Due to their small sizes, compactness, low cost, high sensitivity, high resolution and extraordinary physical properties, nanoresonators have attracted a widespread attention from the scientific community. It is required that the nanoresonators can operate at desired but adjustable resonant frequencies. The impact of NiTi film thickness and dimensions on the first three consecutive resonant frequencies of the cantilever nanobeam is examined. In addition, developed theoretical model can be used as a simple guide for further design of novel tunable cantilever nanoresonators with thin films that cover only partially the entire cantilever length.
Keywords: nanoresonator, resonant frequency, thin film, smart memory alloys, NiTi film. In this section we will try to summarize these problems. An important issue has to be underlined: even if there are many problems, the reaction of scientific community is not uniform. In a very simple scheme we can summarize the guide lines. Many people will agree that modern physics is based on two main pillars: GR and Quantum Field Theory. Each of these two theories has been very successful in its own arena of physical phenomena: GR in describing gravitating systems and non-inertial frames from a classical point of view on large enough scales, and Quantum Field Theory at high energy or small scale regimes where a classical description breaks down.
However, Quantum Field Theory assumes that spacetime is flat and even its extensions, such as Quantum Field Theory in curved spacetime, consider spacetime as a rigid arena inhabited by quantum fields. GR, on the other hand, does not take into account the quantum nature of matter. Therefore, it comes naturally to ask what happens if a strong gravitational field is present at quantum scales. How do quantum fields behave in the presence of gravity? To what extent are these amazing theories compatible? Let us try to pose the problem more rigorously. Firstly, what needs to be clarified is that there is no final proof that gravity should have some quantum representation at high energies or small scales, or even that it will retain its nature as an interaction.
The gravitational interaction is so weak compared with other interactions that the characteristic scale under which one would expect to experience non-classical effects relevant to gravity, the Planck scale, is cm. Such a scale is not of course accessible by any current experiment and it is doubtful whether it will ever be accessible to future experiments either. Let us list some of the most prominent ones here and leave the discussion about how to address them for the next subsection.
Curiosity is probably the motivation leading scientific research. From this perspective it would be at least unusual if the gravity research community was so easily willing to abandon any attempt to describe the regime where both quantum and gravitational effects are important.
The fact that the Planck scale seems currently experimentally inaccessible does not, in any way, imply that it is physically irrelevant. On the contrary, one can easily name some very important open issues of contemporary physics that are related to the Planck scale. A particular example is the Big Bang scenario in which the Universe inevitably goes through an era in which its dimensions are smaller than the Planck scale Planck era. On the other hand, spacetime in GR is a continuum and so in principle all scales are relevant.
From this perspective, in order to derive conclusions about the nature of spacetime one has to answer the question of what happens on very small and very large scales. UV scales: The Quantum Gravity Problem One of the main challenges of modern physics is to construct a theory able to describe the fundamental interactions of nature as different aspects of the same theoretical construct. This goal has led, in the past decades, to the formulation of several unification schemes which, inter alia, attempt to describe gravity by putting it on the same footing as the other interactions.
All these schemes try to describe the fundamental fields in terms of the conceptual apparatus of Quantum Mechanics. This is based on the fact that the states of a physical system are described by vectors in a Hilbert space H and the physical fields are represented by linear operators defined on domains of H. Until now, any attempt to incorporate gravity in this scheme has either failed or been unsatisfactory.
The main conceptual problem is that the gravitational field describes simultaneously the gravitational degrees of freedom and the background spacetime in which these degrees of freedom live. Owing to the difficulties of building a complete theory unifying interactions and particles, during the last decades the two fundamental theories of modern physics, GR and Quantum Mechanics, have been critically re-analyzed. On the one hand, one assumes that the matter fields bosons and fermions come out from superstructures e. Higgs bosons or superstrings that, undergoing certain phase transitions, have generated the known particles.
On the other hand, it is assumed that the geometry e. This interaction necessarily modifies the standard gravitational theory, that is, the Lagrangian of gravity plus the effective fields is modified with respect to the HilbertEinstein one, and this fact can directly lead to the ETGs. From the point of view of cosmology, the modifications of standard gravity provide inflationary scenarios of interest. In any case, a condition that must be satisfied in order for such theories to be physically acceptable is that GR is recovered in the low-energy limit.
Although remarkable conceptual progress has been made following the introduction of generalized gravitational theories, at the same time the mathematical difficulties have increased. The corrections introduced into the Lagrangian augment the intrinsic non-linearity of the Einstein equations, making them more difficult to study because differential equations of higher order than second are often obtained and because it is impossible to separate the geometric from the matter degrees of freedom.
In order to overcome these difficulties and simplify the equations of motion, one often looks for symmetries of the Lagrangian and identifies conserved quantities which allow exact solutions of dynamics to be discovered. The key step in the implementation of this program consists of passing from the Lagrangian of the relevant fields to a point- like Lagrangian or, in other words, in going from a system with an infinite number of degrees of freedom to one with a finite number of degrees of freedom. Fortunately, this is feasible in cosmology because most models of physical interest are spatially homogeneous Bianchi models and the observed Universe is spatially homogeneous and isotropic to a high degree FriedmannLemaitreRobertsonWalker FLRW models.
The need for a quantum theory of gravity was recognized at the end of the s, when physicist tried for the first time to treat all interactions at a fundamental level and to describe them in terms of Quantum Field Theory. Naturally, the. In the first approach applied to Electromagnetism, one considers the electric and magnetic fields satisfying the Heisenberg Uncertainty Principle and the quantum states are gauge-invariant functionals generated by the vector potential defined on three-surfaces of constant time.
