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

CHAPTER 4. CONFORMAL INVARIANCE 54to expect that these particles are the quantized products <strong>of</strong> classical fields. Hence, wemight raise the question if we should not have considered these fields, and their effectson the string action, before we quantized our string action, which is exactly what thischapter investigates.For all intended purposes <strong>of</strong> this chapter, it suffices to look at this problem for thespecific case <strong>of</strong> bosonic string theory.4.1 Conformal Field Theory Jr.Before anything else, it should again be duly noted that the aim <strong>of</strong> this section is not toexplain things, but merely to get a point accross, the point being that conformal fieldtheory is an absolute must for string theory.With that in mind, let us move on. Considering a general coordinate transformationx −→ x ′ , one finds that the metric changes according tog µν (x) −→ g ′ µν(x′ ) = ∂xα∂x ′µ ∂x β∂x ′ν g αβ (x) . (4.1)The group <strong>of</strong> conformal transformations is comprised <strong>of</strong> the subset <strong>of</strong> these generalcoordinate transformations that preserve the angle between two vectors. Concretely, thismeans that they leave the metric invariant, up to a rescaling. 1 Hence, when applying aconformal transformation, the metric changes according tog µν (x) −→ g µν′ (x′ ) = Ω(x)g µν (x). (4.2)The transformations that obey this property are translations, rotations, dilatations, andthe so-called special conformal transformations which vary a vector x µ according toδx µ = b µ x 2 − 2x µ b · x, (4.3)with b µ an infinitesimal translation. If one specializes to the case <strong>of</strong> Minkowski metric,and considers an infinitesimal coordinate transformation x µ −→ x µ + ǫ µ , one can findthe constraint on the parameter in flat space-time to be(δ µν + (D − 2)∂ µ ∂ ν ) ∂ · ǫ = 0, (4.4)with D the dimensionality <strong>of</strong> space-time. For D > 2, the parameter ǫ can be at mostquadratic in x, but if D = 2, as is the case for the string world sheet, this constraint doesnot apply. The consequence there<strong>of</strong> is that the associated generator algebra is infinitedimensional. To see this, and focussing on the string world sheet, introduce coordinatesz = τ − iσ, (4.5)¯z = τ + iσ. (4.6)1 Note that this is not the same as a Weyl rescaling, which does not entail a change <strong>of</strong> coordinates.