Feynman Path Integral Formulation

1.10 String Theory 43Solutions to the massless wave equation in Eq. (1.197) can be obtained in the usualway, by settingX μ (σ,τ) =X μ R (σ − )+X μ L (σ + ) , (1.200)with σ ± ≡ τ ± σ. Then X μ R (σ − ) is the right-moving mode, while X μ L (σ + ) is theleft-moving mode. The boundary conditions depend on whether one has a closedor open string. For closed strings the boundary condition is a periodicity in the σcoordinate,X μ (σ,τ) =X μ (σ + π,τ) . (1.201)For open strings one requires the vanishing of the normal derivative X ′μ = 0atσ = 0,π.One can then expand the solutions to the wave equation of Eq. (1.197) in Fourieramplitudes α μ nX μ R= 1 2 xμ + 1 2 l2 p μ (τ − σ)+ 1 2 il 1∑n≠0n α n μ e −2in(τ−σ) , (1.202)and similarly for X μ L in terms of ˜α n μ . Only an outline of the method will be providedhere, the reader is referred for more detail for example to the recent monograph(Becker, √ Becker and Schwarz, 2007). Here l is the fundamental string length l =2α ′ , and x μ and p μ are the center of mass coordinate and momentum of the string.The reality condition on X μ impliesα μ −n =(α μ n ) † , (1.203)and a similar requirement for the ˜α μ n ’s. To get the correct commutation relations forthe α μ n ’s and ˜α μ n ’s one needs the Poisson brackets for the X μ variables. The onlynon-vanishing one is{Ẋ μ (σ),X ν (σ ′ )} = T −10δ(σ − σ ′ )η μν , (1.204)which implies the commutation relations (via the usual replacement of the Poissonbracket with the commutator {...}→−i[...])fortheα μ n ’s[α μ m,α ν n ]=m δ m+n η μν , (1.205)a similar expression for ˜α μ n , and all other commutators equal to zero. Up to a factorof √ m difference in normalization, these are quite similar to the usual harmonicoscillator operators. But note that due to the appearance of the η μν on the r.h.s. theHilbert space built up from the oscillator operators α μ n is not positive definite.The classical Hamiltonian for the two-dimensional closed string is given byH = 1 2 T ∫ π0dσ(Ẋ ) 2 + X ′ 2= 1 2+∞∑n=−∞α −n α n . (1.206)