This post builds on “From Time to Scale Dynamics”, “Canonical Transformations & Scheme Independence”, and “Irreversible Scale Dynamics”. There we treated the renormalisation‑group (RG) β‑functions as the sole engine of scale evolution:
\[H_{\text{scale}}(g,p)=p_i\,\beta_i(g).\]Today we lift the hood and ask a simple question:
What if the textbook β is just the leading term of a richer, non‑linear flow in $(g_i,p_i)$ phase‑space?
The answer opens an entire landscape of new phenomena—closed orbits, scheme‑independent invariants, and non‑perturbative deformations of critical exponents—while recovering the familiar β‑functions as a special “linear‑momentum” slice.
0 · Why the textbook β is linear
In the minimalist ansatz the momenta $p_i$ enter only once and linearly:
\[H_{\text{scale}}=p_i\,\beta_i(g).\]Differentiating with respect to $p_i$ gives
\[\dot g_i=\partial_{p_i}H_{\text{scale}}=\beta_i(g),\]a direct copy of RG flow. Nothing in symplectic—or even contact—geometry forces us to stop there. We stopped only because perturbation theory at weak coupling typically sets $p_i\approx0$, making higher powers negligible to leading order.
1 · The momentum expansion
The most general analytic scale Hamiltonian is a power series in the canonical momenta:
\[H_{\text{scale}}(g,p)=\sum_{n=1}^{\infty}\frac{1}{n!}\,T^{(n)}_{i_1\dots i_n}(g)\,p_{i_1}\dots p_{i_n}.\tag{1}\]Because the monomial $p_{i_1}\dots p_{i_n}$ is totally symmetric in its indices, only the fully‑symmetric part of every coefficient tensor $T^{(n)}$ can contribute. In particular
\[T^{(1)}_i(g)=\beta_i(g),\qquad T^{(2)}_{ij}=A_{ij}(g)=A_{ji}(g),\qquad T^{(3)}_{ijk}=B_{ijk}(g)=B_{(ijk)}(g),\;\dots\]The symmetry property is crucial: antisymmetric tensors drop out identically and cannot drive the dynamics.
2 · Equations of motion to cubic order
Truncating (1) at $n=3$ and applying Hamilton’s equations yields
\[\begin{aligned} \dot g_i &= \beta_i(g)+A_{ij}(g)\,p_j+\tfrac12 B_{ijk}(g)\,p_jp_k+\mathcal O(p^3),\\[6pt] \dot p_i &= -p_j\,\partial_{g_i}\beta_j-\tfrac12 p_jp_k\,\partial_{g_i}A_{jk}-\tfrac16 p_jp_kp_\ell\,\partial_{g_i}B_{jk\ell}+\mathcal O(p^4). \end{aligned}\tag{2}\]\footnote{Sign convention: throughout we adopt $\dot g_i = \partial_{p_i}H$ and $\dot p_i = -\partial_{g_i}H$, sometimes referred to as the (+ – – –) convention.}
Any initial momentum $p_i(0)\neq0$ therefore feeds back into $\dot g_i$ as long as $A_{ij}\not\equiv0$—a generic situation beyond the linear approximation.
3 · Physics unlocked by higher‑order terms
3.1 Closed orbits and discrete scale invariance
With symmetric but non‑vanishing $A_{ij}$ and/or $B_{ijk}$, the joint $(g,p)$ flow need not terminate at fixed points. Non‑linear couplings between $g$ and $p$ can generate closed orbits—Hamiltonian “limit cycles” in which the system revisits couplings at discrete scale factors. Unlike the incorrect argument in the first post, no antisymmetric part of $A_{ij}$ is required; closed trajectories arise from the conservative two‑degree‑of‑freedom dynamics itself or, if genuine attraction is desired, from the contact‑friction deformation discussed below.
3.2 Scheme dependence revisited
A change of renormalisation scheme is a canonical diffeomorphism that preserves the symplectic form. Once quadratic terms are admitted the family of generating functions $F(g,\tilde p)$ widens, explaining why higher‑loop β‑coefficients mix under scheme change while the one‑loop piece stays universal.
