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Wednesday, October 30, 2024

Is Light Composed of Photons?

A popular channel tries to answer What is a Photon?

The simple answer is that light is composed of photons. A photon is a ball of light. A wise guy commenter gives the more sophisticated answer:

The special relativistic wave equation that accurately describes electrons is the Dirac equation. The Dirac Lagrangian density for electrons has got a local U(1) symmetry because of local causality and local charge conservation (Noether's theorem). U(1) symmetry, because we only ever observe the absolute value squared of the wave function.

This can be modeled in gauge theory as an S^1 fiber bundle (or a U(1) Lie-algebra valued principal g-bundle) over a flat Minkowski spacetime base. Wave functions for the electron field are then sections in this fiber bundle.

To make precise the comparison of geometric data between different spacetime points (gauge covariant derivative), we introduce a connection on this fiber bundle; the electromagnetic vector potential, A_mu (just like the Christoffel symbols/Levi—Civita connection of the tangent bundle in general relativity). Basis vectors/phase can change from place to place either bc. we are using some strange coordinate system (like polar coordinates fx.) or bc. our manifold/bundle is curved (to be precise, the connection is curved). So this connection might have a holonomy/curvature (responsible for geometric Berry phase), just like how spacetime can be curved. In this case, the curvature is caused by the 4-current, just like how spacetime curvature is caused by the stress-energy-momentum tensor. We can take the exterior derivative of this Ehresmann connection 1-form (A_mu), which yields a curvature 2-form, called the electromagnetic/Faraday tensor (or the Riemann curvature tensor in the case of general relativity).

This new field, (A_mu) the vector potential has got its own dynamics. If we derive the equations of motion with the help of the Euler—Lagrange equation, we get back the Lorentz force and Maxwell's equations in the 'classical' case. We can also apply canonical quantization and make the 'A' field values into operators. At low energies, this A field behaves like a quantum harmonic oscillator at each point of spacetime; its energy levels are going to be quantized. The number of quanta in a given frequency mode is what we call the number of photons in that mode (pure numer state/Fock state).

Very good, but it is accurate to say light is composed of photons?

I think not. Light is an electromagnetic wave, and small measurements are quantized. A photon is a measured quanta of light.

You might say, this is like saying a falling tree in the forest does not make a noise if no one listens. Likewise light is not made of photons unless measured.

The difference is that our best theories of trees and sounds say that the tree makes a sound whether anyone listens or not. Our best theories of light do not discretize light until a measurement.

You could say: No, that's wrong, QED uses Feynman diagrams of unobserved particles, including photons.

That is a point, but thinking of light as particles leads to faulty conclusions. QED is really a field theory.

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