Notes on String Theory, Study notes of Relativity Theory

Download Notes on String Theory and more Study notes Relativity Theory in PDF only on Docsity! String Theory Gang Xu of Cornell University a set of notes on string theory based on the string theory class by Liam McAllister ii v Declaration These notes are based on various of resources. The credit will be given in due time. Gang Xu vi vii Acknowledgements Of the many people who deserve thanks, some are particularly prominent: My supervisor. . . x Contents 1 Introduction to String Theory 3 1.1 What does String Theory Describe? . . . . . . . . . . . . . . . . . . . . . 3 1.2 Why would we Want a Theory of Quantum Gravity? . . . . . . . . . . . 4 2 Classical Relativistic String Theory 7 2.1 Adventure of Finding an Action . . . . . . . . . . . . . . . . . . . . . . . 7 2.1.1 First Guess: an Action with a Square Root . . . . . . . . . . . . . 7 2.1.2 A Tougher Task: Looking for an Action without Square Root . . 8 2.2 Symmetries and Boundary Conditions . . . . . . . . . . . . . . . . . . . 13 2.2.1 Symmetries of the Action and Equation of Motion . . . . . . . . . 13 2.2.2 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.2.3 Energy Momentum Tensor and Relation with Noether Theorem . 16 2.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 3 Light Cone Quantization of Bosonic String Theory 21 3.1 The Options of Quantization . . . . . . . . . . . . . . . . . . . . . . . . . 21 3.2 solve the classical theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 List of Figures 27 List of Tables 29 xi xii Chapter 1 Introduction to String Theory “Do you want to go to campus now?” — Flip Tanedo at 3:30am What is string theory? • The quantum theory of one-dimensional objects:strings [needs a picture of string] • Defined as a quantum field theory living on the (1 + 1)-dimensional worldsheet of the string S = ∫ d2σLstring 1.1 What does String Theory Describe? There are many such QFT, i.e. there are many string theories. Sometimes, the strings don’t have to be fundamental: they may arise from wrapped higher-dimensional ob- jects(e.g. membranes) and hence have internal structure.[needs a picture of membrane] We may consider closed strings and open strings.[needs a picture of close string and open string] All closed string theories contain a massless spin-2 excitation. Quantum consistency of string theory requires • more than 3 + 1 dimensions of spacetime: D = 10 superstrings; D = 26 bosonice strings; other values for more exotic theories. 3 4 Introduction to String Theory • that the spacetime metric obeys the Einstein equations Gµν = 8πTµν [needs a picture of mapping from worldsheet to spacetime] Open string theories readily contain nonabelian gauge fields and chiral fermions. Furthermore, open string theories always contain closed strings.[needs a picture of "pant" demonstration] Hence, except in exotic cases, we find that string theory is a theory for quantum gravity. It’s actually a finite one. Moreover, string theory naturally exists in D > 4 and readily contains nonabelian gauge fields+chiral fermions(+SUSY). 1.2 Why would we Want a Theory of Quantum Gravity? Recall in QFT, eg λφ4, we can write the Lagranrian as L = 1 2 (∂φ)2 − 1 2 m2φ2 − λ4φ 4 − ( λ6 M2 )φ6 + · · · (1.1) where the higher order terms will generically exist except for symmetry reasons. As we are very familiar in QFT, the couplings with positive, vanishing, and negative mass dimensions are termed superrenormalizable, renormalizable, nonrenormalizable respec- tively. In gravity, the coupling constant is GN ∼M−2 (can be easily seen from Newtonian Gravity formula for the force F = GNm1m2 r2 with mass dimension 2), which is, from the previous naive argument, highly nonrenormalizable. There is a heuristic argument: point-particle scattering at sufficiently high ener- gies leads to black hole