These procedures are equivalent in Electro-magnetism. The former allows for a well-defined measure, whereas the latter is more convenient for perturbative calculations such as, for example, the computation of the S-matrix in Quantum electrodynamics QED. These methods have been applied also to GR, but many difficulties arise in this case due to the fact that Einsteins theory cannot be formulated in terms of a quantum field theory on a fixed Minkowski background.
To be more specific, in GR the geometry of the background spacetime cannot be given a priori: spacetime is the dynamical variable itself. In order to introduce the notions of causality, time, and evolution, one must first solve the equations of motion and then build the spacetime. For example, in order to know if particular initial conditions will give rise to a black hole, it is necessary to fully evolve them by solving the Einstein equations. Then, taking into account the causal structure of the obtained solution, one has to study the asymptotic metric at future null infinity in order to assess whether it is related, in the causal past, with those initial conditions.
This problem becomes intractable at the quantum level. Due to the Uncertainty Principle, in non-relativistic Quantum Mechanics, particles do not move along well-defined trajectories and one can only calculate the probability amplitude t , x that a measurement at time t detects a particle around the spatial point x.
Similarly, in Quantum Gravity, the evolution of an initial state does not provide a specific spacetime. In the absence of a spacetime, how is it possible to introduce basic concepts such as causality, time, elements of the scattering matrix, or black holes? The canonical and covariant approaches provide different answers to these questions. The canonical approach is based on the Hamiltonian formulation of GR and aims at using the canonical quantization procedure. The canonical commutation relations are the same that lead to the Uncertainty Principle; in fact, the commutation of certain operators on a spatial three- manifold of constant time is imposed, and this three-manifold is fixed in order to preserve the notion of causality.
In the limit of asymptotically flat spacetime, the motion generated by the Hamiltonian must be interpreted as temporal evolution in other words, when the background becomes the Minkowski spacetime, the Hamiltonian operator assumes again its role as the generator of translations. The canonical approach preserves the geometric features of GR without the need to introduce perturbative methods . The covariant approach, instead, employs Quantum Field Theory concepts and methods. The basic idea is that the problems mentioned above can be easily circumvented by splitting the metric g into a kinematical part usually flat and a dynamical part h , as in.
The geometry of the background is given by the flat metric tensor and is the same as in Special Relativity and ordinary Quantum Field Theory, which allows one to define the concepts of causality, time, and scattering. The quantization procedure is then applied to the dynamical field, considered as a small deviation of the metric from the Minkowski background metric. Quanta are discovered to be particles with spin two, called gravitons, which propagate in flat spacetime and are defined by h.
Substituting the metric g into the HilbertEinstein action, it follows that the Lagrangian of the gravitational sector contains a sum whose terms represent, at different orders of approximation, the interaction of gravitons and, eventually, terms describing mattergraviton interaction if matter is present. Such terms are analyzed by using the standard techniques of perturbative Quantum Field Theory. These quantization programs were both pursued during the s and s.
In the canonical approach, Arnowitt et al. In this Hamiltonian formalism, the canonical variables are the three-metric on the spatial submanifolds obtained by foliating the four-dimensional manifold note that this foliation is arbitrary. The Einstein equations give constraints between the three- metrics and their conjugate momenta and the evolution equation for these fields, known as the WheelerDeWitt WDW equation. In this way, GR is interpreted as the dynamical theory of the three-geometries geometrodynamics.
The main difficulties arising from this approach are that the quantum equations involve products of operators defined at the same spacetime point and, in addition, they entail the construction of distributions whose physical meaning is unclear. In any case, the main problem is the absence of a Hilbert space of states and, as consequence, a probabilistic interpretation of the quantities calculated is missing.
The covariant quantization approach is closer to the known physics of particles and fields in the sense that it has been possible to extend the perturbative methods of QED to gravitation. This has allowed the analysis of the mutual interaction between gravitons and of the mattergraviton interactions. The formulation of Feynman rules for gravitons and the demonstration that the theory might be unitary at every order of the expansion was achieved by DeWitt .
Further progress was achieved with YangMills theories, which describe the strong, weak, and electromagnetic interactions of quarks and leptons by means of symmetries. Such theories are renormalizable because it is possible to give the fermions a mass through the mechanism of Spontaneous Symmetry Breaking. Then, it is natural to attempt to consider gravitation as a YangMills theory in the covariant perturbation approach and check whether it is renormalizable. Due to the non- renormalizability of gravity at different orders, its validity is restricted only to the low-energy domain, i.
This implies that the full unknown theory of gravity has to be invoked near or at the Planck era and that, sufficiently far from the Planck scale, GR and its first loop corrections describe the gravitational interactions. In this context, it makes sense to add higher order terms to the HilbertEinstein action as we will do in the second part of this Report.
Besides, if the free parameters are chosen appropriately, the theory has a better ultraviolet behavior and is asymptotically free. Nevertheless, the Hamiltonian of these theories is not bounded from below and they are unstable. In particular, unitarity is violated and probability is not conserved. An alternative approach to the search for a theory of Quantum Gravity comes from the study of the Electroweak interaction.
