# Relationships between QM and QFT Particles

#### Buzz Bloom

Gold Member
It is restricted to particles with sufficiently slow speeds that relativistic effects can be neglected.
Hi Nugatory:

OK. I accept as a definition that that "non-relativistic (NR) QM" excludes photons. I also accept that the currently best understanding of the photon double slit phenomena is based on QED, the QFT relevant to EM.

I am now thinking of a double slit experiment with non-relativistic electrons (NREs). Given that the results of such an experiment involves EM and no relativity, would it be possible to calculate the results based on the Feynman concept being applied to an NRE which travels through all possible paths, i.e., goes through both slits, in reaching its destination? If so, would the use of the Feynman concept by definition make this approach an NR QFT and/or an NR QED? Or are QFTs and QED necessarily by definition relativistic?

Below is a quote from atyy's post #3,
A particle in non-relativistic QM and a particle in relativistic QFT are "essentially" the same. The way one can see this is by reformulating non-relativistic QM of many identical particles as a non-relativistic QFT.​
This seems to imply that a QFT can be either relativistic or NR. It also suggests that using the Feynman concept is also excluded from QM.

What I am trying to understand here is a definitional difference between QM and QFTs. Is it by definition so that QM (1) excludes all but NR particles, and (2) excludes the Feynman concept of a particlal traveling all possible paths? If so, are there any additional definitional distinctions?

Regards,
Buzz

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#### bhobba

Mentor
What I am trying to understand here is a definitional difference between QM and QFTs.
Standard QM is the principles of QM applied to things where position is an observable. If that's the case and you assume the Galilean transformations what you get is the usual QM talked about in books. The detail of this can be found in Chapter 3 of Ballentine - QM - A Modern Development - but it uses advanced math so cant be explained in words.

There is a branch of physics called field theory that deals with fields like EM fields. To handle that with the methods of mechanics ie of particles, you break the field into a large number of blobs and treat each of those blobs like a particle. Then you let the blob size go to zero to get a continuum.

Quantum Field Theory does exactly the same thing. You have a field, break it into blobs, apply the methods of standard QM to those blobs, then take the blob size to zero. That way you get a Quantum Field Theory. Its exactly analogous to classical field theory.

That's the definitional difference. How does all these weird things like 'knots' in the field being particles etc come about? That requires a lot of study and deep math and cant be explained linguistically.

Thanks
Bill

#### Buzz Bloom

Gold Member
Hi Bill:

Thank you very much for your post #26. It is is very helpful.

Next I would like to understand how the definitional differences between QM and QFT in your post relate to (1) relativity and (2) Feyman's concept of multiple paths.

I conclude from earlier discussion in this thread together with your post that only non-relativestic particles have observable positions. I am guessing that I have misunderstood the concept of "observable position". Isn't the position of a photon when it is detected by a detector observed to be at the time of detection at the poistion of the sensitve part of the detector? I am now guessing that having an observable position means that the position is in principle observable at all times, not only at a detector. Is that correct?

I am also guessing that it is possible to use Feyman's concept of multiple paths with NR particles, and doing this could be in the context of QM rather than QFT. Is this correct?

Regards,
Buzz

#### bhobba

Mentor
I conclude from earlier discussion in this thread together with your post that only non-relativestic particles have observable positions.
They can have observable positions in QFT, but the situation is very complex
http://www.mat.univie.ac.at/~neum/physfaq/topics/position.html

In standard QM it is always assumed to have position - in QFT it may or may not.

In books on QFT I have read its not a usual thing that's worried about ie its not important - at least at the level I have investigated - which is not advanced.

Thanks
Bill

#### vanhees71

Gold Member
Hi Nugatory:

OK. I accept as a definition that that "non-relativistic (NR) QM" excludes photons. I also accept that the currently best understanding of the photon double slit phenomena is based on QED, the QFT relevant to EM.

I am now thinking of a double slit experiment with non-relativistic electrons (NREs). Given that the results of such an experiment involves EM and no relativity, would it be possible to calculate the results based on the Feynman concept being applied to an NRE which travels through all possible paths, i.e., goes through both slits, in reaching its destination? If so, would the use of the Feynman concept by definition make this approach an NR QFT and/or an NR QED? Or are QFTs and QED necessarily by definition relativistic?

Below is a quote from atyy's post #3,
A particle in non-relativistic QM and a particle in relativistic QFT are "essentially" the same. The way one can see this is by reformulating non-relativistic QM of many identical particles as a non-relativistic QFT.​
This seems to imply that a QFT can be either relativistic or NR. It also suggests that using the Feynman concept is also excluded from QM.

What I am trying to understand here is a definitional difference between QM and QFTs. Is it by definition so that QM (1) excludes all but NR particles, and (2) excludes the Feynman concept of a particlal traveling all possible paths? If so, are there any additional definitional distinctions?

Regards,
Buzz
Here a lot is very confused. Let's try to sort things out separately.

(a) Quantum field theory vs. quantum mechanics

Quantum field theory is a formulation of quantum theory of many-body systems, for which the particle number is not necessarily conserved. In this sense it is more general than quantum mechanics, which is always about quantum systems with a fixed number of particles. Any problem of quantum mechanics (in this sense) can be formulated in terms of quantum field theory and both theories are completely equivalent in these cases. Quantum field theory can be used as a formulation non-relativistic quantum theory as well as for relativistic. In the relativistic case quantum mechanics is not so well suited, because in relativistic quantum theory you always have some probability to create and destroy particles (or quanta).