formation(unlikely in LHC, though), while string scattering at high energies is much softer(as we will see quantitatively): [needs a picture of scattering of particles and strings] So is that all? A finite theory of Quantum Gravity with some prospect of model- building? NO! The extremely rich structure of string theory has led to important insights into Introduction to String Theory 5 • nonperturbative dualities • gauge theories (eg at strong coupling) • mathematics (eg in algebraic geometry) • black holes (eg microscopic counting of states giving Bekenstein-Hawking entropy. • SYM holography (AdS/CFT correspondence) • theories on branes 8 Classical Relativistic String Theory We can check if this gives the correct action in case of a point particle: ξα ↔ τ , where τ is the worldline parameter. [need a picture of worldline] Then we have h = dXµ dτ dXµ dτ . Using (2.2), we have Spoint = ∫ dτ √ dXµ dτ dXµ dτ (2.3) which agrees with the action from special relativity. More generally, for a theory of relativistic membranes, dimension p + 1, we would have again, hαβ = ∂Xµ ∂ξα ∂Xν ∂ξβ gµν (2.4) as the ”induced metric”, where α, β = 1...p+ 1, and the action is given by S = ∫ dp+1ξ √ − deth (2.5) We’ll come back to this action when we talk about D-branes. 2.1.2 A Tougher Task: Looking for an Action without Square Root The action we just obtained in (2.2) is not easy to quantize. How can we get a simple- looking but equivalent action, so we can quantize that instead? First, we go back to the point particle case to do a warmup. Remind ourselves that the action of a relativistic point particle is written as, S = m ∫ √ ẊµẊµdτ (2.6) where · : d dτ , pay attention that τ is a parameter of the embedding, especiall τ 6= X0. The equation of motion we find from this action (2.6) is mẊµ√ ẊνẊν = const. (2.7) Note that this invariant end the conformal change of parameter τ → λτ . Classical Relativistic String Theory 9 Now let’s try to guess an action without the square root, how about trying a familiar one - kinetic energy plus rest energy: S ′ = ∫ dτ(ẊµẊµ +m2) (2.8) Oops! It is evidently NOT invariant under the parameter change τ → λτ . In order to preserve the ”symmetry” of the original theory, we can introduce a ”compensator” e,an auxilliary field, which transform oppositely under τ → λτ , goes like e → λ−1τ . With this in mind, we can write the second guess as S ′′ = ∫ dτ(e−1ẊµẊµ + em2) (2.9) we can easily see that the power of λ from the ”compensator” e cancels that from τ . In order to check it’s indeed equivalent to the original ugly-square-root action, we should write down the equations of motion for e and Xµ and then integrated out the auxilliary field e. Then we should obtain (2.7). The equation of motion for e is − 1 e2 ẊµẊµ +m2 = 0 (2.10) which gives e = 1 m √ ẊµẊµ (2.11) The equation of motion for Xµ is d dτ (e−1Ẋµ) = 0 (2.12) Now if we subsitute (2.11) into (2.12), we get precisely (2.7). We can view e as an independent ”metric” on the worldline of the particle, and edτ ↔ dτ̂ . That looks very promising: as we wished, we get rid of the square root in the action and the equation of motion doesn’t change! Let’s apply a similar method to our proposed string action we found in 2.1.1, analogue to the world volume to the world sheet: S = −T ∫ d2ξ √ − deth (2.13) 10 Classical Relativistic String Theory with hαβ = ∂Xµ ∂ξα ∂Xν ∂ξβ gµν and T is the tension of the string. This is known as the ”Nambu- Goto” action. Inspired by what we did in the particle case, −m ∫ dτ √ dXµ dτ dXµ dτ → ∫ dτ(e−1ẊµẊµ + em2) (2.14) Polyakov postulate the following change of action for the string theory, −T ∫ d2ξ √ − deth→ −T 2 ∫ d2ξ √ −γγαβ∂αXµ∂βX νηµν (2.15) where gammaαβ is an independent ”metric” on the world sheet, and ∂αX µ∂βX νηµν can be recognized as hαβ, which we defined before as the ”pullback” of the metric of the spacetime. Now we are about to