Nilo Sylvio Costa Serpa, \mathpzc{MSc} - Astronomy

Nilo Sylvio Costa Serpa, - Astronomy

Abstract

This essay aims to summarize the main physical features arising from a new supersymmetric theory of gravitation. Based on preliminary discussions about classical field theory, cosmology, algebra and group theory, and taking formal results and theoretical considerations in comparison with several contributions from great authors, present work deals with gravity inside the limits of a meta-field theory, that is, a non-quantized but consistent representation of supergravity, the supersymmetry between gravitons and gravitinos. The introduction of meta-fields furnishes an independent framework for the study of gravity despite of constraints of quantization, treating the supersymmetric partners as deterministic actors of gravitation and not simply probabilistic entities. I explain my belief that gravitational field, by its own nature, is not quantizable in the same foot as the other fields, what does not means that we can not understand gravity by similar formal veins. Also, present work proposes the implementation of the so-called SCYL program (Supersymmetric Cosmology at Yonder Locals), an attempt to apply supergravity as a meta-field theory to solve problems on astrophysical cosmology. The SCYL program is an effort directed to search convergence between the knowledge brought by cosmology and by supersymmetry to attain more clearness about the structure and evolution of the Universe. As an additional motivation, there are some exercises in the final part of the work to aid fixing of the main concepts.

Keywords: supergravity; graviton; gravitino; field theory; meta-fields; gravitor.

Instituto de Ciências Exatas e Tecnologia

UNIP - Universidade Paulista

SGAS Quadra 913, Conj B

Brasília - DF, Brasil

Abstract

 

Supersymmetry is a beautiful symmetry between bosons and fermions, although there is no evidence of it in Nature. This does not mean that it is not present, but that it must be well hidden.

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Acknowledgements

I thank Dr. José-Abdala Helayel Neto for the stimulating support. Special thanks to Dr. Camilo Tello at INPE - Instituto Nacional de Pesquisas Espaciais - whom contributed with some suggestions. I’ve benefited from conversations with many colleagues, including Julio Peçanha, Marcelo Byrro and Antonio Teixeira. I also would like to thank the encouragement and patience of my family.

Prologue

This essay is a compilation of my ultimate work on the former model for supergravity I proposed at 2002, now reaching its final theoretical outcomes and the top point with the so-called SCYL program (Supersymmetric Cosmology at Yonder Locals). More ontological than phenomenological, present essay also includes some particular notes I made during several classes on supersymmetry and supergravity cursed at CBPF - Centro Brasileiro de Pesquisas Físicas - and from talks with Professors Bert Schroer, Wolfgang Bietenholz and José-Abdala Helayel Neto. Fragments of my lectures on gravitation, wrote before many major advances in the theory, are revised, adapted and resumed to give more clearness for readers that are not entirely familiarized with the principal publication. This study treats mainly the Abelian side of the theory (non-Abelian trend is being taken for another essay). It is not for one that desires to learn field theory and it must be read like a book; it is necessary to be familiarized with basic formalism. I claim attention for the fact that this original theory tries to indicate the true role of gravitino field in the supersymmetric theory of gravity. In particular, the last chapter is my project (the SCYL program) to be sponsored by TechSolarium - Solar Technology -, based in great on the previous chapters.

Conventions and basic definitions

In order to aid the reader, I did here a resume of the principal notations.

  • In the SCYL program I called "cold" the relic gravitinos from early stages of the Universe, and "hot" the gravitinos generated by dense massive objects.

  • Any letter with an overdot (, , etc.) denotes time derivative of the quantity that it represents.

  • Any letter overwrote by a right coma (, , etc.) denotes positional derivative of the quantity that it represents.

Preliminaries

 

Symmetries in modern physics have taken an even stronger role to such an extent that the laws of modern physics cannot even be formulated without the concept of symmetries. To make the framework of local quantum field theory meaningful, symmetries have to be invoked from the very beginning.

Supersymmetry (SUSY) is a Bose-Fermi symmetry referring to the spectrum of coupling energy among particles; it is a device that tries to fulfill a phenomenological gap between the sectors of spectrum related to electroweak interactions and GUT scale (from Gev to Gev). The gap results from the second Higgs quantization, needed in Weinberg-Salam, forcing the introduction of SUSY mechanisms to provide intermediary physics inside those limits. Successive symmetry breaks are in part supplied by gravitic fields that do not couple (at least in thesis) with matter. For a more complete treatment and the necessary field theory background, please, see the standard literature at the end of this work. Supergravity (SUGRA) is the supersymmetry that occurs in gravity. The smallest theory of supergravity relates two types of fields referring to the hypothetical particles graviton and gravitino. The relevance of supergravity to cosmology is that it offers an effective field theory behind the expanding universe and timedependent scalar fields. In particular, as we shall see, the consideration of timedependence as a fundamental feature of the fields in the model I proposed few years ago defines supergravity itself as a theory that describes gravity by cumulative effects on matter.

Gravitorial theory, a sort of supergravity theory, was born to describe supersymmetry in gravity as an outcome of long-time cumulative effects of gravitation. It settles supergravity in a formal "forequantum"   representation as a non-local field theory with local vestiges of the time flowing process, which culminates in scenarios of supersymmetry spontaneous break. We may say it is a "neo-classic"  sight of supergravity, since it does not presuppose probabilistic interpretation of the formalism, either operatorial devices previous and irremediably linked to observers. Even so, theoretical unrolling seems familiar, as gravitors and spinors are similar affinor structures.

