What is String Theory ? - present status from a historical perspective - NTU, 2011 Prologue：What is physics? “The endeavor of exploring, on the basis of empirical facts, the most fundamental laws which exist behind all kinds of phenomena surrounding us.” from What is physics ?, Shin-ichiro Tomonaga, 1979 As the understanding of basic physical laws were developed deeper, our horizons have been broadened, both in macroscopic and microscopic directions. “More is diﬀerent” (P. W. Anderson, 1972) and Deeper is more ! What is the most fundamental problem remaining unsolved in the 20th century physics? 量子力学 相対性理論 Conflict between two central pillars of modern physics, quantum theory and general relativity 相対性理論 量子力学 General Relativity LHS: ‘made of marble‘ Geometry of space-time RH: ‘made of mere wood’ Quantum theory of matter according to Einstein himself. The principles of general relativity are governed entirely by classical concepts of particles and fields. No uncertainties, and no fluctuations at all. “The real goal of my research has always been the simplification and unification of the system of theoretical physics. I attained this goal satisfactorily for macroscopic phenomena, but not for the phenomena of quanta and atomic structure. .................. I believe that despite considerable success, the modern quantum theory is still far from a satisfactory solution of the latter group of problems.” A. Einstein, 1936 Quantum Theory : characterized by fluctuations of physical states superposition principle uncertain relation The constitution and the stability of matter can never be explained without quantum theory. Even the vacuum and the geometry of space-time itself are microscopically fluctuating and superposing among themselves, by pair creation/annihilation of all sorts of elementary particles. Quantum Gravity: quantum theory of space-time geometry creation/annihilation of gravitons If general relativity is combined naively to the ordinary rules ! we encounter serious difficulties. of quantization, P/π Q/2π $ "# Non-renormalizable ultra-violet 2n κ divergences, due to space-timefn (D−2)/2 fluctuations: a n "# n κ a(D−2)/2 $2n fn Information puzzle in (approximate) quantum mechanics around black-holes: This is related to the ultraviolet difficulty, since the wave lengths of signals reaching infinity must be infinitely short at the spacetime horizon. (Time lapse becomes infinitely slow as seen from an outside observer.) a→0 cut-off length scale at small distances a→0 This conflict indicates that General Relativity is not a fundamental theory, but is ・a low-energy effective theory which is valid only at sufficiently long distances, but breaks down microscopically. historical analogy of low-energy effective theory: - four-fermi theory of weak interactions Weinberg-Salam theory - sigma model for low-lying hadrons QCD Quantum theory and general relativistic theory of gravity must be truly unified, by replacing general relativity by a more fundamental microscopic theory of gravity But, the conceptual framework of these theories are quite different : ・the concept of “states” in quantum theory is defined globally on the basis of linear superposition principle. ・space-time and states in general relativity are defined on the basis of patching works of local events . Unfortunately, we do know any valid direct approaches, so far, to the unification of these two different conceptual frameworks in the framework of (quantum) local field theory. And also, old attempts towards non-local field theories suffered from either violation of quantum mechanical unitarity and or Lorentz invariance. String Theory suggests new intrinsic mechanisms for a complete elimination of ultraviolet infinities and for unifying all fundamental interactions including gravity, within the general framework of quantum theory (at least in the perturbation theory). But, unfortunately, is still immature and is in the “making” stage. What is string ? fundamental string (‘ F1’) only has 1 dimensional extension ・mass density : ・ strictly constant string theory the unique fundamental constant of string length constant closed string ： generate gravitational field open string ： generate gauge fields ・closed string interacts with all kinds of strings in a