Outline 1. Introduction 1.1. Historical Overview 1.2. The Theory 2. The Relativistic String 2.1. Set Up 2.2. The Relativistic Point Particle 2.3. The

Transcription

1 Classical String Theory Proseminar in Theoretical Physics David Reutter ETH Zürich April 15, 2013

2 Outline 1. Introduction 1.1. Historical Overview 1.2. The Theory 2. The Relativistic String 2.1. Set Up 2.2. The Relativistic Point Particle 2.3. The General p-brane Action 2.4. The Nambu-Goto Action 2.5. Open and Closed Strings 2.6. The Polyakov Action 2.7. The Equations of Motion 2.8. Equivalence of P and NG action 2.9. Symmetries of the Action 3. Wave Equations and Solutions 3.1. Conformal Gauge & Weyl Scaling 3.2. Equations of Motions & Boundary 3.3. Closed String Solution 3.4. Open String Solution 3.5. Virasoro Constraints 3.6. The Witt Algebra Summary April 15, 2013 Classical String Theory 2/52

3 1. Introduction 1.1. Historical Overview 1969 Nambu, Nielsen and Susskind propose a model for the interaction of quarks - quarks connected by strings. q q 1970 s Quantum Chromodynamics is recognized as the theory of strong interaction. String theory needs 10 dimensions to work, a very unlikely assumption (at this time) for an unneeded theory. String theory went into the dustbin of history. q q q g 1 q q q April 15, 2013 Classical String Theory 3/52

4 1. Introduction 1.1. Historical Overview From all calculations (with closed strings) a massless particle with spin 2 appeares. This is exactly the property of the graviton String theory always contains gravity s After the full discovery of the standard model, it was this fact which made string theory reappear as a promising candidate for a theory of quantum gravitation. April 15, 2013 Classical String Theory 4/52

5 1. Introduction 1.2. The Theory In a nutshell: String theorists propose one dimensional fundamental objects ( strings ) instead of zero dimensional elementary particles. The different elementary particles appear as different oscillation modes of strings. Effects become important for small scale and high energy physics. Planck length ( m) Planck energy ( GeV) April 15, 2013 Classical String Theory 5/52

6 1. Introduction 1.2. The Theory Why String theory? Promising (among other things due to the emergence of the graviton in a natural way) candidate for a theory of quantum gravitation or even a theory of everything. Brought many interesting and important mathematical theories to life. Eludes the problem of renormalization in QFT. April 15, 2013 Classical String Theory 6/52

7 Outline 1. Introduction 1.1. Historical Overview 1.2. The Theory 2. The Relativistic String 2.1. Set Up 2.2. The Relativistic Point Particle 2.3. The General p-brane Action 2.4. The Nambu-Goto Action 2.5. Open and Closed Strings 2.6. The Polyakov Action 2.7. The Equations of Motion 2.8. Equivalence of P and NG action 2.9. Symmetries of the Action 3. Wave Equations and Solutions 3.1. Conformal Gauge & Weyl Scaling 3.2. Equations of Motions & Boundary 3.3. Closed String Solution 3.4. Open String Solution 3.5. Virasoro Constraints 3.6. The Witt Algebra Summary April 15, 2013 Classical String Theory 7/52

8 2. The Relativistic String 2.1. Set Up Start by considering strings as fundamental objects moving in a (D dimensional) Lorentzian spacetime (with Lorentzian metric g µν ) and obeying certain dynamical laws. Goal of this chapter: Find an action principle of the free bosonic string and study its dynamics. Idea Strings behave classically as the one dimensional analogon to point particles. Review point particle action. April 15, 2013 Classical String Theory 8/52

9 2. The Relativistic String 2.2. The Relativistic Point Particle Action should be a functional of the particle s path ( world line ). Natural (and only) candidate: length of the world line S = mc ds. γ Proportionality factor mc follows from classical limit (v 0). April 15, 2013 Classical String Theory 9/52

10 2. The Relativistic String 2.2. The Relativistic Point Particle to do this. In practice, however, world-lines are described as parameterized lines, parameterization is used to compute the action. We parameterize the world-line P of a point particle using a parameter τ (Figure 5 parameter must be strictly increasing as the world-line goes from the initial point x final point x µ f, but is otherwise arbitrary. As τ ranges in the interval [τ i, τ f ] it descr motion of the particle. To have a parameterization of the world-line means that expressions for the coordinates x µ as functions of τ: x µ = x µ (τ). parametrization x µ (τ). induced (1x1) metric on the parametrization domain (pullback of ambient metric g µν ) τ x 0 τ f τ i x µ (τ i ) x µ (τ f ) Γ = xµ τ x ν τ g µν Fig. 5.2 volume (or length) form x 1 (x 2,...) Aworld-linefullyparameterizedbyτ. Allspacetimecoordinatesx µ are functions of x µ : [τ i, τ f ] R D ds = Γ dτ = ẋ µ ẋ µ dτ April 15, 2013 Classical String Theory 10/52