In this approach, gravity is treated neglecting the other fundamental interactions. The unification of the Electromagnetic and the weak interactions suggests that it might be possible to obtain a consistent theory when gravitation is coupled to some kind of matter. This is the basic idea of Supergravity . In this class of theories, the divergences due to the bosons gravitons in this case are canceled exactly by those due to the fermions, leading to a renormalized theory of gravity. Unfortunately, this scheme works only at the two-loop level and for mattergravity couplings.
The Hamiltonian is positive-definite and the theory turns out to be unitary. But, including higher order loops, the infinities re-appear, destroying the renormalizability of the theory. Perturbative methods are also used in String Theories. In this case, the approach is completely different from the previous one because the concept of particle is replaced by that of an extended object, the fundamental string.
The usual physical particles, including the spin two graviton, correspond to excitations of the string. The theory has only one free parameter the string tension and the interaction couplings are determined uniquely. It follows that string theory contains all fundamental physics and it is therefore considered as a candidate for the Theory of Everything. String Theory seems to be unitary and the perturbative series converges implying finite terms.
This property follows from the fact that strings are intrinsically extended objects, so that ultraviolet divergencies coming from small scales or from large transfer impulses, are naturally cured. In other words, the natural cutoff is given by the string length, which is of Planck size lPl.
At scales larger than lPl , the effective string action can be rewritten in terms of non-massive vibrational modes, i. This eventually leads to an effective theory of gravity non-minimally coupled with scalar fields, the so-called dilaton fields. To conclude, let us summarize the previous considerations:. In the quantization program for gravity, two approaches have been used: the covariant approach and the perturbative approach.
They do not lead to a definitive theory of Quantum Gravity. In the low-energy regime with respect to the Planck energy at large scales, GR can be generalized by introducing into the HilbertEinstein action terms of higher order in the curvature invariants and non-minimal couplings between matter and gravity. These lead, at the one-loop level, to a consistent and renormalizable theory. A part the lack of final theory, the Quantum Gravity Problem already contains some issues and shortcomings which could be already addresses by the today physics. We will summarize them in the forthcoming section.
Issues and shortcomings in Quantum Gravity Considered the status of art, are some predictions of Quantum Gravity already available? Can remnants of Planck scale be detected at lower energy couplings and masses? As it is well known, only a fine-tuned combination of the low-energy constants leads to an observable Universe like ours. It would thus appear strange if a fundamental theory possessed just the right constants to achieve this.
Hogan  has argued that Grand Unified Theories constrain relations among parameters, but leave enough freedom for a selection. In particular, he suggests that one coupling constant and two light fermion masses are not fixed by the symmetries of the fundamental theory. The cosmological constant, for example, must not be much bigger than the presently observed value, because otherwise the Universe would expand too fast to allow the formation of galaxies.
The Universe is, however, too special to be explainable on purely anthropic grounds.
We know that the maximal entropy would be reached if all the matter in the observable Universe were collected into a single giant black hole. This entropy would be about , which is exceedingly more than the observed entropy of exp about The probability for our Universe would then be about. From the Anthropic Principle alone one would exp not need such a special Universe. As for the cosmological constant, for example, one could imagine its calculation from a.
At the one-loop level it is sufficient to renormalize only the effective constants Geff and eff which, at low energy, reduce to Newtons constant GN and the cosmological constant. The connection to low energies may nonetheless be non- unique due to the existence of many different possible vacua. Taking the presently observed value for , one can construct a mass according to 1 h 2 15 MeV, G. The observed value of could thus emerge together with medium-size particle mass scales.
Since fundamental theories are expected to contain only one dimensionfull parameter, low-energy constants emerge from fundamental quantum fields. An important example in string theory, is the dilaton field from which one can calculate the gravitational constant. In order that these fields mimic physical constants, two conditions have to be satisfied. First, decoherence must be effective in order to guarantee a classical behavior of the field.
Second, this classical field must then be approximately constant in large-enough spacetime regions, within the limits given by experimental data. The field may still vary over large times or large spatial regions and thus mimic a time- or space-varying constant. The last word on any physical theory has to be spoken by experiments observations. Apart from the possible determination of low-energy constants and their dependence on space and time, what could be the main tests for Quantum Gravity? Black-hole evaporation: A key test would be the final evaporation phase of a black hole.
To this end, it would be useful to observe signatures of primordial black holes. These objects are forming not at the end of stellar collapse, but they can result from strong density perturbations in the early Universe. In the context of inflation, their initial mass can be as small as 1 g. Primordial black holes with initial masses of about 5 g would evaporate at the present age of the Universe. Unfortunately, no such object has yet been observed.
Especially promising may be models of inflationary cosmology acting at different scales . Cosmology: Quantum aspects of gravitational field may be observed in the anisotropy spectrum of the cosmic microwave background. First, future experiments may be able to observe the contribution of the gravitons generated in the early Universe. This important effect was already emphasized in . The production of gravitons by the cosmological evolution would be an effect of Quantum Gravity. Second, quantum-gravitational correction terms from the WheelerDeWitt equation or its generalization in loop quantum cosmology may leave their impact on the anisotropy spectrum.
Third, a discreteness in the inflationary perturbations could manifest itself in the spectrum . Discreteness of space and time: Both in String Theory and Quantum Gravity there are hints of a discrete structure of spacetime.
This quantum foam could be seen through the observation of effects violating local Lorentz invariance , for example, in the dispersion relation of the electromagnetic waves coming from gamma-ray bursts. It has even be suggested that spacetime fluctuations could be seen in atomic interferometry .
