# Basic question about equations of Quantum field theory (QFT)

1. Jul 26, 2014

### fog37

Hello Forum,

The electromagnetic field EM must be treated relativistically because it travels at the speed of light in a vacuum. However, the idea of quantization forces us to treat the field as a quantum mechanical field.

QFT is the answer to that. QFT is quantum mechanics with relativistic features. But we are only talking about special relativity not general relativity, correct? There is no theory that blends QM and GR.

QFT is able to describe the EM field using quantum mechanics. So QFT is relativistic QM. Of course, it can also be applied to other matter fields whose excitations are the electrons, protons, and all other materials particles as long as they travel very fast.

In QFT, are EM fields described using a relativistic version of Maxwell equation or a relativistic version of Schroedinger equation?
What about matter fields? I think the answer for them is the relativistic Schroedinger equation.

What is the electron-positron field? Is it the same thing as the matter field?

What is the Dirac equation? What type of fields does it describe? Is it relativistic and part of QFT?

thanks
fog37

2. Jul 26, 2014

### atyy

Most usually, QFT refers to special relativity. However, gravity can be treated as a spin 2 field on a fixed space-time background. Quantum gravity treated this way is enough for all the stuff we see, but it appears to be inconsistent at high energies. In the latter respect, quantum gravity is not different from quantum electromagnetic fields (quantum electrodynamics or quantum EM), which also appears to be inconsistent at high energies, even though it is a successful theory for all the stuff we see at low energies. http://arxiv.org/abs/1209.3511

QFT can be applied even if the particles travel slowly, since non-relativistic QM can be considered a low energy approximation to QFT.

In quantum electrodynamics, which is the QFT of EM fields, the fields are described using the vector potential form of Maxwell's equations.

The electron-positron field is a matter field (more precisely it is a fermion field), and it is described by the Dirac equation.

3. Jul 26, 2014

### Staff: Mentor

Yes there is:
http://arxiv.org/pdf/1209.3511v1.pdf

Its usually done by the path integral approach.

You start out with <x'|x> then you insert a ton of ∫|xi><xi|dxi = 1 in the middle to get ∫...∫<x|x1><x1|......|xn><xn|x> dx1.....dxn. Now <xi|xi+1> = ci e^iSi so rearranging you get
∫.....∫c1....cn e^ i∑Si.

Focus in on ∑Si. Define Li = Si/Δti, Δti is the time between the xi along the jagged path they trace out. ∑ Si = ∑Li Δti. As Δti goes to zero the reasonable physical assumption is made that Li is well behaved and goes over to a continuum so you get ∫L dt.

Now Si depends on xi and Δxi. But for a path Δxi depends on the velocity vi = Δxi/Δti so its very reasonable to assume when it goes to the continuum L is a function of x and the velocity v.

In this way you see the origin of the Lagrangian. And by considering close paths we see most cancel and you are only left with the paths of stationary action.

The extension to fields is immediate - we replace the particle Lagrangian with a field Lagrangian. The gory detail of how exactly you do this can be found for example in Zee - Quantum Field Theory In A Nutshell:
https://www.amazon.com/Quantum-Theory-Nutshell-Edition-nutshell/dp/0691140340

It's basically a generalisation of a heap of particles connected by springs to conceptually model a field. It's an interesting exercise I did the other day developing it without the aid of such a model.

Its the matter field of electrons and positrons

Its a relativistic version of Schroedinger's equation. It predicted funny stuff like electrons with negative energy which could be accommodated into a reasonable theory by holes, but many viewed it as a bit of a kludge. The matter field approach was cleaner so it won out - but it's now known to basically be the same.

Thanks
Bill

Last edited by a moderator: May 6, 2017
4. Jul 26, 2014

### tom.stoer

You can use a Schrödinger-like equation es well.

$$\mathcal{L}[A,\phi,\psi,\ldots]$$

for gauge fields A (photon, gluon, ...), scalar fields phi (Higgs), spinor field psi (electrons, quarks, ...). Then you derive the canonical conjugate momenta and construct the Hamiltonian density and the Hamiltonian

$$H = \int_{\mathbb{R}^3} d^3x\,\mathcal{H}[A,\phi,\psi,\ldots, \pi_A,\pi_\phi,\pi_\psi,\ldots]$$

The last step (which usually requires approximations) is to find eigenstates for the Schrödinger-like equation

$$(H-E)|E,\ldots\rangle = 0$$

where ... means other quantum numbers like spin, isospin, ...

These eigenstates correspondent to particles like free photons, electrons, ... but also bound states like protons, neutrons, ...