universal manner which is consistent with general relativity at long distances. ・open strings interacts among themselves in a universal manner which is consistent with gauge theory at long distances. classical images of strings in motion time space graviton dilaton In string theory, gravity and gauge interactions are “emergent” phenomena arising from a single dynamical framework of quantum mechanics of relativistic strings : quantum fluctuations of strings quantum fluctuations of space-time and gauge fields and the condensation of various modes of strings induces space-time curvatures, coupling constant, and all allowed `external` fields: the structure to be expected for a backgroundindependent theory. Furthermore, closed strings can also be regarded as being induced from quantum fluctuations (or loop effects) of open strings. related to gauge/gravity correspondence, later “String theory is, at a new level, the realization of old ideas concerning induced gravitation! I cannot refrain from feeling proud on this point!” General Relativity and Gravitation, Vol. 32, No. 2, 2000 A. D. Sakharov (1921-89), 1985 Vacuum Quantum Fluctuations in Curved Space and the Theory of Gravitation† Academician A. D. Sakharov Translated from Doklady Akademii Nauk SSSR, vol. 177, No. 1, pp. 70–71, November 1967. Original article submitted August 28, 1967. In Einstein’s theory of gravitation one postulates that the action of spacetime depends on the curvature (R is the invariant of the Ricci tensor): 1 S(R) = − 16πG ! √ (dx) −g R. (1) Nobel peace prize 1975 If we neglect dilation of violin strings, the degrees of freedom of vibration at each point are ２. 2=3+1-(1+1) space dimensions time The fundamental string live in (critical) 10 (=1+9) dimensional space-time, and the vibrational degrees of freedom are only from transverse directions 8=9+1-(1+1) This is why graviton and gauge ﬁelds can emerge. Potentially, the critical dimension suggests that string theory might be able to explain why our space-time is 4 dimensional, provided that we find appropriate dynamical mechanisms of reducing 10 dimensions to 4. n=0 n=0 sion, of the string sented per unit b 2.1 Discovery of relativistic strings sion, of the s 2.1 (first Discovery of relativistic strings integrals, we have to spec String theory evolved from a proposal made in the late equality, ‘t-channel’ description) or through forma- (first index 0 equalit Brief history of string theory w to is thethe initi evolved from a proposal made the late integrals, 60s forString ation particular 2↔2 scattering amplitude, called the in corresponding oftheory resonance-like states (second equality, ‘s-channel’ tion of reson 2 suppressed incorrespondi the present s ‘Veneziano of mesons which satisfies a special called 60s description). for formula’, a particular 2↔2 scattering amplitude, the description) Here, s and t are Lorentz invariant combinational fig 1968 Veneziano model S is an integral over a 2 symmetry requirement calledofs-tmesons ‘channel’ duality. The Σ i ‘Veneziano formula’, which satisfies a special 2 2suppressed tions of ener tions ofthat energy-momenta s= −(p1 +p-of2Channel )ele,t = −(p +p3form ) , essary fi 2the (s-t) duality takes latter demands the amplitude is composed # is an in symmetry duality. The # requirement called s-t ‘channel’ 3S! Σ and α and Regge behavior and α and α are two parameters. It soon turned out grees of 0 ments such as the formula 2 - narrow-resonance d ξthis L(X, takes the fo = that am latter demands that the amplitude is composed of ele! 1 that this amplitude and its various generalizations can Σ The con approximation ! ! −α s−α −α t−α0 −1 0 −1 be interpret ments as the formula dx x (1 − x) (1) V (s,be t) such = interpreted in terms of the dynamics of relativistic fies the with ! 