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# 9.1: The Program of Renormalization

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$$\newcommand\DDelta ParseError: invalid DekiScript (click for details) Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[16], line 1, column 1 at template() at (Under_Construction/Arovas_Texts/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/09:_Renormalization/9.01:_The_Program_of_Renormalization), /content/body/p[1]/span, line 1, column 23 $$
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$$\newcommand\Vxi ParseError: invalid DekiScript (click for details) Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[20], line 1, column 1 at template() at (Under_Construction/Arovas_Texts/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/09:_Renormalization/9.01:_The_Program_of_Renormalization), /content/body/p[1]/span, line 1, column 23 $$
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$$\newcommand\Vpi ParseError: invalid DekiScript (click for details) Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[22], line 1, column 1 at template() at (Under_Construction/Arovas_Texts/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/09:_Renormalization/9.01:_The_Program_of_Renormalization), /content/body/p[1]/span, line 1, column 23 $$
$$\newcommand\Vvarpi ParseError: invalid DekiScript (click for details) Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[23], line 1, column 1 at template() at (Under_Construction/Arovas_Texts/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/09:_Renormalization/9.01:_The_Program_of_Renormalization), /content/body/p[1]/span, line 1, column 23 $$
$$\newcommand\Vrho ParseError: invalid DekiScript (click for details) Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[24], line 1, column 1 at template() at (Under_Construction/Arovas_Texts/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/09:_Renormalization/9.01:_The_Program_of_Renormalization), /content/body/p[1]/span, line 1, column 23 $$
$$\newcommand\Vvarrho ParseError: invalid DekiScript (click for details) Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[25], line 1, column 1 at template() at (Under_Construction/Arovas_Texts/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/09:_Renormalization/9.01:_The_Program_of_Renormalization), /content/body/p[1]/span, line 1, column 23 $$
$$\newcommand\Vsigma ParseError: invalid DekiScript (click for details) Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[26], line 1, column 1 at template() at (Under_Construction/Arovas_Texts/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/09:_Renormalization/9.01:_The_Program_of_Renormalization), /content/body/p[1]/span, line 1, column 23 $$
$$\newcommand\Vvarsigma ParseError: invalid DekiScript (click for details) Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[27], line 1, column 1 at template() at (Under_Construction/Arovas_Texts/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/09:_Renormalization/9.01:_The_Program_of_Renormalization), /content/body/p[1]/span, line 1, column 23 $$
$$\newcommand\Vtau ParseError: invalid DekiScript (click for details) Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[28], line 1, column 1 at template() at (Under_Construction/Arovas_Texts/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/09:_Renormalization/9.01:_The_Program_of_Renormalization), /content/body/p[1]/span, line 1, column 23 $$
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A statistical mechanical system is defined by a set of degrees of freedom and by a set of coupling constants $$\{K\ns_\alpha\}$$. The degrees of freedom can be discrete, such as Ising spins $$\sigma\ns_i$$, or continuous, such as a field $$\phi(\Br)$$. Additionally, each such system possesses a microscopic length scale $$\ell$$. For discrete, lattice-based systems, this length scale is simply the lattice spacing: $$\ell=a$$. For continuous systems, we can define a microscopic length scale by imposing a cutoff $$\RLambda$$ on the wavevectors we integrate over in all Fourier transforms. That is, we replace $\int\!\!{d^d\!k\over (2\pi)^d}\>F(\Bk) \longrightarrow \int\!\!{d^d\!k\over (2\pi)^d}\>F(\Bk)\,\gla(\Bk)\ ,$ where $$F(\Bk)$$ is any function and $$\gla(\Bk)$$ is the cutoff function. The simplest such case to imagine is a sharp cutoff which is isotropic in wavevector, $$\gla(\Bk)=\RTheta(\RLambda-|\Bk|)$$. Other cutoff schemes, however, are possible, including ‘soft cutoffs’ where $$\gla(\Bk)$$ is smooth. The microscopic length scale is then $$\ell\sim\RLambda^{-1}$$, which is the smallest distance in real space over which the system can independently fluctuate.

The idea behind renormalization is that we can successively winnow degrees of freedom from a system in some exact or approximate way, and in so doing we generate a new version of the system, at a different length scale $$\ell'>\ell$$, and with different couplings $$\{K'_\alpha\}$$. We then iterate this procedure. The result is a set of equations which tell us how the couplings behave under a change of the microscopic length scale. As we shall see, the fixed points of this procedure – where couplings do not change under a change of length scale – are critical points. Such a fixed point is defined by a set of couplings $$\{K^*_\alpha\}$$.

If we denote by $$\CR\ns_b$$ the renormalization procedure $\CR\ns_b\big(\ell\,,\{K\ns_\alpha\}\big) = \big(\ell'\,,\{K'_\alpha\}\big)\ ,$ where $$\ell'=b\,\ell$$, then we have the composition law $$\CR\ns_b\,\CR\ns_{b'}=\CR\ns_{b+b'}$$. The set of transformations $$\{\CR\ns_b\}$$ is collectively referred to as the renormalization group (RG) because of this mathematical structure. It is somewhat of a misnomer, however, since the transformations are only defined for $$b\ge 1$$, which means that there is no inverse operation, and hence no true group structure1. Nevertheless we shall use the RG terminology because it has become universally accepted in the literature.