Quantum field theories are usually built by applying quantization rules to a continuum field theory; this demands the replacement of Poisson brackets by commutators or anticommutators, as we deal with bosonic or fermionic fields respectively. In fact, gravitorial theory is more concerned to supergravity itself, and not to conjectures about the quantum nature of space and time, in the same critic sense pointed by Butterfield and Ishami (2000):

"…general relativity is not just a theory of the gravitational field - in an appropriate sense, it is also a theory of spacetime itself; and hence a theory of quantum gravity must have something to say about the quantum nature of space and time. But though the phrase ’the quantum nature of space and time’  is portentous, it is also very obscure, and opens up a Pandora’s box of challenging notions".

On some sort of theory of quantum gravity they observe that "…it is wrong to try to construct this theory by quantizing the gravitational field, i.e., by applying a quantization algorithm to general relativity (or to any other classical theory of gravity). We shall develop this distinction between the general idea of a theory of quantum gravity, and the more specific idea of quantized version of general relativity…".

Perhaps one may consider a "quantum of spacetime"  as in the times of Zenon of Elea, when Proclo understood a given number as a cutoff on the "compact thickness"  of the continuum of the real numbers; a quantum of spacetime would be a type of occasional physical "cutoff"  in the continuum of the cosmic tapestry. However there is hope that supergravity is a consistent quantum theory of gravity, I believe in a theory essentially "classic"; my concern is to specify a representative frame of the involved symmetries, preceding to statistics or stochastic behavior due to experimental activity, and, thereby, more realistic in the sense of a description of the world that foregoes operatorial interventions (of course we may think, focusing gravitational waves, in "amplitude operators"  within experimental perspectives, as textbooks customarily use to replace momentum by the operator to compute , but this is not yet the case considering the present technology and the fact that the virtual carrier of gravity, the graviton associated to the gravitational wave, is not acted upon by the strong and the electromagnetic interactions). Quantization of gravitorial theory is a matter of lab procedures, and even so there is no certainty whether we are correct to trying to quantize gravity, as exposed above, and whether it would be possible to detect supergravity partners in laboratory.

Why symmetry is important?

The Greeks developed the symmetry concept. To them, symmetry meant proportionality, and, in especial, commensurability, an adequate Latin mirror translation for the original Greek meaning of "syn-metry". So, it is clear that in early times symmetry had spatial meaning. Only when symmetry became a more abstract concept it appeared that invariance through time is part of the symmetry concept too. In fact, symmetry is important because human beings always try to identify regular patterns looking for the best understanding of natural processes. The human brain works in this way, and, although sometimes it may lead to mistakes and erroneous impressions, nature seems to evolve by symmetries or, at least, to induce human brain to see the world as it was made by symmetries. The growing of crystals, the shape of the vertebrates, the snow flocks, the Gaussian distributions applicable to several facts, are good examples of the general tendency to find symmetric patterns. Of course, there are shapes more and more complicated in nature, then with weaker symmetry. But, as everybody, including physicists, has his own idea of what is symmetry, I mean it in the representational framework of group theory.

The notion of symmetry plays a great role in quantum physics. In modern quantum field theories, symmetries are important because we might get information about the system without any knowledge on the real laws behind its dynamics (Wess, 2009). In fact, we start by the implementation of the symmetries themselves to be able to enunciate formally the laws. This way to think a dynamical system is now so deeply ingrain in our minds that we say to understand it if we find an aesthetic symmetry in such system. Supersymmetry is an uncommon deep generalization of symmetry. This generalization was attained by the fact that not only commutators, which the symmetries can be formulated, but also anticommutators are very useful, especially when we deal with particles with half integer spin. Thus, the idea is to formulate a general concept of symmetry, the supersymmetry, in terms of commutators and anticommutators as well. Although there is no experimental confirmation of supersymmetry in nature, it has influenced greatly the theoretical work in high energy physics.

What does quantum mechanics say?

The concept of physical probability was really born with the adventure of quantum mechanics, even though in the core of this discipline it has been treated systematically as the expression of the inexact knowledge. The focal point was to interpret the so called wave function - the amplitude of the wave itself - and Max Born was the prime to achieve it. As we know, is solution of the famous wave equation for one particle with mass , due to Schrödinger,

(1)

where is the potential and is the Planck constant divided by . The functions are in general complex. The connection of such quantities to the "real" world (or, which came to be the same, the acquirement of quantities called "observables") is represented by means of operations such as , where is the complex conjugate of . It prays a fundamental postulate of quantum mechanics that is the density of the probability for a particle of mass to be found at the point , on time . Therefore, the likelihood to locate the particle inside an infinitesimal volumn of space on that time is,

(2)

Only in the case of one particle, the configuration space of the function is isomorphic to the tridimensional space of positions. For two particles, for example, the wave function of the system, , is defined in a configuration space of six dimensions. Since the summation of the probabilities referring to events that are mutually exclusive is 1, it follows,

(3)

