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CAUSAL SPINFOAMS AND THE INTERPRETATION OF SPACETIME
(Comunicatiion presented to the 2nd. Int. Conference on the Ontology of Spacetime)
Miguel Lorente
Department of Physics, University of Oviedo
E-33007 Oviedo, Spain
e-mail: [email protected]
EPISTEMOLOGICAL PRESUPPOSITIONS
We consider three levels in our knowledge of the world: [1]
Level 1: Physical magnitudes, such as distances, intervals, motions, forces ...given by our senses
Level 2: Metrical properties and mathematical relations among them (laws, theories, models)
Level 3: Fundamental concepts, representing the ontological properties of the physical world.
Question: Is the structure of spacetime in level 2 a primitive or derived concept?
Answer: One can consider three different positions [2]
Dualistic theories: Space and time are absolute concepts independent of the bodies (NEWTON
Monistics theories: Spacetime are identified with gravitational fields (EINSTEIN general relativity,
WHEELER geometrodynamics, HAWKING wormholes)
Relational theories: Spacetime is a derived concept corresponding to the relations among elementary beings
(LEIBNIZ) or from quantum processes (WEIZSAECKER)
RELATIONAL THEORIES OF THE STRUCTURE OF SPACETIME
1.
Combinatorial approach:
According to LEIBNIZ
[3] time is the order of those points non existing simultaneously and one is the ratio of the other. Space is
the order of points that exsist simultaneously and are connected by mutual interactions. Space is nothing
more than the set of all points and their relations. In PENROSE model [4] the starting point is the total
angular momentum of some fundamental units, the interaction of which produces a discrete network. An
object is thus located either directionally or positionally in terms of its relations with other objects. One
does not really need a space to begin with. In BUNGE [5] interpretation the fundamental reality is made
out from basic objects, and the structure of spacetime is deduced from the relations among them. In order
to obtain a 3-dimensional physical space one introduces the concept of distance through the relations of
separation between two basic objects. GARCIA-SUCRE [6] uses the concept of preparticle and relations
among them taken from set theory. The structure of spacetime is the collection of physical systems (made
out of preparticles) forming a statistical lattice that approximates the 3-dimensional space.
2.
Causal sets and General Relativity:
The application of spin networks to General Relativity gives rise to causl sets. SORKIN has
Defined it
as a locally finite partially ordered set [7]. According to SORKIN spacetime ceases to exsist
on sufficiently small scales and is superseded by ordered discrete structure, to which the continuum is only
a coarse-grained, macroscopic approximation. Also SMOLIN[8] tells us that EINSTEIN theory of gravity
is a theory of causal structure and the motion of matter is a consequence of alterations in the network of
causal relations
3.
Quantum causal sets.
Some authors have proposed an unification of Quantum Mechanics (QM) and General Relativity (GR) by
associating quantum objects to causal sets. MARKOPOULOU [9] introduces quantum causal histories by
attaching Hilbert spaces to events in a causal set and unitary operators to causal
relations.WEIZSAECKER [10] does not presupposes the exsistence of spacetime, but a finite number of
simple alternatives. They are considered the ultimate objects of the world (urs). The structure of
spacetime is reduced to QM of simple alternatives. Logically from the principles of QM the structure of
spacetime emerges as a derived concepts. LONERGAN [11] has developed a philosophical interpretation
of the structure of spacetime, using the teory of emergent probability as immanent form for the concrete
extensions and durations as materia prima. The probability of such concrete situations plays the rolle of
quantum mechanical objects, the interactions of which produces the network of relations as ontological
background of spacetime.
A critical discussion of the relational theories can be found in EARMAN [15]
OUR MODEL
1.
In our presentation we can distinguish three steps as before:
Combinatorial approach: The ontology of spacetime is founded in the exsistence of
elementay beings interacting among themselves, forming a network of relations that can be represented
by periodic or non periodic (statistical) lattice, on which the motion of particles take place [12]
2.
Discrete manifolds in GR
The network should reproduce the underlying structure of spacetime in GR, such as
Euclidena or hyperbolic tessellation [13] Also in the simplicial spinfoam models
and in the REGGE calculus the spacetime structure, is made out of discrete trianglations
[14]
4.
QM vs. GR
If we accept QM logically and ontologically prior to the structure of spacetime (WEIZSAECKER)
we start with a set of quantum objects endowed with some Hilbert spaces (for instance
simple alternatives represented by SU(2) group) and associate unitary operators to the interaction among
these objects. These interactions follow the probability laws in QM,
giving rise to an infinite of possibilities responsable of the continuity properties of the structure of space
and time.
REFERENCES
1.
M. Lorente, "A causal interpretation ofthe structure of spacetime" in Foundations of Physics (A selection
of papers contributed to the Physical Section of the 7 th International Congress of Logic, Methodology and
Philosophy of Science) Verlag Hoelder, Vienna 1986
2. M. Bunge, "Una teoria relacional del espacio fisico" in Controversias en Fisica, Editorial Tecnos, Madrid
1983, pp. 62-82
3. .G. W. Leibniz, "Initia rerum mathematicarum metaphysica", Mathematische Schriften (G. Gerhardt) VII,
p. 17-39
4. R. Penrose, "Angular Momentum: an approach to combinatorial spacetime" in Quantum Theory and
Beyond (T. Bastin, ed.) Cambridge U. Press, Cambridge 1971, p. 151-181
5. M. Bunge, A. Garcia-Maynez, "A relatinal Theory of Physical Space" Int. J. Theor. Phys.
. 15 (1977) 961-972
6
M. Garcia-Sucre, "A relational theory of spacetime" in Scientific Philosophy Today (J.
Agassi, R. S. Cohen ed.) Reidel 1981, p. 45-69
7. R. Sorkin, "Causal Sets: Discrete Gravity", Notes for Valdivia Summer School 2002,,
arXiv: gr-qc/0309009
8.
L. Smolin, "Three Roads to Quantum Gravity" Basic Books, 2001, F. Markopoulou, L.
Smolin, "Causal evolutiion of spin networks" Nuclear Physics B 508(1997) 409-430
9
F. Markopoulou, "Quantum causal histories", Class. Quantum Grav. 17 (2000)2059-72
10. C. F. Von Weizsaecker, Quantum Theory and the Structure of Spacetime, (6 volumes)
Hanser Verlag, Muenchen 1986
11. B. Lonergan, Insight. A Study of Human Understanding, U. of Toronto, 1992
12. M. Lorente, "Representation s of the discrete inhomogeneous Lorentz group and Dirac wave equation
on the lattice" J. Phys. A. Math. Gen. 32 (1999) 2481-2497.
13. .M. Lorente, "A discrete curvature on a planar graph" arXiv: gr-qc/0512142
14.
15.
.M. Lorente, "Spin Network in Quantum Gravity" arXiv: gr-qc/0512144
J: Earman, World enough and Spacetime. Absolute versus Relational Theories of Space
And Time, MIT Press, Cambridge, Massachusetts 1981