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spin 1/2 stems from e/NULeCannihilation to a quark-antiquark, in the center of mass system. One would naïvely expect that th e quark and the antiquark will emerge from the process moving in opposite directions50on their ways to detectors and will be observed. This is not, however, what happens and instead two narrow jets of hadronsemerge, moving back-yo-back, with the net jet-axis angular distribution consistentwith a spin 1/ 2 character of the quark/antiquark parents sources. The electroweak theory and quantum chromodynamics together constitute the so- called standard model 51with underlying gauge symmetry group SU .3//STXSU.2//STX U.1/. The effective coupling in quantum chromodynamics is expected to become larger at large distances increasing with no bound providing a strong conﬁning forceof quarks and gluons restricting the m within hadrons – a phenomenon that is sometimes referred to as infrared slavery . 52
From our discussion of qua ntum electrodynamics, we r ecall that the effective coupling of a U .1/gauge theory, as an abelian gauge theory, increases with energies. On the other hand, the asymptotic free nature of non-abelian gaugetheories, imply that the effective couplings associated with the groups SU .3/, SU.2/, decrease with energy. Due to the smallness of the U .1/coupling in comparison to the other two at the present low energies, this gives one the hope thatat sufﬁcient high energies these three couplings merge together and the underlyingtheory would be described by one single force. A theory which attempts to unifythe electroweak and the strong interactions is called a grand uniﬁed theory .S u c h theories have been developed 53and the couplings seem to run to merge roughly 48This was discovered by Gross and Wilczek [ 106] and Politzer [ 172]. See also Vanyashin and Terentyev [ 229] for preliminary work on vector bosons.
49Chromodynamics means Colordynamics, and the name Quantum Chromodynamics is attributed to Gell-Mann, see, e.g., Marciano and Pagels [ 158]. 50See, e.g., Fig. 6.7d, and Sect. 6.10.3 51The name “Standard Model” is usually attributed to Weinberg. 52Unfortunately, no complete proof of this is available. 53See, e.g., Georgi et al. [ 94] for pioneering work. See also Beringer et al. [ 16]a n dO l i v ee ta l . [167].
1 Introduction 17 somewhere around 1015–1016GeV . This, in turn, gives the hope of the development of a more fundamental theory in which gravitation, which should be effective at energy scale of the order of Planck energy scalep „c=GN'1019GeV,54or even at a lower energy scale, where G Ndenotes Newton’s gravitational constant, is uniﬁed with the electroweak and strong forces. If the standard model is thelow energy of such a fundamental theory, then the basic question arises as towhat amounts for the enormous difference between the energy scale of sucha fundamental theory (/CAN 10 16–1019GeV) and the deﬁning energy scale of the standard model (/CAN 300 GeV)? This has been termed as the hierarchy problem which will be discussed again later. We will see in V ol. II, in particular, that the abovementioned couplings seem to be uniﬁed at a higher energy scale of the order10 16GeV, when supersymmetry is taken into account, getting it closer to the energy
scale at which gravitation may play an important role. One may generalize the symmetry group of the standard model, and consider transformations which include transformations between quarks and leptons, leadingto a larger group such as, for example, to the SU .5/group, or a larger group, which include SU.3//STXSU.2//STXU.1/. The advantage of having one larger group is that one would have only one coupling parameter and the standard model wouldbe recovered by spontaneous symmetry breaking at lower energies. This opens theway to the realization of processes in which baryon number is not conserved, with abaryon, for example, decaying into lep tons and bosons. T he experimental 55bound on lifetime of proton decay seems to be >1033years and is much larger than the age of the universe which is about 13.8 billion years.56Such a rare event even if it occurs once will give some support of such grand uniﬁed theories.
We have covered quite a large territory and before continuing this presentation, we pose for a moment, at this appropriate stage, to discuss three aspects ofimportance that are generally expected in order to carry out reliable computations in perturbative quantum ﬁeld theory. These are: 1. The development of a powerful and simple formalism for doing this. 2. To show how the renormalization process is to be carried, and establish that the resulting expressions are ﬁnite to any order of perturbation theory. 3. The physical interpretation will be completed if through the process of renor- malization, the initial experimentally unattainable parameters in the theory areeliminated in favor of physically observed ones, which are ﬁnite in number, andare generally determined experimentally as discussed earlier in a self consistentmanner. 54The Planck energy (mass) will be introduced in detail later. 55See, e.g., Olive et al. [ 167].
56A decay of the proton may have a disastrous effect in the stab ility of matter over anti-matter itself in the universe. See, however, the discussion given later on the dominance of matter in the visible universe.
18 1 Introduction Perturbatively renormalizable theories are distinguished from the non-renormaliz- able ones by involving only a ﬁnite number of parameters in the theory that aredetermined experimentally. We discuss each of these in turn. 1. A powerful formalism is the Path Integral one, pioneered by Feynman, 57deﬁning a generating functional for so-called Green functions from which physicalamplitudes may be extracted, and has the general structureR d/SYNŒ¦/c141 e iAction.H e r e d/SYNŒ¦/c141 deﬁnes a measure of integration over classical ﬁelds as the counterparts of the quantum ﬁelds of the theory.58“Action” denotes the classical action. In the simplest case, the measure d /SYNŒ¦/c141 takes the form ˘xd/US.x/as a product of all spacetime points. In gauge theories, due to constraints, the determination ofthe measure of integrations requires special techniques 59and takes on a much more complicated expression and was successfully carried out by Faddeev andPopov in 1967.