Once is not an observable quantity, there is a certain freedom of choice of its form. Besides, the solutions of linear equations, like Schrödinger’s equation, may be multiplied by complex numbers, remaining solutions, so that expression (3) turns possible to choice a correct amplitude factor. The point of view of the physical interpretation sustains that the probability is in fact the reflection of an objective property of the "particle", which is that the possible eigenvalues coexist as propentions in a reference class until a macroscopic intervention (a measurement) takes place. Such intervention changes drastically the original reference class. Let us take a system with states and respectively before and after the experimental intervention. It’s clear that the function is somewhat conjectural here, but, for all theorectical purposes, is ever possible to think this function as a set of states reducible to an unique state (the reduction of the "wave packet"). We must consider the set while not especified any function by the apparatus of measurement, in such manner that we have two distinct instances of the reality, one before and other after the observation. Quantum measurements are represented by a collection ø of operators that act upon the phase space of the system under observation. The subindex labels the possible results of measurement. Let us suppose that the system is in the initial state . The probability for a certain state after the measurement is,

(4)

where is the transposed conjugated of , and . Let it be an orthonormal base. So,

(5)

is a quantum measurement. The intervention of the apparatus modifies the state of the system to,

(6)
(7)
(8)
(9)
(10)

Finally, the implementation of symmetries in generalized quantum mechanical coordinates aaaA symmetry in quantum mechanics is a discrete transformation or a group of continuous transformations that let invariant the Hamiltonian (or the Lagrangian) and the canonical commutation relations of the system. may be represented by a unitary operator in the Hilbert space bbbThe vector space of quantum mechanics is a Hilbert space, that is, an orthonormal vector space in which the vector components are complex scalars; the scalar product satisfies for otherwise ; if and are complex scalars, them
;
the space is complete in the norm .
, so that,

for the groundstate of the Hamiltonian ,

In fact, accordingly the von Neumann theorem, a coordinate transformation that corresponds to a symmetry of the Hamiltonian let invariant the canonical commutation relations of the system and (here is the power of the theorem) may always be implemented by an unitary manner in the Hilbert space of the states. So,

where is an operator that defines a motion constant (thereby furnishing good quantum numbers for the states of the system) so that , and is the set of parameters defining the matrix . Of course, as an effect of the macroscopic intervention, shows some classic traces inherited from the apparatus. But quantum mechanics says nothing about de world out of the experiment. In particular, with respect to gravity, an approach by quantum field theory would need 1) an understandable model of gravitation accordingly some quantization algorithm applied to general relativity, which seems little bearing, and 2) an experimental frame able to reproduce the physical conditions under which the hypothetical quantum nature of gravity may come about, such as in a black hole singularity. In fact, one reason to brush aside an experimental program in this way is the difficulty of formulating quantum theory in a cosmological context in which the observers must be part of the system. Although it appears out of the blue, we may suppose there is a real messenger of gravity and imagine a "metaframe" to render gravitation in a familiar figurative language with no a priori concerns whether the messenger and its supersymmetric partner follow Bose or Fermi statistics beneath lab apparatus. This is my proposal: a supersymmetric meta-field theory on gravity. So, I define meta-field theory as a theory that introduces a supersymmetric metaframe to describe fields as sets of particular transformations between two types of entities, the supersymmetric partners in focus.

It seems to be the best moment to discuss in short semantic features of the explaining representation. An accurate investigation in supergravity, as well in general field theory, is already hard enough to aggregate more difficulties caused by conceptual mistakes. I will try, so much as possible, to refine the presentation of the formalism, provided is a great bind in SUSY literature the time lost with the nearly always confused roll of notations and conventions.

Recollection on general field theory

The classical approach of a system with symmetry is well known in elementary field theory. So, let us consider three complex fields changing beneath a type symmetry accordingly the relations,

  1. ;

  2. ;

  3. .

The dynamics of these fields is assumed to be controlled by the Lagrangian density

(11)

The variation of the Lagrangian is given by,

(12)

From this, we define a density and a current (see the Noether theorem at Chapter 2),

(13)

The derivatives of de Lagrangian are

(14)
(15)
(16)
(17)
(18)

Thus, the total density, that is, the density calculated and summed for all three fields is,

(19)
(20)

The charge in classical level is defined by the integral of for the space-volumn,

(21)

The transition to the formalism of quantized fields redefines the fields as operators in Hilbert space, so that the charge is now,

(22)

must be the operator which the eigenvalues are the values of the individual charges of the fields. All the operators (, , , ) must act on vectors of state constructed from the vacuum by the action of creation operators in terms of which the fields are expanded,

(23)
(24)
(25)

This brief review gives a resume of the basic ideas in general field theory, and it is enough for my pourposes in present essay.

On the invariance of the action in the rigid supersymmetry

I want to exemplify how do we implement a SUSY transformation that lets invariant the action in an essentially temporal and unidimensional conjectured model. Having the usual approaching inspired in Nieuwenhuizen’s work, we generally start from a real bosonic field and a real fermionic field , being the customary independent Grassmann variable. In this explanation, I consider as the unique dimension, since in my theory time has precedence, and only susy generators. Grassmanian quantities obey the rule,

from which,

For all we have,

Also, the Hermitian conjugation on any Grassmanian quantities and reads,

So now, let us take the action given by,

to the simple Lagrangian,

The Klein-Gordon action is expressed by the first term, while the Dirac action by the second. We mix fermions and bosons by the SUSY transformations,

The variation of the Lagrangian gives,


Applying , it follows,

The integration of the action gives,

Neglecting the boundary part at and for , the action is invariant under the proposed SUSY transformations.

