# Is FDTD A Method for Simulating Quantum Field Theory for Photons?

I have been reading a bit about Quantum Field Theory and Photons, I read that the equation(s) for spin 1 massless particles (photons) in Quantum Field Theory are just Maxwell's Equations described in terms of 4-vectors under a Lorenz Gauge Condition. I know about Non - Relativistic Quantum Mechanics, but I don't know much about Quantum Field Theory or Frequency Difference Time Domain (FDTD) simulations, but I wanted to know: can FDTD methods be used to simulate photons (and or electrons, so Quantum Electrodynamics) for Quantum Field Theoretic calculations?

Thanks!

• It is important to make a distinction between "4 vectors" (which would be four vectors that each have three components) and "a 4-vector" (a vector with four components). Only the latter is correct in the current context. Commented Feb 27, 2023 at 22:16
• Sorry my poor English. Editing old comment not allowed then updated; Aside from QFT, solving the electromagnetic problem based on 4-vector within the classical field framework seems to be a more difficult problem. As is well known, mainstream FDTD of electromagnetism chooses electric and magnetic fields as fundamental variables. The first obstacle in using 4-vector is how to achieve Lorentz gauge fixing. Second, the approximate cancellation of longitudinal photon and scalar photon has to be verified numerically. PML in 4-vector is also a problem. Commented Feb 28, 2023 at 1:42
• @WolfgangBangerth The ladder is what I meant, thank you, I just updated it for clarity Commented Mar 1, 2023 at 16:00
• @HEMMI Thanks for the reply, that is a good outline of the challenges I think just based on the little I know about FDTD. It seems like you would need to vary the time steps throughout the Cells in the simulation if that is even possible. Otherwise Im not really sure you can change the Yee Cell to contain a 4th dimension, or what precisely it would be. Commented Mar 1, 2023 at 16:12
• @HEMMI As for lorenz gauge fixing, speculating, might it be possible to use a system of equations to achieve this: start with the unconstrained state put it into a column vector, then use a system of equations to achieve the constrained state as an output column vector? As for the last part, I don't really know what longitudinal photon and scalar photons are, can you please explain? Commented Mar 1, 2023 at 16:17

This does not answer the original question. A reply to the comment field.

I write some things about gauge fixing within my knowledge.

Not in electromagnetic but in elastic problems, incompressible materials such as rubber is a problem type that some authors pay great attention to. The incompressible property is described by the equation $$\text{div}\vec{u}=0$$, where $$\vec{u}$$ is the displacement vector, and Poisson's ratio is 0.5. This feature is not easy to implement within the standard FE method for elastic materials and many advanced authors are trying to solve it. As restraining incompressible elastic problem is a difficult task, I am inspecting that we encounter similar difficulty in electromagnetic gauge fixing.

As for the FEM of low-frequency electromagnetism, the reinvention of the Whitney element (vector finite element, edge element) seems to be successful. If the element shapes are limited to tetrahedron, rectangular box, rectangular prism, the Coulomb gauge $$\text{div}\vec{A}=0$$ is automatic inside the element. I would like to use this method.

Scalar/longitudinal approximate cancellation of photon wave.

First, remember the elastic wave problem (not the photon problem) again. When solving the elastic wave problem, 3-component displacement vectors $$u_x,u_y$$ and $$u_z$$ are usually chosen as fundamental variables. This number 3 corresponds to S-wave (2 components) and P-wave (1 component). Since 3=2+1, it is quite OK, no problem.

However, for high frequency electromagnetic wave problems, if you choose 4-component vector $$A_0,A_x,A_y$$ and $$A_z$$ as the fundamental variable, there should be 4 wave modes: S-wave (2-component), P-wave (1-component) and scarar wave (1-component). But we know that electromagnetic waves are predominantly 2-component transverse waves, so we should hide the remaining 2-components.

As a real-world example of such hiding, the infinitesimal linear wire antenna has an analytic solution; see what I wrote in here; it is taken from Balanis' book. As you can see from equation 4-10a, the longitudinal electric field $$E_r$$ is not zero near a metallic antenna. In deriving 4-10a there is some cancellation of the longitudinal component and the scalar component (I have a note of my own, but it was simply an assertion of Balanis' book description, so I will not write it here). It is an example of the cancellation of the scalar photon and the longitudinal photon. Only the transverse wave survives as it moves away from the metal antenna.

• Thank you for the reply! I know you said it does not answer the original question but its probably the best answer I will get. Thank you! Commented Mar 14, 2023 at 19:30