By M. Aizenman (Chief Editor)

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A word on notation. From now on we will omit the explicit reference to the volume form in the integrals. We shall therefore use the following short-hand notation: V (ω) dx · · · → ··· . 2. The Interaction. Let us now consider the effects of the perturbation S1 . Corresponding to the split (11), the effect of S1 is to introduce two bulk graphs: i V = − Habc x c dζ a ∧ dζ b , 6 i W = − Habc ζ a dζ b ∧ dζ c . 6 (14) (15) We will then consider the following path integral: [dX] e−S0 (X)−S1 (X) f1 (X (τ1 )) · · · fn (X (τn )) [dX] e−S0 (X) [1 + V + W] f1 (X (τ1 )) · · · fn (X (τn )) .

K This implies that4 the number of independent coefficients Wij k is n−1 3 , and that there is a totally antisymmetric tensor, Wij kl , such that Wij k = l Wij kl . Concretely, one can choose Wij kl = 1 Wij k − Wij l + Wikl − Wj kl . n (27) Therefore Eq. (26) can be written as 1 6 Wij kl Sij k − Sij l + Sikl − Sj kl . Sij k Wij kl = i,j,k,l i

Unitary equivalence of temperature dynamics for ideal and locally perturbed fermi-gas. Commun. Math. Phys. 91, no. 4, 301–312 (1983) 5. : Local existence for the Maxwell–Dirac equations in three space dimensions. Comm. Partial Diff. Equs. 21, no. 5–6, 693–720 (1996) 6. : Asymptotic Gaussian property of the solution of the Burgers equation with random initial data. Theory Probab. Appl. 36, no. 2, 217–236 (1991) 7. : Ergodic Theory. New York–Berlin: Springer, 1982 8. : One-dimensional harmonic lattice caricature of hydrodynamics: A higher correction.

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