11 2. The Relativistic String 2.2. The Relativistic Point Particle Therefore, the point particle action is (writing ẋ µ := dxµ dτ ) τf S point particle = mc ẋµẋ µ dτ τ i Better known form: In eigentime parametrization x µ (τ) = (c τ, x(τ)) the action reads S = mc 2 1 v2 c 2 dt. April 15, 2013 Classical String Theory 11/52

12 2. The Relativistic String 2.2. The Relativistic Point Particle S = mc ẋµẋ µ dτ ( ) d mẋ µ = 0. dτ ẋµẋ µ Disadvantages The squareroot is not easy to quantize. The massless case is not covered. The action has primary constraints (constraints not following from the equation of motion). From the definition p µ = L ẋ, it follows directly that p µ p µ µ = m 2 c 2. More about primary constraints in report. April 15, 2013 Classical String Theory 12/52

13 2. The Relativistic String 2.2. The Relativistic Point Particle Trick The Einbein Action: Define the action S = 1 2 τ1 τ 0 dτ e(τ) (e 2 (τ) (ẋ µ (τ)) 2 m 2) where e(τ) is an auxilliary function (which can be varied independently of x µ ). Varying with respect to x µ and e gives δs δe = 0 ẋ2 + e 2 m 2 = 0 δs δx µ = 0 d ( e 1ẋ µ) = 0. dτ April 15, 2013 Classical String Theory 13/52

14 2. The Relativistic String 2.2. The Relativistic Point Particle e = ẋ 2 m ( ) & d ( e 1ẋ µ) = 0 d mẋ µ = 0. dτ dτ ẋµẋ µ The resulting equations of motion are equivalent to the classical point particle equation of motion together with the primary constraint. Found new equivalent action with no squareroot no primary constraint (old primary constraints are turned into equation of motion) April 15, 2013 Classical String Theory 14/52

15 2. The Relativistic String 2.3. The General p-brane Action Generalize: spatial 0-dim point spatial (p-1)-dim objects in a D dimensional spacetime (called brane). represented by its p dimensional worldvolume (p = 1 worldline, p = 2 worldsheet). X µ : U R D April 15, 2013 Classical String Theory 15/52

16 2. The Relativistic String 2.3. The General p-brane Action In analogy to the point particle the reparametrization invariant action of this world volume is S Vol(world volume). If X µ (τ i ) (µ {1,...D}, i {1,..., p}) is a parametrization of the worldvolume of the brane, then the induced metric on the parametrization domain Γ (p) αβ is the pullback of g µν under X µ Γ (p) αβ = Xµ τ α X ν τ β g µν. April 15, 2013 Classical String Theory 16/52

17 2. The Relativistic String 2.3. The General p-brane Action The general p-brane action therefore reads S Brane = T p det( Xµ τ α X ν τ β g µν) d p+1 τ where T p is a proportionality factor. We are now ready to think about strings, or 1-branes. April 15, 2013 Classical String Theory 17/52

18 2. The Relativistic String 2.4. The Nambu-Goto Action The Nambu Goto action is the action for a string or 1-brane: Parametrization (τ, σ) X µ (τ, σ), then S Nambu-Goto = T 0 det(γ (1) c αβ )dτdσ where the proportionality factor T 0 is called string tension. Calculating the determinant of (X µ := Xµ σ, Ẋµ := ( Xµ Γ (1) Ẋ αβ = Ẋ Ẋ X Xµ τ α Xν g τ β µν = ) X Ẋ X X gives τ ) S NG = T 0 c (Ẋ X ) 2 Ẋ 2 X 2 dτdσ April 15, 2013 Classical String Theory 18/52

19 2. The Relativistic String 2.5. Open and Closed Strings Area functional for spatial surfaces For later: Consider two kinds of string, open and closed. Parametrization domain: x 0 = ct < τ < 0 < σ < σ x 2 (x 3,...) Fig. 6.1 Closed Strings: Worldsheet diffeo. to R S 1, periodicity conditions. Convention σ = 2π The world-sheets traced out by an open string (left) and by a closed string (right). Open Strings: x 3 Worldsheet diffeo. to R [0, σ], boundary conditions (later). Convention σ = π x 1 April 15, 2013 Classical String Theory 19/52 x 2

