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By Penrose R.

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**Example text**

Using the identity I 1 [exp ( i sF[-i6/6J(~)] d4y ,J(x) = Fr[-i6/6J(x)]exp ( i /F[-i6/6J] 1 (3-62) the proof of which is left as Exercise 3-8, one finds that the solution of Eq. (3-60) can be written in terms of Z"{nas: [ 19ht[-i6/6J(x)] d4x ] Zo{J) Z{J) = exp i (3-63) Applying K , to 6Z{J)/iFJ(x), and using (3-61) and (3-62), gives: { exp i I YInt[-i6/6J(x)] d4x K,6/i6J(x)Zo{fl = exp( i SYI,, [-i6/6J(x)] 1 d4x J(x)Z"{J) [ ( J -Yht[-i6/6J(x)] =J(x)Z{J) t exp i = J(x)z{J) + Y int [-i6/6J(x)] 1 1 d4x ,J(x) Z0{a Z{J) Equation (3-60) determines 2 up to a multiplicative constant which we choose by imposing the condition Z{J = 0) = 1 To complete the calculation of Z we still have to solve (3-61) for Z o { J } .

From (2-65) we see also that p2 has the same dimensions as k2, [ p 2 ] =L-' (2-72) 2-54 Some Properties of the Free Theory - a Free Euclidean Field Theory in Less than Four Dimensions From (2-65) we see immediately that when p2 0 the field amphtude with k = 0 becomes unstable. In fact, when p2 < 0, the free theory is undefined. We can calculate the correlation function -+ Go(k) = ( *)#(-k) )o Note that Go(k) is, with the definition (2-67) of the Fourier transform, the Fourier transform of the spatial correlation function, namely, which, when V -+ 00, becomes: Go(k) = dk 1~ exp(-ik * r)Go(r) (2-75) As was discussed in Sec.

A N D CRITICAL P H E N O M E S A in (2-77) has to be carried out. The result is In 1 1 = - - Z ln(k2 + p 2 ) + - h(k)G,' (k)h(-k) 2 k 2 k + constant (2-83) Setting h = 0, and using the prescription (2-34) to calculate the specific heat, we find for the leading term in T - To Below four dimensions the specific heat diverges, due to the infrared singularity, and a= 4 (4 - d ) (2-85) There is no divergence when d > 4. But following the constant term, there may be a non-analytic term. It is easy to see that the leading term in the specific heat would come from: In field theory, this dependence of the specific heat on the cut-off, is reflected in the fact that the @2 - (p2 Green function is divergent, even in zeroth order in perturbation theory.

### A lecture on 5-fold symmetry and tilings of the plane by Penrose R.

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