60The path integral expression as it stands, involves continual integrations to be carried out. An equally powerful and quite an elegant formalism is due to Schwinger,61 referred to as the Action Principle or the Quantum Dynamical Principle .F o r quantum ﬁeld theory computations, the latter gives the variation of the so-calledvacuum-to-vacuum transition amplitude (a generating functional): •h0 Cj0/NULias any of the parameters of the theory are made to vary. The latter is then expressedas a differential operator acting on a simple generating functional expressedin closed form. This formalism involves only functional differentiations to becarried out, no functional integrations are necessary, and hence is relativelyeasier to apply than the path integral. We will learn later, for example, that thepath integral may be simply obtained from the quantum dynamical principleby a functional Fourier transform thus invo lving functional integrals. Again the
application of the quantum dynamical principle to the quantization of gaugetheories with underlying constraints require special techniques and it was carriedout in Manoukian [ 150]. All the fundamental interactions in nature are presently described theoretically by gauge theories, involving constraints , and the two approaches of their quantization discussed above will be both treated in this book for pedagogicalreasons and are developed 62 57See Feynman and Hibbs [ 78] for the standard pedagogical treatment. See also Feynman [ 75]. 58We use a general notation /US.x/for the ﬁelds as functions of spacetime variable and suppress all indices that they may carry to simplify the notation. These ﬁelds may include so-called Grassmann ﬁelds. 59Feynman [ 72], DeWitt [ 43,44], and Faddeev and Popov [ 66]. 60Op. cit. 61Schwinger [ 194–197,201]. 62There is also the canonical formalism, see, e.g., Mohapatra [ 161,162] and Utiyama and Sakamoto [ 228].
1 Introduction 19 via the Path Integral [ 66], or via the Action Principle (Quantum Dynamical Principle) [ 150]. 2. Historically, Abdus Salam, was the ﬁrst “architect” of a general theory of renor- malization. In 1951, he carried out a systematic study63of renormalization [ 180], introduced and sketched a subtraction scheme in a general form. Surprisingly, this classic paper was not carefully re examined until much later. This was eventually done in 1976 [ 147] by Manoukian, and inspired by Salam’s work, a subtraction scheme was developed and brought to a mathematically consistentform, and the ﬁniteness of the subtracted, i.e., renormalized, theory was provedby the author 64toanyorder of perturbation theory.65by using, in the process, a power counting theorem established by Weinberg [ 235] for integrals of a special class of functions, thus completing the Dyson-Salam program. The subtractionwas carried out directly in momentum space and no cut-offs were introduced.
Shortly after the appearance of Sala m’s work, two other “architects” of a general theory of renormalization theor y, Bogoliubov and Parasiuk, in a classic paper in 1957 [ 24], also developed a subtraction scheme. In 1966 [ 118], Hepp gave a complete proof of the ﬁniteness of the Bogoliubov-Parasiuk to any orderof perturbation theory, by using in the intermediate stages ultraviolet cut-offs,and in 1969 [ 251], Zimmermann formulated their scheme in momentum space, without cut-offs, and provided a complete proof of ﬁniteness as well, thuscompleting the Bogoliubov-Parasiuk program. This scheme is popularly knownas the BPHZ scheme. The equivalence of the Bogoliubov-Parasiuk scheme, in the Zimmermann form, and our scheme was then proved by Manoukian, 66after some systematic cancelations in the subtractions. This equivalence theorem67uniﬁes the two monumental approaches of renormalization.68 63Salam [ 180], see also Salam [ 179]. 64Manoukian [ 149].
65For a pedagogical treatment of all these studies, see my book “Renormalization” [ 149]. This also includes references to several of my earlier papers on the subject as well as many results related to renormalization theory. 66See Manoukian [ 149]op. cit. 67This result has been also referred to as “Manoukian’s Equivalence Principle”, Zeidler [ 250], p. 972. See also Streater [ 207]. 68I was pleased to see that our equivalence theorem has been also considered, by completely different methods, by Figueroa and Gracia-Bondia [ 80]. For other earlier, and recent, but different, approaches to renormalization theory, see, e.g., Epstein and Glaser [ 65], Kreimer [131 ,132], Connes and Kreimer [ 34,35], and Figueroa and Gracia-Bondia [ 79,81]. See also Landsman [ 135] and Aschenbrenner [ 11].
20 1 Introduction DS : (program )Dyson Salam Author (completion) BP : Bogoliubov - Parasiuk Hepp - Zimmermann (program )Weinberg (completion)Author (equivalence) Fig. 1.8 Developments of the general theory of renormalization from the DS and BP programs. The intricacies of this layout also appear in Zeidler [ 250], pp. 972–975. Regarding the author’s work shown in the above layout and of his completion of the renormalization program stemming out of Salam’s, Streater [ 207] writes: “ It is the end of a long chapter in the history of physics ” The development of the general theory of renormalization from the DS and BP programs may be then summarized as given in Fig. 1.869. 3. The physical interpretation of the theory is completed by showing that the subtractions of renormalization are implemented by counterterms in the theory which have the same structures as the original terms in the theory (i.e., inthe Lagrangian density), 70thus establishing the self-consistency involved in the
elimination of the initial par ameters in the theory in favor of physically observed ones. As mentioned above, for a theory to be renormalizable, i.e., involvingonly a ﬁnite number of parameters determined, in general, experimentally, thecounterterms of the theory must be ﬁnite in number as well. All particles due to their energy content experience the gravitational attraction. Einstein’s theory of gravitation, also referred to as general relativity (GR), is described by a second rank tensor with the energy-momentum tensor of matter as its source from which the energy density of matter may be deﬁned. It maynot be described just by a scalar or just by a vector ﬁeld as they are inconsistentwith experiment. It is easy to see that due to the fact that masses have the samesigns (positive) a theory based on a vector ﬁeld alone will lead to a repulsiverather an attractive gravitational force. 71GR theory predictions are well supported experimentally in our solar system.
