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# 9.S: Summary

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$$\newcommand\Dalpha ParseError: invalid DekiScript (click for details) Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[1], line 1, column 1 at template() at (Under_Construction/Arovas_Texts/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/09:_Renormalization/9.S:_Summary), /content/body/p/span, line 1, column 23 $$
$$\newcommand\Dbeta ParseError: invalid DekiScript (click for details) Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[2], line 1, column 1 at template() at (Under_Construction/Arovas_Texts/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/09:_Renormalization/9.S:_Summary), /content/body/p/span, line 1, column 23 $$
$$\newcommand\Dgamma ParseError: invalid DekiScript (click for details) Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[3], line 1, column 1 at template() at (Under_Construction/Arovas_Texts/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/09:_Renormalization/9.S:_Summary), /content/body/p/span, line 1, column 23 $$
$$\newcommand\Ddelta ParseError: invalid DekiScript (click for details) Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[4], line 1, column 1 at template() at (Under_Construction/Arovas_Texts/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/09:_Renormalization/9.S:_Summary), /content/body/p/span, line 1, column 23 $$
$$\newcommand\Depsilon ParseError: invalid DekiScript (click for details) Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[5], line 1, column 1 at template() at (Under_Construction/Arovas_Texts/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/09:_Renormalization/9.S:_Summary), /content/body/p/span, line 1, column 23 $$
$$\newcommand\Dvarepsilon ParseError: invalid DekiScript (click for details) Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[6], line 1, column 1 at template() at (Under_Construction/Arovas_Texts/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/09:_Renormalization/9.S:_Summary), /content/body/p/span, line 1, column 23 $$
$$\newcommand\Dzeta ParseError: invalid DekiScript (click for details) Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[7], line 1, column 1 at template() at (Under_Construction/Arovas_Texts/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/09:_Renormalization/9.S:_Summary), /content/body/p/span, line 1, column 23 $$
$$\newcommand\Deta ParseError: invalid DekiScript (click for details) Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[8], line 1, column 1 at template() at (Under_Construction/Arovas_Texts/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/09:_Renormalization/9.S:_Summary), /content/body/p/span, line 1, column 23 $$
$$\newcommand\Dtheta ParseError: invalid DekiScript (click for details) Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[9], line 1, column 1 at template() at (Under_Construction/Arovas_Texts/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/09:_Renormalization/9.S:_Summary), /content/body/p/span, line 1, column 23 $$
$$\newcommand\Dvartheta ParseError: invalid DekiScript (click for details) Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[10], line 1, column 1 at template() at (Under_Construction/Arovas_Texts/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/09:_Renormalization/9.S:_Summary), /content/body/p/span, line 1, column 23 $$
$$\newcommand\Diota ParseError: invalid DekiScript (click for details) Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[11], line 1, column 1 at template() at (Under_Construction/Arovas_Texts/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/09:_Renormalization/9.S:_Summary), /content/body/p/span, line 1, column 23 $$
$$\newcommand\Dkappa ParseError: invalid DekiScript (click for details) Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[12], line 1, column 1 at template() at (Under_Construction/Arovas_Texts/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/09:_Renormalization/9.S:_Summary), /content/body/p/span, line 1, column 23 $$
$$\newcommand\Dlambda ParseError: invalid DekiScript (click for details) Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[13], line 1, column 1 at template() at (Under_Construction/Arovas_Texts/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/09:_Renormalization/9.S:_Summary), /content/body/p/span, line 1, column 23 $$
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$$\newcommand\DGamma ParseError: invalid DekiScript (click for details) Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[15], line 1, column 1 at template() at (Under_Construction/Arovas_Texts/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/09:_Renormalization/9.S:_Summary), /content/body/p/span, line 1, column 23 $$
$$\newcommand\DDelta ParseError: invalid DekiScript (click for details) Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[16], line 1, column 1 at template() at (Under_Construction/Arovas_Texts/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/09:_Renormalization/9.S:_Summary), /content/body/p/span, line 1, column 23 $$
$$\newcommand\DTheta ParseError: invalid DekiScript (click for details) Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[17], line 1, column 1 at template() at (Under_Construction/Arovas_Texts/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/09:_Renormalization/9.S:_Summary), /content/body/p/span, line 1, column 23 $$
