The following question refers to ref. 1, the equation are numbered alike with a slightly different notation. The author claims to solve a renormalization group (RG) equation - so the context is physical - using the Method of characteristics but I can't reconnect with the one I find in Mathematics literature (cf. e.g. the wiki link above). The equation to be solved is the following: $$\left[\Lambda\frac{\partial}{\partial\Lambda}+\beta(g_0)\frac{\partial}{\partial g_0}-n\eta(g_0)\right]\Gamma_0(p_i; g_0, \Lambda)=0\tag{9.94}\label{9.94}.$$ They claim to apply the method of characteristics to solve it but here's what they do:
- Introduce a "dilatation parameter" $u$ and two functions theoreof $Z_\text{eff}(u)$ and $g_\text{eff}$(u) defined such that $$u\frac{d}{du}\left[Z_{\text{eff}}^{-n/2}(u)\Gamma_0(p_i;g_\text{eff}(u), u\Lambda)\right]=0\tag{9.97}\label{9.97}.$$
- They write explicitly the derivative in \eqref{9.97} $$Z_{\text{eff}}^{-n/2}(u)\left[\Lambda\frac{\partial}{\partial\Lambda}+u\frac{d}{du}g_\text{eff}(u)\frac{\partial}{\partial g_0}-n\cdot\frac{1}{2}u\frac{d}{du}Z_\text{eff}(u)\right]\Gamma_0(p_i;g_\text{eff}(u),u\Lambda)=0\tag{9.97a}\label{9.97a},$$ which I labeled as \eqref{9.97a} because it is not on the book. Then, they claim that this equation is "compatible" (not sure what they mean) with \eqref{9.94} if the functions $Z_\text{eff}$ and $g_\text{eff}$ satisfy the following differential equations $$\begin{cases} u\frac{d}{du}g_\text{eff}(u)=\beta(g_\text{eff}(u))\qquad g_\text{eff}(1)=g_0 \\ \frac{1}{2}u\frac{d}{du}\log Z_\text{eff}=\eta(g_\text{eff}(u))\qquad Z_\text{eff}(1)=1\end{cases}\tag{9.98-9.99}\label{9.98-9.99}$$
- Finally, using \eqref{9.94}, since it implies that the function inside the brackets is independent of $u$, $$Z_{\text{eff}}^{-n/2}(u)\Gamma_0(p_i;g_\text{eff}(u), u\Lambda)=Z_{\text{eff}}^{-n/2}(1)\Gamma_0(p_i;g_\text{eff}(1), \Lambda)=\Gamma_0(p_i;g_0, \Lambda)$$
So, is this really the method of characteristics? The starting step \eqref{9.97} and that compatibility conditions (which I don't understand yet) look very different from the standard method.
Update
Given the comment by @NinadMunshi, who stated that the method of characteristics is all about writing things in the form $$\text{directional derivative}=0$$ and then suggests that here we only did a clever substitution to do so, I think it's appropriate to be more specific with my doubts:
- Why the whole "compatibility" step? Is it just one of the possible ways to achieve the form \eqref{9.97}?
- The variables are $g$ and $\Lambda$, so I'd expect to have a curve $(g(u),\Lambda(u))$, instead of $(g(u),u\Lambda)$, unless in the second case we mean a fixed $\Lambda$, with some abuse of notation (as if your variables were $x$ and $y$ and you called the characteristics $x(s)=xs$ with constant $x$ instead of a more standard notation like $x(s)=x_0s$).
- The clever substitution has something unusual to it, though. Not only we have our function evaluated on the flow, but also $Z_\text{eff}$ multiplying it.
References
- Quantum Field Theory and Critical Phenomena, Jean Zinn-Justin, 2021, fifth edition. Section 9.11.2-9.12, eqs. (9.91-9.101).