TFPT scientific source excerpt
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000003 | \begin{definition}[Admissible effective average action]
000004 | \label{def:admissible-effective-average-action}
000005 | Let $R_k$ be an admissible infrared regulator commuting with the structural sector selection.
000006 | Define
000007 | \[
000008 | e^{W_k[J]}
000009 | :=
000010 | \int_{\mathcal H_{\mathrm{adm}}}\mathcal D\varphi\,
000011 | \exp\!\Bigl(
000012 | -S_{\mathrm{adm}}[\varphi]
000013 | -\frac12\langle \varphi,R_k\varphi\rangle
000014 | +\langle J,\varphi\rangle
000015 | \Bigr),
000016 | \]
000017 | and set
000018 | \[
000019 | \Gamma_k[\phi]
000020 | :=
000021 | \sup_J\{\langle J,\phi\rangle-W_k[J]\}
000022 | -\frac12\langle \phi,R_k\phi\rangle.
000023 | \]
000024 | The renormalized admissible effective action is
000025 | \[
000026 | \Gamma_{\mathrm{TFPT}}^{\mathrm{ren}}:=\lim_{k\to0}\Gamma_k.
000027 | \]
000028 | \end{definition}
000029 | 
000030 | \begin{theorem}[Exact admissible renormalization-group flow]
000031 | \label{thm:exact-admissible-rg-flow}
000032 | The admissible effective average action satisfies the exact functional flow equation
000033 | \[
000034 | \partial_t \Gamma_k[\phi]
000035 | =
000036 | \frac12\,
000037 | \operatorname{STr}_{\mathcal H_{\mathrm{adm}}}
000038 | \Bigl[
000039 | \bigl(\Gamma_k^{(2)}[\phi]+R_k\bigr)^{-1}
000040 | \partial_t R_k
000041 | \Bigr],
000042 | \qquad
000043 | t:=\log k.
000044 | \]
000045 | Thus the running couplings of the admissible theory are generated dynamically on $\Gamma_k$, not by
000046 | the static projector $\Padm$.
000047 | \end{theorem}
000048 | 
000049 | \begin{proof}
000050 | Differentiate the Legendre transform defining $\Gamma_k$ at fixed mean field $\phi$. Using
000051 | \[
000052 | \frac{\delta^2 W_k}{\delta J\,\delta J}
000053 | =
000054 | \bigl(\Gamma_k^{(2)}+R_k\bigr)^{-1},
000055 | \]
000056 | one obtains
000057 | \[
000058 | \partial_t \Gamma_k
000059 | =
000060 | \frac12 \operatorname{STr}
000061 | \bigl[
000062 | \bigl(\Gamma_k^{(2)}+R_k\bigr)^{-1}\partial_t R_k
000063 | \bigr].
000064 | \]
000065 | The supertrace is taken on the admissible physical sector because the regulated functional integral
000066 | itself is defined there.
000067 | \end{proof}