The key observation, referred to as the principle of equivalence, of Einstein is that at any given point in space and any given time, one may consider a frame in whichgravity is wiped out at the point in question. For example, in simple Newtoniangravitational physics, a test particle placed at a given point inside a freely fallingelevator on its way to the Earth, remains at rest, inside the elevator, for a veryshort time, depending on the accuracy being sought, and, depending on its positionrelative to the center of the Earth, eventually moves, in general, from its originalposition in a given instant. Einstein’s principle of equivalence applies only locally 69The layout in Fig. 1.8is based on Manoukian [ 149], and it also appears in Zeidler [ 250], p. 974. See also Streater [ 207] and Figueroa and Gracia-Bondia [ 80]. 70For a detailed study of this see, Manoukian ([ 148]; Appendix, p. 183 in [ 149]).
71Attempts have been made to include such ﬁelds as well for generalizations of Einstein’s theory, but we will not go into it here.
1 Introduction 21 at a given point and at a given time. At the point in question, in the particular frame in consideration, gravity is wiped out and special relativity survives. Thereconciliation between special relativity and Newton’s theory of gravitation, thenreadily leads to GR, where g ravity is accounted for by the curvature of spacetime and its departure from the ﬂat spacetime of special relativity one has started outwith upon application of the principle of equivalence. By doing this, one is able toenmesh non-gravitational laws with gravity via this principle. Quantum gravity (QG) is needed in early cosmology, black hole physics, and,
in general, to deal with singularities that arise in a classical treatment. QG mustalso address the problem of the background geometry. A common interest infundamental physics is to provide a uniﬁed description of nature which is applicablefrom microscopic to cosmological distances. A fundamental constant of unit oflength that is expected to be relevant to this end is the Planck length as well as thePlanck mass. Out of the fundamental constants of quantum physics „, of relativity c, and the Newtonian gravitational one G N, we may deﬁne a unit of length and mass, the Planck length and Planck mass, respectively, relevant in quantum gravity,through the following ` PDr „GN c3'1:616/STX10/NUL33cm;mPDs „c GN'1:221/STX1019GeV=c2: (1.10) In units„D1,cD1, dimensions of physical quan tities may be then expressed in powers of mass ( ŒEnergy/c141DŒMass/c141;ŒLength/c141DŒMass/c141/NUL1DŒTime/c141;:::), and, as
gravitation has a universal coupling to all forms of energy, one may hope that itmay be implemented within a uniﬁed theory of the four fundamental interactions,with the Planck mass providing a universal mass scale. Unfortunately, it is difﬁcultexperimentally to investigate the quantum properties of spacetime as one would beworking at very small distances. GR predicts the existence of Black holes. Here it is worth recalling of the
detection (“Observational waves from a binary black hole merger”, Phys. Rev. Lett.116, 061102 (1–16) (2016)) by B. P. Abbott et al. of gravitational waves from themerger of two black holes 1.3 billion light-years from the Earth. Recall that a blackhole (BH) is a region of space into which matter has collapsed and out of whichlight may not escape. It partitions space into an inner region which is bounded bya surface, referred to as the event horizon which acts as a one way surface for lightgoing in but not coming out. The sun’s radius is much larger than the critical radiusof a BH which is about 2.5 km to be a black hole for the sun. We will see thatfor a spherically symmetric BH of mass M, the radius of the horizon is given by R BHD2GNM=c2.72 72This may be roughly inferred from Newton’s theory of gravitation from which the escape speed of a particle in the gravitational ﬁeld of a spherically symmetric massive body of mass M,a t a
distance r, is obtained from the inequality v2=2/NULGNM=r<0, and by formally replacing vby
22 1 Introduction One may argue that the Planck length may set a lower limit spatial cut-off. The following formal and rough estimates are interesting. Suppose that by means of a high energetic particle of energy E,hE2i/CANh p2ic2, withhp2ivery large, one is interested in measuring a ﬁeld within an interval of size ıaround a given point in space. Such form of energy acts as an effective gravitational mass M/CANp hE2i=c4 which, in turn distorts space around it. The radius of the event horizon of such agravitation mass Mis given by r BHD2GNM=c2. Clearly we must have ı>rBH, otherwise the region of size ıthat we wanted to locate the point in question will be hidden beyond a BH horizon, and localization fails. Also: hp2i/NAK˝/NUL p/NULhpi/SOH2˛ /NAK„2=4ı2. Hence M/NAK„=2cı, ı>2GNM c2/NAK„GN c3ı; which givesı>rBHDp „GN=c3D`P. Interesting investigations by Hawking73have shown that a BH is not really a
black body, it is a thermodynamic object, it radiates and has a temperature associatedwith it. 74In Chapter 7 in V ol. II , we will see, considering a spherically symmetric BH, that its temperature is given by75 TBHD„c3 8/EMGNMkB: (1.11) where k Bis the Boltzmann constant. Note that a very massive black hole is cold. Recall that entropy Srepresents a measure of the amount of disorder with information encoded in it, and invoking the thermodynamic interpretation of a BH,we may write @S @.Mc2/D1 T; (1.12) which upon integration with boundary condition that for M!0,S!0,g i v e st h e celebrated result SBHDc3kB 4„GNADkBA 4`2 P; AD4/EM/DC22GNM c2/DC32 (1.13) the ultimate speed c to obtain for the critical radius RcriticalD2GNM=c2such that for r<Rcritical a particle cannot escape. 73Hawking [ 114,115]. 74Particle emission from a BH is formally explained through virtual pairs of particles created near
the horizon with one particle falling into the BH while the other becoming free outside the horizon. 75A pedestrian approach in determining the temperature is the following. By comparing the expression of energy expressed in terms of the wavelength of radiation /NAK:EDhc=/NAK, with the expression EDkBT,g i v e s TDhc=kB/NAK. On dimensional grounds /NAK/CAN2GNM=c2,w h i c h gives T/CAN/EM„c3=GNMkB. This is the expression given for the temperature up to a proportionality constant.