However, there exist severe observational constraints . Signatures of higher dimensions: An important feature of String Theory is the existence of additional spacetime dimensions. It is also imaginable that they cause observable deviations from the standard cosmological scenario . Some of these features are discussed in detail in . Of course, there may be other possibilities which are not yet known and which could offer great surprises. It is, for example, imaginable that a fundamental theory of Quantum Gravity is intrinsically non-linear [51,52].
This is in contrast to most currently studied theories of Quantum Gravity, which are linear. Quantum Gravity has been studied since the end of the s. No doubt, much progress has been made since then.
The final goal has not yet been reached. The belief expressed here is that a consistent and experimentally successful theory of Quantum Gravity will be available in the future. However, it may still take a while before this time is reached. In any case, ETGs could constitute a serious approach in this direction. IR scales: Dark energy and dark matter The revision of standard early cosmological scenarios leading to inflation implies that a new approach is necessary also at later epochs: ETGs could play a fundamental role also in this context.
In fact, the increasing bulk of data accumulated over the past few years has paved the way for a new cosmological model usually referred to as the Concordance Model or Cold Dark Matter CDM model. The Hubble diagram of type Ia supernovae hereafter SNeIa measured by both the Supernova Cosmology Project [53,54] and the High-z Team [55,56] up to redshift z 1, was the first piece of evidence that the Universe is currently undergoing a phase of accelerated expansion. When combined with the constraints on the matter density parameter M , these data indicate that the Universe is dominated by an unclustered fluid with negative pressure commonly referred to as dark energy, which drives the accelerated expansion.
This picture has been further strengthened by the recent precise measurements of the CMB spectrum by the WMAP satellite experiment , and by the extension of the SNeIa Hubble diagram to redshifts larger than one . An overwhelming number of papers appeared following these observational evidences, which present a large variety of models attempting to explain the cosmic acceleration. The simplest explanation would be the well known cosmological constant .
As a tentative solution, many authors have replaced the cosmological constant with a scalar field rolling slowly down a flat section of a potential V and giving rise to the models known as quintessence [64,65]. Albeit successful in fitting the data with many models, the quintessence approach to dark energy is still plagued by the coincidence problem since the dark energy and dark matter densities evolve differently and reach comparable values only during a very short time of the history of the Universe, coinciding in order of magnitude right at the present era.
In other words, the quintessence dark energy is tracking matter and evolves in the same way for a long time; at late times, somehow it changes its behavior and no longer tracks the dark matter but begins to dominate in the fashion of a dynamical cosmological constant. This is the coincidence problem of quintessence. Furthermore, the origin of this quintessence scalar field is mysterious, although various usually rather exotic models have been proposed, leaving a great deal of uncertainty on the choice of the scalar field potential V necessary to achieve the late-time acceleration of the Universe.
The subtle and elusive nature of dark energy has led many authors to look for a completely different explanation of the acceleration behavior of the cosmos without introducing exotic components. To this end, it is worth stressing that the present-day cosmic acceleration only requires a negative pressure component that comes to dominate the dynamics late in the matter era, but does not tell us anything about the nature and the number of the cosmic fluids advocated to fill the Universe.
This consideration suggests that it could be possible to explain the accelerated expansion with a single cosmic fluid characterized by an equation of state causing it to act like dark matter at high densities, while behaving as dark energy at low densities. An attractive feature of these models, usually referred to as Unified Dark Energy UDE or Unified Dark Matter UDM models, is that the coincidence problem is solved naturally, at least at the phenomenological level, with such an approach [66,67].
Examples are the generalized Chaplygin gas , the tachyon field , and condensate cosmology . A different class of UDE models with a single fluid has been proposed [71, 72]: its energy density scales with the redshift z in such a way that a radiation-dominated era, followed by a matter era and then by an accelerating phase can be naturally achieved. These models are extremely versatile since they can be interpreted both in the framework of UDE or as two-fluid scenarios containing dark matter and scalar field dark energy.
A characteristic feature of this approach is that a generalized equation of state can always be obtained and the fit to the observational data can be attempted. However, such models explain the phenomenology but cannot be addressed to some fundamental physics. There is another, different, way to approach the problem of the cosmic acceleration. As stressed in , it is possible that the observed acceleration is not the manifestation of yet another ingredient of the cosmic pie, but rather the first signal of a breakdown, in the infrared limit, of the laws of gravitation as we understand them.
From this point of view, it is tempting to modify the EinsteinFriedmann equations to see whether it is still possible to fit the astrophysical data with models containing only standard matter without exotic fluids. Within the same conceptual framework, it is possible to find alternative schemes in which a quintessential behavior is obtained by incorporating effective models coming from fundamental physics and giving rise to generalized or higher order gravity actions  see Refs.
For instance, a cosmological constant may be recovered as a consequence of a non-vanishing torsion field, leading to a model consistent with both the SNeIa Hubble diagram and observations of the SunyaevZeldovich effect in galaxy clusters . SNeIa data could also be efficiently fitted by including in the gravitational sector higher order curvature invariants [82,83,77,84]. These alternative models provide naturally a cosmological component with negative pressure originating in the geometry of the Universe, thus overcoming the problematic nature of quintessence scalar fields.
Cosmological models coming from ETGs are in the track of this philosophy. The variety of cosmological models which are phenomenologically viable candidates to explain the observed accelerated expansion is clear from this short review. This overabundance signals that only a limited number of cosmological tests are available to discriminate between competing theories, and it is clear that there is a high degeneracy of models.