The axiomatics for a supersymmetric gravity

As once told Mario Bunge, "Mathematical forms say by themselves nothing about material reality, and this is just why they may be used (in combination with semantic ’rules’) to say so much about the external world. The eventual objective content that can be poured into mathematical forms lies entirely in the factual (physical, biological,…) meaning attached ad hoc to the symbols appearing in them, that is, in the semantic ’rules’"  (Bunge, 1979). I think that any lecture or class on advanced physics must begin with a great emphasis in this observation in order to safeguard one to overestimate symbols and representations. The development of gravitorial theory is replete of abstractions evoked by the search of representation means for the supersymmetric phenomenal scenario of gravitation. One of the axioms of the theory transcripted bellow shows this fact.

Axiom

Subset of original primitive base

  • - anti-de Sitter space;

  • - affine space;

  • - Clifford algebra.

Proposition

Given beneath symmetry , there exists an affine space such that for some there is an affinor upon which it holds the tautomorphism , or, .

We consider the tautomorphism . It introduces a particular application of invariance valid for - type objects said "gravitors", the column vectors of the complex space upon which act Clifford algebra matrices. This axiom, presuming all the others of the theory, brings great normative meaning for the manner we must deal with gravitors. We note that such entities were arbitrarily symbolized by the sign aided by ’’ and ’’. Such set of signs, being or not associated to sounds, merely serves to give literal form to the ideas. So, while symbol, do not exists else as a possible tool of thinking within a wide semantic framework. But, ultrapassing the mute symbology and arriving to physics, what are gravitors? They are math constructs that need to be expressed in association with particular sentences in order to gain physical significance. Even so, they are only tools of thinking; they have not concrete existence. However the human understanding makes efforts to build some pictures about, we must to pretend nothing more than a logic consistence between the sentences and the transempiric reality they try to explain. Playing the role of tools for the rational activity, those constructs pitch a yarn concerning the world of the exterior things. And the more the "tale"  is good, the more is its likelihood; the more we dismissed the rulers and the clocks, the more we persuade ourselves that only the physicists bestow sense to physics itself. Gravitors and all entities of contemporary physics give us the fugacious impression of naturalness when we divagate on the unfathomable secrets of the Universe.

In the ambit of this ontological discussion, I would like to point out the very well known inquiry about the existence of the graviton. Tony Rothman and Stephen Boughn compiled an outstanding approach on this subject in the article "Can gravitons be detected?"  based on Freeman Dyson’s questioning whether it would be conceivable any experiment to detect a graviton in our quadridimensional Universe. Albeit the incredible frailness of the gravitational interaction, there must be possible to perform a device of detection, a plain of search, otherwise graviton becomes a metaphysical object and we will be forced to decide if it is acceptable to treat as physical this metaphysical entity. The authors say in the introduction of the paper that "For both physical and philosophical reasons the matter turns out to be not entirely trivial, and both considerations require that the rules of the game be defined at the outset. We concede at once that there appear to be no fundamental laws disallowing the detection of a graviton, and so we take the approach of designing thought experiments that might be able to detect one"  (Rothman & Boughn, 2006). I recommend the paper to the reader, but on the face of what I said in last paragraph the only question that makes sense is "what part of the experience may be assigned to the presence of gravitons?"  or "what part of the experience is just representable by the construct ’graviton’?". We see that much more ingeniousness is required than conventional empirism.

   Chapter 1                                                                      

Representation: SUSY change partners in Wick-rotation

  Recalling basic principles, in general relativity a vector is an oriented object perfectly superposed at a particular point in spacetime, in such manner that to compare two vectors at distinct points it is necessary to carry one over to the other in a certain way. Of great physical interest is the transport of a vector along a path with no turning or stretching, the so-called "parallel transport". This concept was introduced still in the early XX century by the Italian mathematician Tulio Levi-Civita, as an outcome of the growing need of formal simplifications. An important feature of this transport in curved spacetimes is that it depends on the path taken. In other words, a curved spacetime is one that, by definition, has different parallel transports between two points according to the possible paths. It means that if a point-particle, moving in the curved spacetime, is affected only by gravity, its velocity vector is parallel transported along the peculiar line it traces in the curved spacetime, that is to say, along the "geodesic".

Metric

The representative spacetime in general relativity is a 4-dimensional Riemannian manifold (in fact, a pseudo-Riemannian manifold). It was Riemann who established the generic concept of multiplicity (Mannigfaltigkeit), later "manifold" in English version. A generic manifold is a topological space which is locally Cartesian, such that the calculus on it presupposes the existence of a smooth coordinate system at some neighborhood of each point. A Riemannian manifold is the pair where is a symmetric positive-definite bilinear form, a tensor field on called "metric" or "distance function", which satisfies the following axioms:

  • for all ;

  • if , então ;

  • for all .

Weak gravitational waves

Now, let us begin with gravitational radiation. Based on the theory of electromagnetic fields, it was proposed that, far from the source, gravitational waves must be described by a solution of a linear approximation of Einstein’s field equations, so that the metric would be,

(26)

where is a flat spacetime metric, and is a correction such that . Applying harmonic coordinates, for , we get,

(27)

and so Einstein’s field equations for a vacuum take the form of the wave equation, that is,

(28)

Known the polarizations of the transversal wave, we have,

(29)

and

(30)

Being , we establish the notation,

(31)

and

(32)

In resume, for a non-massive theory, we assume that a weak perturbation travels across a flat background , far from field sources, as a plane wave in such manner that the metric is . If we introduce a massive graviton, a direct consequence is the speed of propagation depending on the frequency,

(33)

and we fall into a bimetric theory to determine the six polarization modes of the gravitational wave for a massive graviton. The study of such modes is beyond the scope of this work; for all purposes, with the above speed expression in mind, I’ll focalize mainly gravitor’s formal structure associated to the wave propagation.

