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11) Wr(J,0) = fk (e i(j)(15)r ) • Let a: F C be defined by n a(x) = E aor(x,sk) = { 60 - dak k=1 if x = xk otherwise. 13) cy=0 Proof. dxn n! n>1 k=1 [ft k=1 6 j (6x k J=0 ) VVF(J, 0) J=0 n = n E ni1 11 J (xk ) f dx 1 . 7. k=1 k=1 fk ((J, (15)r') n>1 = f2c (e i(jM r) — 1. 1). 13) is obvious from the definition if one sets J(s) = e- da(x). • 30 2. Techniques We shall use the suggestive notation flc (ei( ") r) for Wr (J, (to) since the power series is convergent for all 0 and J. The next theorem contains an alternative formula for the Wick ordered monomials that will be useful in Chap.

We define p - x = pi x i + .. + ,.. 7 and 6(p,q) = Ld6pq . The Fourier Pdxd, fr. dPP(p) = IrdE pEr* P(P) transform of f : F C is f : r* C, given by f(p) = f dx e -i" f (x). r The inverse relation is f(x) = f - dp ei" r i(P). We denote the scalar product for functions i4, 7:b E Cr* by (îko'k)r. 108) Fourier transformation is unitary: (0, Tp) r . = (0, T,b)r. The integral kernel of a linear operator transforms as Â(p,q) = f dx f dy e -i"A(x, y)e i". 109) 20 1. 3(p, q) = 6r. 111) cos(epk )). - k=1 That is, the Laplacian is diagonal in the Fourier basis.

For d = 2, the constant depends on mo > O. 130) we have fP {7)415- Z76 } dd p < 2 ± m0 2 — Cr e,L (X, X) < J {P:iPl< ddp mo 2 ± 3 7 11 2 . 137)). 6 Problems With and Without Cutoffs To see why the short-distance singularity of the continuum covariance is a problem, consider (0(s) 2 )F,À=0 . Icy (0) 0(X) 2 = C r (X , X) —+ . 132) So, the second moment (0(x)0(y)) = C(s, y) has a singularity at s = y (coinciding points). This by itself would not be a problem. The problem is that these singularities cause divergences in the continuum limit of the partition function, and of the correlations, defined above.

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Concepts of force; a study in the foundations of dynamics by Max Jamner

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