Exploring the Connection Between Quantum Mechanics and Quantum Field Theory

[fxdung]

<< Moderator note: Split from https://www.physicsforums.com/threads/why-do-we-need-quantum-mechanics-so-much.859210/ >>

What about particle physics? It bases on QFT therefore on QM.Is that right?

 

[bhobba]

What about particle physics? It bases on QFT therefore on QM.Is that right?

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

Thanks
Bill

 

[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.

 

[bhobba]

Honestly, I do not know why QFT must involve QM

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

 

[fxdung]

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

 

[bhobba]

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

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

 

[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?

 

[bhobba]

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?

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

 

[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.

 

[bhobba]

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.

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:
https://www.physicsforums.com/threads/the-born-rule-in-many-worlds.763139/page-7

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

 

[bhobba]

The commutation relation is of field but not of of operators of quantum particle.

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

 

[Orodruin]

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.

 

[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?

 

[bhobba]

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?

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

 

[bhobba]

The path integral is a fundamental concept from QM and also at the heart of QFT.

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

 

[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?

 

[bhobba]

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?

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

 

[fxdung]

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

 

[bhobba]

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

See the link I gave previously to post 137 where I give the proof in modern form based on POVM's.

See also:
http://kiko.fysik.su.se/en/thesis/helena-master.pdf

It's the work of the great mathematician Andrew Gleason:
http://www.ams.org/notices/200910/rtx091001236p.pdf

Thanks
Bill

 

[A. Neumaier]

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.

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.

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:
https://www.physicsforums.com/threads/the-born-rule-in-many-worlds.763139/page-7

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.

 

[fxdung]

Can we deduce QM from QFT meaning if QFT is true then QM must be true?

 

[bhobba]

Can we deduce QM from QFT meaning if QFT is true then QM must be true?

Of course. The following gives the detail:
https://www.amazon.com/dp/9812381767/?tag=pfamazon01-20

Also see:
http://www.colorado.edu/physics/phys5250/phys5250_fa15/lectureNotes/NineFormulations.pdf

See the second quantisation formulation which is basically QFT.

I do not agree with Dr Neumaier on this. Every textbook I have read on QFT says different - but then again they are not advanced. I will leave arguing about it to those that know more like Orodruin.

My understanding though is THE QFT textbook by Wienberg takes a different approach, but although I have the books they are beyond my current level - maybe one day when I get the time.

Thanks
Bill

 

[Demystifier]

The correct chain of deductions, which is well understood and works fine, is
relativistic QFT -> non-relativistic QFT -> non-relativistic QM

One may also want to take a different path
relativistic QFT -> relativistic QM -> non-relativistic QM
but it turns out that such a chain of deductions is not so well defined and depends on some interpretational subtleties.
For some of the subtleties and (my) attempts to resolve them see
http://arxiv.org/abs/quant-ph/0609163
http://arxiv.org/abs/0705.3542
http://arxiv.org/abs/1205.1992

 

[fxdung]

Which books say about the first chain of deductions?

 

[Demystifier]

Which books say about the first chain of deductions?

I don't know a book which says it explicitly, but implicitly you can find it in
https://www.amazon.com/dp/019969933X/?tag=pfamazon01-20
After reading this book, you should be able to make such a chain of deductions by yourself.

 

[Demystifier]

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.

Would you agree with me (see post #23) that non-relativistic QFT can be derived from relativistic QFT? Further, would you agree with me that non-relativistic QM can be derived from non-relativistic QFT?

With this chain of reasoning one avoids dealing with the problematic relativistic QM. For instance, one never needs to introduce a notion of a relativistic position operator.

 

[fxdung]

What is about if I think that QM is ''a small part'' of a larger theory that is QFT?

 

[vanhees71]

I'd say QFT is the comprehensive concept. Any quantum theoretical description can be formulated in terms of QFT. Non-relativistic QM of systems with fixed particle number is a special case and easy to derive from QFT.

