Mathematical signature of photon's and electron's field functions

[birulami]

Is is correct to say that the photon, as an electromagnetic wave, is represented by a function $$\def\R{\mathbb{R}}f:\R^4\to \R^3\times \R^3$$ indicating that each point in 4D spacetime gets assigned a pair of vectors for the electric and the magnetic component respectively?

What are the physical units of the two resulting vectors?

Does the electron also have a field representation? What is its signature? Can I have an example field function? What are the physical units of the resulting values?

Thanks,
Harald.

 

[Xezlec]

I don't think "Classical Physics" is the place to post questions about photons... is it? Photons are quantum.

Is is correct to say that the photon, as an electromagnetic wave, is represented by a function $$\def\R{\mathbb{R}}f:\R^4\to \R^3\times \R^3$$ indicating that each point in 4D spacetime gets assigned a pair of vectors for the electric and the magnetic component respectively?

A photon is represented by a quantum mechanical wave function, I believe. This wave function is complex-valued at every point in 4D spacetime. It's not a 3D vector. The wave function is related to electric and magnetic fields, but I don't claim to know exactly how.

However, it is true that all electromagnetic waves in the classical sense are completely described by the function you've written. Just don't mention photons

What are the physical units of the two resulting vectors?

In classical physics, the units of the electric field (I mean the E-field) are Volts per meter (or Newtons per Coulomb, they mean the same thing). The units of the magnetic field (I mean the B-field) are Teslas.

Note that there are alternate forms of these. Sometimes people talk about the electric field in terms of the D-field, which is the "electric displacement field". This is measured in Coulombs per square meter. And sometimes people talk about the magnetic field in terms of the H-field, which may be called the "auxiliary field". This is in units of Amperes per meter.

Does the electron also have a field representation?

Again, electrons are described by a wave function in quantum physics. That's about all I know. Classically, electrons are just electrons.

 

[tacman]

Hello Harald,

Thank you for your question. The mathematical signature of photon and electron field functions is a very interesting topic in physics. Let me address your questions one by one.

Firstly, it is correct to say that the photon, as an electromagnetic wave, is represented by a function f: ℝ^4 → ℝ^3 × ℝ^3. This function maps a point in 4-dimensional spacetime (x, y, z, t) to a pair of vectors representing the electric and magnetic components of the photon's electromagnetic field at that point. The electric and magnetic components are represented by 3-dimensional vectors in ℝ^3.

The physical units of these vectors depend on the choice of units for the electric and magnetic fields. In the SI system, the electric field is measured in volts per meter (V/m) and the magnetic field is measured in tesla (T). In the CGS system, the electric field is measured in statvolts per centimeter (statV/cm) and the magnetic field is measured in gauss (G). So, the physical units of the two resulting vectors will depend on the choice of units for the electric and magnetic fields.

Moving on to your question about the electron, yes, it also has a field representation. The electron's field is described by the Dirac equation, which is a relativistic wave equation. The signature of the electron's field is a spinor field, which is a mathematical object that represents the electron's intrinsic spin. An example field function for the electron's field can be written as ψ: ℝ^4 → ℂ^4, where ℂ^4 represents a 4-dimensional complex vector space.

The physical units of the resulting values will again depend on the choice of units for the electron's field. In the SI system, the electron's field is measured in joules per meter cubed (J/m^3) and in the CGS system, it is measured in ergs per centimeter cubed (erg/cm^3).

I hope this helps to clarify the mathematical signature of photon and electron field functions. Please let me know if you have any further questions.