Physics is full of levels of description – atoms, molecules, fluids; spins, domains, magnets. We usually fix the level and then ask how the variables at that level evolve in time. The idea explored here is the reverse:

Let the scale flow, and ask how the Hamiltonian changes along that flow.

Doing so may give us a unified language for:

• the renormalization group (RG) in quantum field theory,
• effective theories in condensed-matter physics,
• cross-scale phenomena such as turbulence or criticality.

Our goal is not to build the full mathematical machinery today, but to sketch a minimal working picture that a BSc-level reader can follow.

1. Analogy: Newtonian motion ↔ “scale motion”

The key swap is

\[t \longrightarrow \tau \equiv \ln\!\frac{\mu}{\mu_0}, \qquad x \longrightarrow H,\]

so scale $\mu$ plays the role of “time”.

2. A toy model: ballistic motion in scale

Suppose we postulate the scale Lagrangian

\[\mathcal L(H,\dot H)=\tfrac12 A\,\dot H^{\,2}-\tfrac12 B\,H^{2},\]

where the dot means $\displaystyle \dot H \equiv \frac{dH}{d\tau}$ and $A,B$ are positive constants. The Euler–Lagrange equation gives

\[A\,\ddot H + B\,H = 0,\]

whose solution is sinusoidal in $\tau$. Translating back, the Hamiltonian oscillates as we zoom in and out—hardly realistic, but it shows:

• There exists an action principle for $H(\mu)$.

• The corresponding conserved quantity is

  \[\mathcal E=\tfrac12 A\,\dot H^{\,2}+ \tfrac12 B\,H^{2}.\]

Interpretation. $\mathcal E$ measures how the shape of the Hamiltonian changes with scale. If $\mathcal E=0$ the Hamiltonian sits at a fixed point; if $\mathcal E>0$ it “moves” through Hamiltonian space as we coarse-grain or fine-grain.

3. Connection to the renormalization group

For real quantum or statistical systems we already have equations of scale motion: the beta functions

\[\frac{dg_i}{d\tau}=\beta_i(\{g\}),\]

where ${g_i}$ are couplings in the Hamiltonian. The proposal is to embed these first-order RG equations into a second-order, Hamilton-like system by

1. Doubling variables. Treat $g_i$ as coordinates and introduce conjugate “momenta” $p_i$.

2. Choosing a scale Hamiltonian

   $$ \boxed{H_\mathrm{scale}(\{g\},\{p\}) = \sum_i p_i\,\beta_i(\{g\})\; +\; F(\{g\})} $$

   whose first Hamilton equation reproduces the beta functions.

3. Interpreting $p_i$. They measure the sensitivity of the effective action to changes in the couplings – a notion already familiar from functional RG and holographic RG.

Validity check. Canonical Hamiltonian flow is time-reversible, but RG flow is not (coarse-graining loses information). The trick above gets around this by enlarging phase space; the $(g_i,p_i)$ trajectories can be reversible even though the projection onto ${g_i}$ is not. Similar enlargements appear in:

• Hamiltonian RG (Wegner, Ann. Phys. 1974)
• Holographic RG (de Boer, Verlinde & Verlinde, JHEP 2000)
• Classical-statistical RG (Polonyi, Central Eur. J. Phys. 2003)

4. Why idealize to “conservative” scale flow?

Realistic RG trajectories dissipate information just as real mechanical systems dissipate energy through friction. Yet idealizing to an energy-conserving, symplectic picture has paid off spectacularly in mechanics; think of planetary orbits or accelerator design. Likewise, an idealized symplectic RG framework can:

• package many beta functions into one Hamiltonian object,
• reveal hidden invariants (e.g. C-functions or a-theorems),
• offer geometric intuition (flows on phase space vs. flows on coupling space).

Even if dissipation must be re-introduced later, starting from the conservative limit often reveals the dominant skeleton of the dynamics.

5. The Hamilton-Jacobi Formulation

We can take this framework one step further by introducing a Hamilton-Jacobi formulation. In this approach, we define an effective action $S(g,\tau)$ such that:

\[p = \frac{\delta S(g,\tau)}{\delta g}\]

The Hamilton-Jacobi equation then takes the form:

\[\frac{\partial S(g,\tau)}{\partial \tau} + H\left(g, \frac{\delta S(g,\tau)}{\delta g}\right) = 0\]

For our scale Hamiltonian, this becomes:

\[\frac{\partial S(g,\tau)}{\partial \tau} + \frac{\delta S(g,\tau)}{\delta g} \cdot \beta(g) + F(g) = 0\]

This equation encodes how the effective action changes under scale transformations. If we set $F(g) = 0$ for simplicity, then along the RG trajectories, the effective action $S$ remains constant, which can be interpreted as a form of “conserved scale information.”