1 Introduction 23 referred to as the Bekenstein-Hawking Entropy formula76of a BH. This relation relates quantum gravity to information theory. This result is expected to hold in any consistent formulation of quantum gravity, and shows that a BH has entropy unlikewhat would naïvely expect from a BH with the horizon as a one way classical surfacethrough which information is lost to an external observer. The proportionality of theentropy to the area rather than to the volume of a BH horizon should be noted. Italso encompasses Hawking’s theorem of in crease of the area with time with increase of entropy. We will discuss the Bekenstein-Hawking Entropy formula below inconjunction with loop quantum gravity and string theory. Now we turn back to the geometrical description of gravitation given earlier, and introduce a gravitational ﬁeld to account fo r the departure of the curved spacetime
metric from that of the Minkowski one to make contact with the approaches ofconventional ﬁeld theories, dealing now with a ﬁeld permeating an interaction between all dynamical ﬁelds. The quantum particle associated with the gravitationalﬁeld, the so-called graviton, emerges by considering the small ﬂuctuation of themetric, associated with curved spacetime of GR about the Minkowski metric, asthe limit of the full metric, where the gravitational ﬁeld becomes weaker and the particle becomes identiﬁed. This allows us to determine the graviton propagatorin the same way one obtains, for example, the photon propagator in QED, andeventually carry out a perturbation theory as a ﬁrst attempt to develop a quantumtheory of gravitation. In units of„D1;cD1, Newtons gravitational constant G N, in 4 dimensional spacetime, has the dimenionality ŒGN/c141DŒMass/c141/NUL2, which is a dead give away of
the non-renormalizabilty of a quantum theory of gravitation based on GR. The non-renormalizability of the theory is easier to understand by noting that the divergences, in general, tend to increase as we go to higher orders in the gravitational couplingconstant without a bound, implying the need of an inﬁnite of parameters need tobe ﬁxed experimentally 77and hence is not of any prac tical value. Some theories which are generalizations of GR, involving higher order derivatives, turn out to berenormalizable 78but violate, in a perturbative setting, the very sacred principle of positivity condition of quantum theory. Unf ortunately, such a theory involves ghosts in a perturbative treatment, due to the rapid damping of the propagator at highenergies faster than 1=energy 2, and gives rise, in turn, to negative probabilities.79 One is led to believe that Einstein’s general relativity is a low energy effective
theory as the low energy limit of a more com plicated theory, and as such it provides a reliable description of gravitation at low energies. Moreover, one may argue thatthe non-renormalizability of a quantum theory based on GR is due to the fact thatone is trying to use it at energies which are far beyond its range of validity. As a 76Bekenstein [ 14]. 77Manoukian [ 149]a n dA n s e l m i[ 9]. 78Stelle [ 204]. 79Unitarity (positivity) of such theories in a non- perturbative se tting has been el aborated upon by Tomboulis [ 224].
24 1 Introduction matter of fact the derivatives occurring in the action, in a momentum description via Fourier transforms, may be considered to be small at sufﬁciently low energies. Inview of applications in the low energy regi me, one then tries to separate low energy effects from high energy ones even if the theory has unfavorable ultraviolet behaviorsuch as in quantum gravity. 80Applications of such an approach have been carried out in the literature as just cited, and, for example, the modiﬁcation of Newton’sgravitational potential at long distances has been determined to have the structure U.r/D/NULG Nm1m2 rh 1C˛GN.m1Cm2/ c2rCˇGN„ c3r2i ; (1.14) for the interaction of two spin 0 particles of masses m1andm2.H e r e˛;ˇ; are dimensionless constants,81and the third term represents a quantum correction being proportional to„. Conventional quantum ﬁeld theory is usually formulated in a ﬁxed, i.e., in, a
priori, given background geometry such as the Minkowski one. This is unlike theformalism of “Loop Quantum Gravity” (LQG) also called “Quantum Field Theoryof Geometry”. The situation that we will encounter in this approach is of a quantum ﬁeld theory in three dimensional space , which is a non-perturbative background independent formulation of quantum gravity. The latter means that no speciﬁcassumption is made about the underlying geometric structure and, interestinglyenough, the latter rather emerges from the th eory. Here by setting up an eigenvalue equation of, say, an area operator, in a quantum setting, one will encounter agranular structure of three-dimensional space yielding a discrete spectrum for areameasurements with the smallest possible having a non-zero value given to be of theorder of the Planck length squared: „G N=c3/CAN10/NUL66cm2.82The emergence of
space in terms of “quanta of geometry”, providing a granular structure of space, isa major and beautiful prediction of the theory. The 3 dimensional space is generatedby a so-called time slicing procedure of spacetime carried out by Arnowitt, Deserand Misner. 83The basic ﬁeld variables in the theory is a gravitational “electric” ﬁeld, which determines the geometry of such a 3 dimensional space and naturally emergesfrom the deﬁnition of the area of a surface in such a space, and its canonical conju-gate variable is referred to as the connection. By imposing equal time commutationrelation of these two canonically conjugate ﬁeld variables, the quantum version ofthe theory arises, and the fundamental problem of the quantization of geometryfollows. The basic idea goes to Penrose [ 171] whose interest was to construct the concept of space from combining angular momenta. It is also interesting that theproportionality of entropy and the surface ar ea of the BH horizon in the Bekenstein-
Hawking Entropy formula has been derived in loop quantum gravity. 84 80Donoghue [ 55–57] and Bjerrum-Bohr et al. [ 19], Bjerrum-Bohr et al. [ 20]. 81Recent recorded values are ˛D3,a n dˇD41=10/EM Bjerrum-Bohr et al. [ 19]. 82Rovelli and Smolin [ 174], Ashtekar and Lewandoski [ 12], and Rovelli and Vidotto [175 ]. 83Arnowitt et al. [ 10]. 84See, e.g., Meissner [ 160]a n dA n s a r i[ 8].
