By I. T. Todorov, D. Ter Haar
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Extra resources for Analytic Properties of Feynman Diagrams in Quantum Field Theory
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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