The phenomenological spacetime and quantization

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The ”phenomenological”space time and quantization.Marcelo BOTTA CANTCHEFF 1FaMAF,Medina Allende y Haya de la Torre,Ciudad Universitaria,5000C′o rdoba,Argentina Fax nr:+54-51-334054Personal address:General Paz 5403-D,5000Cordoba,Argentina.Abstract In order to obtain a well de?ned quantum gravity we de?ne the space-time in relation to the ”phenomenology”of the physical interactions.Under

certain conditions the Feynman Rules may be the literal description of this phenomenology;however we shall to speculate with this in general .Besides,we comment the reasons that give to the gravitational ?eld a privileged sit-uation over the others.

The usual meaning of spacetime will be reviewed.The idea is simple:in general, only certain subset of the current spacetime is involved in the evolution of a given physical system(we are interested in quantum systems)or observations;only this has e?ective physical signi?cance and its description is required.furthermore, to consider the rest of the events can to introduce di?culties unnecessarily,in particular,in order to obtain a well de?ned quantum gravity.Each”e?ective”event must be associated with a”transformation”(or observation)of a physical system.The spacetime(the set of the events)would be not given before the?elds, it develops together with the interactions and obviously,the causal structure of the spacetime is in correspondence with the causal connections between the ele-mental phenomena that compound the evolution of the full system.We call to this concept:”Phenomenological Space Time(P.S.T.)”.Notice that naturally, within this context the space time need not to be continuous[1].

The mechanism proposed is:”if some full dynamic scheme for the?elds is as-sumed,the P.S.T.relative to a particular process,will be the set of support-points on which it developes,in consistence with that dynamic.”

This concept could be successfully applied in the quantization only if a”mod-i?cation”of the usual dynamic for the phenomena is assumed;the canonical formulation of quantum?eld theory[2]2requires a continuum background and is not clear the meaning of an event in the sense P.S.T.

Thus,the question is:what does the phenomenology mean,in general?;what is, precisely a transformation of a system in the framework of quantum?eld theory? We attempt to introduce the answer.

For this,it is convenient to formulate the dynamic in terms of an general expres-sion for the transition amplitude between two quantum states,which in general will be some functional of the space time points involved in the process where the

?elds develope;for example the sum over histories.Then we shall to de?ne the M-spacetime such that the transition amplitude let”well behaved”.It signify to allow only a spacetime such that it avoid the singularities and so,the problems of in?niteness of quantum gravity would be solved by construction3

Now,we illustrate these ideas assuming as generic the”scenery”where the per-turbation theory is built:

In The Covariant Perturbation Method(C.P.M.)[3],the gravity is treated4as a spin-2?eldγab on a background5(R4,ηab=diag(?1;1;1;1)).The full metric is g ab≡ηab+γab.The Lagrangian for gravity coupled with the”matter?elds”is obtained replacingηab→g ab and?a→?a6.

whereγab is not assumed be small in a set?inclouded in a compact subset of R4 and g ab satisfy the exact Einstein’s equation;in the spacetime complement7?0,γab is small and the the interactions are considered negligible.

The commutation relations and the causality conditions are de?ned with respect toηab.Notice that the role ofηab is analogous to an external electromagnetic?eld in Q.E.D.;it must be treated classically.

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Then formally,one can to construct the Feynman’s Rules(F.R.)in the con?gura-tion space.The F.R.,illustrated by the Feynman’s Diagrams,are constructed as a ”pictorial”description of the mathematics involved in the perturbation schemme; they allow an interpretation of the fundamental interactions strongly intuitive, and it lead us to conjeture that in this context,they must not be interpreted only as a formal illustration and gives a precise meaning to the”phenomenology”mentioned above

In order to apply the mechanism of suggested by the P.S.T and the complete

agreement of the predictions of perturbative scheme with the existing experi-ments allows us to assume8:

(#)”the F.R.(in the Con?guration Space9)are not a mathematical arti?cer, they represent the fundamental structure of the processes”.

So,the canonical approach”will be an approximation”which can be recovered in the”classical”limit,when the quantum e?ects of the gravity are negligibles.

In this scheme a”transition”(virtual)between two quantum states is described pictorially,and now literally,with a vertex of a Feynman Diagram;then”for each space time point there exist a set of such vertex associated to the full process occurring in whole the spacetime.”

This is the new precise structure of a spacetime event in terms of the physical processes such as we thought.

