I have been working on a theoretical framework that draws on concepts from conformal field theory (CFT) to describe fluid dynamics and could potentially be well suited for describing certain turbulent flows. In this approach, I use scalar and vector potentials to capture both conservative and non-conservative aspects of fluids, aiming to simplify complex fluid phenomena with a Maxwell-like formalism.

Overview of the Theory

In this model, fluid flows are described using real scalar $\phi$ and vector potentials $V^\mu = (p, v)$. The key feature is the introduction of a minimal coupling term $(D_\mu \phi)(D^\mu \phi)$, where $D_\mu$ represents a covariant derivative coupling $\phi$ to $V^{\mu}$. Notably, this term is conformally invariant, meaning it respects the symmetry under conformal dilatations (Weyl transformations) $\sigma = e^{\alpha}$. The full list of transformations is:

$\phi \rightarrow \sigma \phi$

$D^{\mu}\phi \rightarrow \sigma D^{\mu}\phi$

$V^{\mu} \rightarrow V^{\mu} + \alpha^{\mu}$

$g_{\mu \nu} \rightarrow \sigma^{-2}g_{\mu \nu}$

To further characterize the fluid dynamics, I define a vorticity tensor $\Omega_{\mu \nu}$, analogous to the electromagnetic field tensor in electrodynamics. Instead of $E$ and $B$ fields, I define the transport field as $U = (v \cdot \nabla)v = -\nabla p - \frac{\partial v}{\partial t}$ and vorticity $W = \nabla \times v$. For the total Lagrangian for the system, $\mathcal{L}_{T}$, I included a kinetic term for the vorticity, $\frac{1}{4} \Omega_{\mu \nu} \Omega^{\mu \nu}$. However, this kinetic term explicitly breaks the conformal symmetry, leading to potentially interesting physical consequences.

$\frac{1}{4} \Omega_{\mu \nu}\Omega^{\mu \nu} \rightarrow \sigma^{-4} \frac{1}{4} \Omega_{\mu \nu}\Omega^{\mu \nu}$

Equations of Motion and Symmetry Breaking

The equations of motion derived from this framework are:

$D_{\mu}D^{\mu} \phi = 0$, which is conformally invariant, and $\partial_\mu \Omega^{\mu \nu} = j^\nu$, where the current $j^\nu$ transforms as $j^\mu \rightarrow \sigma^{2} j^\mu$. This equation is not conformally invariant, indicating a breaking of the symmetry. $\partial_{\mu}\Omega^{\mu \nu}$ and $j^{\mu}$ do not transform the same under $\sigma$.

This explicit symmetry breaking due to the kinetic term seems to introduce a fixed scale into the theory, which in general is crucial for describing fluid phenomena that exhibit characteristic lengths, such as vortex dynamics and turbulence.

Furthermore, the conserved quantity $\frac{Q}{\rho} = \oint U \cdot dS$ acts as an analog to electric charge in electromagnetism, with units corresponding to mass flow rate (kg/s$^2$). This "charge" represents sources or sinks of advective acceleration in the fluid, similar to how electric charge induces an electromagnetic field.

I am particularly interested in understanding whether this explicit symmetry breaking is physically significant. How does this compare to explicit symmetry breaking in other physical theories, and what implications might it have for understanding complex fluid phenomena like vortex shedding and turbulence? Typically the only examples I find are in the context of quantum field theory, not classical field theory.

Any insights or feedback from the community would be greatly appreciated!

Edit:

We want our scalar field $\phi$ to be symmetric under local scaling transformations of the form $\phi \rightarrow \sigma \phi$. To ensure this invariance we introduce the covariant derivative \begin{equation} D^{\mu} = \partial^{\mu} - V^{\mu} \end{equation} where the field $V^{\mu} = (p, v)$ is the solenoidal velocity and it’s associated pressure. The covariant derivative couples $\phi$ to $V^{\mu}$ while still respecting the scaling symmetry

\begin{equation*} D^{\mu}(\sigma \phi) = \sigma D^{\mu} \phi. \end{equation*}

We can then modify our Lagrangian for the theory $\mathcal{L}(\phi, \partial_{\mu}\phi) \rightarrow \mathcal{L}(\phi, D_{\mu}\phi)$ so the Lagrangian for this free field theory (minimal coupling) is then \begin{equation*} \mathcal{L}= (D_{\mu}\phi)(D^{\mu}\phi) \end{equation*} which have the equations of motion $D_{\mu}D^{\mu}\phi = 0$ or \begin{equation} \partial_{\mu}\partial^{\mu}\phi - V_{\mu}\partial^{\mu}\phi - V^{\mu}\partial_{\mu}\phi-\partial_{\mu}V^{\mu}\phi + V^{\mu}V_{\mu}\phi = 0. \end{equation}

There are some coupling and potential terms that emerge characterized by the field $V^{\mu}$ and parameter $\alpha$. Perhaps these are the origin of the potentials you were talking about?

I think an example might help elude what the parameter $\sigma$ could represent physically. The most tractable example I have is the simple case of acoustic waves, as they are easily represented with a scalar field $\phi$ in one spatial dimension $x$.

First I want to clarify that we are fixing the background field $V^{\mu}$ with $V^{\mu} \approx \delta V^{\mu} = \alpha^{\mu}$ were $\alpha^1$ is a constant which effectively suppresses the vorticity.

The equations of motion in $(t, x)$ reduce to

\begin{equation*} \frac{\partial^2 \phi}{\partial t^2} - \frac{\partial^2 \phi}{\partial x^2} - 2\alpha \frac{\partial \phi}{\partial x} + \alpha^{2}\phi = 0 \end{equation*}

which is the Webster horn equation but with an extra term $\alpha^2 \phi$ which has the effect of phase shifting the solutions. We can recognize the symmetry operator here as $\sigma = e^{\alpha x}$ representing the geometry (curvature in this case) of the exponential horn. This is reflected in the canonical Webster Horn equation, where the equations of motion are invariant under any choice of initial cross-sectional area due to the term $\sigma^{-1} \frac{d \sigma}{d x}$. If the cross sectional area can be represented by $S(x) = S_0 \sigma = S_0 e^{\alpha x}$ then the term will always reduce to $\alpha$ no matter the choice of $S_0$. This type of self-similarity is expected in a theory that is scale invariant.

As far as why do all this when Helmholtz-decomposition is perfectly adequate? Why not? Recasting theories in different frameworks can add new perspectives and potentially insights into known phenomenon. I have gained some weird deep intuition about certain principles while engaging in this representation and have found myself baffled many times. I also intend to use this framework as the foundation for future work in emergent fluid-like structures, but I am not nearly there yet.