By I. T. Todorov, D. Ter Haar

ISBN-10: 0080165443

ISBN-13: 9780080165448

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Ferreira and C. 1272[astro-ph] (unpublished). 5. M. Ba˜nados, A. Gomberoff and D. Rodrigues (unpublished). 6. N. Boulanger, S. Cnockaert and M. Henneaux, JHEP 0306, 060 (2003) [arXiv:hep-th/ 0306023]. 7. S. Deser and G. W. Gibbons, Class. Quant. Grav. 15, L35 (1998) [arXiv:hep-th/9803049]. 8. E. S. Fradkin and A. A. Tseytlin, Ann. Phys. 162, 31 (1985). 9. S. B. Giddings, Phys. Lett. B 268, 17 (1991). 10. G. T. Horowitz, Class. Quant. Grav. 8, 587 (1991). 11. A. Y. Kamenshchik, U. Moschella and V.

In the modern particle physics these two theories are very intimately connected even though the direct physical consequences of them look quite different. 3 The Hamiltonian as a Quadratic Form The two theories share also another unique property. We note that the free Hamiltonian for the (N = 4, d = 4) Yang–Mills Theory H0 = d 4 x d 4 θ d 4 θ¯ φ¯ a 2 ∂ ∂¯ a φ , ∂ +2 (49) 44 L. Brink can be rewritten as a quadratic form H0 = 1 √ ( W0 , W0 ), 2 2 (50) using the inner product notation ( φ , ξ ) ≡ 2i 1 d 4x d 4θ d 4 θ¯ φ¯ + ξ , ∂ (51) where φ and ξ are chiral superfields and W0a = ∂ q¯ φ a , ∂+ + (52) is a fermionic superfield, the free dynamical supersymmetry variation of the superfield (SU(4) spinor indices are summed over).

C. 4 ∂ + O(κ 3 ), (136) while the scalar supergravity Lagrangian (108) becomes 1 i jkl + − i jkl C (∂ ∂ − ∂ ∂¯ )C 24 κ2 klmn pqi j pqi j + Ci jkl C (∂ +Cmnpq ∂ −C + ∂ −Cmnpq ∂ +C 96 pqi j pqi j −∂ Cmnpq ∂¯ C − ∂¯ Cmnpq ∂ C ) + O(κ 3 ). LS = − (137) 60 L. Brink Both contain the light-cone time derivative ∂ − in their interactions. In order to have a Hamiltonian without this derivative we eliminate it by the field redefinitions C i jkl = D i jkl − κ2 1 D pq[i j ∂ + Dkl]mn D pqmn 4 ∂+ 1 ∂ + B[i j + (Dkl]mn ∂ + Bmn ) ∂ + 3κ2 2∂+ + 1 3 κ 2 i jklrstu ε ∂ + Brs + (Dtumn ∂ + Bmn ) + O(κ 3 ), + 2 · 4!

### Analytic Properties of Feynman Diagrams in Quantum Field Theory by I. T. Todorov, D. Ter Haar

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