In each order,the S-matrix expansion is expressed[2]by:

i n

S(n)[?]≡

8of course,under the conditions stablished above.

9perhaps it is a unnatural restriction in(#),because the construction of F.R.in momentum space is not direct,they assume the momentum conservation in each vertex;but if?is discrete the Noether’s theorem cannot be applied because the notion of the operations of symmetry on the spacetime changes.

10T is the time ordered product

That is expressed in the following de?nition;

(*)Given a Fock-Space F(built with the solutions of free?elds),and a”initial”state|ψin>∈F,then:”for a quantum transition(|ψin>→|ψout>∈F);M is a subset of R4such that,each corresponding element of the S-matrix expansion[2] remains?nite in each order”;and the vector S|ψin>have norm unity.

C.P.M.has two high di?culties;?rst,the in?niteness in the perturbation theory for gravity;we sketch to show that”there exists at least a trivial spacetime M such that the S-matrix expansion is?nite for all quantum process”:

the minimal spacetime M=?0where the S-matrix is trivially well de?ned: <ψout|S[Φ]|ψin>=<ψout|ψin>,and the interaction region?0,would be the empty setΦ.

Thus,in principle,we could”to enlarge?0”adding points such that S remains ?nite.

Clearly,two spacetimes will be equivalents when they have equal distribution of probabilities given by the S-matrix.

Note that,in order to de?ne more precisely M;we have adopted the”extra”restriction:”M,is some subset of R4”.Also we could to assume that M is max-imal in the following sense;that it cannot be included into any other M′(both included in R4)satisfying(*)”.Which ensures that when the quantization of the gravity is not considered and the theory is renormalizable;the perturbation series is?nite in each order thus,we don’t need to constrain the set of events to avoid the in?nities.M agrees with the continuum background in consistence with the canonical?eld theory which do not includes gravity(for example Q.E.D.on Minkowski space time).However it can be not su?cient to determinate completly the spacetime M11;the inclusion de?nes a”partial order”and each class totally ordered may to have an maximal element12

Other alternatives can be proposed;for example,perhaps one could to de?ne other scheme with a natural probability for each spacetime which satis?es(*),in this case we do not need to determinate M univocally.

The second problem in C.P.M.is about the ambiguity in the causal structure of the space time;the causality conditions holds with respect to the background metricηab rather than the true g ab but in the free region?0,g ab is accuratelyηab, and the causal structure of both metric agree.Besides we hope that in general,?contains a?nite number of elements13when the gravity is considered;if this is true:both causal structure agree in all space time region in elaborate but nonethe-less well posed sense:the notion of the causal structure is based on the type of the curves;let a timelike curveγon the background,thenγintersects to?in a discrete set of points thus,the proper time ofγdo not changes when the true metric is considered.It is to say:”the metric g ab in?(discrete)is ill de?ned; however is possible to de?ne the causal structure with respect to the background (R4,ηab)without ambiguity”.

Notice,on the other hand,that there is an algebraic explanation of the privileged situation of the metric?eld over the others:the quantized gravitational?eld de-termines the the spacetime points as consecuence of(#):

The expression(1)requires a metric g ab in each point x to”contract”the tensorial (or spinorial14)indexes of the?elds interacting15.Thus g ab must be present in each term of the Langrangian,it to say that the metric g ab”necessarily”interacts

with all the other?elds in each spacetime point.Each one of the”events”is a vertex with at least one”gravitonic line”on it.

This appears as extremly trivial way to avoid the problem of non renormalizability of C.P.M.however;a assumption like(#)does justify it!

The most signi?cant aspect of this scenery is to?x the attention on the evolution of the?elds which”develops”the spacetime.This is a ontologic requirement, believed only a philosophical question without relevance in physics.

Surely there is no an unique way to develop this.

Finally,

the P.S.T.depends critically of the”phenomenology”assumed,later we spec-ulate that:

”the F.R.constitue the fundamental entity”,and(#)is the fundamental(and strongly intuitive)description of all quantum process”not only in the conditions of perturbation theory.

Clearly,this could does a?ect more strongly the spacetime structure and the phys-ical laws.

References

[1]Bombelli,Lee,Meyer,Sorkin;Phys.Rev.Lett,59,521(1987).

[2]F.Mandl and G.Shaw,Quantum Field Theory,edited by John Wiley(1984).

[3]Robert M.Wald,General Relativity(14.1),edited by The University of

Chicago(1984).

[4]Botta Cantche?,unpublished.+