## 13T1 Theorem Replacing fields with the solutions of their field equations preserves the master equation

Let $S(\Phi,U,K,K_{U})$ denote the solution of the master equation $(S,S)=0$, where $\{\Phi ^{A},U\}$ are the fields and $\{K_{A},K_{U}\}$ are the sources coupled to the $\Phi ^{A}$- and $U$-gauge transformations. If we replace $U$ with the solution $U^{*}(\Phi ,K,K_{U})$ of the $U$-field equations $$\frac{\delta _{r}S}{\delta U}=0,$$ then the action S^{*}(\Phi ,K,K_{U})=S(\Phi... read more

## 12T1 Theorem Procedure to convert the functional integral to the conventional form

Consider a functional integral $\mathcal{I}=\int [\mathrm{d}\varphi ]\hspace{0.02in}\exp \left( -S(\varphi)+\int J\left( \varphi -bU\right) \right) ,$ where $U(\varphi ,bJ)$ is a local function of $\varphi$ and $J$, and $b$ is a constant. Then there exists a perturbatively local change of variables $\varphi =\varphi (\varphi ^{\prime },b,bJ)=\varphi ^{\prime }+\mathcal{O}(b),$ expressed as a series expansion in... read more

## 06T1 Theorem Terms quadratically proportional to the field equations and field redefinitions

Consider an action $S$ depending on fields $\phi_{i}$, where the index $i$ labels both the field type, the component and the spacetime point. Add a term quadratically proportional to the field equations $S_{i}\equiv \delta S/\delta \phi _{i}$ and define the modified action $$\phantom{(1)}\qquad\qquad\qquad S^{\prime }(\phi _{i})=S(\phi _{i})+S_{i}F_{ij}S_{j}, \qquad\qquad\qquad (1)$$ where $F_{ij}$... read more

## 05T1 Theorem Maximum poles of Feynman diagrams

The maximum pole of a diagram with $V$ vertices and $L$ loops is at most $1/\varepsilon^{m(V,L)}$, where $m(V,L)=\min (V-1,L).$ The result holds in dimensional regularization, where $\varepsilon = d-D$, $d$ is the physical dimension and $D$ the continued one. Moreover, vertices are counted treating mass terms and the other... read more

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### Book

14B1 D. Anselmi
Renormalization

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Last update: May 9th 2015, 230 pages

Contents: Preface | 1. Functional integral | 2. Renormalization | 3. Renormalization group | 4. Gauge symmetry | 5. Canonical formalism | 6. Quantum electrodynamics | 7. Non-Abelian gauge field theories | Notation and useful formulas | References

Course on renormalization, taught in Pisa in 2015. (More chapters will be added later.)