Recap. In the post “From Time to Scale Dynamics” we traded the ordinary notion of time for the log‑scale $\tau = \ln(\mu/\mu_0)$ and promoted the Hamiltonian itself to the role of a dynamic coordinate. Writing $(g_i , p_i)$ for positions (couplings) and their conjugate momenta, we obtained a symplectic picture of RG flow:
\[\dot g_i = \frac{\partial H_{\text{scale}}}{\partial p_i}, \qquad \dot p_i = -\frac{\partial H_{\text{scale}}}{\partial g_i},\]with the minimalist choice
\[H_{\text{scale}} = p_i \, \beta_i(g).\]Today we push the analogy further. In ordinary mechanics, canonical transformations change coordinates while preserving Hamilton’s equations. Exactly the same machinery lets us understand
Why changing the renormalisation scheme leaves the physics intact.
The payoff is a deeper grasp of universality plus a new collection of tools for discovering invariants and symmetries in QFT. Let’s go exploring!
1 · A puzzle of uniqueness
Different renormalisation prescriptions—dimensional vs. cut‑off, minimal subtraction vs. momentum subtraction, on‑shell vs. off‑shell—yield different‑ looking $\beta$‑functions. Yet measurable quantities (masses, S‑matrix elements, critical exponents, …) coincide. How can that be guaranteed?
The textbook answer involves lengthy checks order‑by‑order in perturbation theory. Scale dynamics offers a one‑line explanation: a change of scheme is nothing but a canonical transformation in the phase‑space $(g,p)$ of scale flow.
2 · Canonical transformations: a refresher
A map $(q,p) \rightarrow (Q,P)$ is canonical when it leaves the symplectic 2‑form invariant:
\[\omega = \sum_i dq_i \wedge dp_i = \sum_i dQ_i \wedge dP_i.\]A convenient way to construct such maps is with a generating function. Choose $F(q,P)$ and define
\[p_i = \frac{\partial F}{\partial q_i}, \qquad Q_i = \frac{\partial F}{\partial P_i}.\]Why does this preserve $\omega$? (technical aside)
Taking the exterior derivative of $p_i,dq_i - P_i,dQ_i = dF$ and wedging both sides with the coordinate differentials gives
\[\sum_i dq_i \wedge dp_i = \sum_i dQ_i \wedge dP_i,\]so the symplectic form is unchanged — precisely the canonical condition. That result survives verbatim when $(q,p)$ are replaced by $(g,p)$.
3 · Scheme changes as canonical transformations
A renormalisation‑scheme change is just the point transformation
\[\tilde g_i = f_i(g),\]with the conjugate momenta determined by
\[\tilde p_i = \sum_j \frac{\partial g_j}{\partial \tilde g_i}\,p_j.\](The Jacobian above is the inverse of $\partial \tilde g/\partial g$.) That relation is the generating‑function prescription with $F(g,\tilde p)=f_i(g),\tilde p_i$.
Under the transformation the β‑functions obey
\[\tilde \beta_i(\tilde g) = \sum_j \frac{\partial \tilde g_i}{\partial g_j} \,\beta_j\bigl(g(\tilde g)\bigr),\]which is exactly the textbook formula for scheme dependence — but here it is just the chain rule applied inside a canonical map.
4 · Example: Minimal subtraction → physical subtraction in $\phi^4$
Take the scalar $\phi^4$ theory in $d=4-\varepsilon$. Between the minimal subtraction (MS) coupling $g$ and a physical scheme (PS) coupling $G$ we may choose
\[G \;=\; g\,[1 + a g + b g^2 + \mathcal O(g^3)].\]In MS, the two‑loop β‑function is
\[\beta_{\text{MS}}(g) = -\varepsilon g + A g^2 - B g^3 + \mathcal O(g^4).\]Applying the canonical prescription yields
\[\beta_{\text{PS}}(G) = -\varepsilon G + A G^2 -\bigl[\,B - aA\,(\varepsilon - A G)\bigr]G^3 + \mathcal O(G^4).\]Setting $\varepsilon!=!0$ (critical dimension) we see the one‑loop coefficient $A$ is invariant; the two‑loop coefficient shifts by terms proportional to $a$ but never changes sign, explaining why the first two coefficients are universal.
5 · QCD teaser: “optimal” resummation
The QCD β‑series in $\overline{\text{MS}}$ is
\[\beta_{\overline{\text{MS}}}(g) = -b_0 g^2 - b_1 g^3 - b_2 g^4 + \cdots.\]A canonical transformation of the form $\tilde g = g,[1 + c_1 g]$ turns this into
\[\beta_{\text{opt}}(\tilde g) = \frac{-b_0 \tilde g^2}{1 + c_1 \tilde g} + \mathcal O(\tilde g^4),\]in which the troublesome $b_2$ term is resummed into the denominator. How is $c_1$ fixed? By matching the $\tilde g^3$ coefficient; expanding the RHS and comparing with the original series gives
\[\boxed{\;c_1 = \frac{b_2}{b_1}\;}.\]Footnote: this matching is just the statement that canonical maps preserve physics — we equate the two three‑loop coefficients and solve for $c_1$.
6 · Invariants from the generating function
Given any generating function $F$ for a scheme change, the combination
\[I = H_{\text{scale}}\bigl(g,p\bigr) - \tau\,\frac{\partial H_{\text{scale}}}{\partial \tau}\]Poisson‑commutes with $H_{\text{scale}}$ in every scheme and is therefore a scheme‑independent conserved quantity. (With the minimal Ansatz, $I$ simply equals the “scale energy” $p_i\beta_i$.)
More generally, if the β‑functions are a gradient, $\beta_i = G^{ij}\partial_j C(g)$ with flat $G^{ij}$, the flow is Liouville‑integrable. Canonical symmetry then guarantees $\dim(g)$ independent integrals; the famous Zamolodchikov $c$‑function in 2D CFT is one such $C(g)$.
7 · Scale‑dynamics toolkit
| Classical mechanics | Scale dynamics |
|---|---|
| Symplectic form $dq\wedge dp$ | $dg\wedge dp$ |
| Canonical map | Scheme change |
| Generating function $F(q,P)$ | $F(g,\tilde p)$ bridges regularisations |
| Hamilton–Jacobi eq. | Relates couplings across schemes |
| Adiabatic invariant | Scheme‑independent integral of motion |
8 · Research frontiers
- Strong‑coupling tricks: find canonical maps that accelerate convergence near fixed points.
- Dualities: test whether Seiberg‑like dualities can be recast as canonical transformations.
- Exact RG: translate Wetterich’s functional flow into canonical language and search for hidden integrals.
- Quantisation of scale space: promote $g$ and $p$ to operators; different schemes may correspond to ordering choices.
9 · Conclusion
Interpreting scheme changes as canonical transformations:
- Provides a geometric guarantee that physics is preserved.
- Identifies which combinations of couplings and β‑coefficients are universal.
- Supplies practical tools for optimising perturbative series and exposing hidden structure.
Next time we’ll connect these ideas to information geometry and trace how information is lost—or preserved—as we flow across scales.
This series is exploratory; feedback and critiques are warmly welcomed.