# I Relation between QM and QFT

1. Feb 24, 2016

### fxdung

Last edited by a moderator: Feb 24, 2016
2. Feb 24, 2016

### bhobba

Of course. I simply gave the first example that popped into my head.

Thanks
Bill

3. Feb 24, 2016

### fxdung

Honestly, I do not know why QFT must involve QM.I see QM and QFT do not contradict with each other.But I can not deduce QM from QFT.

4. Feb 24, 2016

### bhobba

QFT is just an application of QM which is a general overarching theory.

The way you get QFT is divide a field into a lot of blobs, apply QM to those blobs, then let the blob size go to zero.

Thanks
Bill

5. Feb 24, 2016

### fxdung

In QFT text books they all say about S-matrix through Feynman diagrams,Green Functions,cross section....But why they do not say any about QM?

6. Feb 24, 2016

### bhobba

Right at the start they derive QFT the way I described. The rest is just mathematical development of it.

There is also an equivalent approach applying QM principles with relativity.

Thanks
Bill

7. Feb 24, 2016

### fxdung

In QFT they base on commutator of Field operator and Momentum of Field operator,then deduce the creation and annihalation operators.Where is the principles of QM?

8. Feb 24, 2016

### bhobba

Right at the start where the field is quantised using the standard QM commutation relations:
http://www.damtp.cam.ac.uk/user/tong/qft/two.pdf

Thanks
Bill

9. Feb 24, 2016

### fxdung

In QM there is a relation between particle and wave function,in QFT there is a relation between particle and field.If QFT involve QM,where is the relation between field and wave function.In Tong's lecture notes,the commutation relation is of field but not of of operators of quantum particle.

10. Feb 24, 2016

### bhobba

I was going to give a detailed response but a couple of things changed my mind.

First - this is getting way off topic - it needs a separate thread.

Second I think you need to think about things a bit more. In particular you need to understand what the foundational axiom of QM is. See post 137:

After that you should be able to see exactly what a wave-function is and why its not a principle of QM - hint position is not necessarily an observable. Also reading chapter 3 of Ballentine will help.

I will contact the moderators and get them to create a new thread.

Thanks
Bill

11. Feb 25, 2016

### bhobba

Like a lot of more advanced treatments aimed at graduates steps are left out. Here the step of dividing the field into blobs, treating each blob as a particle then taking the limit is left out.

Thanks
Bill

12. Feb 25, 2016

### Orodruin

Staff Emeritus
Feynman diagrams and Green functions as applied to QFT are just tools developed to compute the path integral. The path integral is a fundamental concept from QM and also at the heart of QFT. You can compute things such as cross sections also in QM. QFT is just QM applied to an infinite number of coupled harmonic oscillators.

Also note that both Feynman diagrams and Green functions are not QFT or QM specific. They can be applied in several different fields and are just tools for solving differential equations. Thinking they are particular to QFT is like thinking integration can only be used in classical mechanics. Also, they are both perfectly applicable to solving problems in non-relativistic QM.

13. Feb 25, 2016

### fxdung

Where is the Second Axiom of QM(say :there is a operator P of state that the average of an observation O is Tr(PO)) in QFT?

14. Feb 25, 2016

### bhobba

It applies to the blobs but is not used as far as I know later - at least I haven't seen it. One can almost certainly find a use for it - its just at my level of QFT I haven't seen it. Some others who know more may be able to comment. BTW the link I gave which proved Gleason showed its not really an axiom - but rather a consequence of non-contextuality - but that is also a whole new thread.

Thanks
Bill

Last edited: Feb 25, 2016
15. Feb 25, 2016

### bhobba

Indeed it is.

Its not hard to relate it to the two axioms from Ballentine and the link I gave.

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.

Its a bit of fun working through the math with Taylor approximations seeing its quite a reasonable process.

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.

Its is also a very common and elegant way of developing QFT. But is equally applicable to standard QM.

Thanks
Bill

16. Feb 25, 2016

### fxdung

And the probability character presents in expression of S-matrix: <out state/in state>,in cross section,in Green functions...,this character implies the Second Axiom.Is that right?

17. Feb 25, 2016

### bhobba

The second axiom of QM is not really an axiom - its a consequence of non-contextuality. That's why Gleason is so important but usually left out of even advanced treatments - which is a pity. So the answer is no.

Thanks
Bill

18. Feb 25, 2016

### fxdung

I have not know non-contextuality and Gleason.Which books say about those topics?

19. Feb 25, 2016

### bhobba

20. Feb 25, 2016

### A. Neumaier

I think you have a very valid point. QM and QFT are closely related, but there is no simple relationship between them, and this shows in the very different treatment they get in textbooks. In particular, the relationship is far more complicated than bhoppa paints it.
QM in the conventional axiomatic form is about small quantum systems and their interaction with external measurement devices. This is reflected in the fact that the axioms explicitly involve statements about the measurement process. The states of a system evolve by unitary time evolution whose generator is a Hamiltonian given by an explicit expression in a small set of basic observables.

In relativistic QFT one specifies a field operator at every point in space-time. Therefore there is no place for an outside observer to make a measurement with a classical apparatus. Therefore the conventional axioms for QM say nothing about relativistic QFT. The states of a system evolve by unitary time evolution whose generator is given (except for free fields) by a highly implicit construction involving renormalization.

On the other hand, QM and QFT share a common mathematical structure. In both theories, there is a*-algebra of quantities unitarily represented by operators on a Hilbert space, and a cone of states, positive linear functionals on this *-algebra. In both theories, Lie algebra techniques and hence commutation rules play an important role for the construction of the representations needed. Moreover, the asymptotic limit of the time evolution for $t\to\pm\infty$ leads in both cases to an S-matrix interpretable in terms of asymptotic free fields, one for each bound state. These asymptotic free fields have a many particle interpretation, which gives the link between QFT and QM.

It is not very difficult (though time-consuming, since there are lots of applications) to verify that this is the only link between QFT and QM exploited in the applications.

Last edited: Feb 25, 2016