cftsolver - A Classical Field Theory Solver

I've always been a visual learner. We can solve Quantum Field Theory (QFT) by doing quantum mechanics - computing the S-matrix from the path integral, a sort of overlap integral between two definite states. What about solving the underlying classical field theory before quantizing? This tool aims to visualize interacting theories by solving the Euler-Lagrange equations of motion (EoM) explicitly for a set of input field configurations. This is effectively a system of coupled partial differential field equations on a lattice.

Requirements

• OpenGL 4.6

TODO

• implement field classes
• euler lagrange equations of motion: scalar QED + gauge fixing
• add modules for 1-D and 2-D solvers
• test OpenGL

Long term:

• test scattering
• test decays
• make interactive in web browser embedding

Modules

Scalar-Yukawa Theory

Let's first start with a simple theory of a massive, real scalar field $\phi$ without any interaction. It's Lagrangian can be written as $$\mathcal{L} = \frac{1}{2}(\partial_\mu \phi) (\partial^\mu \phi) - \frac{1}{2}m^2 \phi^2$$ With the Euler-Lagrange equations, this leads to the Klein-Gordon equation of motion for the field $\phi$,

$$(\Box + m^2) \phi = 0$$

If we want to simulate this with finite difference methods, there are a few routes to take. The simplest is to open up the KG equation and discretize the derivatives with central difference, working for now in 1 space and 1 time dimension:

$$\partial_{t}^2 \phi \to \frac{\phi(t_{i+1}) - 2 \phi(t_{i}) + \phi(t_{i-1})}{\Delta t^2}$$ $$\partial_{x}^2 \phi \to \frac{\phi(x_{i+1}) - 2 \phi(x_{i}) + \phi(x_{i-1})}{\Delta x^2}$$

Where we are now describing the field over a N-dimensional space lattice with coordinates $x_{i}$ of spacing $\Delta x$ and time steps $\Delta t$ over time coordinates $t_{i}$. This permits us to solve for the "push" of $\phi$ to the next time step;

$$\phi(x_{i},t_{i+1}) \simeq 2 \phi(x_{i},t_{i}) - \phi(x_{i},t_{i-1}) + (\partial^2_{x_{i}}\phi(x_{i},t_{i}) - m^2\phi(x_{i},t_{i})) (\Delta t)^2$$

From this equation we can immediately recognize that, just like solving any PDE, we require some boundary condition information; to kick things off we require data on the fields at two time slices -- the present slice and the one before -- and we also need spatial boundary conditions at the edges of our discretized space in order to compute the values of the field and its derivatives.

Scalar QED

wip

Scalar QED with Massive Photons

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Scalar QED with a Higgs Potential

wip