 

[fxdung]

What are the axioms of QFT?

 

[vanhees71]

There are the Wightman axioms, but QFT is not really fully mathematically strict.

https://en.wikipedia.org/wiki/Wightman_axioms

 

[A. Neumaier]

Would you agree with me (see post #23) that non-relativistic QFT can be derived from relativistic QFT?

Only in a very vague sense.

Nonrelativistic QFT is usually taken to be the statistical mechanics of gases, liquids, and solids made of nuclei and electrons, with electromagnetic interaction modeled as external field only. (To handle photons needs at least a partially relativistic setting.) As such it shares the abstract features of relativistic QFT, except that it takes the limit $c\to\infty$ to simplify the dynamics. However, the fields appearing in nonrelativistic QFT (the electron field and one interacting spacetime field for every nuclide appearing in the model - or an external periodic potential if none is modeled) are very different from those appearing in relativistic QFT (one space-time field for every elementary particle). I haven't seen any derivation of the former from the latter. There is a chain of reasoning going from quarks to hadrons to nuclides, considered as asymptotic free fields, but as far as I have seen none that would allow me to say that interacting nonrelativistic QFT is derivable from the relativistic version. It is regarded as an effective theory for the latter, but not because of a derivation but based on plausibility reasoning only.

Further, would you agree with me that non-relativistic QM can be derived from non-relativistic QFT?

No; I completely disagree!

It only works in the opposite direction, presented in all textbooks on statistical mechanics, by a two-step process of generalization and abstraction during which some features of QM are lost. First one generalizes the setting of QM by turning the number of particles - which in QM is a parameter only - into an operator acting on Fock space whose spectrum are the nonnegative numbers. Then one gets rid of all measurement issues by replacing the Born rule by the definition of ensemble expectations via $\langle X\rangle:=\mbox{tr} ~\rho X$, which no longer refers to observation and measurement. This allows one to consider arbitrarily large systems - which constitutes the second generalization - and the thermodynamic limit of infinite volume (which is needed to make it a QFT proper). Then one is at the level of field expectations and field correlations, which are the subject of QFT. Note that the notions of observation and measurement - the most controversial features of QM - are lost during this abstraction process.

Because of this loss, one can go only part of the way back if one tries to reverse the direction, going from nonrelativistic QFT to QM. One can consider a fixed number of particles and restrict to the eigenspace of the number operator with fixed eigenvalue $N$. This produces (restricting for simplicity to a single scalar field) the Hilbert space of totally symmetrized wave functions in $N$ 3-dimensional position coordinates $x_i$. On this Hilbert space, only those operators (constructed from the field operators in Fock space) have a meaning that commute with the number operator. This is not enough to construct position and momentum operators for the individual particles but only for their center of mass. One sees already here that one needs to make additional assumptions to recover traditional quantum mechanics.

Worse, since in the QFT description both observers and measurements are absent, one has to introduce observers and measurements and their properties by hand! In particular, the Born rule of QM, that tells what happens in a sequence of ideal measurements, must be postulated in addition to what was inherited from QFT! Unless the concept of observers and measurement are fully defined in quantum mechanical terms so that one could deduce their properties. While this seems not impossible, it certainly hasn't been done so far!

 

[Demystifier]

This is not enough to construct position and momentum operators for the individual particles but only for their center of mass.

Interesting point! I need to think about it.

 

[fxdung]

Please tell a bit more about the Born rule of QM,about the sequence of ideal measurements...

 

[Demystifier]

limit of infinite volume (which is needed to make it a QFT proper)

I don't think that this is really essential for QFT.

 

[Demystifier]

Worse, since in the QFT description both observers and measurements are absent, one has to introduce observers and measurements and their properties by hand!

With that I strongly disagree. Yes, observers and measurements in standard formulation of quantum theory are introduced in a rather ad hoc way, but that refers equally to non-relativistic QM, non-relativistic QFT and relativistic QFT.