Understanding Fields in Quantum Mechanics: Electrons, Waves, & Particles

[gk007]

What is a field in quantum mechanics (not the classical version)? And when it is said that an electron is an "excitation state of a field", does that mean that electrons are created by wave or disturbances in a field? Also, is there a different type of field for each fundamental particle, or can it be simplified to one big field?

 

[tom.stoer]

What is a field in quantum mechanics

You have to study quantum field theory.

What one does is (roughly speaking) the following. One takes the Dirac field (mathematically it is a classical field), takes it's Fourier transform and translates the Fourier components b(p), b*(p) and d(p) d*(p) into operators. This step is called quantization. For each three-momentum p there are these operators which are related to the creation and annihilation operators in case of the harmonic oscillator. That means a plane wave with a certain momentum p is "created" in the Hilbert space using a creation operator. Attention: there is not only one pair, but two pairs for each p.

Also, is there a different type of field for each fundamental particle, or can it be simplified to one big field?

One needs a field for each particle, e.g. one field for the photon (4-potential), one for the electron and the positron, one for the quarks (the different colors are treated via indices, so the field becomes a 4-spinor with an additional color-index i=1..3), one for the gluon (4-potential now with a color index a=1..8) etc.

Finding one big field is the dream of theoretical physicists in the context of a "theory of everything". String theory (a much debated, partial controversial issue) comes rather close to this dream, as there is only one string.

 

[gk007]

OK, thanks for clearing everything up :)

 

[tom.stoer]

really, everything?

 

[khemist]

Do you have any papers or references to someone performing those operations? Id be interested to see...

 

[tom.stoer]

Every book on quantum field theory will do.

I recommend
- Ryder
- Weinberg
- Srednicki (draft: http://www.physics.ucsb.edu/~mark/ms-qft-DRAFT.pdf; ; chapter 3 Canonical Quantization of Scalar Fields)