1 0 open strings ! ! provided α open strings propagating space-time, ent top 0 = 1. L −α s−α −1 −α t−α0 −1 0 = g (X)∂ The analogo which can equally be described by exchanges of particles µν dx x (1) V (s, t) = 4 (1 − x) with analogous amplitudes which correspond to closed Rieman betweenThe two interacting strings 0 particles where (ξ1 , ξ2 ) with z were = ξ1 poles at strings were dlesexam and ∞also constructed. ∞ For " " dimensional coordinates pa which can equally be described by exchanges of particles rn (s) rn (t) # 2 V (s, t) = = (2) (n − 1)/α a For example, the pole singularities at s or t = m = χ(Σ) ≡ face Σ. The space-time co n 2 2 between two #interacting particles t − m s − mn where n µ (ξ1 , ξ n=0 n=0 of strings sented states by fields X (ξ) (µw (n − 1)/α are interpreted as representing possible Rieman ∞ ∞ " " dimensional 2 number of d r (s) r (t) (first equality, ‘t-channel’ description) or through formaindex 0 being the time n n of strings with definite (mass) . There are an infinite mation V (s, t) = (second equality, = ‘s-channel’ (2) face Σ. The and rotation tion of resonance-like states is the metric tensor of t 2 2 t − m s − m spectrum ofn=0 relativistic open strings number of them corresponding to various vibrational inserted n n n=0 that forψ description). Here, s and t are Lorentz invariant combina- tional field variable infi sented bycom and rotational modes of strings. Actually, spective Nambu, Susskind, ..... 2Nielsen, 2 it also turned string are tions(first of energy-momenta s = −(p1description) +p2 ) , t = −(por essary fields, index which us 2 +p 3) , equality, ‘t-channel’ through forma0theor bei 3 thatα0for completely consistent formulations of quantum ing infin andtion α# and are two parameters. It soon turned out grees of freedom, such as s sions must of resonance-like states (second equality, ‘s-channel’ is the metr 5 string theory, necessary that the space-time dimen- sions), strip that this amplitude and it itsisvarious generalizations can The constant gs , of called st 26, Similar formula (Virasoro, Shapiro), description). Here, s and t are Lorentz invariant combina- tional fieldor be interpreted in terms of the dynamics of relativistic fies the of final) Riemann sions must be at some particular value (critical dimenex fermi 2 2weightsheet) of energy-momenta s = −(p essary fields corresponding to closed strings 1 +p2 )α0, t==1.−(pent 2 +ptopologies. 3) , opentions strings propagating space-time, provided It is well o can elimina sions), 26, or if we want to include space-time (and worldtively, Y. Nambu (1921~ ) # 3 4 and α and α are two parameters. It soon turned out grees of2 clas free 0 The analogous amplitudes Riemann surfaces are 6 which correspond to closed (mass) with sheet) fermions consistently, at 10. In the latter case we This In the 1970s, the string theory made its development from the S-matrix formula to a dynamical theory of relativistic strings. Various ideas which would become the basis for later developments was emerging from the studies of its connection to local field theory. 1 Gravity from strings: personal reminiscences of early developments Tamiaki Yoneya Institute of Physics, University of Tokyo Komaba, Meguro-ku, Tokyo 153-8902, Japan Abstract I discuss the early developments of string theory with respect to its connection with gauge theory and general relativity from my own perspective. The period covered is mainly from 1969 to 1974, during which I became involved in research on dual string models as a graduate student. My thinking towards the recognition of string theory as an extended quantum theory of gravity is described. Some retrospective remarks on my later works related to this subject are also given. (arXiv:0911.1624,) to appear in The Birth of String Theory, Cambridge Univ. Press. 1970 ~ 1978 Initial developments (models for hadronic interactions) Nambu-Goto action Light-cone quantization, no-ghost theorem, critical dimensions (26 or 10) Ultraviolet finiteness (modular invariance) Neveu-Schwarz-Ramond model (inclusion of “G”-partiry and fermionic degrees of freedom) Space-time supersymmetry Developments related to field-theory /string connection (‘70s) `Fishnet’ diagram interpretation, Nielsen-Olesen vortex Derivation of gauge theory, general relativity and supergravity from strings in the zeroslope limit gravity and unification Construction of various supersymmetric gauge and gravity theories String picture from strong-coupling lattice gauge theory t` Hooft’s large N limit 1984~1989 First revolution Green-Schwarz anomaly cancelation Five consistent perturbative string vacua (I, IIA, IIB, 2xHetro) in 10D Compactifications, new connections to mathematics CFT technique, renormalization group interpretation 1990~1994 