Gravitons and gravitinos

Gravitons, the hypothetical gauge bosons, messenger particles of gravity, are thought here as being associated to affinors deduced of an adS (anti-de Sitter) specific application. In the same way are thought not less hypothetical gravitinos, their supersymmetric fermionic gauge partners, commonly represented by Majorana vector spinor fields of spin 3/2. Gravitinos are expected to be present in all local supersymmetric models, which are regarded as the more natural extensions of the standard model of high energy physics. In the framework of minimal supergravity, the gravitino mass is, by construction, expected to lie around the electroweak scale, that is, in the range.

Supersymmetry describes fermions and bosons in a unified way as partners of a supermultiplet. Such multiplets necessarily have a decomposition in terms of boson and fermion states of different spins. So, the supergravity multiplet consists of the graviton and its superpartner, the gravitino (in fact, the gravitino multiplet contains (1; 3/2) and (-3/2; -1), that is a gravitino and a gauge boson; on the other hand, the graviton multiplet includes (3/2; 2) and (-2; -3/2), corresponding to the graviton and the gravitino). Really the graviton spin 2 derives from the rank 2 of the metric tensor which describes the gravitational field. At first look, gravitino could have spin 5/2 as often as 3/2, but the advantage to choose spin 3/2 is the absence of the goldstino in supersymmetry breaking theories.

After formal considerations (Serpa, 2002), resulting components for gravitons and gravitinos, accordingly gravitorial representation, are respectively:

(34)
(35)

where

(36)

and

(37)

with the customary Pauli matrices,

(38)

We suppose the states of graviton are "mirrored" in states of gravitino, always in pairs, beneath adS Clifford algebra (gravitinos correspond to transformations of graviton affinors), so that,

(39)

ou

(40)

being and its inverse elements of the algebra. These transformations are of type . We suppose too, remembering that nothing compel us to use anticommuting quantities to formulate symmetry between bosons and fermions, a superspace with four gravitorial coordinates in addition to the ordinary . Also, as schematized in figure 2, there is a translation in spacetime when a particle (graviton) transforms into its superpartner (gravitino) and then back into the original particle (graviton) (this is a fundamental property of supersymmetry). Such translation probably occurs in a jump between different metrics, which would be a reason of the difficulty to prove the existence of supergravity. The content of equations (39) and (40) is in fact similar to the content of the supersymmetric transformations generated by quantum operators that transmute fermionic states in bosonic ones and vice-versa:

(41)

The column-vectors above must obey a selective criterion derived from the theory in order to preserve solely the best logical representatives. This criterion is divided in three rules (the "iron trigon"):
1. Column vectors are linked to the adS Clifford Algebra,

(42)
(43)
(44)
(45)
(46)

by the tautomorphism , from which

(47)

2. Pair-wise selected column-vectors are physically representative iff each vector is a Wick-rotation of the other;
3. Column-vectors are elements of the semi-Abelian group (special group of gravitors), deduced from the 16-second matrices of the vectors,

where the four orange matrices integrate the column-generators of the group. The vectors , which indeed form the group , compose by themselves the group with a "feedback" property: . A program in R-language was made to test group elements and their creation by the generators. For example, with the group operation "" on all ordered ,

(48)

There are several motivations for the use of adS Clifford algebra, but it is enough to stand out that if one chooses a matrix representation for the generators of a simple acting on spinor or gravitor space , so is also generated by the transposed matrixes ( is uniquely defined)cccFor and being components of a vector , ( is a dimensional vector space endowed of a scalar product ), Cartan’s equation says, for a Dirac spinor , .. Mathematical symmetries like this are very advantageous to deal with physical ones and they are generally taken with natural good receptivity. Besides, 1) Clifford algebras usually furnish spinorial representations of rotation groups and 2) supergravity does not exist in de Sitter space (Pilch, Sohnius & Nieuwenhuizen, 1985), two sufficient reasons by which one applies adS Clifford algebra for supergravity with gravitorial affinors. Not so facile, however, is to use and justify physically the Wick-rotation. Its application is in such manner confuse that in many works I was even unable to resolve whether the authors were discussing having in mind Lorentzian or Euclidean signature; indeed, I could not see any clear justification with physical significance to introduce imaginary rotations in those discussions. Excepting the few one can find on Wick-rotation applied to the momentum variable in Green’s functions, any remarkable reference has been done, over all about the possible roles fulfilled by bosons and fermions in Wick-rotation (concerning peculiarly the fermions, this was noted also by Nieuwenhuizen and Waldron)dddP. van Nieuwenhuizen, A. Waldron, "A Continuous Wick Rotation for Spinor Fields and Supersymmetry in Euclidian Space", hep-th/9611043, proceedings of the string conference held at Imperial College, London, 1996.. Even in early quantum physics, the imaginary unit is hollow of physical significance. As an example, concerning Pauli matrixes, we see that physicists like to put them in one-to-one correspondence with orthogonal directions in Euclidean 3-space, expressing their orthogonality by the Grassmannian outer product . Thereby, the product reflects the identity between , as the pseudo-scalar unit for Euclidean 3-space, and a trivector created by the outer product of the orthogonal vectors , and . Everything is accepted tacitly as a mere formal result. Also in twistor theory, no direct physical interpretation is generally assigned to the complex coordinates. By the way, recalling Clifford algebras, we see that a reflection related to a plane orthogonal to is given by (see tautomorphism at Axiom 1) in spinor space, and for time-like we must substitute by to satisfy the imposition of identity for squared reflections, indeed a beautiful feature but much more connected to mathematical modeling than to physical requests (I am trying simply to show where we may be more emphatic about physics). Finally, discussing Higgs mechanism for gravity and considering a Lorentz violating spectrum in a model for non-massive gravitons, scientists are once more laconic about the imaginary frequency at very low momenta. Only in 1977 there appear an interesting work of the French physicist and philosopher of science Jean Émile Charon, Theorie de la Relativité Complexe, in which he proposes a complex quadridimensional riemannian structure to the physical space with a metric,