Let us remark that both the SNeIa Hubble diagram and the angular-sizeredshift relation of compact radio sources  are distance-based probes of the cosmological model and, therefore, systematic errors and biases could be iterated. With this point in mind, it is interesting to search for tests based on time-dependent observables.
For example, one can take into account the lookback time to distant objects since this quantity can discriminate between different cosmological models. The lookback time is observationally estimated as the difference between the present-day age of the Universe and the age of a given object at redshift z. This estimate is possible if the object is a galaxy observed in more than one photometric band since its color is determined by its age as a consequence of stellar evolution.
Hence, it is possible to obtain an estimate of the galaxys age by measuring its magnitude in different bands and then using stellar evolutionary codes to choose the model that best reproduces the observed colors [86,87]. Coming to the weak-field limit, which essentially means considering Solar System scales, any alternative relativistic theory of gravity is expected to reproduce GR which, in any case, is firmly tested only in this limit and at these scales . Even this limit is a matter of debate since several relativistic theories do not reproduce exactly the Einsteinian results in their Newtonian limit but, in some sense, generalize them.
As was first noticed by Stelle , R2 -gravity gives rise to Yukawa-like corrections to the Newtonian potential with potentially interesting physical consequences. For example, it is claimed by some authors that the flat rotation curves of galaxies can be explained by such terms . In general, any relativistic theory of gravitation yields corrections to the weak-field gravitational potentials e. Furthermore, the newborn gravitational lensing astronomy  is providing additional tests of gravity over small, large, and very large scales which will soon provide direct measurements of the variation of the Newton coupling , the potential of galaxies, clusters of galaxies, and several other features of self-gravitating systems.
This short overview shows that several shortcomings point out that GR cannot to be the final theory of gravity notwithstanding its successes in addressing a large amount of theoretical and experimental issues. ETGs could be a viable approach to solve some of these problems at IR and UV scales without pretending to be the comprehensive and self- consistent fundamental theory of gravity but in the track outlined by GR and then in the range of gauge theories.
This review paper is mainly devoted to the theoretical foundation of ETGs trying to insert them in the framework of gauge theories and showing that they are nothing else but a straightforward extension of GR. The cosmological phenomenology and the genuinely astrophysical aspects of ETGs are not faced here. We refer the readers to the excellent reviews and books quoted in the bibliography [96,78,97,79,77,98,65,]. Modern gauge theory has emerged as one of the most significant developments of physics of XX century. It has allowed us to realize partially the issue of unifying the fundamental interactions of nature.
We now believe that the Electromagnetism, which has been long studied, has been successfully unified with the nuclear weak interaction, the force to which radioactive decay is due. What is the most remarkable about this unification is that these two forces differ in strength by a factor of nearly This important accomplishment by the WeinbergSalam gauge theory , and insight gained from it, have encouraged the hope that also the other fundamental forces could be unified within a gauge theory framework.
At the same time, it has been realized that the potential areas of application for gauge theory extended far beyond elementary particle physics. Although much of the impetus for a gauge theory came from new discoveries in particle physics, the basic ideas behind gauge symmetry have also appeared in other areas as seemingly unrelated, such as condensed matter physics, non- linear wave phenomena and even pure mathematics.
This great interest in gauge theory indicates that it is in fact a very general area of study and not only limited to elementary particles. Gauge invariance was recognized only recently as the physical principle governing the fundamental forces between the elementary particles. Yet the idea of gauge invariance was first proposed by Hermann Weyl in when only the electron and proton  were known as fundamental particles. It required nearly 50 years for gauge invariance to be rediscovered and its significance to be understood.
The reason for this long hiatus was that Weyls physical interpretation of gauge invariance was shown to be incorrect soon after he had proposed the theory. Gauge invariance only managed to survive because it was known to be a symmetry of Maxwells equations and thus became a useful mathematical help in order to simplify equations and thus became a useful mathematical device for simplifying many calculations in the electrodynamics.
In view of present success of gauge theory, we can say that gauge invariance was the classical case of a good idea which was discovered long before its time. In this section, we present a brief summary of gauge theory in view of the fact that any theory of gravity can be considered under the same standard.
The early history of gauge theory can be divided into old and new periods where the dividing can be set in the s. The most important question is what motivated Weyl to propose the idea of gauge invariance as a physical symmetry? How did he manage to express it in a mathematical form that has remained almost the same today although the physical interpretation has altered radically?
And, how did the development of Quantum Mechanics lead Weyl himself to a rebirth of a gauge theory? The new period of gauge theory begins with the pioneering attempt of Yang and Mills to extend gauge symmetry beyond the narrow limits of Electromagnetism . Here we will review the radically new interpretation of gauge invariance required by YangMills theory and the reasons for the failure of original theory.
By comparing the new theory with that of Weyl, we can see that many original ideas of Weyl have been rediscovered and incorporated into the modern theory . In these next subsection, our purpose is to present an elementary introduction to a gauge theory in order to show that any relativistic theory of gravity is a gauge theory. In physics, gauge invariance also called gauge symmetry is the property of a field theory where different configurations of the underlying fundamental but unobservable fields result in identical observable quantities.
A theory with such a property is called a gauge theory. A transformation from a field configuration to another is called a gauge transformation. Modern physical theories describe nature in terms of fields, e. On the other hand, the observable quantities, namely the ones that can be measured experimentally as charges, energies, velocities, etc. This and any kind of invariance under a transformation is called a symmetry.