Development of “old” matrix models Double scaling limit c=1 strings, 2D gravity, ‘non-critical’ strings topological field theories and strings 1995~1999 Second revolution discovery of D-branes statistical interpretation of black-hole entropy in the BPS or near-BPS limits conjecture of M-theory New matrix models, supermembranes, M(atrix) theory conjecture, ..... AdS/CFT correspondence, ....... The most important development after 1998 : General idea of gauge-gravity correspondence has been flourishing. starting from Maldacena’s paper in 1997 unification of two old ideas on strings from the 70s ? hadronic strings for quark confinement from gauge theory string theory for ultimate unification as an extension of general relativity Gauge/Gravity correspondence The fact that gravity (general relativity) and gauge theory are united in a single framework leads to an entirely new connection between them, giving nontrivial predictions to the gaugetheory side from gravity side (and vice versa) a realization of `holography’: quantum gravity in the bulk must be formulated using only the degrees of freedom on the boundary. (‘t Hooft, Susskind, .....) open-closed string duality in string perturbation theory (simplest one-loop case) D-branes creation and annihilation of open strings exchange of closed strings (including graviton exchange) analytic continuation effective theory =gauge theory effective theory =gravity description on the boundary description in the bulk ators with the exponents η = 1 + 4!/5 Gravity Correspondence ! Example: 1999) on"the correlation functions for a strong-coupling 1 (Sekino-T.Y. +ij Predictions Str F X · · · X ≡ ij i il gaugeJltheory dimensional super Yang-Mills theory, the so-called `Matrix 1 ,i1 ,··· ,i(0+1 1. theory’) has l N Overview: recently been checked through Monte Carlo simulations Matrix theory 2. D-partilce interactions 0.01 (!O(t)O(t " )# ∼ (|t − t " |−η )) Three typical examples: 0.0001 ! 3. Strings as wrapped membranes J + -type operators with the exponents η = 1 + 4!/5 1e-006 +ij Jl ,i1 ,··· ,il + J1 , N=3, !=12 -4 1.8 3.15*10 /t + J2 , N=3, !=12 -5 2.6 1.68*10 /t + J3 , N=3, !=16 -6 3.4 1.24*10 /t + J4 , N=3, !=16 5.16*10-8 +/t4.2 1e-008 !4. Strong" 1 Fij Xi1 · · · Xil ≡ Str coupling behavior of N Matrix theory " )# ∼ " |−η )) 5. " " − η Three typical examples: (!O(t)O(t (|t − t Three typical examples: (!O(t)O(t )# ∼ (|t − t |Concluding )) remarks: 0.01 ! J -type operators with the exponentscan ηwe= 1 + 4!/5 1e-012 ! J + -type operators with the exponents η = 1 + 4!/5 1e-010 0.1 1 t make progress towards M-theory? 0.0001 ! 1 +ij !≡ Str F X" 1 +ij J 1e-006 ij i1 =J1, 2, 3, 4)≡are plotted (in l ,i1 ,··· ,il Fij X Str · · · X il N i1 l ,i1 ,··· ,il · · · Xil point functions !Ji+ (t)Ji+ (0)# (i N for N = 3, β = 4, Λ = 12 for i = 1, 2 and1e-008 Λ= 16Rev. forLett. J104 Hanada-Nishimura-Sekino-T.Y.. Phys. (2010)151601 , N=3, !=12 3.15*10 /t −1/3 see also arXiv:1108.5153[hep-th] , N=3, !=12 J . nit of time length is λ 1.68*10 /t 1 2 + -4 1.8 + -5 2.6 + J , N=3, !=16 " The meaning of string theory ・encompasses almost all relevant ideas and/or methods devised in the past towards unification ・provides an entirely new scheme of unifying all interactions including gravity (motion or spectra and interaction are completely unified) ・provides a microscopic explanation of black hole entropy in terms of quantum statistical language using D-branes (albeit in some special cases) ・resolves non-renormalizable divergences in terms of intrinsic non-locality in a way which is consistent with unitarity X2× cance tates. It seems reasonable to expect that this effect completely 1 X3× The characterization of short distance structure of X1×1 ance effect with respect to the center-of-mass coordinates of string · p = 4 g = 0 g = 1 g = 2 2g c X space-time embodied in string theory relates to one of 2×2 hese appreciably to physical processes. Si 4 ghigher = 0 gmodes = 1 contribute g=2 2g X3×3 · · mostof fundamental questionscorresponding of quantum gravity. magnitude the spatial extension to a large energy · XN × 2 sim · ∆E, we are led to a remarkably s expected to behave as ∆X ∼ " · Ω = Ω1 + Ω2 , Ω1 ∩ Ωs2 = ∅ · a possible expression : directions · Ω Ω1 +of Ω2magnitude , Ω1 ∩ Ω2∆X = ∅ for fluctuations along spatial