By Warren Siegel

"For an individual drawn to the quickly constructing box of string conception (graduate scholars with the right kind physics background), Siegel's e-book is a very sturdy advent. " B R Parker selection, united states, 1989 "Siegel has performed a huge function within the software of BRST formalisms for quantization to thread concept ... Siegel's ebook is of significant curiosity when you intend to do study in string box theory." O W Greenberg Foundations of Physics, united states, 1991

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GENERAL BRST OSp(1,1|2) will play a central role in the following chapters: In chapt. 4 it will be used to derive free gauge-invariant actions. A more general form will be derived in the following sections, but the methods of this section will also be used in sect. 3 to describe Lorentz-gauge quantization of the string. 4. From the light cone In this section we will derive a general OSp(1,1|2) algebra from the light-cone Poincar´e algebra of sect. 3, using concepts developed in sect. 6. We’ll use this general OSp(1,1|2) to derive a general IGL(1), and show how IGL(1) can be extended to include interactions.

GENERAL BRST F αβ = 0 → pα Aβ = − 21 {Aα , Aβ } = −Aα Aβ ∇α φ = 0 → pα φ = −Aα φ . ) Deﬁning “ | ” to mean |xα =0 , we now interpret Aa | as the usual gauge ﬁeld, iAα | as the FP ghost, and the BRST operator Q as Q(ψ|) = (pα ψ)|. ) Then ∂ α ∂ β = 0 (since α takes only one value and ∂ α is anticommuting) implies nilpotence Q2 = 0 . 5) is the auxiliary ﬁeld. 6b) where we have generalized to graded algebras with graded commutator [ , } (commutator or anticommutator, as appropriate). 6b) are the usual group structure constants times δ-functions in the coordinates.

5b), and ﬁeld strengths with graded commutators (commutators or anticommutators, according to the statistics). 2a) so that gauge-invariant quantities can be constructed only from the usual F ab . 2a) (consider {∇α , ∇β } and [∇α , ∇a ] acting on φ). These constraints can be solved easily: F αa = 0 → pα Aa = [∇a , Aα ] , 32 3. GENERAL BRST F αβ = 0 → pα Aβ = − 21 {Aα , Aβ } = −Aα Aβ ∇α φ = 0 → pα φ = −Aα φ . ) Deﬁning “ | ” to mean |xα =0 , we now interpret Aa | as the usual gauge ﬁeld, iAα | as the FP ghost, and the BRST operator Q as Q(ψ|) = (pα ψ)|.