1 Introduction 25 Supersymmetry is now over 40 years old. Supersymmetry provides a symmetry between fermions and bosons. Borrowing a statement made by Dirac, speaking of theories, in general, it is a theory with mathematical beauty .85The name “Supersymmetry” for this symmetry is attributed to Salam and Strathdee as itseemed to have ﬁrst appeared in the title of one of their papers [ 184]. An abbreviated name for it is SUSY , as some refer to this symmetry. The latter is not onlybeautiful but is also full of thought-provoking surprises. To every degree of freedomassociated with a particle in the standard model, in a supersymmetric version,there corresponds a degree of freedom associated with a partner, referred to as asparticle, with the same mass and with opposite statistics to the particle. 86Unlike
other discoveries, supersymmetry was not, a priori, invented under pressure set byexperiments and was a highly intellectual achievement. Theoretically, however, itquickly turned out to be quite important in further developments of quantum ﬁeldtheory. For one thing, in a supersymmetric extention of the standard model, theelectroweak and strong effective couplings do merge at energies about 10 16GeV , signalling the possibility that these interactions are different manifestations of asingle force in support of a grand uniﬁed theory of the fundamental interactions.Also gravitational effects are expected to be important at the quantum level at thePlanck energy of the order 10 19GeV , or possibly at even lower energies, giving the
hope of having a uniﬁed theory of the four fundamental interactions at high energies.Supersymmetry leads to the uniﬁcation of coupling constants. SUSY also tends to“soften” divergences of a theory in the sense that divergent contributions originatingfrom fermions tend to cancel those divergent contributions originating from bosonsdue to their different statistics. One of the important roles that supersymmetry may play in a supersymmetric extension of the standard problem is in the so-called hierarchy problem. The basic
idea of a facet of this is the following. A fundamental energy scale arises in thestandard model from the vacuum expectation value of the Higgs boson which setsthe scale for the masses in the theory, such as for the masses of the vector bosons.It turns out that the masses imparted to the initially massless vector bosons, forexample, via the Higgs mechanism, using the parameters in the Lagrangian densityare in very good agreement with experimental results. On the other hand, if oneintroduces a large energy scale cut off /DC4/CAN10 15GeV , of the order of a grand uniﬁed energy scale, or the Planck energy scale 1019GeV , at which gravitation may play a signiﬁcant role, to compute the shift of the squared-mass of the Higgs boson, as ascalar particle, due to the dynamics (referred to as radiative corrections), it turns outto be quadratic 87in/DC4, which is quite large for such a large cut-off. This requires that the bare mass squared of the Higgs boson to be correspondingly large to cancel
85Here we recall the well known statement of Dirac, that a theory with mathematical beauty is more likely to be correct than an ugly one that ﬁts some experimental data [ 53]. 86This is such that the total number of fermion degrees of freedom is equal to the total bosonic ones. 87See, e.g., Veltman [ 230].
26 1 Introduction such a quadratic dependence on /DC4and obtain a physical mass of the Higgs boson of the order of magnitude of the minute energy88./CAN102GeV/, in comparison, characterizing the standard model, and this seems quite unnatural for the cancelation of such huge quantities.89This unnatural cancelation of enormously large numbers has been termed a facet of the hierarchy problem. Supersymmetric theories have,in general, the tendency to cancel out such quadratic divergences, up to possibly ofdivergences of logarithmic type which are tolerable, thus protecting a scalar particlefrom acquiring such a large bare mass. So supersymmetry may have an importantrole to play here. SUSY relates fermions to bosons, and vice versa, and hence a generator is required which is of fermionic type, that is, it carries a spinor index as in theDirac ﬁeld 90to carry out a transformation fermion $boson. Since the spins of
fermions and bosons are different, this nece ssarily means that such a generator does notcommute with the angular (spin) momentum operator as supersymmetry unites particles of the same mass and different spins into multiplets. Bosons and fermionshave, in general, different masses, which means that SUSY is to be spontaneouslybroken if such a symmetry is to have anything to do with nature. If supersymmetrybreaking sets at such an energy scale as 1 TeV or so, then some of the lowest masssuperpartners may be hopefully discovered. 91 Of particular interest was also the development of the superspace concept as an extension of the Minkowski one, where one includes an additional degree offreedom usually denoted by 92/DC2D./DC2a/to the space coordinates .t;x/,w h i c h turns out to be quite convenient in deﬁning and setting up SUSY invariant integralssuch as the action of a dynamical system. 93To describe dynamics this, in turn,
necessitates to introduce superﬁelds of different types94as functions of these variables. The extension of the algebra of the Poincaré group to a superalgebra was ﬁrst carried out by Gol’fand and Likhtman in [ 100] to construct supersymmetric ﬁeld theory models, and with the implementation of spontaneous symmetry breaking 88Aad et al. [ 1] and Chatrchyan et al. [ 31]. 89As mentioned earlier, the question, in turn, arises as to what amounts for the enormous difference between the energy scale of grand uniﬁcation and the energy scale that characterizes the standard model. 90This point is of importance because an earlier attempt by Coleman and Mandula [ 33] to enlarge the Poincaré group did not work. They considered only so-called “Bose” generators (that is tensors,and not spinors) in their analysis. 91Perhaps an optimist would argue that since antiparticles corresponding to given particles were
discovered, the discovery of superpartners associated with given particles would not be out ofthe question either. The underlying symmetries involved in these two cases are, however, quite ofdifferent nature. 92This is called a Grassmann variable. 93See Salam and Strathdee [ 185,186]. 94Details on superﬁelds will be given in Vol. II. The explicit expression of the pure vector superﬁeld has been recently obtained in Manoukian [ 155].
