Equations Without Equations: Towards Formalizing Physicists' Reasoning

Not all mathematical solutions to physical equations are physically meaningful: e.g., if we reverse all the molecular velocities in a breaking cup, we get pieces self-assembling into a cup. The resulting initial conditions are ``degenerate'': once we modify them, self-assembly stops. So, in a physical solution, the initial conditions must be ``non- degenerate''. A challenge in formalizing this idea is that it depends on the representation. Example~1: we can use the Schr\"odinger equation ${\rm i}\hbar\displaystyle\frac{\partial\Psi}{\partial t}= -\displaystyle\frac{\hbar^2}{2m}\Delta\Psi+V(\vec r)\Psi$ (1) to represent $V(\vec r)$ as $F(\Psi,\ldots)$. The new equation $dF/dt=0$ is equivalent to (1) but now $V(\vec r)$ is in the initial conditions. Example 2: for a scalar field $\varphi$, we describe a new ``equation'' which is satisfied iff $\phi$ satisfies the Euler-Lagrange equations for some Lagrangian $L(\varphi,\varphi_{,i}\varphi^{,i})$. So, similarly to Wheeler's cosmological ``mass without mass,'' we have ``equations without equations.''