Schwinger model

QFT for dummies.

Quantum field theory suffers from something called an ultraviolet divergence problem (of which the brilliantly named ultraviolet catastrophe is a special case). Basically, as the theory goes to short distances/high energy, infinities begin to appear. That is a problem. The standard solution is to employ perturbation theory to regularize the integrals and then renormalize them to find the correct answer, but for the computationally minded, there is an alternative regularization scheme, namely lattice regularization. In this approach, we avoid short distances by treating all physical variables as existing on a lattice with a finite spacing. The physical solution is recovered by taking the limit as the lattice spacing goes to zero.

Although lattice regularization is most notably employed in quantum chromodynamics (where perturbative methods are less puissant), it has also been used to study the Schwinger model, which is quantum electrodynamics in one dimension (so, no magnetic field, no photons).

We solve the time dynamics of the Lattice Schwinger Model using exact diagonalization for some initial state (state (t) = state (t=0) exp(-i H t)). Below, the initial state is an electron at site 1 and a positron at site 12. They cannot stay still (due to the uncertainty principle) and in this example we use closed boundary conditions, so they move towards each other, passing and then bouncing off the far boundary.

Although this system is very idealized, it can be used to investigate interesting phenomena. For instance, the flux unwinding of inflatons (modeled by 1D electron/positron pairs) is of interest to some theories of cosmic inflation. In the below figure, the electric field starts with a positive value, after which electron/positron pairs form and unwind the field, causing it to reach a negative value.

We also investigate the phenomenon of quantum decoherence. We show that decoherence succeeds for a relativistic QFT (Schwinger model) coupled to two non relativistic quantum systems. The two quantum systems are meant to model the detector and an air molecule, and both are required for decoherence, which arises from tripartite density matrices. We are also able to measure the speed of decoherence in the QFT, as shown below:

Don’t worry about the colors, but the lines show the speed of light, the speed at which the signal that decoherence has occurred at sites 7-8 propagates through the rest of the system, and the speed of the charges. The decoherence signal travels faster than the charges, but slower than lightspeed!