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Home , Action (physics)

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 : the Nambu-Goto string action The first thing we need to understand is what the configuration of a string on looks like. At 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 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 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 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 that they form is θ, then the area that the parallelogram subtends is equal to δA = |vτ ||vσ| cos(θ) (3) Now, the angle between the vectors can be calculated using the dot product as follows |v · v | sin(θ) = τ σ (4) |vτ ||vσ| √ Now we use the relation sin(θ) = 1 − cos2 θ to rewrite the area element as follows p 2 2 2 δA = |vτ | |vσ| − (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 √ δA = pdet g∗dσdτ = g∗d2σ (8)

Note:The notation g∗ in the square root is the 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 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 of the boundary for example. We will not deal with this situation in what follows. We will concentrate on having a good for the area action.

2 The

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 x|mu(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 √ 1√ 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 √ ∂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 √ ∗ ∗ 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. When we push- forward this vector to spacetime, we find that ∂σ is pointing only in the space direction orthogonal to the time slicing we chose at the beginning. When we change coordinates on the worldsheet, keeping τ fixed, the vector field ∂τ actually changes. Remember that d ∂ = | , τ dτ σfixed so when we change σ → σ˜(σ, τ), we can change the vector field ∂τ because the notion of keeping σ fixed has changed. Writing on the worldsheet the coordinate change carefully we have that ∂τ ∂σ ∂σ ∂ = ∂ + ∂ = ∂ + ∂ (23) τ˜ ∂τ˜ τ ∂τ˜ σ τ ∂τ˜ σ so that ∂τ picks a term proportional to the space direction. Since the induced metric on the worldhseet has Lorentzian signature and ∂σ is spacelike, we can always find a f(σ, τ) such that

∂τ + f(σ, τ)∂σ ⊥ ∂σ (24)

∂σ and then we can solve ∂τ˜ = f(σ, τ) by the implicit function theorem. It is this orthogonality that implies that gind(∂τ , ∂σ) = 0 can be imposed everywhere.

6 In this setup, when we push-forward ∂τ , it is timelike in Minkowsky space. This means that at fixed σ, the point with the same label of sigma, but slightly later in τ 0 = τ + δτ is in the strict future of the point labeled by σ, τ. Not only this, it is uniquely characterized by corresponding to the direction that has the longest starting at σ, τ, and ending sometime later inside the worldsheet at τ + δτ. Even after doing this, there is still some remnant of reparametrization invariance by transformations σ0(σ). We can use this further so that gσσ = 1 in the initial time-slice, but not everywhere. Equivalently, we can get it in such a form that (x0)2 = 1 at t = 0. In this form, dσ ' ds, is the arc-length along the string, measured at equal time slices in the initial time slice. The action in this particular system takes the form Z q 2 S = −T 1 − vT dtds (25) where vT is the component of the orthogonal to the string (space) 0 ˙ direction. vT is transverse because of the constraint (~x )(~x) = 0. There is no inner motion along the length of the string (so called longitudinal polar- ization of modes). This is natural too: it is a effect of reparametrization invariance. Repeat: There is no longitudinal mode, only the transverse veloc- ity enters. A longitudinal velocity is an artifact of changing parametrizations of the worldsheet and has no content As such, we can imagine that for each little piece of string ds, there is a rest T ds, and it moves at a vT . The action looks as if we are just summing the particle action over these bits. (Remember thatxx ˙ 0 = 0 means that the velocity vector is transverse to the string bit direction), but this is a bit if a fake, as each bit would not usually retain it’s rest mass. If we don’t fix gσσ at the initial slice (not a good idea since in general gσσ will be time dependent), then the action is written as Z q 2 0 S = −T 1 − vT |~x |dtdσ (26) and |~x0|dσ is the density of string length per unit sigma. Now we can consider better options for gauge fixing σ. If the string is open, we fix σ so that the range of σ always lies in the 0, π range at t = 0. This range will be preserved if the boundary conditions of the string are time independent (ends fixed somewhere lets say, because then the vector

7 ∂t ' ∂τ |σ=0,π would already be orthogonal to ∂σ), so the value of σ would be preserved by our gauge choice. This is also true for some directions being Dirichlet and some being Neumann boundary conditions, so long as these conditions are time independent (we say that the string ends on a static brane). For closed strings the range is chosen from (0, 2π) with periodic boundary conditions. This would also tell us that the (infinitesimal) future of 0 and 2π match, because the future point is defined by the longest proper time inside the worldsheet from a given starting point and ending at a slightly later time. From here, we can try to solve the equations of motion. However, the action is still not simple enough to solve the equations of motion directly. We can check however, that given a specific form of Xµ(σ, τ), that it solves the equations of motion.

4 The meaning of T

We now want to make sure that T can indeed be interpreted as the mass per unit length of a string. To this effect notice that the following is a solution of the equations of motion:

X0 = τ, X1 = σ, X2,...D−1 = 0 (27)

1 1 with fixed boundary conditions X (τ, 0) = 0, X (τ, σmax) = L. This is basically a string of length L. From a direct computation, we get that vT = 0 (The transverse directions are x2,...XD−1, which are not moving). If we compute the energy, we obtain it from a Legendre transform Z E = dσΠX˙ − L (28) where L is the lagrangian density. Since this solution has X˙ = 0, and |X~ 0| = 1 for the dynamical variables (the ones that contribute to a transverse velocity), we find that Z E = − dσL = TL (29)

This is, the energy is equal to the total length of the string, times T . This is what we called the string tension: it gets interpreted as the energy per unit length of a string.