20 2. The Relativistic String 2.6. The Polyakov Action In principle this is all, we have postulated the dynamics of the string and could start solving the Nambu Goto action. Problem: Nambu-Goto has a square root There are primary constraints (the so called Virasoro constraints later) Trick: Find an action, which has no squareroot, no primary constraint and is equivalent to the NG action same as einbein trick for the relativistic particle... April 15, 2013 Classical String Theory 20/52

21 2. The Relativistic String 2.6. The Polyakov Action The Polyakov action is defined as S P = T d 2 σ hh αβ α X µ β X ν g µν = T 2 2 d 2 σ hh αβ Γ αβ h αβ is a symmetric (2x2) tensor, which can be varied independently of X µ. It is taken as an intrinsic metric tensor on the parametrization domain with inverse h αβ. h αβ is the two dimensional analogon to the einbein function e. The Polyakov action has no primary constraints. April 15, 2013 Classical String Theory 21/52

22 2. The Relativistic String 2.7. The Equations of Motion To do: Find equations of motion and proof that they are equivalent to NG (+ primary constraints). Varying ( htαβ ( hh ) δs = T d 2 σ δh αβ + 2 αβ α β X µ δx µ) + bnd term where T αβ is the energy momentum tensor, the factor appearing when varying S with respect to h αβ or as a functional derivative T αβ = 1 T 1 δs h δh αβ = 1 2 αx µ β X µ 1 4 h αβh γδ γ X µ δ X µ = 1 2 Γ αβ 1 4 h αβh γδ Γ γδ. April 15, 2013 Classical String Theory 22/52

23 2. The Relativistic String 2.8. Equivalence of Polyakov and Nambu-Goto Action The equations of motion are T αβ = 0 α ( hh αβ β X µ) = 0 see blackboard (equivalence of actions) Only for the two dimensional case the actions are equivalent. Remark: Only for minimizing h αβ fixed, we have S NG = S P (in general not the same expression). April 15, 2013 Classical String Theory 23/52

24 2. The Relativistic String 2.9. Symmetries of the Action Global Symmetries: Poincare Invariance δx µ = a µ νx ν + b µ where a µ ν = a µ ν δh αβ = 0 a general infinitesimal Poincare transformation. X µ transforms as expected as a vector, h αβ as a scalar. April 15, 2013 Classical String Theory 24/52

25 2. The Relativistic String 2.9. Symmetries of the Action Local Symmetries: Reparametrization Invariance Follows directly from the choice of action. Formally φ : σ α σ α a reparametrization of the parametrization domain. With ξ α := δφ α and δx µ = ξ α α X µ δh αβ = ξ γ γ h αβ + α ξ γ h γβ + β ξ γ h αγ. Weyl Symmetry a bit more concealed symmetrie, invariance of the action wrt. Weyl scaling of the worldsheet metric h. δx µ = 0 where Λ is an arbitrary function. δh αβ = 2Λh αβ April 15, 2013 Classical String Theory 25/52

26 Outline 1. Introduction 1.1. Historical Overview 1.2. The Theory 2. The Relativistic String 2.1. Set Up 2.2. The Relativistic Point Particle 2.3. The General p-brane Action 2.4. The Nambu-Goto Action 2.5. Open and Closed Strings 2.6. The Polyakov Action 2.7. The Equations of Motion 2.8. Equivalence of P and NG action 2.9. Symmetries of the Action 3. Wave Equations and Solutions 3.1. Conformal Gauge & Weyl Scaling 3.2. Equations of Motions & Boundary 3.3. Closed String Solution 3.4. Open String Solution 3.5. Virasoro Constraints 3.6. The Witt Algebra Summary April 15, 2013 Classical String Theory 26/52

27 3. Wave Equations and Solutions 3.1. Conformal Gauge and Weyl Scaling Goal of this chapter: Simplify equations of motion using the symmetries of the action and find solutions. Using reparametrization invariance we can simplify h αβ. Claim: For any two dimensional Lorentzian (meaning signature (-1,1)) metric h αβ one can find coordinates σ 1, σ 2, such that h αβ = Ω(σ 1, σ 2 )η αβ where η αβ is the Minkowski metric (and Ω a scalar function). April 15, 2013 Classical String Theory 27/52