3.3 Non‑perturbative deformations of critical exponents
Near a fixed point $(g_i^,p_i^)=(g_i^*,0)$, linearising (2) gives
\[\mathcal M=\begin{pmatrix} \partial_{g_j}\beta_i & A_{ij}\\[4pt] 0 & -\partial_{g_i}\beta_j \end{pmatrix}_{(g^*,0)}.\]Because $\mathcal M$ is block‑triangular, its eigenvalues are those of the diagonal blocks: the standard critical‑exponent matrix $\partial_{g}\beta$ and its negative transpose. The off‑diagonal block $A_{ij}$ alters eigenvectors (Jordan structure) but does not shift eigenvalues at linear order. Observable shifts in critical exponents therefore originate from higher‑order (non‑linear) corrections—exactly what the extended Hamiltonian is equipped to capture.
4 · Worked example: $\phi^4$ in $d=4-\varepsilon$
The one‑loop β‑function is
\[\beta(g)= -\varepsilon g + \lambda g^2.\]Suppose integrating out irrelevant operators induces a quadratic tensor
\[A_{gg}=c\,g.\]The truncated Hamiltonian is then
\[H=p\,(-\varepsilon g + \lambda g^2)+\tfrac12 c\,g\,p^2.\tag{3}\]With a single coupling this is a one‑degree‑of‑freedom Hamiltonian system, so phase‑space volume is conserved and closed orbits—when they appear—are neutrally periodic, not attracting.
Along a constant‑$H$ trajectory the quadratic polynomial in $p$ has discriminant
\[D=(\lambda g^{2}-\varepsilon g)^{2}+2c\,g\,H.\tag{4}\]Closed orbits exist whenever $D$ changes sign between two turning points $g_{\min},g_{\max}$.
Direct integration shows three qualitative regimes:
- Small $\lvert c \rvert$ — trajectories drift slowly toward the Wilson–Fisher fixed point.
- Moderate $\lvert c \rvert$ — the non‑linear $p^{2}$ term bends the constant‑$H$ curves into closed loops, realising discrete scale invariance.
- Large $\lvert c \rvert$ — kinetic energy dominates and trajectories run away.
The exact period of a closed orbit is an elliptic integral. For $\lvert c \rvert \ll1$ it reduces to the leading approximation
\[\tau\;\approx\;\frac{2\pi}{\sqrt{c\,\lambda\,\varepsilon}}\;\Bigl[1+\mathcal O(c^{0})\Bigr].\tag{5}\]5 · When does the linear truncation hold?
| Regime | Justification for dropping $p^2$ and higher |
|---|---|
| Perturbative UV | Bare actions start with $p_i\approx0$; symmetry and loop factors typically suppress $A_{ij}$ and higher. |
| Near a stable fixed point | Momentum decays exponentially, so $A_{ij}p_j\ll\beta_i$. |
| Contact friction $\gamma>0$ | As shown in the “Irreversible” post, $p$ is damped when $\gamma>\lambda_{\text{RG}}$. |
| Strong coupling / Irrelevant fixed points | Higher‑order momentum terms often dominate, so the linear truncation fails altogether; a resummed or non‑perturbative treatment is mandatory. [Clarified wording] |
6 · Outlook & open questions
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Quantising scale mechanics The canonical structure begs for quantisation. Does promoting $p_i$ to differential operators reproduce known functional‑RG equations or unveil new, genuinely quantum scale phenomena?
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Contact‑geometry deformations Dissipation (the “$\gamma$‑term”) converts closed orbits into stable limit cycles. Mapping the transition from conservative to contact flows may shed light on irreversibility in real QFTs.
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Numerical experiments Lattice implementations could inject controlled momentum packets into RG flows and track the emergence (or absence) of discrete scale invariance.
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Scheme‑independent invariants Beyond the β‑vector‑field, what additional integrals of motion survive arbitrary canonical transformations? First candidates are the Casimirs built from symmetric tensors $T^{(n)}$.
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Bridging to holography In AdS/CFT the radial Hamiltonian already contains higher‑order momenta. Do the closed orbits described here parallel holographic RG cycles?
Key takeaway
The familiar β‑function sits atop a tower of higher‑momentum couplings. Ignoring them is safe in the perturbative UV and near attractive fixed points—but can erase whole classes of scale phenomena whenever momentum feedback becomes non‑negligible.
Future posts will extend the analysis to multi‑field models and explore how contact‑geometry friction selects which higher‑order terms actually matter in real QFTs.