1 Introduction 27 by V olkov and Akulov in [ 232]. In [243 ], Wess and Zumino also,95independently, developed supersymmetric models in 4 dimensions, and this work has led to an avalanche of papers on the subject and to a rapid development of the theory. Inparticular, supersymmetric extensions of the standard model were developed, 96 supergravity, as a supersymmetric extension of gravitational theory, was alsodeveloped. 97Unfortunately, things do not seem to be much better for supergravity, as far as its renormalizability is concerned.98 Now we come to String Theory. String Theory is a theory which attempts to provide a uniﬁed description of all the fundamental interactions in Nature and,in particular, give rise to a consistent theory of quantum gravity. A string is afundamental one dimensional extende d object, and if it has to do with quantum gravity, it is, say, of the order of the Plank length ` PDp GN„=c3/CAN10/NUL33cm,
involving the three fundamental constants: Newton’s gravitational constant G N,t h e quantum unit of action „, and the speed of light c. Since no experiments can probe distances of the order of the Planck length, such a string in present day experimentsis considered to be point-like. When a string, whether closed or open, moves inspacetime, it sweeps out a two dimensional surface referred to as a worldsheet.String Theory is a quantum ﬁeld theory which operates on such a two dimensional worldsheet . This, as we will see, has remarkable consequences in spacetime itself, albeit in higher dimensions. Particles are identiﬁed as vibrational modes of strings,and a single vibrating string may describe several particles depending on itsvibrational modes. Strings describing bos onic particles are referred to as a bosonic
strings, while those involving fermionic ones as well are referred to as superstrings.The remarkable thing is that the particles needed to describe the dynamics ofelementary particles arise naturally in the mass spectra of oscillating strings, and are not, a priori, assumed to exist or put in by hand in the underlying theories.Thedimensionality of the spacetime in which the strings live are predicted by the underlying theory as well and are necessa rily of higher dimensions than four for
consistency with Lorentz invariance of spacetime at the quantum level, consistingof a dimensionality of 26 for the bosonic strings and a spacetime dimensionality of10 for the superstrings. The extra dimensions are expected to curl up into a spacethat is too small to be detectable with present available energies. For example thesurface of a hollow extended cylinder with circular base is two dimensional, withone dimension along the cylinder, and another one encountered as one moves onits circumference. If the radius of the base of the cylinder is relatively small, thecylinder will appear as one dimensional when viewed from a large distance (lowenergies). Accordingly, the extra dimensions in string theory are expected to be 95These basic papers, together with other key ones, are conveniently collected in Ferrara [ 71]. 96See Fayet [ 67], Dimopoulos and Georgi [ 45]. 97See Freedman, van Nieuwenhuizen and Ferrara [84 ], Deser and Zumino [ 42].
98See, e.g., Deser et al. [ 41], Deser [40 ], Stelle [ 205, 206], and Howe and Stelle [ 124].
28 1 Introduction small and methods, referred to as compactiﬁcations,99have been developed to deal with them thus ensuring that the “observable” dimensionality of spacetime is four. Superstring theories involve fermions and are thus relevant to the real world, but there are, however, several superstring theories. Also unlike the loop quantum gravity, which provides a background independent formulation of spacetime withthe latter emerging from the theory itself, as discussed earlier, the strings instring theories are assumed to move in a pre-determined spacetime, and thus spacetime plays a passive role in them. 100A theory, referred to as M-Theory,101 based on non-perturbative methods, is envisaged to unify the existing superstringstheories into one single theory, instead of several ones, and be related to them by various limiting and/or transformation rules, referred to as dualities, 102and is of 11
dimensional spacetime. M-Theory is believed to be approximated by 11 dimensionalsupergravity, 103and the spacetime structure is envisaged to emerge from the theory as well. Bosonic strings involve tachyonic states. This is unlike the situation insuperstring theories in which supersymmetry plays a key role in their deﬁnitions,and a process referred to as a GSO proj ection method, ensuring the equality of the degrees of freedom of bosonic and fermionic states, as required by supersymmetry,in turn implies that no tachyonic states appear in the theory. 104 String theory was accidentally discove red through work carried out by Veneziano in 1968 when he attempted to write down consistent explicit expressions of meson-meson scattering amplitudes in strong interactions physics. 105This was, of course
before the discovery of QCD. With the many excited states of mesons and baryons(resonances), it was observed experimentally that there exists a linear relationshipbetween spin Jand the mass Msquared of a resonance given by a linear relationship with a universal slope WdJ dM2D˛0;˛0Š1GeV/NUL2; (1.15) deﬁning so-called Regge trajectories. Veneziano postulated and wrote down ascattering amplitude of meson – meson scattering: p 1.m1/Cp2.m2/!p3.m3/C p4.m4/, which, in particular, showed that the amplitude involves the exchange of an inﬁnite number of particles (corresponding to arbitrary integer spins). This is unlikethe situation in conventional ﬁeld theory as QED or the standard model, where theyinvolve the exchange of a ﬁnite number of particles to any given order. String theoryshares this property of the Veneziano a mplitude. As a matter of fact the Veneziano 99An idea used by Kaluza and Klein in their attempt to unify gravity and electromagnetism in a 5
dimensional generalization of general relativity. 100See also Horowitz [ 122]. 101Townsend [ 227], Witten [ 244], and Duff [ 58]. 102Duff [ 58] and Schwarz [ 191]. 103Cremmer et al. [ 37]. 104The GSO method of projection was proposed in Gliozzi et al. [ 98,99]. 105Veneziano [ 231], see also Lovelace and Squires [ 145] and Di Vecchia [ 54].
