By Perrine C. D.

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Hypothesis (HS) is typically verified using the techniques of Appendix C. 3 Let : · :S denote Wick ordering with respect to the covariance S. a) Prove that if bH :bJ :S dµS (b) ≤ F|H|+|J| for all H, J ∈ r≥0 Mr then bH :bJ :zS dµzS (b) ≤ Hint: first prove that |z| F |H|+|J| bH :bJ :zS dµzS (b) = z (|H|+|J|)/2 for all H, J ∈ r≥0 Mr bH :bJ :S dµS (b). 21. 6 D W (α+1)F Assume Hypotheses (HG) and (HS). Let α ≥ 2 and W ∈ AC0 obey ≤ 1/3 . 11. 7 D W (α+1)F Assume Hypotheses (HG) and (HS). Let α ≥ 2 and W ∈ AC0 obey ≤ 1/3 .

35. Hypothesis (HS) is typically verified using the techniques of Appendix C. 3 Let : · :S denote Wick ordering with respect to the covariance S. a) Prove that if bH :bJ :S dµS (b) ≤ F|H|+|J| for all H, J ∈ r≥0 Mr then bH :bJ :zS dµzS (b) ≤ Hint: first prove that |z| F |H|+|J| bH :bJ :zS dµzS (b) = z (|H|+|J|)/2 for all H, J ∈ r≥0 Mr bH :bJ :S dµS (b). 21. 6 D W (α+1)F Assume Hypotheses (HG) and (HS). Let α ≥ 2 and W ∈ AC0 obey ≤ 1/3 . 11. 7 D W (α+1)F Assume Hypotheses (HG) and (HS). Let α ≥ 2 and W ∈ AC0 obey ≤ 1/3 .

DµC (ψ) ≤ 2n E(k) . ψ (x , κ ) · · · ψ . σi,1 i,1 i,1 σi,ei (xi,ei , κi,ei ) . dµC (ψ) ≤ 2 Here E(k) denotes the norm of the matrix Eσ,σ (k) 38 σ,σ ∈S dk (2π)d+1 E(k) (m+n)/2 dk (2π)d+1 as an operator on 2 Σi ei 2 C|S| . Proof: Define (i, µ) 1 ≤ i ≤ n, 1 ≤ µ ≤ ei X= A (i, µ), (i , µ ) = Cσi,µ ,σi ,µ (xi,µ , xi ,µ ) Let Ψ (i, µ), κ , (i, µ) ∈ X, κ ∈ {0, 1} be generators of a Grassmann algebra and let dµA (Ψ) be the Grassmann Gaussian measure on that algebra with covariance A. This construction has been arranged so that ψσi,µ (xi,µ , κi,µ )ψσi ,µ (xi ,µ , κi ,µ ) dµC (ψ) = Ψ (i, µ), κi,µ Ψ (i , µ ), κi ,µ ) dµA (Ψ) and consequently n i=1 .