do the same thing again: write down the equations of motion for γαβ and Xµ, and convince ourselves that the Polyakov action of the string theory (2.15) agrees with the Nambu-Goto action (2.13). Since γαβ is a (1+1)d metric as a genuine physical field, does that mean we are going to do (1 + 1)d General Relativity coupled to the fields Xµ? The answer is, fortunately, no. The Einstein-Hilbert term SEH = 1 2k2 ∫ d2ξ √ −γR doesn’t exist and in fact, GR in (1 + 1)d is nearly trivial. A side story [need a box probably]: GR in (1+1)d. From SEH = 1 2k2 ∫ d2ξ √ −γR , we get the Einstein equations(or just pull out of your memory) Rαβ − 1 2 gαβR = Tαβ (2.16) where the energy-momentum tensor Tαβ is taken to be 0 in vacuum. But we know the Riemann tensor has the property: Rαβγδ = −Rβαγδ = −Rαβδγ = Rγδαβ (2.17) since αβγδ ∈ {0, 1}, we can write Rαβγδ = fεαβεγδ (2.18) Classical Relativistic String Theory 13 Now we can calculate √ −h, which shows up in the Nambu-Goto action (2.13). From (2.36), we have deth = ( 1 2 γγδhγδ) 2 det γ (2.37) using h and γ are 2× 2 matrix and 1 2 γγδhγδ is just a proportional number, which gives us √ −h = 1 2 γγδhγδ √ −γ (2.38) Now we use (2.36) again, we have hαβ√ −h = γαβ√ −γ (2.39) which gives us γαβhαβ = √ −h√ −γ γαβγαβ = 2 √ −h√ −γ (2.40) Now we plug (2.40) back in our Polyakov action (2.15), we obtain Sp = −T 2 ∫ d2ξ √ −γγαβhαβ = −T ∫ d2ξ √ −h = SNG (2.41) We successfully get rid of the square root on our coordinate fields Xµ in Nambu-Goto action and from now on we will use Polyakov action. Hooray! 2.2 Symmetries and Boundary Conditions 2.2.1 Symmetries of the Action and Equation of Motion Now let’s now think about this action before trying to study the full classical equation of motion: what symmetries we have and what kind of boundary conditions we have to impose? 14 Classical Relativistic String Theory Let’s rewrite the Polyakov action again with spacetime metric gµν , −T ∫ d2ξ √ − deth→ −T 2 ∫ d2ξ √ −γγαβ∂αXµ∂βX νgµν (2.42) List of the symmetries: • Poincare symmetry: the action is invariant under Xµ(τ, σ) → Λµ νX ν(τ, σ) + aµ, because the only part of the action that depends on this transformation is ∂αX µ∂βX νgµν → Λµ ρ∂αX ρΛν σ∂βX σgµν (2.43) Note that Λµ ρΛν σgµν = gρσ, so the action is not changed.(The indices α, β are on the worldsheet, has nothing to do with Poincare symmetry in the spacetime). • Reparametrization/general coordinate/diffeomorphism: the action is invariant un- der τ → τ ′(τ, σ) , σ → σ′(τ, σ) , ∂ξ′γ ∂ξα ∂ξ′δ ∂ξβ γ′γδ(τ ′, σ′) = γαβ(τ, σ) (2.44) The last transformation gaurantees that γαβ∂αX µ∂βX ν will not change, and the integral measure ∫ d2ξ √ −γ follows the usual invariance of General Relativity.(Note that this time the index µ, ν don’t transform.) • Weyl invariance: under rescaling of γ′αβ(τ, σ) = e2ω(τ,σ)γαβ(τ, σ), the action is also invariant because √ −γ scales as e2ω(τ,σ) and γαβ scales as e−2ω(τ,σ) and all the other terms don’t change. This is an additional redundancy. Note that the Poincare is an ”internal” also global symmetry while the diffeomor- phism and Weyl are local symmetries. Now let us go back to the equations of motions (2.24) and (2.25). The former has been calculated in the previous subsection and the result is (2.35). Now let us calculate the latter, the Euler Lagrangian equation for Xµ, which is much easier using (2.42), from which we immediately have δSp δXµ = 0 , δSp δ∂αXµ = −T 2 √ −γ2γαβ∂βX νgµν (2.45) which together give us ∂α( √ −γγαβ∂βXµ) = 0 (2.46) Classical Relativistic String Theory 15 Recall that the definition of Laplacian in the curved space is ∇2f ≡ 1 √ g ∂a( √ ggab∂bf) (2.47) Thus the Euler Lagrangian equation is nothing but √ −γ∇2Xµ = 0 (2.48) This looks just like the wave equation we get in classical mechanics except that now we have to use the Laplacian operator in the curved space. In the next subsection, we will find