1 Introduction 29 amplitude may be derived from string theory. Nambu [164 ], Nielsen [ 166]a n d Susskind [ 211] have shown that the famous expression of the amplitude postulated by Veneziano may be interpreted as a quantum theory of scattering of relativistic strings. Although, a priori, this was assumed to describe a strong interaction process,Yoneya [ 248], and Scherk and Schwarz [ 190] made use of the fact that string theory (involving closed strings) contains a spin 2 massless state, which was identiﬁedwith the elusive graviton, in addition to a whole spectrum of other excitation modes,to propose that string theory provides a framework for the uniﬁcation of generalrelativity and quantum mechanics. As early as 1971, Neveu and Schwarz [ 165], and Raymond [ 173] included fermions in their analyses, which eventually led to the notion of superstrings, and during a short period of time, several types 106of superstrings were introduced in the literature.
Due to the assumed non-zero extensions of strings, it is hoped that they provide, naturally, an ultraviolet cut-off /ETX/CAN.`P//NUL1and render all processes involving strings ultraviolet ﬁnite. This is unlike conventional quantum ﬁeld theory inter-actions where all the quantum ﬁelds are multiplied locally at the same spacetimepoints, like multiplying distributions at the same point, and are, in this sense, quitetroublesome. In string theory, two strings with given vibrational modes, identifying two given particles, may combine forming one string with an arbitrary number ofdifferent vibrational modes associated with a myriad number of particles, deﬁninggeneralized 3-vertices. The combined string may again split into two strings withassociated vibrational modes, identiﬁed appropriately with two more particles,describing a scattering process of 2 particles !2 particles. Thus interactions involve string worldsheets of various topologies arise.
Other extended objects are also encountered in string theory called branes which, in general, are of higher spatial dimensions than one, with the string deﬁned as a onedimensional brane. For example, an open string, satisfying a particular boundarycondition, referred to as a Dirichlet boundary condition, speciﬁes a hypersurface, referred to as a D brane, on which the end points of the open string reside. On theother hand, the graviton corresponds to a vibrational mode of closed strings, andsince the latter, having no ends, may not be restricted to a brane and moves awayfrom it. This might explain the weakness of the gravitational ﬁeld, if our universeis a 3 dimensional brane embedded in a higher dimensional spacetime. Masslessparticles encountered in string theory are really the physically relevant ones becauseof the large unit of mass .` P//NUL1/CAN1019GeV in attributing masses to the spectrum of massive particles.107As we will see a massless particle may acquire mass if,
106Green and Schwarz [102 ,103] and Gross et al. [ 107,108]. 107A systematic analysis of all the massless ﬁeld excitations encountered in both bosonic and superstrings are investigated in Manoukian [ 152–154], in their respective higher dimensional spacetimes, and include the determinations of the degrees of freedom associated with them. Note that in four dimensional spacetime the number of degrees of freedom (spin states) of non-scalarﬁelds is always two. This is not true in higher dimensional spacetime. For example, the degrees offreedom associated with a massless vector particle is 8 in 10 dimensions, while for the graviton is
30 1 Introduction for example, the end points of the open string are attached to two different branes, instead of a single brane. We will learn the remarkable facts that E instein’s general relativity as well as Yang-Mills ﬁeld theory may be obtained from string theory. Interesting high energy scattering amplitudes have been computed in string theory over the years,108which provide a hint that space may not be probed beyond the Planck length – a result shared with “loop quantum gravity”. It is worthmentioning that the Bekenst ein-Hawking Entropy relation has been also derived in string theory. 109 In recent years much work has been done, which is worth mentioning here but rather brieﬂy, indicating that general relationships may exist between ﬁeldtheories and string theories, and consequently considerable attention was giventrying to make such a statement more and more precise, with the ultimate hopeof providing, in turn, a consistent and acceptable quantum theory of gravitation
relevant to our world but much work still remains to be done. In particular,much study has been made to study the equivalence relation between certain fourdimensional gauge theories and superstring theories, referred to as the AdS/CFTcorrespondence, where AdS space stands for anti-de-Sitter space, and CFT standsfor conformal ﬁeld theory. 110Such correspondences have been also referred to as Gauge/Gravity duality, as well as Maldacena duality, a duality which wasﬁrst proposed by Maldacena. 111Without going into technical details, the aim of this work is to show, for example, the existence of an equivalence relationbetween a certain supersymmetric SU .N/Yang-Mills ﬁeld theory in 4 dimensional
Minkowski spacetime, and a superstring theory in a 5 dimensional AdS space,having one additional dimension to the Minkowski one, and with the 5 dimensionsof the AdS space supplemented by 5 extra dimensions deﬁned by a ﬁve-sphere,making up the 10 dimensions of superstrings mentioned earlier. The interest in thiswork is that it deals with a connection between string theory (involving gravity) and 35, as shown later in Chapter 3of Vol. II. In 4 dimensions, their degrees of freedom are, of course, two. 108See, e.g., Amati et al. [ 3,4] and ’t Hooft [ 219]. 109See, e.g., Strominger and Fava [ 208] and Horowitz et al. [ 123]. 110AdS space and CFT symmetry may be introduced as follows. AdS space, in Ddimensions, may be deﬁned in terms of coordinates zD.z0;z1;:::; zD/NUL1;zD/satisfying a quadratic equationPD/NUL1 kD1.zk/2/NUL.z0/2/NUL.zD/2D/NUL R2, for a given constant R2, embedded in a .DC1/dimensional space with interval squared d s2DPD/NUL1
jD1dzj2/NULdz02/NULdzD2. On the other hand a D/NULSphere is deﬁned in terms of coordinates y1;:::; yDC1satisfying a quadratic equationPDC1 jD1.yj/2D/SUB2 for a given constant /SUB. The conformal group, as applied in 4 dimensional Minkowski spacetime, is deﬁned by a scale transformation x/SYN!/NAKx/SYN, and a so-called special (conformal) transformation x0/SYN x02Dx/SYN x2Ca/SYN; for a constant 4-vector a/SYN, in addition to the Poincaré ones. 111Maldacena [ 146]. See also Witten [ 245], Gubser et al. [ 109], and Aharoni et al. [ 2].
