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String Theory Niklas Beisert ITP, PSI Summer School More than Higgs { Effective Theories for Particle Physics Lyceum Alpinum, Zuoz 21 August 2014 Introduction You've heard a lot about particle physics in the past few days: • particles of the standard model; • particles that have long been predicted, and that have recently been observed; • particles that have long been predicted, and that have recently not been observed; • further conjectured particles, that have not been observed yet, and probably never will be observed, because they probably do not exist. This talk: String Theory Zuoz 2014, Niklas Beisert 2 Effective Theory Why talk about String Theory? Here?! • string theory is a model of particle physics . • . at low energies (compared to Planck scale). • it predicts new particles and features, many of them, . • . at least at excessively high energies. • so we can only dream about them. After this talk, please! Many results in string theory use effective theory concepts. • In fact, if you believe string theory describes nature, all that you've heard about here is nothing but an effective theory of string theory. but still a long way to go. Zuoz 2014, Niklas Beisert 3 Overview Therefore let me introduce string theory and some of its basic concepts: • What is String Theory? • Problem of Quantum Gravity. • String Theory Basics. • String Theory and Gravity. • D-branes. Material from a one-semester lecture course . Zuoz 2014, Niklas Beisert 4 I. What is String Theory? Zuoz 2014, Niklas Beisert 5 Einstein with a violin? Zuoz 2014, Niklas Beisert 6 Nothing to do with violins! Zuoz 2014, Niklas Beisert 7 What is String Theory? String Theory: [Wikipedia] • In physics, string theory is a theoretical framework in which the point-like particles of particle physics are replaced by one-dimensional objects called strings. • String theory aims to explain all types of observed elementary particles using quantum states of these strings. • In addition to the particles postulated by the standard model of particle physics, string theory naturally incorporates gravity, and so is a candidate for a theory of everything, a self-contained mathematical model that describes all fundamental forces and forms of matter. • Besides this hypothesized role in particle physics, string theory is now widely used as a theoretical tool in physics, and has shed light on many aspects of quantum field theory and quantum gravity. Zuoz 2014, Niklas Beisert 8 What else is String Theory? • incredibly exciting (for some), • rich in hidden, exceptional and beautiful structures, • popular: I Brian Greene, The Elegant Universe (book, TV) I Woody Allen, Whatever Works (movie) I ... Zuoz 2014, Niklas Beisert 9 Cartoon: Witten's Dog from Futurama, Episode 11: Mars University Zuoz 2014, Niklas Beisert 10 String Theory vs. Loop Quantum Gravity Except from \The Big Bang Theory" Episode 2.2 { \The Codpiece Topology" Zuoz 2014, Niklas Beisert 11 So What is String Theory? Cartoon on String Theory xkcd webcomic: 171 Zuoz 2014, Niklas Beisert 12 Strings String theory is a physical model whose fundamental objects are strings. Strings have the following properties: • spatially 1D extended • with tension • without inner structure • open or closed Zuoz 2014, Niklas Beisert 13 Spacetime Diagrams When painting the path of a string one gets 2D surfaces in spacetime: compare: string open rotating closed particle motion string string string Zuoz 2014, Niklas Beisert 14 String Interactions fundamental interactions between strings: open closed mixed compare: strings strings strings point-particle Benefits of string theory (precitive power): There are merely two constants of nature, namely • string tension (within a string), • string coupling (coupling between strings). Zuoz 2014, Niklas Beisert 15 Cartoon: Witten's Dog from Futurama, Episode 11: Mars University Zuoz 2014, Niklas Beisert 16 Why String Theorie? Cartoon on String Theory xkcd webcomic: 171 Zuoz 2014, Niklas Beisert 17 String Spectrum How does this relate to particle physics? Limit of large string tension: many types of particles! 2 2 2 linear spectrum mn = m0 + n · ∆m . spectrum of hadronic excitations similar! String theory was a promising candidate. Later better explanation: quark model, QCD, SM. Instead string theory incorporates a consistent quantum gravity! Zuoz 2014, Niklas Beisert 18 II. Problem of Quantum Gravity Zuoz 2014, Niklas Beisert 19 Einstein Gravity Einstein's general relativity described by Hilbert action 1 Z S = d4x p− det g R[g]: GR 16πG • fundamental field is metric gµν measuring distances. µν µν ρ • R = g Rµν[g] = g R µρν[g] is curvature scalar. • coupling to matter via diffeo.