out what boundary conditions we have to impose on these embedding function Xµs. 2.2.2 Boundary Conditions In order to find the necessary boundary conditions, we vary the action with respect to Xµ → Xµ + δXµ. The action should be invariant under such a translational transfor- mation, and the change is of course δSp = −T 2 2 ∫ dτ ∫ dσ √ −γγαβ∂αXµ∂β(δXµ) (2.49) We will use a very common technique, integration by parts, to extract the boundary terms, let us recall this technique briefly: for O being an arbitrary operator, we have∫ dτO∂τδXµ = − ∫ dτ(∂τO)δXµ +OδXµ| τf τi (2.50) similarly, we have ∫ dσO∂σδXµ = − ∫ dσ(∂σO)δXµ +OδXµ|σ=π σ=0 (2.51) The change in the action is then δSp = T ∫ dτdσ(∂β √ −γγαβ∂αXµ)δXµ − T ∫ dτdσ∂β( √ −γ∂βXµδXµ) (2.52) We will set δXµ|τi = δXµ|τf = 0 as in a single particle case. Then the boundary term is −T ∫ dτ √ −γ∂σXµδXµ|σ=π σ=0 (2.53) 18 Classical Relativistic String Theory Under this transformation, we can calculate our change in Lagranrian to be, from now on, for simplicity, we will drop the index a, δL = ∂L ∂φ δφ+ ∂L ∂∂αφ δ∂αφ (2.65) = ∂L ∂φ εβ∂βφ+ ∂L ∂∂αφ ∂αε β∂βφ = εβ∂βL = εβ∂αδ α βL Now from (2.63), we have ∂αj α = ∂α ( ∂L ∂∂αφ εβ∂βφ− εβδαβL ) (2.66) which gives us jαβ = ∂L ∂∂αφ ∂βφ− δαβL (2.67) and jαβ ⇔ Tαβ. It is also useful to notice that under constant spacetime translations,Xµ → Xµ + εµ, that is δXµ(τ, σ) = εµ. Since the shift is constant and the action does not de- pend on the spacetime coordinates, recall that it depends on only the derivative of the spacetime coordinates, V µ = 0 and εµjαµ = ∂L ∂∂αXµ δXµ = εµ ∂L ∂∂αXµ ⇔ jαµ = ∂L ∂∂αXµ (2.68) We can write out the components respect to the worldsheet, jτµ = ∂L ∂Ẋµ and jσµ = ∂L ∂X′µ . As we learned in the field theory, space integral of the time component gives charge Q = ∫ dd−1j0. In this case we have Qµ = ∫ dσjµτ , and Qµ is the spacetime momentum. Now let us calculate the worldsheet energy momentum tensor for our Polyakov action using Noether procedure which was derived for actions that do not depend explicitly on the coordinates in (2.67) with of course the general field φ replace by the mapping function Xµ, Tαβ = ∂L ∂(∂αXµ) ∂βXµ − γαβL (2.69) Classical Relativistic String Theory 19 with L = −T 2 ∂γXν∂γXν . Thus we obtain Tαβ = −T ( ∂αXµ∂βXµ − 1 2 γαβ∂γXµ∂γXµ ) (2.70) since γαβγ αβ = 2, it is evident that γαβT αβ = 0, that is Tαβ is traceless. Let us now compare it with the definition of the general relativity Tαβ ≡ C√ −γ δS δγαβ where recall we already calculate it in (2.35). Now it is easy to see we will obtain the same expression for the energy momentum tensor as the one we obtained through Noether procedure with a proportional constant 2π if we choose C = 4π. 2.3 Summary So let us summarize what we learned from classical relativistic strings. First, we find an action proportional to the world volume of the string worldsheet vol(Φ(Σ)), the Nambu- Goto action, SNG = −T ∫ d2ξ √ −h, where hαβ ≡ gµν∂αX µ∂βX ν is the pullback of the metric to the worldsheet. We get rid of the annoying square root by introducing a metric on the worldsheet γαβ and obtain Polyakov action Sp = −T 2 ∫ dξ2 √ −γγαβhαβ, and the two actions are equivalent each other by making use of the equation of motion of γαβ. The following properties of the action will be useful in the quantization process of this action, • Equations of the motion – For embedding functions Xµ, we obtain the wave equation ∇2Xµ = 0 – For the auxiliary metric γαβ, we have Tαβ = 0 since δSp δγαβ = 0 • Symmetries: – D-dimensional Poincare symmetry – 2d diffeomorphism inherited from the general relativity. – Weyl rescaling symmetry. • Boundary conditions: – Neumann boundary conditions: ∂σXµ(τ, 0) = ∂σXµ(τ, l) = 0 or – Periodic boundary conditions: φ(τ, l) = φ(τ, 0) for φ = Xµ, ∂σXµ, and √ −γ or – Dirichlet conditions: Xµ(τ, 0) = Xµ 0 , Xµ(τ, l) = Xµ l 20 Light Cone Quantization of Bosonic String Theory 23 String Theory Gang Xu of Cornell University a set of notes on string theory based on the string theory class by Liam McAllister ii v Declaration These notes are based on various of resources. The credit will be given in due time. Gang Xu vi vii Acknowledgements Of the many people who deserve thanks, some are particularly prominent: My supervisor. . . x Contents 1 Introduction to String Theory 3 1.1 What does String Theory Describe? . . . . . . . . . . . . . . . . . . . . . 