In the post “From Time to Scale Dynamics” we treated scale evolution as perfectly conservative: a Hamiltonian $H_{\text{scale}}$ moved through symplectic phase‑space $(g_i,p_i)$ just like a planet through ordinary space.
Real renormalization‑group (RG) flows, however, are irreversible. Coarse‑graining shreds microscopic detail; only critical data survive. Here we embed that one‑way character directly into the geometry.
Key idea. Enlarge phase space from symplectic to contact by adding one extra coordinate $s$ that measures lost information. Contact flows naturally spiral into fixed points, matching the physical arrow of renormalization.
0 Lightning recap (conservative case)
With couplings $g_i$ and conjugate momenta $p_i$, the symplectic scale Hamiltonian was
\[H_{\text{scale}}(g,p)=p_i\,\beta_i(g)+F(g),\]generating reversible equations
\[\dot g_i = \beta_i(g), \qquad \dot p_i = -p_j\,\partial_{g_i}\beta_j-\partial_{g_i}F.\tag{1}\]The two‑form $\omega = \mathrm dg_i\wedge \mathrm dp_i$ is preserved, so time reversal simply flips $p\to-p$.
1 Why symplectic fails in RG
- Irreversibility. RG trajectories approach fixed points only for increasing $\tau=\ln(\mu/\mu_0)$. The reverse path is ill‑defined.
- Strict monotones. In 2D CFT the Zamolodchikov $C$-function decreases; in $d>2$, $a$- or $F$-functions do the same. A conservative $H_{\text{scale}}$ cannot reproduce that.
- Entropy production. Integrating out short modes raises the Shannon entropy of the blocked field.
Contact geometry is the minimal extension that accommodates all three properties.
2 Contact primer in one paragraph
In Darboux coordinates $(g_i,p_i,s)$ a contact one‑form is
\[\eta = \mathrm ds - p_i\,\mathrm dg_i.\tag{2}\]A contact Hamiltonian $K(g,p,s)$ generates the vector field
\[X_K = \partial_{p_i}K\,\partial_{g_i} -\bigl(\partial_{g_i}K + p_i\,\partial_s K\bigr)\partial_{p_i} +\bigl(p_i\,\partial_{p_i}K - K\bigr)\partial_s.\tag{3}\]Unlike the symplectic case, the Lie derivative obeys
\[\mathscr L_{X_K}\eta = -\partial_s K\;\eta,\]meaning $\eta$ rescales rather than staying exact—precisely the “slack” dissipative systems need.
3 Building an irreversible scale Hamiltonian
Add a linear “friction” term to the conservative $H_{\text{scale}}$:
\[\boxed{K(g,p,s)=p_i\,\beta_i(g)+F(g)\; -\; \gamma s},\qquad \gamma>0.\tag{4}\]Plugging (4) into (3) gives the contact RG equations
\[\boxed{\begin{aligned} \dot g_i &= \beta_i(g),\\[4pt] \dot p_i &= -p_j\,\partial_{g_i}\beta_j - \partial_{g_i}F\; + \;\gamma p_i,\\[4pt] \dot s &= p_i\,\beta_i(g) - K = -F(g) + \gamma s. \end{aligned}}\tag{5}\]Interpretation.
- The $g$-projection reproduces the standard beta functions.
- The extra $+\gamma p_i$ term tries to amplify $p$; decay or growth depends on competition with the $-p_j\partial_{g_i}\beta_j$ piece.
- With $F=0$, $\dot s = \gamma s$ so $s(\tau)=s(0)\,e^{\gamma\tau}$; entropy grows unboundedly as expected from continual coarse‑graining.
4 A strict monotone: $C_{\text{mon}} = K$
For any Darboux contact flow one has the identity:
\[\dot K = -\,\partial_s K\;K. \tag{6}\]With $\partial_s K = -\gamma$ this becomes
\[\dot K = -\gamma\,K.\]Hence
\[C_{\text{mon}}(\tau) = K(\tau), \qquad \dot C_{\text{mon}} = -\gamma\,C_{\text{mon}} < 0. \tag{7}\]$C_{\text{mon}}$ therefore decreases strictly along every contact-RG trajectory and is stationary only at fixed points—exactly the behaviour of Zamolodchikov’s $C$ in 2D and of the $a$/$F$ functions in higher dimensions.
5 Example: $\phi^4$ in $d=4-\varepsilon$
The one‑loop beta function reads
\[\beta(g)= -\varepsilon g + A g^{2},\qquad A>0.\]Set $F=0$. Equations (5) give
\[\dot g = -\varepsilon g + A g^{2},\qquad \dot p = p\,(\varepsilon-2Ag)+\gamma p,\qquad \dot s = \gamma s.\tag{8}\]Linearising around the Wilson–Fisher fixed point $g^{*}=\varepsilon/A$:
\[\dot p = -\bigl(\lambda_{\text{RG}}-\gamma\bigr)\,p,\qquad \lambda_{\text{RG}}\equiv\varepsilon.\tag{9}\]- If $\gamma < \lambda_{\text{RG}}$ the momentum decays exponentially and the flow spirals into $(g^{*},0,s!\to!\infty)$.
- If $\gamma > \lambda_{\text{RG}}$ the $+\gamma p$ term wins initially, inflating $p$, but $C_{\text{mon}}$ still shrinks because $\dot C_{\text{mon}}=-\gamma C_{\text{mon}}$ independently of $p$. Numerical trajectories show outward loops that nevertheless converge toward the fixed point in the $g$-$p$ plane while $s$ diverges.
6 Scale Noether-like quantity
Consider the transformation $(g,p,s)\mapsto(g+\delta g,\,p,\,e^{-\lambda},\,s+\lambda p!\cdot!\delta g)$ that rescales $\eta$ by $e^{-\lambda}$. For an exactly marginal direction $\delta g$ (so $\beta_{\delta g}=0$) one finds
\[I(\tau)=e^{\lambda\tau}\,p\!\cdot\!\delta g,\qquad \dot I = -\lambda e^{\lambda\tau}K. \tag{10}\]| Because $K\to0$ while $C_{\text{mon}}$ decreases, $ | I | $ grows toward a finite limit—the corrected statement once signs are consistent. |
7 Practical blueprint (updated)
| Step | Action | Outcome |
|---|---|---|
| 1 | Provide beta functions $\beta_i(g)$. | Source data |
| 2 | Pick $\gamma>0$ and (optionally) $F(g)$. | Strength of irreversibility |
| 3 | Build $K=p_i\beta_i - \gamma s + F$. | Contact Hamiltonian |
| 4 | Integrate (5) numerically. | Spiral trajectories |
| 5 | Plot $C_{\text{mon}}=-K$. | Strictly falling curve |
| 6 | Compare $C_{\text{mon}}$ with known $C$/$a$/$F$. | Consistency check |
8 Outlook
- Beyond one loop. Higher‑order $F(g)$ modifies $\dot s$ but leaves $\dot C_{\text{mon}}=-\gamma C_{\text{mon}}$ intact.
- Tensor networks. On a lattice, $\Delta s$ matches the log of discarded bond dimensions, quantifying information loss per RG step.
- Quantum contact flows. Geometric quantisation turns $K$ into a non‑Hermitian operator, linking open‑system dynamics with RG.
9 Take-home message
Replacing symplectic by contact scale dynamics nails the essentials: irreversible momenta, a strictly decreasing $C$-like quantity, and an intrinsic arrow of renormalization.
Feedback and wild ideas remain most welcome—the next instalment will add stochastic forces and explore fluctuation theorems in scale space.