8 Now let us do small fluctuations around this solution. A small fluctuation requires adding transverse motion (keeping the boundary condition fixed for the time being). A small fluctuation is an infinitesimal change δx⊥, δx˙ ⊥ for 2,...,D−1 ˙ ~ 0 2 ~ 0 2 X . These contribute both to vT = δX⊥ and to |X | = 1 + |δX⊥| . Taylor expanding, we get that 1 1 1 1 L = −T (1 − |δ~X˙ |2 + |δX~ 0 |2) = −T + T |δ~X˙ |2 − T |δX~ 0 |2 + ... (30) 2 ⊥ 2 ⊥ 2 ⊥ 2 ⊥ This can be compared to the Lagrangian of a non-relativisitc string. The first term is a constant. This is the bare potential energy per unit length of ˙ 2 the string. The term with squared is~X⊥| , and is usually identified with the mass density per unit length. This is equal to T . The term that is quadratic in gradients is again equal to T and is the non-relativistic string tension. The equations of motion derived for the fluctuations are the wave equation. This is a wave equation for transverse fluctuations. The speed of sound is equal to one: this means that the fluctuations on the string propagate at the speed of light within the string.

5 Lightcone gauge

The first thing we notice is that the equations of motion of the string look very similar to massless fields propagating on the worldsheet. This is, the equation of motion of X looks as follows √ ∗ ∗αβ µ ∂α g g ∂βX = 0 (31) We would have obtained the same equations of motion from an action 1 Z √ g∗g∗αβ∂ Xµ∂ X = 0 (32) 2 β α µ if g∗ could be considered as a fixed background metric. These equations are those of a massless particle moving in a two dimen- sional worldsheet. Therefore it is convenient to go to a lightcone coordinate system, where light propagation is simplest. This is, we want to consider a coordinate system on the worldsheet that is adapted to the nature of massless particles. These special coordinates on the worldsheet define a lightcone coordinate system ds2 ∼ f(σ)dσ+dσ−

9 • One can easily picture a construction of such a coordinate system. Take any point of the worldsheet. The fact that the signature is Lorentzian means that for this point the lightcone consists of two lines (moving to the left and right at the speed of light). We use these two special curves to build a reference frame. Go to the left (future) and mark along the left null curve by some parametrization. Do a similar parametrization on the right null curve towards the future. At each point on these two special curves choose the other future-pointing null curve. These two curves will meet at some point p, so we define the coordinates of p by the projections on the two parametrizations. Call these σ±. It is clear from the construction that changing the coordinates of p along σ+ moves us along a null curve. Therefore the metric has no (dσ+)2 component. Similarly for dσ−

Also notice that in this coordinate system we have the property that

√ 0 1 −g∗g∗αβ ∼ (33) 1 0 so it is as if the metric were flat. This is, the equations of motion will take the form µ ∂+∂−X = 0 (34) The wave equation in two dimensions is easily solved. The most general solution is of the form

µ + − µ + µ − x (σ , σ ) = xL(σ ) + xR(σ ) (35) where xL,R are waves moving to the left or right on the worldsheet. We now ± write these in Lightcone formalism, where we have x and X⊥ 0 1 Since g ∼ f(σ±), we find that it must be the case that ind 1 0

+ − g++ = gσ+σ+ = 0 = −∂+x ∂+x + ∂+x⊥∂+x⊥ (36) + − g−− = gσ−σ− = 0 = −∂−x ∂−x + ∂−x⊥∂−x⊥ (37)

These are constraints that need to be satisfied in order to have a solution of the string equations. These equations are called the Virasoro constraints.

10 Now, the choice of lightcone gauge is that X+ = Cτ = C(σ+ + σ−)/2 This defines almost uniquely the coordinates σ+ and σ−. Notice that in this gauge the Virasoro constraints simplify:

− 0 = −C∂+x + ∂+x⊥∂+x⊥ (38)

This can be rewritten as 1 ∂ x− = ∂ x ∂ x (39) + C + ⊥ + ⊥

+ − so if we fix xL⊥(σ ), xR⊥(σ ) to our favorite set of functions, we can integrate to obtain x−(σ±) up to a constant of integration. This means that if the equations of motions for x⊥ are satisfied, we can construct a one parameter solution of the equations of motion in lightcone gauge and we’re “done” solving the classical string motion. This parameter is an initial condition: the center of mass coordinate of the string in the direction x−. To fix the constant C, we specify that the string coordinate σ = (σ+ −σ−) runs from (0, π) or from (0, 2pi) depending on if we have an open versus a R + closed string. In that case we have that dσ∂τ X = Cπ or C2π. This will be seen to be proportional to the P + (where all these factors of 2π etc. in the definition of the string tension will actually start to matter). The periodicity in σ will then induce periodicities in σ|pm that will need to be explored.

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