6. A Concrete Example: The $\phi^4$ Theory

Let’s apply our framework to the $\phi^4$ theory in $d=4-\varepsilon$ dimensions, a standard model in quantum field theory. The beta function at one-loop order is:

\[\beta(g) = -\epsilon g + Ag^2\]

where $A$ is a positive constant. This theory has the well-known Wilson-Fisher fixed point at $g^* = \epsilon/A$.

In our scale-Hamiltonian approach:

\[H(g,p) = p \cdot (-\epsilon g + Ag^2)\]

Hamilton’s equations give us:

\[\frac{dg}{d\tau} = -\epsilon g + Ag^2 \quad \text{(reproducing the standard RG flow)}\] \[\frac{dp}{d\tau} = -p \cdot (-\epsilon + 2Ag) = p \cdot (\epsilon - 2Ag)\]

At the fixed point $g^* = \epsilon/A$:

\[\frac{dp}{d\tau}\Big|_{g=g^*} = p \cdot (\epsilon - 2A \cdot \frac{\epsilon}{A}) = -\epsilon p\]

This shows that as we approach the fixed point, the conjugate momentum $p$ decays exponentially with scale. This can be interpreted as a “loss of memory” of microscopic details as we move to larger scales—precisely what we expect from effective field theories.

7. Step-by-step blueprint for readers to try

1. Pick a simple theory – e.g. the $\phi^4$ model in $d=4-\varepsilon$.
2. Write its beta function $\beta(g)$ at one-loop.

3. Introduce a conjugate momentum $p$ and scale Hamiltonian

   \[H=p\,\beta(g).\]
4. Derive Hamilton’s equations and check that $\dot g=\beta(g)$.
5. Study the $p$-equation; interpret decay or growth near fixed points.
6. Look for conserved quantities (possibly after adding a suitable $F(g)$).
7. Generalize to multiple couplings and compare with standard RG results.

Each step is no harder than undergraduate analytical mechanics once the beta function is known.

8. Physical Interpretation and Key Insights

What does this mathematical framework tell us about the physical world?

1. Scale Invariance as a Fixed Point: Just as equilibrium states in temporal dynamics correspond to fixed points where forces balance, scale invariance corresponds to fixed points in the RG flow where the system looks identical at different scales.

2. Conservation Laws Across Scales: Just as time-translation symmetry leads to energy conservation, certain scale transformations might preserve their own invariants, revealing new conservation principles.

3. Extended Phase Space: By introducing conjugate momenta to couplings, we gain a richer description of scale evolution, potentially revealing hidden structure in the space of physical theories.

4. Effective Field Theories: This formalism offers a new perspective on why effective field theories work so well—they can be understood as approximations that preserve certain scale invariants while allowing irrelevant information to decay.

9. Relations to existing work

10. Limitations & open questions

• Irreversibility. How precisely does information loss arise when the enlarged $(g,p)$ flow is reversible? One option is to treat physical RG as a contact (dissipative) deformation of the symplectic structure.
• Non-perturbative regimes. The simple Hamiltonian ansatz may fail when strong coupling generates multiple relevant operators.
• Physical interpretation of $p_i$. Is there an observable associated with these “momenta”, or are they purely auxiliary?

Exploring these will decide whether “scale Hamiltonians” become more than a pedagogical tool.

11. Take-home message

Thinking of the Hamiltonian itself as a coordinate that moves when we zoom between scales unifies two pillars of physics:

• Mechanics: time evolution generated by a Hamiltonian
• Renormalization: scale evolution generated by beta functions

The proposed framework is elementary yet opens doors to geometric insights and cross-disciplinary analogies. Whether or not every detail survives deeper scrutiny, the perspective alone can sharpen our intuition about why physics looks different at different magnifications.

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Note: This blog post presents a theoretical framework that is still under development. While it builds on established concepts in physics, the specific formalism described here should be considered exploratory.