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where terms not written out involve higher derivatives of m(x), each with its own coupling constant. The coefficient of the first term is chosen to be ~ to fix the scale of the order parameter. The coupling constants are functions of the cutoff A, which eventually tends to infinity. Our goal is to obtain finite physical answers in that limit. To that end, we continue the coarse-graining, which introduced A in the first place, to ever longer wavelengths. As before, the RG transformation is defined as coarse-graining followed by rescaling, to make the system look like the original one. We shall define the procedure in "momentum space," the k space of Fourier

Fig. 18.5 Coarse-graining in momentum space and in real space. In the former, one effectively lowers the cutoff. In the latter, one blots out finer details, enlarging the effective lattice spacing.

analysis. Figure 18.5 shows a comparison between coarse-graining in momentum and in "real" space: Instead of expanding the unit cell in real space, we lower the effective cutoff in momentum space. In terms of the Fourier transforms k) of the order parameter defined in (17.8), the Hamiltonian takes the form

E[m]

FIGURE 9.25 One of the big winners of the telecom bubble of 1999/2000 was Lucent (symbol LU). Once the bubble burst, heralded by the rounded top, LU descended from $50 to nearly $0.

(18.53)

j dm(k) dm*(k) e-E[m]

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You can use indents as another way to control the horizontal positioning of text in a document. Indents are simply margins that affect individual lines or paragraphs. You might use an indent to distinguish a particular paragraph on a page for example, a long quote. Word offers several tools for setting indents. For example, the Home tab on the Ribbon contains buttons for quickly increasing and decreasing indents by a predefined amount. You can make more precise changes to indent settings in the Paragraph dialog box. Finally, you can use the Word ruler to set indents.

(18.54)

where the prime indicates that the product is to be taken only over half of k space (because we are dealing with a real order parameter). We could also disregard the prime, but take the square root of the answer. The RG transformation consists of "integrating out" the k values within a shell between the radii A and Alb (b > 1), and then rescaling. Specifically the procedure consists of three steps:

1. Integration. Define a new Hamiltonian E' by "integrating out" the k values whose magnitude lies between A and Alb (b > 1):

(18.55)

where the constant Q is a function of A and all the coupling (k) with constants. The new Hamiltonian E' [m 1depends only on m Ikl < Alb. Apart from that, it has the same form as (18.53) with new coupling constants:

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