28 3. Wave Equations and Solutions 3.1. Conformal Gauge and Weyl Scaling Using Weyl Scaling: gauge away Ω (set Ω = 1). Having used all symmetries, we are in a gauge with h αβ = η αβ Such a choice of coordinates is called a conformal gauge. We will from now on work in these coordinates (calling them τ and σ). April 15, 2013 Classical String Theory 28/52

29 3. Wave Equations and Solutions 3.2. Equations of Motions and Boundary Conditions In conformal gauge, the Polyakov action takes a simple form S P = T 2 σ 0 dσ τf The variations fulfill τ i dτ η αβ α X µ β X µ = T 2 σ 0 dσ δx µ (σ = 0, σ) arbitrary for open strings δx µ (σ + 2π) = δx µ (σ) for closed strings δx µ (τ i ) = δx µ (τ f ) = 0 τf τ i (Ẋ2 ) dτ X 2 Varying with respect to X µ (last term vanishes for closed string): δs P = T σ 0 dσ τf τ i dτ δx µ ( σ 2 τ 2 ) τf Xµ T dτ σ X µ δx µ ] σ=σ σ=0 τ i April 15, 2013 Classical String Theory 29/52

30 3. Wave Equations and Solutions 3.2. Equations of Motions and Boundary Conditions This results in the (wave-) equation of motion and boundary conditions ( 2 τ σ 2 ) X µ = 0 X µ (σ + 2π) = X µ (σ) closed string X µ (σ = 0, π) = 0 open string. Neumann condition for open string (=free ends). Dirichlet (δx µ (σ = 0, σ = σ) = 0 would have worked too (fixed ends)). Should not forget to impose the vanishing of the energy momentum tensor T αβ = 0 ( Ẋ ± X ) 2 = 0 Virasoro constraint April 15, 2013 Classical String Theory 30/52

31 3. Wave Equations and Solutions 3.3. General Solution for the Closed String Equations of motion and boundary conditions closed string: wave equation: ( 2 τ σ 2 ) X µ = 0 periodicity condition: X µ (σ + 2π, τ) = X µ (σ, τ) Virasoro constraints: (Ẋ ± X ) 2 = 0 In light cone coordinates σ ± = τ ± σ ± = 1 2 ( τ ± σ ) the wave equation reads : + X µ = 0 April 15, 2013 Classical String Theory 31/52

32 3. Wave Equations and Solutions 3.3. General Solution for the Closed String Solution to wave equation: X µ (σ, σ + ) = X µ R (σ ) + X µ L (σ+ ) X R and X L are arbitrary functions only dependent on boundary conditions left - and right movers. Closed String Besides periodicity no boundary condition X R and X L are independent. This is not the case for open strings Neumann boundary condition connects them (open string = standing waves reflected) April 15, 2013 Classical String Theory 32/52

33 3. Wave Equations and Solutions 3.3. General Solution for the Closed String Claim (proof in report): The 2π periodicity of X µ is equivalent to X µ R (σ ) and + X µ L (σ+ ) are 2π periodic with the same zero-mode. Expand in fourier series: X µ R (σ ) = 1 4πT + X µ L (σ+ ) = 1 4πT n= n= α µ n e inσ α µ n e inσ+, Constants are choosen for convenience. α n µ and α µ n are generally independent (exception: α µ 0 = αµ 0 ). April 15, 2013 Classical String Theory 33/52

34 3. Wave Equations and Solutions 3.3. General Solution for the Closed String Integrating these expressions oscillator expansion (setting p µ = 4πT α µ 0 = 4πT α µ o ): and together X µ R (σ ) = 1 2 xµ + 1 4πT pµ σ + X µ L (σ+ ) = 1 2 xµ + 1 4πT pµ σ + + i 4πT n 0 i 4πT n 0 1 n αµ n e inσ 1 n αµ n e inσ+ X µ (σ, τ) = x µ + 1 2πT pµ τ + i 1 ( α µ }{{} 4πT n n e inσ + α µ n e inσ) e inτ n 0 center of mass motion }{{} oscillation of the string April 15, 2013 Classical String Theory 34/52

35 3. Wave Equations and Solutions 3.3. General Solution for the Closed String Properties: X µ real implies x µ, p µ are both real (α n µ ) = α µ n (α µ n ) = α µ n. x µ is the center of mass of the string at τ = 0: 1 2π 2π 0 dσ X µ (σ, 0) = x µ The canonical τ-momentum is P µ τ = L = ( T (Ẋ2 X 2)) = TẊ µ Ẋ µ Ẋ µ 2 therefore the total momentum of the string is P µ c.o.m = 2π 0 dσ P µ τ = p µ April 15, 2013 Classical String Theory 35/52

