By U.S. Department Of Energy

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**Example text**

Eliminating Gij by their ﬁeld equations gives back the hamiltonian form. Many examples will be given in chapts. , we require H0 = 0 because the proper time is not physically observable). Here we give an interesting example in D=2 which will also be useful for strings. 14) is the derivative with respect to the 1 space coordinate (which acts as the index M or i from above). From the algebra of P ± A′ , it’s easy to check, at least at the Poisson bracket level, that the G algebra closes and H0 is invariant.

6b). 6a), these properties also hold if that term is dropped. 2. 6a) still gives Q2 = 0. 6a) is still nilpotent, it requires extra terms in order to allow gauge ﬁxing this invariance of the ghosts. In some cases (see sect. 4) this requires an inﬁnite number of new terms (and ghosts). In general, the procedure of adding in the additional ghosts and invariances can be tedious, but in sect. 4 we’ll ﬁnd a method which automatically gives them all at once. The gauge-ﬁxed action is required to be BRST-invariant.

6). 1a) where Aα now includes both ghost and antighost. The appearance of the NL ﬁeld is due to the ambiguity in the constraint F αβ = p(α Aβ) +· · ·. The remaining (anti)BRST transformation then follows from further diﬀerentiation: {pα , pβ }Aγ = 0 → 1 pα B = − 21 [Aα , B] + i 12 Aβ , {Aα , Aβ } . 1b) 36 3. 3) ∂C iβ) where ( ) means index symmetrization. These operators form an ISp(2) algebra J αβ = C i (α consisting of the translations Qα and rotations J αβ on the coordinates xα : {Qα , Qβ } = 0 , [J αβ , Qγ ] = −C γ(α Qβ) [J αβ , J γδ ] = −C (γ(α J β)δ) , .

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