he=order XN ×N uncertainty relation of space and time · participating within the time interval ∆T = ∆t of interactions: · ∆E∆t ! h φ+ 2 > ∆X∆T " ∆E∆t ! h ∼ s. φ+ φ− φ+ F h ∼ ∆X t is natural ∆E to call this2 relation ∆t =the ∆T ‘space-time uncertainty relatio ! s h this relation + mphasized that is not a modification of the φusual F φ− unce ∆E ∼ ∆X 2 ∆t = ∆T ∆T ∼ scale along the lo ∗) s be ・This!can interpreted the we use units in which ¯ Throughout the present as paper, h = 1, c = 1. space-time representation of the world-sheet conformal symmetry which governs the quantum dynamics of strings and is responsible for the reduction of degrees of freedom both in UV and IR regions. ∆T ∼ scale along the longitudinal ∆X ∼ scale along the ∆X ∼ scale along the transverse d To summarize: string theory enables us to derive general relativity and gauge interaction naturally from quantum mechanics of fundamental strings. In this sense, string theory provides the ingredients which are necessary for achieving the final unification. (This is already a grand prediction!) At the present stage of development, however, it is still an immature ‘theory’. ・we only have a perturbative definition for constructing scattering amplitudes. ・we do not know the primordial principles behind it. ・we cannot yet make any definite predictions to the real world. The present immaturity of string theory does not mean that the theory cannot make any predictions in principle. There is no other competing theory, exhibiting such a tight structure of self-consistency and hence in principle of highest predictability at least potentially. The present status of string theory is somewhat similar to that of early quantum theory before 1924. We do not yet have true languages for describing string theory. It is unclear to what extent various recent ideas (say, the “string landscape”, “world-brane scenarios”, ...) for model building survive when we would finally find such true languages. We need non-perturbative definition of string theory such that its whole apparatus could be put on computers (remember how lattice gauge theory has been benefitting QCD). We have to be patient and to learn more, in view of such a grandiose synthesis we are looking for ! On the experimental side, hopefully, various increasingly precise data from cosmological observations (background fluctuations, dark matter, dark energy, etc) would provide us key basis for the future development. Epilogue: Historical remarks ・ Dirac’s anticipation of strings in 1955 “The lines would then be the elementary concept in terms of which the whole theory of electrons and the electromagnetic field would have to be built up. Closed lines would be interpreted as photons and open lines would have their ends interpreted as electrons or positrons. ...” P. A. M. Dirac (1902-1984) GAUGE-INVARIANT FORMULATION OF QUANTUM ELECTRODYNAMICS1 Canadian Journal of Physics, 33(1955) 650 ABSTRACT Electrodynamics is formulated so as t o be manifestly invariant under general gauge transformations, through being built up entirely in terms of gauge-invariant dynamical variables. The quantization of the theory can be carried out by the usual rules and meets with the usual difficulties. I t is found that the gauge-invariant operation of creation of an electron involves the simultaneous creation of an electron and of the Coulomb field around it. The requirement of manifest gauge invariance preventsonefrom using the concept of a n electron separated from its Coulomb field. H. Yukawa (1907-1981) was also one of main advocates of non-local field theory from the late 1940s. 「素粒子を点と思っていたのでは、 てん で話にならない」 It is pointless to conceive elementary particles as mere points. Einstein: General Relativity Quantum Mechanics Kaluza Klein Supergravity Weyl Gauge Principle Super Symmetry Standard Model Quantum Field Theory Black hole Unitarity puzzle (information problem) Yang-Mills Theory UV problem Nonrenormalizability Superstring M-theory Gauge/string Correspondence ‘holography’ Web of Unification Perhaps, we are at the dawn of a new era of physics. The Night sky in Zermatt ©T.Y. 1983 What would be the appearance of string theory after the next 30 years? Thanks ! Matterhorn ©T.Y., 1983

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