(49)

so that,

(50)

with

(51)

Charon argues, among other ideas, that only such a complex space turns possible to extend general relativity to quantum field domain and justify the four complex extra components (including time) as a way to assign physical quantities (for example, the action associated to the spin) to the mathematical point of spacetime eeeJ. Charon, "Theorie de la Relativité Complexe", Albin Michel, Paris (1977).. The theory did not gain the merited attention, but it is a fact that Charon gave physical significance to imaginary dimensions.

John G. Taylor, in his good times, ascribed clear physical sense to imaginary quantities when he wrote the third chapter of "The New Physics", entitled "Faster Than Light". Telling us about Einstein’s famous article of 1905, we may read at page 94 of the Spanish version: "…si acelerar una partícula hasta la velocidad de la luz exige una cantidad infinita de energía, acelerarla por encima de este valor requerería una energía imaginaria. Una cantidad imaginaria está formada por el producto de un número real y la raíz cuadrada de menos uno. Aun cuando esta cantidad puede manejarse sin mayor problema como símbolo, en la realidad no resulta posible medirla" (Taylor, 1974). Here, an imaginary physical quantity is one to which there is no sense to apply rules or clocks; is one to which observational operations are not defined. Even so, it is considerably present within the frame that explains the world. Book criticisms apart, few times I saw so much clearness in a simple communication of an idea not so simple.

Not exactly in the same reasoning line, but in a certain way similar to mine, Nieuwenhuizen and Waldron propose "a continuous Wick-rotation for Dirac, Majorana and Weyl spinors from Minkowski spacetime to Euclidean space, which treats fermions on the same footing as bosons" (Nieuwenhuizen & Waldron, 1996). They emphasize that the study do not focuses the Wick-rotation of the momentum variable but a Wick-rotation of the field theory itself. After some observations, they were leaded to suggest for a Dirac spinor the Wick-rotation,

(52)
(53)

in which is the Euclidian Dirac spinor and a diagonal matrix with entries that acts only Wick-rotating time sector (the exponents and are elements of the Euclidean Clifford algebra). Resembling argumentations are applied to Majorana and Weil spinors. In analogous sense, as we may define a dual field and a dual function , so that and constitute a Legendre transformation or, which came to be the same, a duality symmetry,

(54)

we have in gravitorial theory a symmetry,

(55)

for gravitinos or,

(56)

for gravitons. What I will defend here is that Wick-rotation, a particular kind of abstract representation that serves very well to explain isometric transformations with physical significance, puts face-to-face, in a simple manner, the state-object and its qualitative change. It preserves a great mathematical inheritance, as we deal with the so intuitive and powerful idea of rotation group but, at the same time, warrants that "something" is no more the "same thing" from the viewpoint of system’s physics under isometric isomorphism. When we Wick-rotate a bosonic representation, we are bringing a fermionic one but in a unique affine frame because . Graviton and gravitino share that affine space in pairs, as the object and its image.

   Chapter 2                                                                      

Lagrangian density and non-locality: backward to relics and beyond

 

Discovery of an elementary spin 3/2 particle in the laboratory

would be a triumph for supergravity because the only consistent field theory for interacting spin 3/2 fields is supergravity.

Any cosmological theory is suported by the gravitation theory. Gravity is the only relevant force in the scale of galaxy clusters and beyond. The gravitation theory can be constructed in different ways and this is still a source of puzzles, mainly in discussions about quantization of gravity and unification of the forces. In fact, there are three main approaches to relativistic gravity theories:

  • gravity is a property of spacetime itself, the geometry of curved spacetime;

  • gravity is a kind of matter within the spacetime (the relativistic field theory in flat spacetime);

  • gravity is the effect of the direct interaction between ponderable particles.

