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The Nambu-Goto String Action the first Thing We Need to Understand Is What the Configuration Space of a String on Spacetime Looks Like

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The Nambu-Goto String Action the first Thing We Need to Understand Is What the Configuration Space of a String on Spacetime Looks Like Week 2 Reading material from the books • Zwiebach, Chapter 6, 7, 8, 9 • Polchinski, Chapter 1 • Becker, Becker, Schwartz, Chapter 2 • Green, Schwartz, Witten, chapter 2 1 The relativistic string 1.1 Classical dynamics: the Nambu-Goto string action The first thing we need to understand is what the configuration space of a string on spacetime looks like. At time t, we should be able to identify the location of the string by various functions xi(σ; t) and this should be all of it, where σ is either a periodic variable, or a segment depending on wether we have an open string or a closed string. However, this is a redundant description of the system, because if we change σ ! σ0(σ; t) we get that the string at time t is located at exactly the same geometric locus. Notice that although this is a convenient way to parametrize the trajec- tory of the string, it is rather hard to pass from one relativistic coordinate system to another, because under relativistic transformations the xi mix with t and writing this new information in terms of t0 is in general hard. A much more convenient point of view is to understand things from a more geometrical perspective: The string motion sweeps a two dimensional surface in spacetime. We can parametrize this trajectory by D functions of two auxiliary vari- ables σ; τ: xµ(σ; τ) where σ has the topology of either a segment, or a circle. Now it is easy to do Lorentz transformations, because we can just act linearly on the coordinates xµ and carry along the paramerization in terms of the worldsheet coordinates σ; τ. 1 This choice makes the Lorentz symmetry more manifest. However, we have gone from D − 1 functions of σ; t to D functions of σ; τ. As in the point particle case, it would seem that in this way we are making the configuration space bigger. Clearly, we could choose another parametrization of the surface, in terms of σ0; τ 0, and we would be describing the same surface in different "internal coordinates". It should be the case that the dynamics does not care for which "inter- nal coordinate" system we use. Therefore the dynamical principle should be invariant under reparametrizations of σ; τ. This will in the end be used to re- move the extra redundancies introduced by adding an arbitrary parametriza- tion of the surface. We usually formulate dynamics by a least action principle. We want this action principle to be relativistically invariant (thus making Lorentz symmetry manifest). We should also make a null hypothyesis: only the geometric embedding of the surface in spacetime matters to define the action. (We mean here that we make minimal input on the dynamical theory). Finally, we would want the wordvolume description to be local on the embedding. This is, we should be able to formulate the action principle as Z µ µ S ∼ dσdτL(x (σ; τ);@σ,τ x (σ; τ)) and possibly higher derivatives with respect to τ; σ. Making the geometric hypothesis for the dynamics is simple: In an euclidean geometry, we would assign the total action to the total area of the surface. This is geometric and local. It is important to realize how to measure the local area in \general coor- dinates σ; τ". A small displacement in σ, dσ, translates into a small displacement in the Euclidean geometry @xi vi = δxi = dσ (1) σ σ @σ This is a small vector displacement in the geometry. Similarly, we can define a small displacement due to variation of the τ coordinate. @xi vi = δxi = dτ (2) τ τ @τ 2 These two vectors define a small parallelogram in the embedding. If the angle that they form is θ, then the area that the parallelogram subtends is equal to δA = jvτ jjvσj cos(θ) (3) Now, the angle between the vectors can be calculated using the dot product as follows jv · v j sin(θ) = τ σ (4) jvτ jjvσj p Now we use the relation sin(θ) = 1 − cos2 θ to rewrite the area element as follows p 2 2 2 δA = jvτ j jvσj − (vσ · vτ ) (5) We can now introduce the following symmetric tensor on the worldsheet to simplify the formula: ∗ gρθ = (@ρ~x)(@θ~x) (6) This metric tensor g∗ is induced from the embedding (similar to the pullback for forms). With this definition the above formula for the area looks as follows p ∗ ∗ ∗2 δA = gσσgττ − gστ dσdτ (7) and this can also be expressed