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$$\newcommand\Vmu ParseError: invalid DekiScript (click for details) Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[18], line 1, column 1 at template() at (Under_Construction/Arovas_Texts/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/09:_Renormalization/9.S:_Summary), /content/body/p/span, line 1, column 23 $$
$$\newcommand\Vnu ParseError: invalid DekiScript (click for details) Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[19], line 1, column 1 at template() at (Under_Construction/Arovas_Texts/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/09:_Renormalization/9.S:_Summary), /content/body/p/span, line 1, column 23 $$
$$\newcommand\Vxi ParseError: invalid DekiScript (click for details) Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[20], line 1, column 1 at template() at (Under_Construction/Arovas_Texts/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/09:_Renormalization/9.S:_Summary), /content/body/p/span, line 1, column 23 $$
$$\newcommand\Vom ParseError: invalid DekiScript (click for details) Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[21], line 1, column 1 at template() at (Under_Construction/Arovas_Texts/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/09:_Renormalization/9.S:_Summary), /content/body/p/span, line 1, column 23 $$
$$\newcommand\Vpi ParseError: invalid DekiScript (click for details) Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[22], line 1, column 1 at template() at (Under_Construction/Arovas_Texts/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/09:_Renormalization/9.S:_Summary), /content/body/p/span, line 1, column 23 $$
$$\newcommand\Vvarpi ParseError: invalid DekiScript (click for details) Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[23], line 1, column 1 at template() at (Under_Construction/Arovas_Texts/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/09:_Renormalization/9.S:_Summary), /content/body/p/span, line 1, column 23 $$
$$\newcommand\Vrho ParseError: invalid DekiScript (click for details) Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[24], line 1, column 1 at template() at (Under_Construction/Arovas_Texts/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/09:_Renormalization/9.S:_Summary), /content/body/p/span, line 1, column 23 $$
$$\newcommand\Vvarrho ParseError: invalid DekiScript (click for details) Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[25], line 1, column 1 at template() at (Under_Construction/Arovas_Texts/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/09:_Renormalization/9.S:_Summary), /content/body/p/span, line 1, column 23 $$
$$\newcommand\Vsigma ParseError: invalid DekiScript (click for details) Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[26], line 1, column 1 at template() at (Under_Construction/Arovas_Texts/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/09:_Renormalization/9.S:_Summary), /content/body/p/span, line 1, column 23 $$
$$\newcommand\Vvarsigma ParseError: invalid DekiScript (click for details) Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[27], line 1, column 1 at template() at (Under_Construction/Arovas_Texts/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/09:_Renormalization/9.S:_Summary), /content/body/p/span, line 1, column 23 $$
$$\newcommand\Vtau ParseError: invalid DekiScript (click for details) Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[28], line 1, column 1 at template() at (Under_Construction/Arovas_Texts/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/09:_Renormalization/9.S:_Summary), /content/body/p/span, line 1, column 23 $$
$$\newcommand\Vupsilon ParseError: invalid DekiScript (click for details) Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[29], line 1, column 1 at template() at (Under_Construction/Arovas_Texts/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/09:_Renormalization/9.S:_Summary), /content/body/p/span, line 1, column 23 $$
$$\newcommand\Vphi ParseError: invalid DekiScript (click for details) Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[30], line 1, column 1 at template() at (Under_Construction/Arovas_Texts/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/09:_Renormalization/9.S:_Summary), /content/body/p/span, line 1, column 23 $$
$$\newcommand\Vvarphi ParseError: invalid DekiScript (click for details) Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[31], line 1, column 1 at template() at (Under_Construction/Arovas_Texts/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/09:_Renormalization/9.S:_Summary), /content/body/p/span, line 1, column 23 $$
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## References

• M. Kardar, Statistical Physics of Fields (Cambridge, 2007) A superb modern text, with many insightful presentations of key concepts.
• M. Plischke and B. Bergersen, Equilibrium Statistical Physics ($$3^{rd}$$ edition, World Scientific, 2006) An excellent graduate level text. Less insightful than Kardar but still a good modern treatment of the subject. Good discussion of mean field theory.
• G. Parisi, Statistical Field Theory (Addison-Wesley, 1988) An advanced text focusing on field theoretic approaches, covering mean field and Landau-Ginzburg theories before moving on to renormalization group and beyond.
• J. P. Sethna, Entropy, Order Parameters, and Complexity (Oxford, 2006) An excellent introductory text with a very modern set of topics and exercises.

## Endnotes

1. A more mathematically rigorous name would be the renormalization monoid.
2. Our choice of what terms to put in the exponent in the second line below is dictated by global $${\mathbb Z}_2$$ symmetry. Once we sum over $$\sigma_0$$, the result should be invariant under simultaneous reversal of $$\sigma_{1-4}$$.