# Relation between QM and QFT.

1. Nov 23, 2005

### Son Goku

I've read on a few websites, as well as having heard from my professors that the relation between QM and QFT is that QM is the (0+1) approximation of QFT.

Does mean that QM ignores the spatial degrees of freedom of field, or is there something else to it, or have I got it all completely wrong.

Basically I'm asking in what situations does QFT reduce to QM.

2. Nov 23, 2005

### marlon

QFT is the unification of special relativity and QM.

In QM the basic ingredient are wavefunctions.
In QFT the basic ingredient are fields of which the fluctuations correspond to particles.

To describe interactions one uses perturbation theory (in the case of a small coupling constant, ie interactions are not too strong) in both theories. When going from the initial state (just before the interaction) to the final state (just after the interaction) one passes through the socalled intermediate states. These states violate the HUP for a very short while. In QM, the virtual states really are just a sequence of possible states one has to pass in order to go from the initial state to the final state. In QFT, these states correspond to virtual particles because these states are described in terms of fluctuating fields.

Take out the special relativity part and you are back in QM, if the number of particles in the quantumstate remains fixed. Read the EDIT

regards
marlon

EDIT : i forgot to add this : In quantum mechanics, physics is described in terms of wavefunctions that correspond to quantum states of a system. It is important to realize that there is a basic demand for this theory to work : a fixed number of particles. However, for the systems in which particles are created and destroyed, the above demand is not respected. Well, the solution is the introduction of quantum field theory, ie : second quantization...So the first quantization goes from classical field theory to QM (with fixed amount of particles) and the second quantization goes from QM to QFT (variable amount of particles through creation and annihilation.)

In short : classical field -->"first quantization" -->wavefunction (QM)--> "second quantization" --> Quantum fields (QFT)

Last edited: Nov 23, 2005
3. Nov 23, 2005

### Lonewolf

And what about the QFT treatment of BCS? Hardly uses a relativistic quantum field. Condensed matter uses non-relativistic quantum fields all the time.

4. Nov 23, 2005

### Son Goku

Thanks very much for the quick reply.
Just with regard to your last point, I've heard of texts refer to relativistic quantum mechanics, is this a combination of QM and SR which still leaves you with QM, but is flawed in some manner, so you need to introduce QFT anyway?

In other words you can combine QM and SR into relativistic QM, but their true unification is QFT.

5. Nov 23, 2005

### marlon

Indeed, in order to answer i wrote an EDIT in my first post.

marlon

6. Nov 23, 2005

### Lonewolf

And an excellent answer it was

7. Nov 23, 2005

### marlon

What texts ? It is difficult to answer this question if i do not have the context within which such a statement is made.

Again, it is not just that QFT = QM + special relativity. Read the EDIT in my first post. There is this important aspect of "fixed number of particles in a quantumstate". Bsides creation and annihilation operators exist both in QM and QFT, but in QM they raise or lower the energy of a state, in QFT these operators actually can create an extra electron (this does NOT happen in QM) for example. This happens in beta decay and also explains why beta decay CANNOT be explained in terms of QM.

marlon

8. Nov 23, 2005

Staff Emeritus
The phrase "relativistic quantum mechanics" used to be applied to Dirac's theory.

9. Nov 23, 2005

### Son Goku

Yes, Dirac's equation was being refered to.
That explains it, thanks.