Chapter 8 World-sheet Currents For physical insight, physicists often turn to ideas of symmetry and invari- ance. Symmetry properties of dynamical systems and conserved quantities are closely related. We will learn that in string theory there are currents that flow on the two-dimensional world-sheet traced out by the string in spacetime. The conserved charges associated with these currents are key quantities that characterize the free motion of strings. We will discover a nice physical in- terpretation of the objects Pτ and Pσ which we encountered earlier. 8.1 Electric charge conservation We begin our study by reviewing the physics and the mathematics of charge conservation in the context of Maxwell theory. This classic example will help us to develop a more general understanding of the concept of conserved currents. In electromagnetism, the conserved current is the four vector jα =(cρ,j ), where ρ is the charge density and j is the current density. Why do we say that jα a conserved current? By definition, jα is a conserved current because it satisfies the equation α ∂αj =0. (8.1.1) Any four-vector which satisfies this equation is called a conserved current. The term “conserved current” is a little misleading, but it is a convention. More precisely, we should say that we have a conserved charge,because it is really the charge associated with the current that is conserved. Let us see how this conservation arises. 173 174 CHAPTER 8. WORLD-SHEET CURRENTS When we separate the space and time indices in (8.1.1) we get ∂j0 ∂ j0 + ∂ ji = + ∇·j =0. (8.1.2) 0 i ∂x0 Why is this equation a statement of charge conservation? In electromag- netism, the total charge Q(t)inavolume V is just the integral of the charge density ρ over the volume: j0(t, x) Q(t)= ρ(t, x) d3x = d3x. (8.1.3) V V c Up to a constant, the charge is the integral over space of the first component of the current. Its time derivative is given by dQ ∂j0 = d3x. (8.1.4) 0 dt V ∂x Using equation (8.1.2), we can write dQ = − ∇·jd3x. (8.1.5) dt V Letting S denote the boundary of V , the divergence theorem gives: dQ = j · da. (8.1.6) dt S This equation encodes the physical statement of charge conservation: the charge inside a volume V can only change if there is an appropriate flux of current across the surface S bounding the volume. In many cases, we take V to be so large that the current j vanishes on the surface S.Inthese cases, dQ =0. (8.1.7) dt The charge Q is then time-independent and is said to be “conserved”. 8.2 Conserved charges from Lagrangian sym- metries One of the most useful properties of Lagrangians is that they can be used to deduce the existence of conserved quantities. Conserved quantities can 8.2. CONSERVED CHARGES FROM LAGRANGIAN SYMMETRIES175 be quite helpful in understanding the dynamics of a system. In this section we begin our work in the context of Lagrangian mechanics, learning how to construct the conserved quantity associated to a symmetry. We then turn to Lagrangian densities, and show how to construct the conserved current associated to a symmetry. Let L(q, q˙; t)beaLagrangian that depends on a coordinate q,avelocityq ˙, and possibly, time. Consider, moreover, a variation of the coordinate of the form q(t) → q(t)+δq(t) , (8.2.1) where δq(t)issome specific infinitesimal variation. Infinitesimal variations can be written as δq(t)=h(q(t), q˙(t); t) , (8.2.2) where is an infinitesimal constant and h(q, q˙; t)isafunction. As a result of the variation in (8.2.1), there is a corresponding variation in the velocityq ˙: d(δq(t)) q˙(t) → q˙(t)+ . (8.2.3) dt Suppose you wish to determine how L(q, q˙; t)changes as a result of the vari- ations (8.2.1) and (8.2.3). Because δq is infinitesimal, the variation of the Lagrangian consists only of terms which are linear in δq.Ifthose terms vanish, the Lagrangian is said to be invariant. We claim that if the Lagrangian is indeed invariant under the variations (8.2.1) and (8.2.3), then the quantity Q, defined by ∂L Q≡ δq , (8.2.4) ∂q˙ is conserved in time for physical motion. That is, for any motion q(t) satis- fying the equations of motion, the ‘charge’ Q is a constant: dQ =0. (8.2.5) dt Note that the on the left-hand side of (8.2.4) cancels with the in the definition (8.2.2) of δq. 176 CHAPTER 8. WORLD-SHEET CURRENTS To prove the conservation of Q, consider the