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Scale Transformations of Fields and Correlation Functions

The number d φ is called the field dimension of the free field φ(x). Its value (7.4) is a direct consequence of the naive dimensional properties of x and φ. In the units used in this text, in which we have set kBT = 1 [recall the convention stated before Eq. (2.6)], the exponential in the partition function is e, so that E[φ] is a dimensionless quantity. The coordinates have the dimension of a length. This property is expressed by the equation [x] = L. The field in (7.5) has then a naive dimension (also called engineering or technical dimension) [φ] = L 0 φ . To establish contact with the field theories of elementary particle physics, we shall use further natural units in which c = h̄ = 1. Then the length L is equal to an inverse mass μ (more precisely, L is the Compton wavelength L = h̄/mc associated with the mass m) . It is conventional to specify