1 Introduction 31 supersymmetric gauge theories. This brings us into contact with the holographic principle, in analogy to holography in capturing 3 dimensional images of objectson a 2 dimensional (holographic) plate, 112showing that an equivalence relation exists between the 3 and the 2 dimensional set-ups. The 4 dimensional quantumﬁeld theory is like a hologram capturing information about the higher dimensionalquantum gravity theory. In this case the SU( N) theory provides a holographic description of gravitational ﬁeld. This is in analogy to black hole entropy withits encoded information being proportional to the area rather than to the volumeof the region enclosed by the horizon. Perhaps holography is a basic property ofstring theory and one expects that much has to be done before developing a realisticquantum gravity, and in turn provide a b ackground independent formulation for string theory. The holographic principle was ﬁrst proposed by ’t Hooft. 113
We close this chapter by commenting on two symmetries which seem to be observed in Nature, that is of the CPT symmetry and of the Spin & Statisticsconnection and of their relevance to our own existence. We will see how these symmetries arise from quantum ﬁeld theory in Sect. 4.10 and Sect. 4.5, respectively. CPT taken in any order, seems to be an observed symmetry in Nature, where C stands for charge conjugation with which par ticles are replaced by their antiparticles and vice versa, P represents space reﬂ ection, while T denotes time reversal. Local Lorentz invariant quantum ﬁeld theories preserve (Sect. 4.10)t h e C P T symmetry. Experimentally, symmetry violations are well known to occur whenone restricts to one or to the product of two transformations in CPT in dealingwith some fundamental processes. For example the violation of parity was alreadyestablished in 1957 114as well as the violation of charge symmetry.115Later, in
1964 CP violation, and hence also of T, was observed in neutral Kaon decays.116 The CP transformation and C, provide the fundamental relations between matterand anti-matter. The question then arises as to why we observe, apart in acceleratorexperiments, only one form (matter) than the other form in the visible universe –a key criterion for our own existence . If an equal amount of matter and anti-matter was produced at some stage then why, our visible universe is matter dominated.Sakharov in 1967 [ 177] proposed that a key reason for this is CP violation. In more details to explain this asymmetry, he proposed that (1) baryon number is notconserved. (This is supported by recent gra nd uniﬁed ﬁeld theories,) (2) CP and C are violated, (3) the universe has gone through a phase of extremely rapid expansionto avoid the pairing of matter and anti-matter. The violation of such symmetries, at 112Recall that the twodimensional holographic plate which registers the interference of reﬂected
light off an object and an unperturbed Laser beam stores information of the shape of the three dimensional object. As one shines a Laser beam on it an image of the three dimensional object emerges. 113’t Hooft [ 220], see also especially Thorn [ 223], as well as the analysis with further interpretations by Susskind [ 212]. See also Bousso [ 27]. 114Wu et al. [ 246], Garwin et al. [ 91], and Friedman and Telegdi [ 86]. 115Garwin et al. [ 91]. 116Christenson et al. [ 32].
32 1 Introduction the microscopic level, and their consequences on a macroscopic scale is certainly intriguing. Clearly, the “Spin & Statistics” connection, of which the Pauli exclusion principle is a special case applicable to spin 1/ 2 particles, is important not onlyin physics but in all of the sciences, and is relevant to our own existence. Forone thing, the periodic table of elements in chemistry is based on the exclusionprinciple. In simplest terms, the upshot of this is that half-odd-integer spin ﬁeldsare quantized by anti-commutators, while integer spins ﬁelds are quantized bycommutators. This result is of utmost signiﬁcance for our existence. As a matter offact the Pauli exclusion principle is not only sufﬁcient for the stability of matter 117 in our world but it is also necessary.118In the problem of stability of neutral
matter, with a ﬁnite number of electrons per atom, but involving several nuclei, andcorrespondingly a large number of electrons N, the stability of matter, based on the Pauli exclusion principle, or instability of so-called “bosonic matter”, in which theexclusion principle is abolished, rests rather on the following. For “bosonic matter”,the ground state energy E N/CAN/NUL N˛, with˛>1 ,119where.NCN/denotes the number of the negatively charged particles plus an equal number of positivelycharged particles. This behavior for “bosonic matter” is unlike that of matter, withthe exclusion principle, for which ˛D1. 120A power law behavior with ˛>1 implies instability as the formation of a single system consisting of .2NC2N/ particles is favored over two separate systems brought together each consistingof.NCN/particles, and the energy released upon collapse of the two systems into one, being proportional to Œ.2N/ ˛/NUL2.N/˛/c141, will be overwhelmingly large
for realistic large N, e.g., N/CAN1023. Dyson [ 61], has estimated that without the exclusion principle, the assembly of two macroscopic objects would release energycomparable to that of an atomic bomb, and such “matter” in bulk would collapseinto a condensed high-density phase and our world will cease to exist. 121Ordinary matter, due to the exclusion principle, occupies a very large volume. This pointwas emphasized by Ehrenfest in a discussion with Pauli in 1931 122on the occasion of the Lorentz medal to this effect: “We take a piece of metal, or a stone. Whenwe think about it, we are astonished that this quantity of matter should occupy solarge a volume”. He went on by stating that the exclusion principle is the reason:“Answer: only the Pauli principle, no two electrons in the same state”. In this regard, 117For a pedagogical treatment of the problem of “stability of matter” and related intricacies, see Manoukian [ 151], Chapter 14. 118Lieb and Thirring [ 144] and Thirring [ 222].
119Dyson [ 61], Lieb [ 143], and Manoukian and Muthaporn [ 156]. 120Lieb and Thirring [ 144] and Thirring [ 222]. 121In the Preface of Tomonaga’s book on spin [ 226], one reads: “The existence of spin, and the statistics associated with it, is the most subtle and ingenious design of Nature – without it the whole universe would collapse”. 122See Ehrenfest [62 ].
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