Euler-Lagrange equations: d ∂L ∂L − =0. (8.2.6) dt ∂q˙ ∂q Since the Lagrangian does not change under the coordinate and velocity variations (8.2.1) and (8.2.3), we must have ∂L ∂L d δq + (δq)=0. (8.2.7) ∂q ∂q˙ dt We now prove our claim by direct calculation of the time derivative of Q: dQ d ∂L ∂L d ∂L ∂L d = δq + (δq)= δq + (δq)=0, (8.2.8) dt dt ∂q˙ ∂q˙ dt ∂q ∂q˙ dt where we used (8.2.6) and (8.2.7). We see that our formulation of charge conservation is indeed correct. Let’s apply this new perspective on charge conservation to a Lagrangian that depends only on the velocity: L = L(˙q). How can we vary q but leave L(˙q) unchanged? One way is to apply q(t) → q(t)+, where is any constant. Correspondingly, we see thatq ˙(t) → q˙(t)+d/dt =˙q(t). Becauseq ˙ does not change, L(˙q)does not change. Making use of (8.2.4), we find ∂L ∂L ∂L Q= δq = −→ Q = . (8.2.9) ∂q˙ ∂q˙ ∂q˙ We recognize Q as the momentum associated to the coordinate q. This quan- tity is conserved because the position q does not appear in the Lagrangian. Note that the conservation equation dQ/dt =0coincides with the Euler- Lagrange equation (8.2.6). This example illustrates a familiar result in La- grangian mechanics: if the Lagrangian of a system does not contain a given coordinate, the momentum conjugate to that coordinate is conserved. For 1 2 a free nonrelativistic particle, for example, L = 2 m(˙q) .Inthis case Q = mq˙. Now let’s consider Lagrangian densities with symmetries. While a sym- metry of a Lagrangian guarantees the existence of conserved charges,asym- metry of a Lagrangian density guarantees the existence of conserved currents. We write the action as the integral of the Lagrangian density over the full set of coordinates ξα of some relevant “world”: 0 1 k a a S = dξ dξ ···dξ L(φ ,∂αφ ) . (8.2.10) 8.2. CONSERVED CHARGES FROM LAGRANGIAN SYMMETRIES177 Here k denotes the number of space dimensions of the world. The world could be the full Minkowski spacetime, some subspace thereof, or, for example, the two-dimensional parameter space of the string world-sheet. The fields φa(ξ) are functions of the coordinates, and ∂φa ∂ φa = , (8.2.11) α ∂ξα are derivatives of the fields with respect to the coordinates. Each value of the index a corresponds to a field. Consider now the infinitesimal variation φa(ξ) → φa(ξ)+δφa(ξ) , (8.2.12) a and the associated variation for the field derivatives ∂αφ . The infinitesimal variations are conveniently written as a i a δφ = hi (φ, ∂φ) , (8.2.13) where i are a set of infinitesimal constants, and, for brevity, we ommitted a i all indices in the arguments of hi .Wehave included an index in because the variation may involve several parameters. A spacetime translation, for example, would involve as many parameters as there are spacetime dimen- sions. Since the index i is repeated in (8.2.13), it is summed over. You must distinguish clearly the various types of indices we are working with: α index to label world coordinates ξα, or vector components, i index to label parameters in the symmetry transformation, (8.2.14) a index to label various fields in the Lagrangian. If L is invariant under (8.2.12) and the associated variations of the field α derivatives, then the quantities ji defined by ∂L i jα ≡ δφa , (8.2.15) i a ∂(∂αφ ) are conserved currents: α ∂α ji =0. (8.2.16) We will prove this shortly. In (8.2.15) the repeated field index a is summed over. If the index i is present in (8.2.13), then we have several currents 178 CHAPTER 8. WORLD-SHEET CURRENTS indexed by i, one for each parameter of the variation. The components of the currents, as many as the number of dimensions in the world, are indexed by α.Itisimportant not to confuse the very different roles that these two kinds of indices play: α ji : i labels the various currents, α labels the components of the currents. (8.2.17) We showed in section 8.1 that a conserved current gives rise to a conserved charge. The charge was the integral over space of the zeroth component of the current. It follows that the currents jα give rise to the conserved charges i 1 2 ··· k 0 Qi = dξ dξ dξ ji . (8.2.18) We have as many conserved charges as there are parameters in the symmetry transformation. Quick Calculation 8.1. Verify that (8.2.16) implies that dQ i =0, (8.2.19) dξ0 α when the currents ji vanish sufficiently rapidly at spatial infinity.