame of reference, and they can be intuitively understood by taking into account that even macroscopic instruments must react according to quantum laws if they are of any use for measuring atomic phenomena. Bohr has illustrated this by many instructive examples. The determination of the distance R½ of two events needs two neighboring space-time measurements; how could they be made with macroscopic instruments if the distance is of atomic size?
If one looks at this question from the standpoint of momenta, one encounters another paradoxical situation. There is of course a quantity analogous to R, namely P: p2 = pkpk = E2 - p2, where Pk = (p, E) represents momentum and energy. But this is not a continuous variable as it represents the square of the rest mass. A determination of P means therefore not a real measurement but a choice between a number of values corresponding to the particles with which one has possibly to do. (...) It looks therefore, as if the distance P in momentum space is capable of an infinite number of discrete values which can be roughly determined while the distance R in coordinate space is not an observable quantity at all.
This lack of symmetry seems to me very strange and rather improbable. There is strong formal evidence for the hypothesis , which I have called the principle of reciprocity, that the laws of nature are symmetrical with regard to space-time and momentum-energy, or more precisely, that they are invariant under the transformation
xk -> pk, pk -> -xk (I.1)
The most obvious indications are these. The canonical equations of classical
mechanics are indeed invariant under tile transformation (I.1), if only the first 3 components of the 4-vectors xk and Pk are considered. These equations hold also in the matrix or operator form of quantmn mechanics. The commutation rules
xk pl - pl xk =i hbar delta kl (I.3)
and the components of the angular momentum,
mkl = xk pl - xl pk (1.4)
show the same invariance, for all 4 components. These examples are, in my opinion, strongly suggestive, and I have tried for years to reformulate the fundamental laws of physics in such a way that the reciprocity transformation (I.1) is valid (...). I found very little resonance in this endeavor; apart from my collaborators, K. Fuchs and K. Sarginson, the only physicist who took it seriously and tried to help us was A. Lande (...). But our efforts led to no practical results; there is no obvious symmetry between coordinate and momentum space, and one had to wait until new experimental discoveries and their theoretical interpretation would provide a clue. (...) There must be a general principle to determine all possible field equations, in particular all possible rest masses.(...) I shall show that the principle of reciprocity provides a solution to this new problem - whether it is the correct solution remains to be seen by working out all consequences. But the simple results which we have obtained so far are definitely encouraging (...).'
2 The very problem of a serious contradiction between quantum theory and relativity was addressed again, in 1957, by E.P. Wigner in a remarkable paper 'Relativistic Invariance and Quantum Phenomena', [Wigner, 1957]. Wigner starts with the assertion that 'there is hardly any common ground between the general theory of relativity and quantum mechanics'. He then goes on to analyze the limits imposed on space-time localization of events by quantum theory to conclude that:
'The events of the general relativity are coincidences, that is collisions between particles. The founder of the theory, when he created this concept, had evidently macroscopic bodies in mind. Coincidences, that is, collisions between such bodies, are immediately observable. This is not the case for elementary particles; a collision between these is something much more evanescent. In fact, the point of a collision between two elementary particles carl be closely localized in space-time only in case of high energy collisions.'
Wigher analyzes the quantum limitations on the accuracy of clocks, and he finds that "a clock, witha running time of a day and an accuracy of 10-8 second, must weigh almost a gram-for reasons stemming solely from uncertainty principles and similar considerations". 4
2. Reciprocity the Twistor Way
Algebraic description of the conformal domain D4
4. Conclusions: quantum conformal oscillator
1 He failed, but
many years later Armand Wyler [Wyler, 1968,1969,1971] obtained a
reasonable value by playing, as we shall see~ with a similar
geometrical idea. Wyler failed however in another respect: he was
unable to formulate all the principles that are necessary to justify
his derivation . His work was criticized (cf. [Robertson, 1971;
Gilmore, 1971; Vigier, 1976]), his ideas not understood, his name
disappeared from the lists of publishing scientists.
2 It must be said that later on, in his autobiographic book 'My life and my views', [Born, 1968], Born hardly devoted more than a few lines to the principle of reciprocity. Apparently he was discouraged by its lack of success in predicting new experimental facts.
3 I will return to this conclusion when interpreting space-time as the Shilov boundary of the conformal domain D4.
4 In 1986 Karolyhazy et al. in the paper 'On the possible role of gravity in the reduction of the wave function', [Karolyzy, 1986], presented another analysis of the imprecision in space-time structure imposed by the quantum phenomena. They proposed 'to put the proper amount of haziness into the space time structure'. Their ideas, as well as the ideas of a "stochastic space-time" most notably represented by E. Prugovecki (cf. [Prugovecki, 1991]) and references therein) all point in a similar direction.
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Last modified on: April 20, 2000.