3 1.2 Why would we Want a Theory of Quantum Gravity? . . . . . . . . . . . 4 2 Classical Relativistic String Theory 7 2.1 Adventure of Finding an Action . . . . . . . . . . . . . . . . . . . . . . . 7 2.1.1 First Guess: an Action with a Square Root . . . . . . . . . . . . . 7 2.1.2 A Tougher Task: Looking for an Action without Square Root . . 8 2.2 Symmetries and Boundary Conditions . . . . . . . . . . . . . . . . . . . 13 2.2.1 Symmetries of the Action and Equation of Motion . . . . . . . . . 13 2.2.2 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.2.3 Energy Momentum Tensor and Relation with Noether Theorem . 16 2.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 3 Light Cone Quantization of Bosonic String Theory 21 3.1 The Options of Quantization . . . . . . . . . . . . . . . . . . . . . . . . . 21 3.2 solve the classical theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 List of Figures 27 List of Tables 29 xi xii Chapter 1 Introduction to String Theory “Do you want to go to campus now?” — Flip Tanedo at 3:30am What is string theory? • The quantum theory of one-dimensional objects:strings [needs a picture of string] • Defined as a quantum field theory living on the (1 + 1)-dimensional worldsheet of the string S = ∫ d2σLstring 1.1 What does String Theory Describe? There are many such QFT, i.e. there are many string theories. Sometimes, the strings don’t have to be fundamental: they may arise from wrapped higher-dimensional ob- jects(e.g. membranes) and hence have internal structure.[needs a picture of membrane] We may consider closed strings and open strings.[needs a picture of close string and open string] All closed string theories contain a massless spin-2 excitation. Quantum consistency of string theory requires • more than 3 + 1 dimensions of spacetime: D = 10 superstrings; D = 26 bosonice strings; other values for more exotic theories. 3 4 Introduction to String Theory • that the spacetime metric obeys the Einstein equations Gµν = 8πTµν [needs a picture of mapping from worldsheet to spacetime] Open string theories readily contain nonabelian gauge fields and chiral fermions. Furthermore, open string theories always contain closed strings.[needs a picture of "pant" demonstration] Hence, except in exotic cases, we find that string theory is a theory for quantum gravity. It’s actually a finite one. Moreover, string theory naturally exists in D > 4 and readily contains nonabelian gauge fields+chiral fermions(+SUSY). 1.2 Why would we Want a Theory of Quantum Gravity? Recall in QFT, eg λφ4, we can write the Lagranrian as L = 1 2 (∂φ)2 − 1 2 m2φ2 − λ4φ 4 − ( λ6 M2 )φ6 + · · · (1.1) where the higher order terms will generically exist except for symmetry reasons. As we are very familiar in QFT, the couplings with positive, vanishing, and negative mass dimensions are termed superrenormalizable, renormalizable, nonrenormalizable respec- tively. In gravity, the coupling constant is GN ∼M−2 (can be easily seen from Newtonian Gravity formula for the force F = GNm1m2 r2 with mass dimension 2), which is, from the previous naive argument, highly nonrenormalizable. There is a heuristic argument: point-particle scattering at sufficiently high ener- gies leads to black hole formation(unlikely in LHC, though), while string scattering at high energies is much softer(as we will see quantitatively): [needs a