-covariant derivative: @µ ! Dµ[g]. Gravity has a non-linear action translating to infinitely many vertices p 3=2 2 SGR = + G + G + G + G + :::: Not a fundamental problem; moreover G is small (gravity is weak). Zuoz 2014, Niklas Beisert 20 Non-Renormalisability Additional terms in gravity action conceivable Z S ∼ d4xR + ∗R2 + ∗R3 + ∗D2R2 + :::: Translates to additional interaction vertices p 3=2 S ∼ G + (G + c4) + (G + ∗) + :::: Additional constants required to absorb divergences in loops, e.g. 2 3 (G + c4) + G + G + :::: Problem: Infinitely many constants needed to absorb all divergences! Zuoz 2014, Niklas Beisert 21 String Theory and Quantum Gravity String theory promises to solve the problem of quantum gravity: • string excitation spectrum contains massless spin-2 particles; • they behave as Einstein gravitons to first approximation; • string theory is finite, no divergences (of the above kind); • string theory has just two fundamental coupling constants. Is all well now!? Beware of the vacuum. Other Features. • String spectrum contains many other particles (gauge bosons, . ); but still unclear how to extract the standard model. • GUT-like groups SU(5), SO(10),..., E8 appear. • String theory is supersymmetric; need to break at low energies. • Exciting geometrical / mathematical concepts arise. Zuoz 2014, Niklas Beisert 22 III. String Theory Basics Zuoz 2014, Niklas Beisert 23 Particle Worldline Action Start with an analogous system: Relativistic Particle Embedding of worldline µ D−1;1 X (τ): R ! R : Proper time action Z Z Z q dXµ S ∼ m ds := m dτ jX_ j := m dτ −X_ µX_ ; X_ µ = : µ dτ Equations of motion m P_ µ = 0 with P µ = X_ µ: jX_ j Note: momentum P µ and equation of motion covariant under reparametrisation τ ! τ 0(τ). Zuoz 2014, Niklas Beisert 24 New Particle Worldline Action Can use an alternative action with additional field e(τ): R ! R Z 1 −1 _ µ _ 1 2 S ∼ dτ − 2 e X Xµ + 2 em : Equations of motions X_ µ P_ µ = 0 with P µ = and jX_ j = em: e First equation linear in X_ µ and equivalent to above (with second). Second equation is algebraic in e! • e does not carry additional (quantum) degrees of freedom; • recover the original action upon substituting e = jX_ j=m. Z m jX_ j Z _ µ _ 2 _ S ∼ dτ − X Xµ + m = dτ mjXj: 2jX_ j 2m Zuoz 2014, Niklas Beisert 25 Nambu{Goto Action Repeat steps for a string embedding µ 1;1 D−1;1 X (τ; σ): R ! R Proper area action (Nambu{Goto) Z 2 p µ S ∼ T d ξ − det γ with γαβ := @αX @βXµ: Remarks: • γαβ[pX] is induced metric on worldsheet; • d2σ − det γ is worldsheet \area" element; • T is string tension. Equation of motion extremises worldsheet area p αβ µ @α − det γ γ @βX = 0: Equation is invariant under worldsheet reparametrisation ξα 7! ξ0α(ξ). Zuoz 2014, Niklas Beisert 26 Polyakov Action Nambu{Goto equations of motion non-linear; difficult!! Introduce worldsheet metric gαβ T Z S ∼ d2ξp− det g gαβ @ Xµ @ X : 2 α β µ Again, two equations of motion p αβ µ 1 γδ @α − det g g @βX = 0; γαβ[X] − 2 gαβg γγδ[X] = 0: Remarks: • first equation linear in X; • second equation algebraic in g; worldsheet metric proportional to induced metric: gαβ ∼ γαβ[X]; • equations equivalent to original ones. Zuoz 2014, Niklas Beisert 27 Symmetries and Conformal Gauge Action and equations of motion invariant under: µ µ ν µ • target space Poincar´etransformations X (ξ) ! M νX (ξ) + B . • worldsheet diffeomorphisms ξα 7! ξ0α(ξ); • worldsheet metric rescaling gαβ(ξ) ! α(ξ)gαβ(ξ); 2 + 1 fields carry redundant information (gauge). Conformal Gauge: Exploit redundancy to set gαβ = ηαβ (3 eq.). Reduced action and equations of motion: T Z S ∼ d2ξ ηαβ @ Xµ @ X ; ηαβ@ @ Xµ = 0: 2 α β µ α β Almost simple linear system, Xµ are massless waves on worldsheet. Do not forget other equation! Virasoro constraint (non-linear): 1 γδ Tαβ[X] := γαβ[X] − 2 ηαβη γγδ[X] = 0: Zuoz 2014, Niklas Beisert 28 String Modes General solution for X by massless plane waves Z dk Xµ(ξ) = aµ(k) eikσ+ijkjτ + aµ(k)∗ e−ikσ−ijkjτ : 2jkj Need to impose boundary conditions. E.g. closed string: Xµ(τ; σ + 2π) = Xµ(τ; σ): Periodicity forces k to be integer: string Fourier modes µ 1 µ µ p τ X αL;n αR;n Xµ(ξ) = xµ + + Im p e−in(τ+σ) + p e−in(τ−σ) : 2πT n=1 πT n πT n Solutions parametrised by: • centre of gravity position and momentum xµ, pµ; µ • string mode excitation amplitudes αL/R;n, n = 1; 2;:::. Zuoz 2014, Niklas Beisert 29 Quantisation We want to quantise the system • a free particle (xµ, pµ) µ µ,∗ • a set of harmonic oscillators (αR=L;n, αR=L;n) Canonical quantisation: µ ν µν µ ν;∗ µν x ; p = iη ; αR=L;m; αR=L;n = mδm;nη : States are tensor products of • momentum eigenstate jP i; • excitation eigenstate jNi = (αy)N j0i; for each string mode. String ground state j0; P i = jP i ⊗ j0i ⊗ j0i ⊗ ::: µ µ µ • Almost point particle α ≈ 0: X ≈ x +pp τ=2πT .
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