36 3. Wave Equations and Solutions 3.4. General Solution for the Open String Equations of motion and boundary conditions open string: ( wave equation: 2 τ σ 2 ) X µ = 0 boundary condition: σ X µ σ=0,π = 0 Virasoro constraint: (Ẋ ± X ) 2 = 0 Similar analysis leads to oscillator expansion: X µ (τ, σ) = x µ + 1 πt pµ τ + i 1 }{{} πt n αµ n e inτ cos(nσ) n 0 center of mass motion }{{} oscillation of the string with x µ, p µ real and (α n µ ) = α µ n. Left and right movers are not independent anymore. April 15, 2013 Classical String Theory 36/52

37 3. Wave Equations and Solutions 3.5. Virasoro Constraints We have found the general solution for the wave equation under consideration of boundary conditions. We still have to impose the Virasoro constraints (Ẋ ± X ) 2 = 0 or Tαβ = 0. Where did they come from again? primary constraints of NG action equation of motion for h αβ in the P action expressed as vanishing of the energy momentum tensor T αβ equivalent to ( Ẋ ± X ) 2 = 0. Can be seen as the string analogon to p µ p µ = m 2 c 2. April 15, 2013 Classical String Theory 37/52

38 3. Wave Equations and Solutions 3.5. Virasoro Constraints Adviseable to discuss Light cone coordinates a bit further: The conformal metric looked like h αβ = η αβ. Therefore the light cone metric is η ++ = η = 0 and η + = η + = 1 2. The energy moment tensor T αβ = 1 2 αx µ β X µ 1 4 h αβh γδ γ X µ d X µ becomes in this coordinates with h αβ = η ± T ++ = 1 2 ( +X) 2 T = 1 2 ( X) 2 T + = T + = 0 April 15, 2013 Classical String Theory 38/52

39 3. Wave Equations and Solutions 3.5. Virasoro Constraints What restriction impose the Virasoro constraints on α n µ, α µ n? ( 0 =! T = 1 2 ( X) 2 = 1 1 ) 2 α n µ e inσ 2 4πT = 1 8πT =: 1 4πT k= n= n= α µ n α kµ e i(n+k)σ = }{{} m=n+k 1 8πT L m e imσ where L m := 1 2 m 0! = T ++ = 1 4πT L m e imσ+ where L m := 1 2 m α n α m n e imσ n m n= n= α n α m n α n α m n April 15, 2013 Classical String Theory 39/52

40 3. Wave Equations and Solutions 3.5. Virasoro Constraints Because m L me imσ! = 0 for all σ, all so called Virasoro modes L m must vanish. For the closed string one hast therefore to impose the additional constraints on the modes L m = L m! = 0 We have found the most general solution for the classical relativistic closed string: X µ (σ, τ) = x µ + 1 2πT pµ τ + i ( 4πT n 0 1 n α µ n e inσ + α µ n e inσ) e inτ with L m := 1 2 n= α n α m n, L m = 1 2 n= α n α m n fulfilling L m = L m = 0. April 15, 2013 Classical String Theory 40/52

41 3. Wave Equations and Solutions 3.5. Virasoro Constraints For the open string the calculation is analog. Because X R and X L are not independent any more one has only one additional constraint T ++ = 1 4πT m L m e imσ+ T = 1 4πT L m e imσ m where L m = 1 2 n α n α m n! = 0 April 15, 2013 Classical String Theory 41/52

42 3. Wave Equations and Solutions 3.6. The Witt Algebra For later quantization: Algebraic considerations of the classical problem. The Poisson bracket for coordinate fields X µ (σ), their conjugate momentum fields P µ (σ) and functionals f (X, P), g(x, P) is defined using the functional derivative δf δg {f, g} P = dσ δx µ (σ) δp µ (σ) δf δg δp µ (σ) δx µ (σ) April 15, 2013 Classical String Theory 42/52

43 3. Wave Equations and Solutions 3.6. The Witt Algebra Interested in τ- propagation: X µ (σ, τ) is seen as a field in σ conjugate field momentum Then Π µ (σ, τ) = L Ẋ µ (σ, τ) = TẊ µ (σ, τ) {X µ (σ, τ), X ν (σ, τ)} = 0 {Π µ (σ, τ), Π ν (σ, τ)} = 0 {X µ (σ, τ), Π ν (σ, τ)} = g µν δ(σ σ ) Next step: Calculate Hamiltonian April 15, 2013 Classical String Theory 43/52

