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.

1. Introduction


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:


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.


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)


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:


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

2.1. Interpretation

3. 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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 Born, M., 'A suggestion for unifying quantum theory and relativity', Proc. Roy. Soc., A165:291-303, 1938

Born, M., 'Reciprocity and the Number 137. Part I', Proc. Roy. Soc. Edinburgh, 59:219-223, 1939

Born, M., 'Reciprocity Theory of Elementary Particles', Rev. Mod. Phys., 21:463-473, 1949

Born, M., 'My Life and my Views', Charles Scribner's Sons, New York, 1968

Chavel, I., 'Riemannian Symmetric Spaces of Rank One' Marcel Dekker, Inc. New York 1972

Coquereaux, R. and A. Jadczyk, 'Conformal theories, curved phase spaces, relativistic wavelets and the geometry of complex domains', Rev. Math. Phys., 2:1-44, 1990

Gilmore, R., 'Scaling of Wyler's Expression for cF, Phys. Rev. Lett., 28:462-464 Jadczyk, A., 'Geometry of indefinite metric spaces', Rep. Math. Phys, 1:263-276, 1971.

Karolyhazy, F., A. Frenkel, and B. Lukacs, 'On the possible role of gravity in the reduction of the wave function', in Quantum Concepts in Space and Time, Ed. by R. Penrose and C. J. Isham, Clanderon Press, Oxford 1986

Kim, Y.S. and M.E. Noz, Phase Space Picture of Quantum Mechanics World Scientific, Singapore 1991

Kobayashi, S., and K. Nomizu, 'Foundations of Differential Geometry, Volume II' Inter-science Publ., New York-London-Sydney 1969

Karpio, A., A. Kryszen and A. Odzijewicz, 'Two-twistor conformal Hamiltonlan spaces', Rep. Math. Phys. , 24:65 80, 1986

Mulak, W., 'Quantum SU(1,1) Harmonic Oscillator'. in Proc. XXV Syrup. Math Phys., Torun 1992, Preprint IFT UWr, Wroclaw 1993 - to appear

Odzijewicz, A. , 'A Model of Conformal Kinematics', Int. J. Theor. Phys. ,15: 576-593, 1976

Odzijewicz, A. , 'On reproducing kernels and quantization of states', Commun. math. Phys., 114:577-597, 1988.

Penrose, R., and W. Rindlet, 'Spinors and Spacc-Time. Volume 2: Spinor and Twistor Methods in Space-Time Geometry', Cambridge University Press, 1986

Prugovecki, E., 'Geometro Stochastic Locality in Quantum Spacetime and Quantum Diffusions', Found. Phys., 21:93 124

Robertson,B., 'Wyler's Expression for the Fine-Structure Constant alpha, Phys. Rev. Lett., 27:1545-1547

Unterberger,A., 'Analyse Harmonique et Analyse Pseudo-differentielle du Cone de la Lumiere', Asterisque, 156, 1987

Vigier,J-P., 'On the geometrical quantization of the electric charge in five dimensions'and its numerical determination as a consequence of asymptotic SO(5,2) group invariance, in Coll. Int. CNRS, 237:411 418, 1976

Wigner,E.P., 'Relativistic Invariance and Quantum Phenolnena', Rev. Mod. Phys., 29:255-269, 1957

Wigner,E.P., 'Geometry of Light Paths bctween Two Material Bodies', J. Math. Phys.,2:207 211, 1961

Wyler,A., 'On the Conformal Groups in the Theory of Relativity and their Unitary Representations', Arch. Rat. Mech. and Anal.,31:35-50, 1968

Wyler,A., L'espace symetrique du groupe des equations de Maxwell' C. R. Acad. Sc. Paris,269:743 745

Wyler,A., 'Les groupes des potentleis de Coulomb et de ¥ukava', C..R. Acad. Sc. Paris,271:186 188





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