exam2 review - MASSACHUSETTS INSTITUTE OF TECHNOLOGY...

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MASSACHUSETTS INSTITUTE OF TECHNOLOGY Physics Department 8.251: String Theory for Undergraduates Prof. Barton Zwiebach April 30, 2007 REVIEW NOTES FOR TEST 2 Notes by Alan Guth I. World-Sheet Currents: Noether’s Theorem: Suppose than an action of the form S = 0 1 . . . k L ( φ a , ∂ α φ a ) (1.1) is invariant under an infinitesimal variation of the fields φ a ( ξ ) φ a ( ξ ) + δφ a ( ξ ) , with δφ a ( ξ ) = i h a i ( φ a , ∂ α φ a ) (1.2) in the sense that the Lagrangian density is changed at most by a total derivative, δ L = i Λ α . (1.3) ∂ξ α i Then the currents j i α ( ξ ) defined by i j i α ( α L φ a ) δφ a i Λ α i (1.4) are conserved: α j i α = 0 5) (for each i ). (1. This implies that the corresponding charges, Q i = 1 . . . k j i 0 ( ξ ) , (1.6) are independent of time. The proof is constructed by replacing δ L in Eq. (1.3) by an expansion in terms of the derivatives of L and the variation of the fields (1.2), and then using the Lagrangian equations of motion.
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8.251 REVIEW NOTES FOR TEST 2, SPRING 2007 p. 2 SEC. I: WORLD-SHEET CURRENTS World-Sheet Application 1: Conservation of Momentum: Identifying ξ 0 τ and ξ 1 σ , we can apply this theorem to the string world-sheet. Defining X ˙ µ τ X µ and X µ σ X µ , the string action can be written S = T 0 τ f σ 1 ( X ˙ · X ) 2 ( X ˙ ) 2 ( X ) 2 = 0 1 L ( 0 X µ , ∂ 1 X µ ) , (1.7) c τ i 0 which is invariant under the symmetry δX µ ( τ, σ ) = µ , (1.8) which describes a uniform spacetime translation of the string coordinates X µ . (Note that this is really a family of D symmetries, one for each value of the spacetime index µ . But I will continue to describe it as one symmetry, in the sense that it forms one multiplet of symmetries.) The corresponding conserved current is j µ α ( j µ 0 , j µ 1 ) = L , L = P µ τ , P µ σ . (1.9) ∂X ˙ µ ∂X µ More compactly, j µ α = P µ α . (1.10) The conserved charge is the total spacetime momentum of the string: σ 1 p µ = P µ τ ( τ, σ ) . (1.11) 0 The above expression gives the conserved momentum as an integral over a line of constant τ , but the reparameterization invariance of the string suggests that there is nothing special about such a line. In fact we found that the conservation equation α j µ α = 0 implies, with the use of the two-dimensional divergence theorem, that we can write p µ = P µ τ − P µ σ , (1.12) γ where γ describes a general curve. For open strings γ must begin at one end of the string and end at the other, and for closed strings it must wind once around the world-sheet.
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8.251 REVIEW NOTES FOR TEST 2, SPRING 2007 p. 3 SEC. I: WORLD-SHEET CURRENTS World-Sheet Application 2: Lorentz Symmetry and its Currents: Lorentz transformations can be described by δX µ = µν X ν , where µν = νµ . (1.13) The string Lagrangian is invariant under this symmetry, and with Noether’s theorem one obtains the conserved world-sheet current M α P α X ν P α , where α = 0 . (1.14) µν = X µ ν µ α M µν The conserved charge is
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