44 3. Wave Equations and Solutions 3.6. The Witt Algebra The τ Hamiltonian is H = With σ 0 dσ Ẋ µ Π µ L = T 2 σ 0 dσ ( Ẋ 2 + X 2) Ẋ 2 = ( + X + X) 2 = 2T T X X X 2 = ( + X X) 2 = 2T T 2 + X X. Therefore the Hamiltonian (for τ propagation) is H = 2T σ 0 dσ (T ++ + T ) April 15, 2013 Classical String Theory 44/52

45 3. Wave Equations and Solutions 3.6. The Witt Algebra Closed String Open String T ++ = 1 4πT L m e imσ+ m T ++ = 1 4πT L m e imσ+ m T = 1 4πT L m e imσ m T = 1 4πT L m e imσ m Hamiltonian Hamiltonian H = L 0 + L 0 Goal: Calculate brackets of the Virasoro modes. H = L 0 April 15, 2013 Classical String Theory 45/52

46 3. Wave Equations and Solutions 3.6. The Witt Algebra Closed String: From full solution X µ (σ, τ) = x µ + 1 2πT pµ τ + Π µ (σ, τ) = TẊ µ = 1 T 2π pµ + 4π 2π i 4πT n 0 1 ( α µ n n e inσ + α µ n e inσ) e inτ ( α µ n e inσ + α µ n e inσ) e inτ calculate x µ = 1 2π 0 X µ (σ, 0)dσ, p µ = 2π 0 Π µ (σ, 0)dσ and 4πT 2π i X µ (σ, 0)e inσ dσ = 1 ( α µ 2π 0 n n α µ n) 1 4π 2π Π µ (σ, 0)e inσ dσ = α n µ + α µ n 2π T 0 n 0 April 15, 2013 Classical String Theory 46/52

47 3. Wave Equations and Solutions 3.6. The Witt Algebra With these formulas calculate (for example) {x µ, p ν } = 1 2π { = 1 2π = 1 2π 2π 0 2π 0 2π 0 X µ (σ, 0) dσ, 2π dσ dσ 0 2π In this way one finds (δ m+n := δ m+n,0 ) 0 2π 0 Π ν (σ, 0) dσ } dσ {X µ (σ, 0), Π ν (σ, 0)} dσ g µν δ(σ σ ) = g µν {α µ m, α ν n} = {α µ m, α ν n} = imδ m+n g µν {α µ m, α ν n} = 0 {x µ, p ν } = g µν April 15, 2013 Classical String Theory 47/52

48 3. Wave Equations and Solutions 3.6. The Witt Algebra Therefore using L m = 1 2 n α n α m n, L m = 1 2 n α n α m n : {L m, L n } = i(m n)l m+n {L m, L n } = i(m n)l m+n {L m, L n } = 0 Witt algebra ( quantize to get Virasoro algebra). The Virasoro modes L m, L m generate an infinite dimensional Lie algebra (Witt algebra) of conserved charges respecting the closed string boundary condition. April 15, 2013 Classical String Theory 48/52

49 3. Wave Equations and Solutions 3.6. The Witt Algebra For the open string the Virasoro modes L m = 1 2 the same commutation relation n α nα m n fulfill {L m, L n } = i(m n)l m+n April 15, 2013 Classical String Theory 49/52

50 3. Wave Equations and Solutions 3.6. The Witt Algebra One question is left: Why did the infinite dimensional Witt algebra appear in this classical calculation? Consider the unit circle S 1 and the group of diffeomorphisms on it. A diffeomorphism θ θ + a(θ) is generated by the operator D a = ia(θ) d dθ. A complete basis for such operators is given by D n = ie inθ d dθ fulfilling the commutator relation [D n, D m ] = i(m n)d m+n. We see: The Witt algebra is simply the Lie algebra of the group of diffeomorphisms on the circle! April 15, 2013 Classical String Theory 50/52

51 Summary We defined Nambu-Goto action in analogy to the point particle.... found equivalent Polyakov action with nicer properties.... discussed symmetries and found equations of motion.... used conformal gauge to simplify equations of motion.... derived general solutions for open and closed strings.... realized that the Virasoro modes generate the Witt algebra. April 15, 2013 Classical String Theory 51/52

52 Questions? Thank you for your attention! April 15, 2013 Classical String Theory 52/52