For example, in classical Electromagnetism the Electric field, E, and the magnetic field, B, are observable, while the underlying and more fundamental electromagnetic potentials V and A are not. Under a gauge transformation which jointly alters the two potentials, no change occurs either in E or B or in the motion of charged particles.
In this example, the gauge transformation was just a mathematical feature without any physical relevance, except that gauge invariance is intrinsically connected to the fundamental law of charge conservation. As shown above, with the advent of Quantum Mechanics in the s, and with successive Quantum Field Theory, the importance of gauge transformations has steadily grown.
Gauge theories constrain the laws of physics, because of the fact that all the changes induced by a gauge transformation have to cancel each other out when written in terms of observable quantities. Over the course of the 20th century, physicists gradually realized that all forces fundamental interactions arise from the constraints imposed by local gauge symmetries, in which case the transformations vary from point to point in space and time. Perturbative quantum field theory usually employed for scattering theory describes forces in terms of force mediating particles called gauge bosons.
The nature of these particles is determined by the nature of the gauge transformations. The culmination of these efforts is the Standard Model, a quantum field theory explaining all of the fundamental interactions except gravity. In , people thought that only two fundamental forces of nature existed, Electromagnetism and Gravitation. In that same year, a group of scientists also made the first experimental observation of starlight bending in the gravitational field of the sun during a total eclipse .
The brilliant confirmation of Einsteins General Theory of Relativity inspired Weyl to propose his own revolutionary idea of gauge invariance in To see how this came about, let us first briefly recall some basic ideas on which Relativity was built. The fundamental concept underlying both Special and GR is that are no absolute frames of reference in the Universe. The physical motion of any system must be described relatively to some arbitrary coordinate frame specified by an observer, and the laws of physics must be independent of the choice of frame. In Special Relativity, one usually, defines convenient reference frames, which are called inertial, in motion with uniform velocity.
For example, consider a particle which is moving with constant velocity v with respect to an observer. Let S be the rest frame of the observer and S be an inertial frame which is moving at the same velocity as the particle. The observer can either state that the particle is moving with velocity v in S or that it is at rest S. The important point to be noted from this trivial example is that the inertial frame S can always be related by a simple Lorentz transformation to the observer frame S. The transformation depends only on the relative velocity between the and observers, not on their positions in spacetime.
The particle and observer can be infinitesimally close together or at opposite ends of the Universe; the Lorentz transformation is still the same. Thus the Lorentz transformation, or rather the Lorentz symmetry group of Special Relativity, is an example of global symmetry. In GR, the description of relative motion is much more complicated because now one is dealing with the motion of a system embedded in a gravitational field. As an illustration, let us consider the following gedanken experiment for measuring the motion of a test particle which is moving through a gravitational field.
The measurement is to be performed by a physicist in an elevator. The elevator cable as broken so that the elevator and the physicist are falling freely . As the particle moves through the field, the physicist determines its motion with respect to the elevator. Since both particle and elevator are falling in the same field, the physicist can describe the particle motion as if there were no gravitational field. The acceleration of the elevator cancels out the acceleration of particle due to gravity. This example of the Principle of Equivalence, follows from the well-known fact that all bodies accelerate at the same rate given the gravitational field e.
Let us now compare the physicist in the falling elevator with the observer in the inertial frame in Special Relativity. It could appear that the elevator corresponds to an accelerating or non-inertial frame that is analogous to the frame S in which the particle appeared to be at rest. However, it is not true that a real gravitational field does not produce the same acceleration at every point in space.
As one moves infinitely far away from the source, the gravitational field will eventually vanish. Thus, the falling elevator can only be used to define a reference frame within an infinitesimally small region where the gravitational field can be considered uniform. Over a finite region, the variation of the field may be sufficiently large for the acceleration of the particle not to be completely canceled. We see that an important difference between Special Relativity and GR is that a reference frame can only be defined locally or at a single point in a gravitational field.
This creates a fundamental problem. To illustrate this difficult point, let us now suppose that there are many more physicists in nearby falling elevators. Each physicist makes an independent measurement so that the path of particle in the gravitational field can be determined. The measurements are made in separate elevators at different locations in the field.
Clearly, one cannot perform an ordinary Lorentz transformation between the elevators. If the different elevators were related only by Lorentz transformation, the acceleration would have to be independent of position and the gravitational field could not decrease with distance from the source. Einstein solved the problem of relating nearby falling frames by defining a new mathematical relation known as connection. To understand the meaning of a connection, let us consider a four-vector A which represents some physically measured quantity. Now suppose that the physicist in the elevator located at x observes that A changes by an amount dA and a second physicist in a different elevator at x observes a change in dA.
In Special Relativity, the differential dA is also a vector like A itself. The relation 2. We can no longer assume that the transformation from x to x is linear in GR. Thus, we must write for dA the general expression. Such terms are actually quite familiar in physics.
They occur in curvilinear coordinate systems. These curvilinear coefficients are denoted by the symbol. They are also called affine connections or Christoffel symbols . It is important to note that the gravitational connection is not simply the result of using a curvilinear coordinate system. The value of the connection at each point in spacetime is dependent on the properties of the gravitational field. The field is important in the determination of the relative orientation of the different falling elevators in the same way that the upward direction on the surface of the earth varies from one position to another.
