DBI Analysis of Open String Bound States on Non-compact D-branes

CHAPTER 2. BOSONIC STRINGS 26to use Eq. 2.34, this readsẊ µ R = l s∑α µ ne −2in(τ−σ) , (2.78)Ẋ µ L = l s∑˜α µ ne −2in(τ+σ) , (2.79)X µ′R = −l s∑α µ ne −2in(τ−σ) , (2.80)X µ′L = l s∑˜α µ ne −2in(τ+σ) . (2.81)Next, we compute the square <strong>of</strong> these derivatives.)Ẋ 2 = Ẋ2 R + Ẋ2 L + 2(ẊR · Ẋ L⎧( )⎨2 ( ) ⎫2⎬ = ls∑α 2 µ ⎩ne −2in(τ−σ) +∑˜α ne µ −2in(τ+σ) ⎭{( ) ( )}+ 2ls∑α 2 n e −2in(τ−σ) ·∑˜α n e −2in(τ+σ)(2.82)(X′ ) 2 ( )= X′ 2 ( )R + X′ 2 (L + 2 X′R · X L)′⎧( )⎨2 ( ) ⎫2⎬ = ls∑α 2 µ ⎩ne −2in(τ−σ) +∑˜α ne µ −2in(τ+σ) ⎭{( ) ( )}− 2ls∑α 2 n e −2in(τ−σ) ·∑˜α n e −2in(τ+σ)(2.83)When summing Eqs. 2.82 and 2.83, we see that the crossterms will cancel each other,and that the squared terms will just gain a coefficient <strong>of</strong> two:⎧( )⎨2 ( ) ⎫2⎬ Ẋ 2 + X ′2 = 2ls∑α 2 µ ⎩ne −2in(τ−σ) +∑˜α ne µ −2in(τ+σ) ⎭ . (2.84)A general term in these squared sums can be written as(α m · α n )e −2i(τ−σ)(m+n) , (2.85)and this expression is integrated in Eq. 2.77. Recalling that∫ π0dσ e 2iσ(m+n) = πδ m+n , (2.86)