picture of scattering of particles and strings] So is that all? A finite theory of Quantum Gravity with some prospect of model- building? NO! The extremely rich structure of string theory has led to important insights into Introduction to String Theory 5 • nonperturbative dualities • gauge theories (eg at strong coupling) • mathematics (eg in algebraic geometry) • black holes (eg microscopic counting of states giving Bekenstein-Hawking entropy. • SYM holography (AdS/CFT correspondence) • theories on branes 8 Classical Relativistic String Theory We can check if this gives the correct action in case of a point particle: ξα ↔ τ , where τ is the worldline parameter. [need a picture of worldline] Then we have h = dXµ dτ dXµ dτ . Using (2.2), we have Spoint = ∫ dτ √ dXµ dτ dXµ dτ (2.3) which agrees with the action from special relativity. More generally, for a theory of relativistic membranes, dimension p + 1, we would have again, hαβ = ∂Xµ ∂ξα ∂Xν ∂ξβ gµν (2.4) as the ”induced metric”, where α, β = 1...p+ 1, and the action is given by S = ∫ dp+1ξ √ − deth (2.5) We’ll come back to this action when we talk about D-branes. 2.1.2 A Tougher Task: Looking for an Action without Square Root The action we just obtained in (2.2) is not easy to quantize. How can we get a simple- looking but equivalent action, so we can quantize that instead? First, we go back to the point particle case to do a warmup. Remind ourselves that the action of a relativistic point particle is written as, S = m ∫ √ ẊµẊµdτ (2.6) where · : d dτ , pay attention that τ is a parameter of the embedding, especiall τ 6= X0. The equation of motion we find from this action (2.6) is mẊµ√ ẊνẊν = const. (2.7) Note that this invariant end the conformal change of parameter τ → λτ . Classical Relativistic String Theory 9 Now let’s try to guess an action without the square root, how about trying a familiar one - kinetic energy plus rest energy: S ′ = ∫ dτ(ẊµẊµ +m2) (2.8) Oops! It is evidently NOT invariant under the parameter change τ → λτ . In order to preserve the ”symmetry” of the original theory, we can introduce a ”compensator” e,an auxilliary field, which transform oppositely under τ → λτ , goes like e → λ−1τ . With this in mind, we can write the second guess as S ′′ = ∫ dτ(e−1ẊµẊµ + em2) (2.9) we can easily see that the power of λ from the ”compensator” e cancels that from τ . In order to check it’s indeed equivalent to the original ugly-square-root action, we should write down the equations of motion for e and Xµ and then integrated out the auxilliary field e. Then we should obtain (2.7). The equation of motion for e is − 1 e2 ẊµẊµ +m2 = 0 (2.10) which gives e = 1 m √ ẊµẊµ (2.11) The equation of motion for Xµ is d dτ (e−1Ẋµ) = 0 (2.12) Now if we subsitute (2.11) into (2.12), we get precisely (2.7). We can view e as an independent ”metric” on the worldline of the particle, and edτ ↔ dτ̂ . That looks very promising: as we wished, we get rid of the square root in the action and the equation of motion doesn’t change! Let’s apply a similar method to our proposed string action we found in 2.1.1, analogue to the world volume to the world sheet: S = −T ∫ d2ξ √ − deth (2.13) 10 Classical Relativistic String Theory with hαβ = ∂Xµ ∂ξα ∂Xν ∂ξβ gµν and T is the tension of the string. This is known as the ”Nambu- Goto” action. Inspired by what we did in the particle case, −m ∫ dτ √ dXµ dτ dXµ dτ → ∫ dτ(e−1ẊµẊµ + em2) (2.14) Polyakov postulate the following change of action for the string theory, −T ∫ d2ξ √ − deth→ −T 2 ∫ d2ξ √ −γγαβ∂αXµ∂βX νηµν (2.15) where gammaαβ is an independent ”metric” on the world sheet, and ∂αX µ∂βX νηµν can be recognized as hαβ, which we defined before as the ”pullback” of the metric of the spacetime. Now we are about to do the same thing again: write down the equations of motion for γαβ and Xµ, and convince ourselves that the Polyakov action of