The analogy with curvilinear coordinate systems merely indicates that the mathematical descriptions of free-falling frames and curvilinear coordinates are similar. Einstein generalized this similarity and arrived at the revolutionary idea of replacing gravity by the curvature of spacetime . Let us briefly summarize the essential characteristics of GR that Weyl would have utilized for his new gauge theory.
First of all, GR involves a specific force, gravitation, which is not studied in Special Relativity. However, by studying the properties of coordinates frames just as in Special Relativity, one learns that only local coordinates can be defined in a gravitational field. This local property is required by the physical behavior of the field and leads naturally to the idea of a connection between local coordinate frames.
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Thus the essential difference between Special Relativity and GR is that the former is a global theory while the latter is a local theory. This local property was the key to Weyls gauge theory . In Section 3 we will develop extensively this idea. Weyl went a step beyond GR and asked the following question: if the effects of a gravitational field can be described by a connection which gives the relative orientation between local frame in spacetime, can other forces of nature such as Electromagnetism also be associated with similar connections?
Generalizing the idea that all physical measurements are relative, Weyl proposed that absolute magnitude or norm of a physical vector also should not be an absolute quantity but should depend on its location in spacetime. A new connection would then be necessary in order to relate the lengths of vectors at different positions.
This is the scale or gauge invariance. It is important to note here that the true significance of Weyls proposal lies on the local property of gauge symmetry and not in a special choice of the norm or gauge as a physical variable. The assumption of locality is a powerful condition that determines not only the general structure but many of the detailed features of gauge theory.
Weyls gauge invariance can be easily expressed in mathematical form . Let us consider a vector at position x with norm given by f x. The factor S x is defined for convenience to equal unity at the position x. The derivative S is the new mathematical connection associated with the gauge change. Weyl identified the gauge connection S with the electromagnetic potential A.
Since the forms of 2. Unfortunately, it was soon pointed out that the basic idea of scale invariance itself would lead to conflict with known physical facts . Some years later, Bergmann noted, that Weyls original interpretation of gauge invariance would also be in conflict with Quantum Theory.
Since the wavelength is determined by the particle mass M, it cannot depend on position and thus contradicts Weyls original assumption about scale invariance. Despite the initial failure of Weyls gauge theory, the idea of a local gauge symmetry survived. It was well known that Maxwells equations were invariant under a gauge change. However, without an acceptable interpretation, gauge invariance was regarded as only an accidental symmetry of Electromagnetism. The gauge transformation property in Eq.
Only the electric and magnetic fields were considered to be real and observable. Gauge symmetry was retained largely because it was useful for calculations in both classical and quantum electrodynamics. In fact a lot of problems in electrodynamics can often be most easily solved by first choosing a suitable gauge, such as the Coulomb gauge or Lorentz gauge, in order to make the equations more tractable . It is clear that the electromagnetic interaction of charged particles could be interpreted as a local gauge theory within the context of Quantum Mechanics.
In analogy with Weyls first theory, the phase of a particle wavefunction can be identified as a new physical degree of freedom which is dependent on the spacetime position. The value can be changed arbitrarily by performing purely mathematical phase transitions on the wavefunction at each point. Therefore, there must be some connections between phase value nearby points. The role of this connection is payed by the electromagnetic potential.
This strict relation between potential and the change in phase is clearly demonstrated by the AharonovBohm effect . Thus by using the phase of wavefunction as the local variable instead of the norm of a vector, Electromagnetism can be interpreted as a local gauge theory very much as Weyl envisioned.
Gauge transformations can be viewed as merely phase changes so that they look more like a property of Quantum Mechanics than Electromagnetism. In addition, the symmetry defined by the gauge transformations does not appear to be natural. The set of all gauge transformations forms a one-dimensional unitary group known as the U 1 group. This group does not arise from any form of coordinate transformation like the more familiar spin-rotation group SU 2 or Lorentz group.
Thus, one has lost the original interpretation proposed by Weyl of a new spacetime symmetry. The status of gauge theory was also influenced by the historical fact that Maxwell had formulated Electromagnetism long before Weyl proposed the idea of gauge invariance. Therefore, unlike the GR, the gauge symmetry group did not play any essential role in defining the dynamical content of Electromagnetism. This sequence of events was to be completely reversed in the development of modern gauge theory [,,].
In , Yang and Mills proposed that the strong nuclear interaction can be described by a field theory like Electromagnetism. They postulated that the local gauge group was the SU 2 isotopic-spin group. This idea was revolutionary because it changed the very concept of the identity of an elementary particle. If the nuclear interaction is a local gauge theory like Electromagnetism, then there is a potential conflict with the notion of how a particle state.
For example, let us assume that we can turn off the electromagnetic interaction so that we cannot distinguish the proton and neutron by electric charge. We also ignore the small mass difference. We must then label the proton as the up state of isotopic spin 12 and the neutron as the down state. But if isotopic spin invariance is an independent symmetry at each point in spacetime, we cannot assume that the up state is the same at any other point. The local isotopic spin symmetry allows to choose arbitrarily which direction is up at each point without reference to any other point.
Given that the labeling of a proton or a neutron is arbitrary at each point, once the choice has been made at one location, it is clear that some rule is then needed in order to make a comparison with the choice at any other position. The required rule, as Weyl proposed originally, is supplied by a connection. A new isotopic spin potential field was therefore postulated by Yang and Mills in analogy with the electromagnetic potential. However, the greater complexity of the SU 2 isotopic-spin group as compared to the U 1 phase group means that YangMills potential will be quite different from the electromagnetic field.
