$$
m \sim L^{-\bar{\beta} / \nu} \sim V^{-\bar{\beta} / \nu d}
$$
where we have used the notation $\tilde{\beta}$ for the critical exponent of the magnetization and $L$ denotes the linear extension of the spin system. The total magnetization at $\beta_{c}$ will be
$$
M=m V \sim V^{1-\tilde{\beta} / \nu d} .
$$
$$
p \sim m, \quad p V \sim M
$$
The largest spin clusters at the critical point will be fractal. In peculation theory one defines the fractal dimension $D$ of a cluster by
$$
p L^{d}=L^{D}, \quad \text { i.e. } \quad p V=V^{D / d} .
$$
We conclude that $D$ is related to $\tilde{\beta}$ by
$$
\frac{D}{d}=1-\frac{\tilde{\beta}}{\nu d} \quad\left(=\frac{15}{16} \quad \text { for the Ising model }\right)
$$
7CCMMS32 COURSE NOTES ：
$$
\left[-\frac{\hbar^{2}}{\mathcal{E}} \kappa^{2} G^{a b} \frac{\delta^{2}}{\delta q^{a} \delta q^{b}}+\sqrt{g} U\right] \Phi_{\mathcal{E}}[q]=0
$$
associated with the parameter $\varepsilon$. Finally, if we also impose the mass-shell constraint
$$
\boxplus \Psi=-\mathcal{M}^{2} \Psi
$$
then the only physical states are those with $\varepsilon=M^{2}$, and the (classically indeterminate) constant $M$ can be absorbed, via
$$
\hbar_{e f f}=\frac{\hbar}{\mathcal{M}}
$$
into a rescaling of Planck’s constant.