the string theory (2.15) agrees with the Nambu-Goto action (2.13). Since γαβ is a (1+1)d metric as a genuine physical field, does that mean we are going to do (1 + 1)d General Relativity coupled to the fields Xµ? The answer is, fortunately, no. The Einstein-Hilbert term SEH = 1 2k2 ∫ d2ξ √ −γR doesn’t exist and in fact, GR in (1 + 1)d is nearly trivial. A side story [need a box probably]: GR in (1+1)d. From SEH = 1 2k2 ∫ d2ξ √ −γR , we get the Einstein equations(or just pull out of your memory) Rαβ − 1 2 gαβR = Tαβ (2.16) where the energy-momentum tensor Tαβ is taken to be 0 in vacuum. But we know the Riemann tensor has the property: Rαβγδ = −Rβαγδ = −Rαβδγ = Rγδαβ (2.17) since αβγδ ∈ {0, 1}, we can write Rαβγδ = fεαβεγδ (2.18) Classical Relativistic String Theory 13 Now we can calculate √ −h, which shows up in the Nambu-Goto action (2.13). From (2.36), we have deth = ( 1 2 γγδhγδ) 2 det γ (2.37) using h and γ are 2× 2 matrix and 1 2 γγδhγδ is just a proportional number, which gives us √ −h = 1 2 γγδhγδ √ −γ (2.38) Now we use (2.36) again, we have hαβ√ −h = γαβ√ −γ (2.39) which gives us γαβhαβ = √ −h√ −γ γαβγαβ = 2 √ −h√ −γ (2.40) Now we plug (2.40) back in our Polyakov action (2.15), we obtain Sp = −T 2 ∫ d2ξ √ −γγαβhαβ = −T ∫ d2ξ √ −h = SNG (2.41) We successfully get rid of the square root on our coordinate fields Xµ in Nambu-Goto action and from now on we will use Polyakov action. Hooray! 2.2 Symmetries and Boundary Conditions 2.2.1 Symmetries of the Action and Equation of Motion Now let’s now think about this action before trying to study the full classical equation of motion: what symmetries we have and what kind of boundary conditions we have to impose? 14 Classical Relativistic String Theory Let’s rewrite the Polyakov action again with spacetime metric gµν , −T ∫ d2ξ √ − deth→ −T 2 ∫ d2ξ √ −γγαβ∂αXµ∂βX νgµν (2.42) List of the symmetries: • Poincare symmetry: the action is invariant under Xµ(τ, σ) → Λµ νX ν(τ, σ) + aµ, because the only part of the action that depends on this transformation is ∂αX µ∂βX νgµν → Λµ ρ∂αX ρΛν σ∂βX σgµν (2.43) Note that Λµ ρΛν σgµν = gρσ, so the action is not changed.(The indices α, β are on the worldsheet, has nothing to do with Poincare symmetry in the spacetime). • Reparametrization/general coordinate/diffeomorphism: the action is invariant un- der τ → τ ′(τ, σ) , σ → σ′(τ, σ) , ∂ξ′γ ∂ξα ∂ξ′δ ∂ξβ γ′γδ(τ ′, σ′) = γαβ(τ, σ) (2.44) The last transformation gaurantees that γαβ∂αX µ∂βX ν will not change, and the integral measure ∫ d2ξ √ −γ follows the usual invariance of General Relativity.(Note that this time the index µ, ν don’t transform.) • Weyl invariance: under rescaling of γ′αβ(τ, σ) = e2ω(τ,σ)γαβ(τ, σ), the action is also invariant because √ −γ scales as e2ω(τ,σ) and γαβ scales as e−2ω(τ,σ) and all the other terms don’t change. This is an additional redundancy. Note that the Poincare is an ”internal” also global symmetry while the diffeomor- phism and Weyl are local symmetries. Now let us go back to the equations of motions (2.24) and (2.25). The former has been calculated in the previous subsection and the result is (2.35). Now let us calculate the latter, the Euler Lagrangian equation for Xµ, which is much easier using (2.42), from which we immediately have δSp δXµ = 0 , δSp δ∂αXµ = −T 2 √ −γ2γαβ∂βX νgµν (2.45) which together give us ∂α( √ −γγαβ∂βXµ) = 0 (2.46) Classical Relativistic String Theory 15 Recall that the definition of Laplacian in the curved space is ∇2f ≡ 1 √ g ∂a( √ ggab∂bf) (2.47) Thus the Euler Lagrangian equation is nothing but √ −γ∇2Xµ = 0 (2.48) This looks just like the wave equation we get in classical mechanics except that now we have to use the Laplacian operator in the curved space. In the next subsection, we will find out what boundary conditions we have to impose on these embedding function Xµs. 2.2.2 Boundary Conditions In order to find the