**BORN'S RECIPROCITY IN THE CONFORMAL DOMAIN**

ARKADIUSZ JADCZYK

*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

Abstract:

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:

'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 x^{k}
: (x,t) for which the quadratic form *R = x ^{k}x_{k} *=
t

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 R^{½} 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: p^{2 }= *p ^{k}p_{k} *=
E

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

x^{k} -> p^{k},
p^{k} -> -x^{k }(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 x^{k}and P^{k}are considered. These equations hold also in the matrix or operator form of quantmn mechanics. The commutation rules

x^{k} p_{l}
- p_{l} x^{k} =i hbar delta^{ k}_{l (I.3)}

and the components of the angular momentum,

*m _{kl} = x*

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}

^{2.1. Interpretation}

^{3.
Algebraic description of the conformal domain D}_{4}

_{4.
Conclusions: quantum conformal oscillator}

_{}

Footnotes:

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.

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