No matter the choice, it is important to look upon that up to now relativistic gravity has been tested experimentally only in weak field approximation. All the well known relativistic effects as the delay of light signals, the gravitational lensing, the gravitational frequency shift, the pericenter advance, the rotational effects and the gravitational radiation from binary systems may be obtained from any reasonable gravitational theory. Also is important to regard that the shapes of the observable systems at large scales or even at galaxy scale are results of the long time cumulative action of gravity, which points to the relevance of the time in gravity phenomenon. Viewed by this angle, gravity is more a cosmic measure of the matter evolution all along the time. First of all, in gravity theories one must consider that an ordinary rotation of a macroscopic device can be enough to disturb ontological status of a particle by emission of gravitational waves. This fact serves to detach the relevance that may have one of the simplest examples of a symmetry transformation. Besides, we know that orthogonal groups - space entries and time entries - are of great usefulness because they describe Lorentz symmetry in spacetimes having diagonal metric with eigenvalues and eigenvalues (or vice-versa); on the other hand, I am interested in Wick-rotations to represent the global entail between supergravity partners, and so I am not limiting the theory to an extended local supersymmetry with an real internal symmetry.

As I first note, Wick rotations were introduced more like an ingenious math trick to regularize quantum field theories than a formal representation of a physical fact or property. Integrations on meromorphic functions over spacetimes of Minkowski, where the action appears in path integrals as , are frequently faced from Feynman diagrams, and show divergence. Making the action imaginary, thus, becoming the metric Euclidean, the analytic continuation of gains a real negative exponent, forcing path integral to converge. So, the Euclidean (E) transformation may be seen as,

where we understand that , and assume the Wick rotation,

In gravitorial theory, Wick rotation acts to modify the physical status of the field; it does not enter the theory to remove mathematical divergences. Rather than viewing it like a trick for the convergence of the path integral, Wick rotation is treated as a necessary feature of the supersymmetric physics of gravity in the form of an enantiomorphic transformation.

Fig.1- Scheme of two strings fusion as a cobordism. According to the theory, two chords, one fermionic and other bosonic, fuse with one another in order to make a loop of interaction. However I remember being so magnetized by string theories in the early 1990’s, it is not the case to be here exhaustive about this subject. I’ll just handle some introductory features related to the main thesis in view, remitting the reader to the last chapter of this work.

The generators of SUSY are elements of the Clifford Algebra and, at the same time, elements of the orthogonal group that represent Wick-rotations when acting on gravitors; by its turn, the subset of selected gravitors is formed from elements of the group which have the property to obey tautomorphism condition for some component of the algebra , except trivial quadratic-type forms . Since the reasons for a Wick-rotation go far beyond the simple operational commodity of an artifice, I must clarify its introduction with logic consistence. In the early studies about gravitors, the former Lagrangian density that controls the interactions of the fields was, assuming all needed extra-constants included into kets,

(57)

with typical monoms of field products. Further application of the Noether theorem to all degrees of freedom and some theoretical considerations leaded, within a topology , to

(58)

and to

(59)
(60)

with

(61)

Besides that Lagrangian of interactions, I considered another to exhibit a "gauge" field endowed of mass coupled to gravitino, such as,

(62)

from which

(63)

Recalling conventions, a) in some situations it is convenient to express gravitational wave amplitude by the corresponding energy density flow in or other bulk (this is not obligatory since we know the metric of spacetime), and b) relevant physical quantities have units which are powers of mass (for , length is because is a length, etc). The fields and , as coordinates of the whole system, are related to gravitons and gravitinos respectively. The field is the gravitorial gauge inscription of the mass contribution outer to the adS zone (fig. 2) with appearing due to and its coupling to other fields. The field is an auxiliary non-coupled field. Time integrals applied denote strong interference of system’s history on local field inhomogeneities. Equations (59) and (60) show symmetries arising from stream evanescing superpositions as of the Noether current given by,

(64)

Never is overmuch to remind the content of Noether’s theorem. For a system with Lagrangian density of the type , a continuous symmetry of generates an equation of continuity , where and are functionals of , so that is a constant of motion. Accordingly Noether’s theorem, from the point of view of my theory, the fields and their derivatives or integrals are such that current decais to faster than as . This is known as Noether’s theorem for classic fields. We’ll verify this property ahead (exercise 1). As pointed out by O’Raifeartaigh, "The Noether theorem gives the general relationship between symmetries and conservation laws. […] Thus to every symmetry there corresponds a conserved quantity and conversely" (O’Raifeartaigh, 1997). This is clearly understood from (59) and (60), provided we focus time translations; Hamiltonian approach of system’s dynamics also points out this evidence. Let being given by,

(65)

Hamiltonian density is defined as

(66)

Now, is precisely the content between brackets when we apply Noether’s theorem in Lagrangians like (62). Let us take . For we get:

(67)
(68)

The same reasoning applied to the system of equations originated by furnishes the symmetries (59) and (60). The conserved current there implies an inverse relation between and its Wick-rotation. And more: a non-local algebraic structure . Modern physicists do not like so much to deal with non-locality. There is no doubt it is a very strange attitude, since non-local phenomenology is very well-known in classical physics. In real space, incompressible turbulence is non-local as sound speed is infinite in incompressible fluids. In Fourier space, the equation also points to non-locality; on the other hand, the shell-to-shell energy transfer is local. The truth is a dialectical combination of both features. I can accept that locality is a logic necessity of the so-local human understanding. It makes possible the work of practical experimenters in order to obtain reproducible results. What I can not accept is the anthropocentric doctrine that build the world image by laboratory limitations. In present theory, supersymmetric transformations between gravitons and gravitinos occur beneath extreme and unstable conditions at the vicinity of a strong gravitational source. The vicinity is described by a very confined adS manifold where a tangent mass retention field , the filtrino, acts to intermediate the mass exchange with Minkowsky space. The instability of adS world models (Coleman & Luccia, 1980) is precisely the physical "spice" for a certain virtual mechanism of equilibrium with evasion of gravitons from the neighborhood of gravitational sources by adS - Wick-rotations of gravitinos (remember the "iron trigon"). This is quite consistent with decay law of gravitino’s influence, since adS zone tends to remain next to the source. As strong gravitational sources are in general very far, it would be a mistake to disregard historical roots in present ontological status of a graviton, whose lifetime is presumed very large but unknown. At the laboratory, the maximum one may suppose, if such is possible and after modest precautions to isolate experiments, is a situation in which similar initial conditions could be reproduced; thus, we gain,