in a more concise notation as p δA = pdet g∗dσdτ = g∗d2σ (8) Note:The notation g∗ in the square root is the scalar determinant of the metric, this is abuse of notation, g∗ is also the metric two tensor with indices omitted. Usually the context dictates the interpretation, or one can use bold- face for tensors In a Lorentzian geometry we can use an analytic continuation of the area formula above to define a similar quantity. To calculate the area of a surface in Euclidean geometry, we have an induced metric on the surface, from the embedding. The total area in the Lorentzian case can be calculated by using the area density in terms of the induced metric, with an extra minus sign to take into account the different signature of the induced metric (Lorentzian) Z S = −T d2σp− det(g∗) (9) 3 The induced metric is given still by the same formula dxµ dxν ds2 = g∗ dσρdσθ = g dσρdσθ (10) ind ρ,θ µν dσρ dσθ (We use metric conventions with (-,++++++) signature, and the in- duced metric has the same signature, so det(g∗) is negative.). Because S has units of Energy times time (dimensionless in natural units), we need to add dimensionful quantities to S to make it match: This defines the string tension T . Notice that this action is reparametrization invariant: we can change σ; τ to σ0; τ 0(σ; τ) and it would take the same form in the new coordinate system. This is because we are using the natural volume density defined by the metric as our action. This is a symmetry of the problem that we can use to define the spatial and time coordinate on the worldsheet however we want and we will exploit this later on. If the string is closed, we are done: we use periodic boundary conditions. If the string is open, there could be an additional geometric term in the action on the boundary of the string that measures the length of the boundary for example. We will not deal with this situation in what follows. We will concentrate on having a good variational principle for the area action. 2 The equations of motion We begin with Z Z S = −T d2σp− det(g∗) = −T d2σL (11) where L is the Lagrangian density. Now, we want to compute the Euler- Lagrange equations of motion. Since the action only depends on derivatives of the xjmu(σ), the Euler-Lagrange equations of motion read as follows @L @α µ = 0 (12) @x,α µ µ Here we are using the common (Einstein) notation where @αx (σ) := x,α. To compute the expression @L µ (13) @x,α 4 we do it as follows, following the chain rule ∗ @L @L @gβγ µ = ∗ µ (14) @x,α @gβγ @x,α where we first vary as if the g∗ are independent variables. With the formula for a variation of a determinant δ det M = det MTr(M −1δM) (15) we obtain readily that p 1p δ( −g∗) = −g∗(g∗)βγδg∗ (16) 2 βγ and using that ∗ @gβγ ν α ν α µ = ηµν@βx δγ + ηµν@γx δβ (17) @x,α we get that p @L ∗ ∗ βα ν µ = −g (g ) ηµν@βx (18) @x,α To write this one -liner we have used the freedom to relabel indices, and the two terms that show up are identical. This gives the factor of 2 that cancels the factor of 1=2 from the square root. This looks simple only because we are using the symbols gαβ and it's in- verse as if they are variables. In practice they \look horrible" as raw expres- µ sions in @αx . The full Euler-Lagrange equations of motion are therefore p ∗ ∗ βα ν @α −g (g ) ηµν@βx = 0 (19) 3 Static gauge 0 Let us choose the parametrization x (σ; τ) = τ = t. (Notice that @σt = 0). Then we have that 2 i 2 2 i i0 i0 2 2 gind = −dτ + (x _ ) dτ + dτdσx_ x + (x ) dσ (20) 5 Now, we also have that 2 det(gind) = gττ gσσ − gτσ (21) = (−1 + (x _ i)2)(x0)2 − (x0x_)2 (22) So the action has been simplified a little bit, but it is still not ideally simpli- fied. We still have the freedom to make changes of σ ! σ(τ; σ). We can use this freedom so that we can in general fix the internal coordinate system of 0 the string so that gτσ = 0, this is, we can arrange it so that x x_ = 0. Let us understand how that works. We have to think of xµ(σ; τ) as functions on the worldsheet that do not depend on the internal coordinates of the surface. When we claim that one of the coordinates is set to x0 = τ, then we have that dx0 = dτ is a preferred 1-form on the worldsheet (because it is determined from the embedding and the operation of taking differentials is independent of coordinates). Since dτ(@σ) = 0 (standard formula for differentials being the dual basis to vectors), this prescription actually defined a preferred vector direction on the worldsheet: but the trick is that it is not @τ that ends up fixed, but the @σ direction that got fixed.
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