necessary boundary conditions, we vary the action with respect to Xµ → Xµ + δXµ. The action should be invariant under such a translational transfor- mation, and the change is of course δSp = −T 2 2 ∫ dτ ∫ dσ √ −γγαβ∂αXµ∂β(δXµ) (2.49) We will use a very common technique, integration by parts, to extract the boundary terms, let us recall this technique briefly: for O being an arbitrary operator, we have∫ dτO∂τδXµ = − ∫ dτ(∂τO)δXµ +OδXµ| τf τi (2.50) similarly, we have ∫ dσO∂σδXµ = − ∫ dσ(∂σO)δXµ +OδXµ|σ=π σ=0 (2.51) The change in the action is then δSp = T ∫ dτdσ(∂β √ −γγαβ∂αXµ)δXµ − T ∫ dτdσ∂β( √ −γ∂βXµδXµ) (2.52) We will set δXµ|τi = δXµ|τf = 0 as in a single particle case. Then the boundary term is −T ∫ dτ √ −γ∂σXµδXµ|σ=π σ=0 (2.53) 18 Classical Relativistic String Theory Under this transformation, we can calculate our change in Lagranrian to be, from now on, for simplicity, we will drop the index a, δL = ∂L ∂φ δφ+ ∂L ∂∂αφ δ∂αφ (2.65) = ∂L ∂φ εβ∂βφ+ ∂L ∂∂αφ ∂αε β∂βφ = εβ∂βL = εβ∂αδ α βL Now from (2.63), we have ∂αj α = ∂α ( ∂L ∂∂αφ εβ∂βφ− εβδαβL ) (2.66) which gives us jαβ = ∂L ∂∂αφ ∂βφ− δαβL (2.67) and jαβ ⇔ Tαβ. It is also useful to notice that under constant spacetime translations,Xµ → Xµ + εµ, that is δXµ(τ, σ) = εµ. Since the shift is constant and the action does not de- pend on the spacetime coordinates, recall that it depends on only the derivative of the spacetime coordinates, V µ = 0 and εµjαµ = ∂L ∂∂αXµ δXµ = εµ ∂L ∂∂αXµ ⇔ jαµ = ∂L ∂∂αXµ (2.68) We can write out the components respect to the worldsheet, jτµ = ∂L ∂Ẋµ and jσµ = ∂L ∂X′µ . As we learned in the field theory, space integral of the time component gives charge Q = ∫ dd−1j0. In this case we have Qµ = ∫ dσjµτ , and Qµ is the spacetime momentum. Now let us calculate the worldsheet energy momentum tensor for our Polyakov action using Noether procedure which was derived for actions that do not depend explicitly on the coordinates in (2.67) with of course the general field φ replace by the mapping function Xµ, Tαβ = ∂L ∂(∂αXµ) ∂βXµ − γαβL (2.69) Classical Relativistic String Theory 19 with L = −T 2 ∂γXν∂γXν . Thus we obtain Tαβ = −T ( ∂αXµ∂βXµ − 1 2 γαβ∂γXµ∂γXµ ) (2.70) since γαβγ αβ = 2, it is evident that γαβT αβ = 0, that is Tαβ is traceless. Let us now compare it with the definition of the general relativity Tαβ ≡ C√ −γ δS δγαβ where recall we already calculate it in (2.35). Now it is easy to see we will obtain the same expression for the energy momentum tensor as the one we obtained through Noether procedure with a proportional constant 2π if we choose C = 4π. 2.3 Summary So let us summarize what we learned from classical relativistic strings. First, we find an action proportional to the world volume of the string worldsheet vol(Φ(Σ)), the Nambu- Goto action, SNG = −T ∫ d2ξ √ −h, where hαβ ≡ gµν∂αX µ∂βX ν is the pullback of the metric to the worldsheet. We get rid of the annoying square root by introducing a metric on the worldsheet γαβ and obtain Polyakov action Sp = −T 2 ∫ dξ2 √ −γγαβhαβ, and the two actions are equivalent each other by making use of the equation of motion of γαβ. The following properties of the action will be useful in the quantization process of this action, • Equations of the motion – For embedding functions Xµ, we obtain the wave equation ∇2Xµ = 0 – For the auxiliary metric γαβ, we have Tαβ = 0 since δSp δγαβ = 0 • Symmetries: – D-dimensional Poincare symmetry – 2d diffeomorphism inherited from the general relativity. – Weyl rescaling symmetry. • Boundary conditions: – Neumann boundary conditions: ∂σXµ(τ, 0) = ∂σXµ(τ, l) = 0 or – Periodic boundary conditions: φ(τ, l) = φ(τ, 0) for φ = Xµ, ∂σXµ, and √ −γ or – Dirichlet conditions: Xµ(τ, 0) = Xµ 0 , Xµ(τ, l) = Xµ l 20 Light Cone Quantization of Bosonic String Theory 23 24 Colophon This thesis was made in LATEX 2ε using the “hepthesis” class [?]. 25 28 List of Tables 29