(69)

as well as

(70)

and the reduced total Lagrangian,

(71)

But this sounds more as a Gedankenexperiment than a realistic one. The reality is, unfortunately, implacable. The weakness of gravitational force is extraordinary, and physicists are not able to directly verify the existence of gravitational waves, much less to simulate initial conditions for gravitorial transformations; gravitino interactions are of the rate of Newton’s extremely small constant . What makes time to be prevalent here is that gravity tends to manifest its effects only after long time of cumulative events; just because this feature we do not perceive gravitons or gravitational waves. The locality of human experience is inadequate to reproduce what occurred so long ago in extreme mass clustering conditions. Gravity is not a common force but a force that represents time strength in cosmic evolution at large scales. We need to admit the eventual impossibility of complete unification; we need to treat gravity contrarily to the usual misconception of another ordinary quantizable physical manifestation well localized in spacetime. The sole one can wait for is unification of rational thinking and, in a certain way, a lato sensu unification of applied general tools, perhaps the only kind we can expect to see beneath the logical objections pointed out by Ishan and colleagues. Transparence of gravitation with respect to all other forces seems to confirm this point of view.

In addition, it is more logic to imagine that gravitation operates by a timelike field evolution (non-local), so that we can actually measure only residual quantities associated to spacelike field evolution (local). To do this, we must find an appropriate "gauge", which gives a local response after incorporate the inheritance of the whole process (next section). Now, consider an operator acting on and such that

(72)

and

(73)

From (60) we know that holds the relation,

(74)

To connect both actions based upon figure 2, we must remind that gravitino local loss of mass under filtrino field rebates the mass attached to graviton, from which,

(75)

where is the retaining mass due to field, . Thus,

(76)

Operator acts to ad a kinematical relation between graviton and gravitino if we consider massive gravitons in Wick-rotation at adS zone.

From "non-local gauge" to "local coupling gauge"

 

The final goal of supersymmetric meta-field theory is to provide

a consistent representation of gravity and at the same time to explain gravitation without any interference of the observer on the observed facts.

The introduction of a non-local inheritance factor as a device to include far-off interferences that may exist in certain phenomena is due to Vito Volterra, at the early twenty century, with his famous equation to the growth problem,

(77)

Not much has been made to make good use of this very logical idea in the so-called "hereditary mechanics", a name coined by Emile Picard (to him, modern generations owe a famous work in three volumes, the Traité d’Analyse, 1891-96). Accordingly this great French mathematician, in all the study of classical mechanics "the laws which express our ideas on motion have been condensed into differential equations, that is to say, relations between variables and their derivatives.

Fig. 2 - Diagram of adS zone boundary interchange. Gravitinos lose mass smoothly to pass to Friedmann-Lemaître-Robertson-Walker (FLRW) space as gravitons . On the inverse way, gravitons gain mass to cross back to gravitational source vicinity as gravitinos . From the viewpoint of a quantum field theory, with a vanishing cosmological constant we have a supersymmetry breakdown scale as a vacuum expectation value of auxiliary field just placed at the boundary. The width of adS zone depends on source intensity. The field plays a similar part of the Higgs mechanism. I adopted a spherical Lemaître-Tolman (LT) interface between adS and FLRW zones for further considerations. The fundamental idea is to apply the Swiss cheese inhomogeneous model to the Universe until some scale limit to be determined. Several LT bubbles may contain a central massive adS zone that determines the intensity and diameter of the filtrino field boundary. The final hypothesis is that the gravitino survives mainly within the limits of LT zones, that is, it is most likely to find gravitinos inside LT zones.

We must not forget that we have, in fact, formulated a principle of nonheredity, when we suppose that the future of a system depends at a given moment only on its actual state, or in a more general manner, if we regard the forces as depending also on velocities, that the future depends on the actual state and the infinitely neighboring state which precedes. This is a restrictive hypothesis and one that, in appearance at least, is contradicted by the facts. Examples are numerous where the future of a system seems to depend upon former states. Here we have heredity. In some complex cases one sees that it is necessary, perhaps, to abandon differential equations and consider functional equations in which there appear integrals taken from a distant time to the present, integrals which will be, in fact, this hereditary part" (Picard, 1907). An absolute gravitation local theory seems to me so absurd as the old idea of absolute space. There is no sense regarding the sky without its evolutional non-local essence. Although the disregard of inheritance factors is in part consequence of an exaggeration of simplification, non-locality phobia in quantum field theory is very related with the fear to lose Lorentz and gauge invariance, both well preserved with local variables. We must remember that cubic string field theory (analogous to Chern-Simons gauge theory) fffChern-Simons gauge theory introduces a gauge field