Physics Reports 509 (2011) 167–321_extended Theories of Gravity
In Electromagnetism, the potential provides a connection between the phase values of the wavefunction at different positions. In the YangMills theory, the phase is replaced by a more complicated local variable that specifies the direction of the isotopic spin. An obvious way to relate the up states at different locations x and y is to ask how much the up state at x needs to be rotated so that it is oriented in the same direction as the up state at y. This suggests that the connection between isotopic spin states at different points must act like isotopic spin rotation itself.
In other words, if a test particle in the up state at x is moved through the potential field to position y, its isotopic spin direction must be rotated by the field so that it is pointing in the up direction corresponding to the position y. We can immediately generalize this idea to states of arbitrary isotopic spin.
Since the components of an isotopic spin state can be transformed into another one by elements of the SU 2 group, we can conclude that the connection must be capable of performing the same isotopic spin transformations as the SU 2 group itself. This idea that the isotopic spin connection, and therefore the potential, acts like the SU 2 symmetry group is the most important result of the YangMills theory. This concept lies at the heart of the local gauge theory. It shows explicitly how the gauge symmetry group is built into the dynamics of the interaction between particles and fields.
How is ti possible for a potential to generate a rotation in an internal symmetry space? To answer this question, we must define the YangMills potential more carefully in the language of the rotation group. A three-dimensional rotation R of a wavefunction is written as. Let us compare this rotation with the phase change of wavefunction after a gauge transformation.
The rotation has the same mathematical form as the phase factor of the wavefunction. However, this does not mean that the potential itself is a rotation operator like R. We noted earlier that the amount of phase change must also be proportional to the potential in order to ensure that Schrdinger equation remains gauge invariant. To satisfy this condition, the potential must be proportional to the angular momentum operator L in 2.
This relation indicates that the YangMills is not a rotation, but rather is a generator of a rotation. For the case of Electromagnetism, the angular momentum operator is replaced by a unit matrix and Ai x is just proportional to the phase change. We can immediately deduce some interesting properties of the YangMills potential. We can associate this formal operation with a real process where a neutron absorbs a unit of isotopic spin from the gauge field and turns into a proton.
This example indicates that the YangMills gauge field must itself carry electric charge unlike electromagnetic potential. The YangMills field also differs in other respects from the electromagnetic field but they both have one property in common, namely, they have zero mass. The zero mass of the photon is well known from Maxwells equations, but local gauge invariance requires that the mass of the gauge potential field be identically zero for any gauge theory.
The reason is that the mass of the potential must be introduced into a Lagrangian through a term of the form. This guarantees that the correct equation of motion for a vector field will be obtained from the EulerLagrange equations. Unfortunately, the term given by 2. The special transformation property of the potential will introduce extra terms in 2. Thus, the standard mass term is not allowed in the YangMills gauge field must have exactly zero mass like the photon. The YangMills field will therefore exhibit long-range behavior like Coulomb field and cannot reproduce the observed short range of the nuclear force.
Since this conclusion appeared to be an inescapable consequence of a local gauge invariance, the YangMills theory was not considered to be an improvement on the already existing theories of the strong nuclear interaction. Although the YangMills theory field in its original purpose, it established the foundation for modern gauge theory. The SU 2 isotopic-spin gauge transformation could not be regarded as a mere phase change; it required an entirely new interpretation of a gauge invariance. Yang and Mills showed for the first time that local gauge symmetry was a powerful fundamental principle that could provide new insight into the newly discovered internal quantum numbers like isotopic spin.
In the YangMills theory, isotopic spin was not just a label for the charge states of particles, but it was crucially involved in determining the fundamental form of the interaction . The YangMills theory revived the old ideas that elementary particles have degrees of freedom in some internal space. By showing how these internal degrees of freedom could be unified in a non-trivial way with the dynamical motion in spacetime, Yang and Mills discovered a new type of geometry in physics.
The geometrical structure of a gauge theory can be seen by comparing the YangMills theory with of GR. The essential role of the connection is evident in both gauge theory and relativity. There is an analogy between non-inertial coordinate frame and gauge theory but the local frame has to be is located in an abstract space associated with the gauge symmetry group. To see how the gauge group defines an internal space, let us examine the examples of the U 1 phase group and the SU 2 isotopic spin group.
For the U 1 group, the internal space consists of all possible values of the phase of the wavefunction. These phase values can be interpreted as angular coordinates in a two-dimensional space. The internal symmetry space of U 1 thus looks like a ring, and the coordinate of each point in this space is just the phase value itself. The internal space defined by SU 2 group is more complicated because it describes rotation in a three-dimensional space.
We recall that the orientations of an isotopic spin state can be generated by starting from a fixed initial isotopic direction, which can be chosen as the isotopic spin up direction, and then rotating to the desired final direction. The values of the three angles which specify the rotation can be considered as the coordinates for a point inside an abstract three-dimensional space. Each point corresponds to a distinct rotation so that the isotopic spin states themselves can be identified with the points in this angular space.
Thus, the internal symmetry space of the SU 2 group looks like the interior of a three- dimensional sphere. The symmetry space of a gauge group provides the local non-inertial coordinate frame for the internal degrees of freedom. To an imaginary observer inside this internal space, the interaction between a particle and an external gauge field looks like a rotation of the local coordinates.
The amount of the rotation is determined by the strength of the external potential, and the relative change in the internal coordinates between two spacetime points is just given by the connection as stated before.