The big arena of non-relativistic QFT is condensed-matter physics, where you often use the concept of "quasi-particles". This technique has been invented by Landau in low-temperature physics of liquid helium. The point is that formally you can describe collective excitations of a many-body system in an analogous way as dilute gases. The "particle-like excitations" are then described as an almost ideal gas of quasi-particles. This is in some way a misnomer, because it's just the mathematical analogy which coined this name quasi-particles. An example are phonons in solid-state physics. They are quantized collective vibrations (like sound waves propagating in the classical picture). The point however is that you can describe bulk properties like the specific heat of the crystal using these quantized collective vibrations, i.e., a set of independent harmonic oscillators (which are mathematically precisely analogous to the description of non-interacting particles in quantum field theory). Now the lattice vibrations are perturbations of the lattice and these perturbations can interact with the electrons and also among themselves, and these "interactions" between electrons and quasiparticles or among the quasiparticles themselves can be treated with perturbation theory. You have corresponding Feynman diagrams and everything very much like in relativistic quantum field theory used in particle physics.

(b) Feynman's path-integral formulation of quantum theory

Feynman's path-integral formulation is just another way to express the same quantum theories discussed above. In quantum mechanics you have functional integrals over trajectories in phase space to begin with. It can be derived from the usual operator formulation of quantum theory. In non-relativistic QM often you have a Hamiltonian which looks like $H=p^2/(2m)+V(x)$ (single relativistic particle moving in a potential force field, e.g., an electrostatic field). Now, the only path integrals you can really solve analytically are path-integrals where the integrand is a Gaussian, and this is what happens in this case: You can integrate analytically over the momentum part of the phase-space trajectory and are left with a path-integral over configuration space, and this is what Feynman figured out in his PhD thesis (published in Review of Modern Physics).

The path-integral technique is also applicable in quantum-field theory, but there you integrate over field configurations not particle trajectories. At the end it's just another formulation of quantum-field theory in the operator language.

Which method to use, depends on the problem you working on. Often the path-integral formalism can be simpler than the operator formulation or the other way around.

#### Buzz Bloom

Gold Member
Hi Bill and vanhees:

Thank you both very much for your very helpful posts. I think I now have an understanding about the differences and similaities regarding the concept of "particle" between QM and QFT at a sufficiently "intermediate" level appropriate to my current interest and mental capabilities.

Regards,
Buzz

#### DrDu

The main difference between non-relativistic and relativistic QFT is that the former can be shown to exist while the latter doesn't.
Galilean covariance guarantees the conservation of mass and particle number while in relativistic QFT's even the slightest interaction leads to the creation of an infinity of particles which then live in a completely different Hilbert space. Up to now, this spoils any consistent QFT. All we have are some perturbation series aka Feynman diagrams.

#### Buzz Bloom

Gold Member
in relativistic QFT's even the slightest interaction leads to the creation of an infinity of particles which then live in a completely different Hilbert space. Up to now, this spoils any consistent QFT.
Hi @DrDu:

Thanks for you post. It makes a correction to what I previously thought I understood.

As I understand the quote above, you are saying that while non-relativistic QM and QFT are consistent, relativistic QFT is not. I conclude from this, and from the earlier discussion in this trhread, that relativity doesn't work well regarding both QM and QFT. For QM, relativity doesn't apply at all, even for photons. Does your post imply that QFT also doesn't deal with photons adequately, ir is it just relativistic non-zero mass particles that cause QFT problerms.

Regards,
Buzz

#### DrDu

The problem is with interacting particles. Hence, it also applies to photons, as photons also interact with electrons and other charged particles. From what I remember, the problems are even more severe for massless particles than for massive ones. The problem of showing that the Yang-Mills theory - which is thought to be maybe the most well behaved QFT - actually exists as a well defined mathematical theory is one of the millenium problems of the Clay institute:

https://en.wikipedia.org/wiki/Yang–Mills_existence_and_mass_gap

#### atyy

There are rigourously constructed interacting relativistic QFTs in less than 1+1D and 2+1D. The question is open in 3+1D.

In rigourous interacting relativistic QFT, there are not particles in any fundamental sense. But in non-rigourous language that physicist use for interacting relativistic QFT, there are particles.

#### DrDu

I don't think this is mainly a question of mathematical rigor. The Wightman axioms are very physical.
If a reasonable physical QFT would exist, we should be able to write down its Hamiltonian.

#### Buzz Bloom

Gold Member
Hi DrDu and atyy:

Thank you both for your most recent posts. Just when I thought I had established for myself a satisfactory understanding of the relationships betweeen QM and QFT for my mental capacity to grasp, you have added some more concepts. I now feel prompted to ask additional questions.

I gather that QFTs have a close connection with particle physics, includng the Lie groups used to express field/particle properties. I get the impression that QM, specifically with its limitation rearding particle number conservation, doesn't have much or anything to do with particle physics. Is this correct?

The question is open in 3+1D
From this quote, atyy, I guess that QM applies OK to a 3D-space, 1D-time spacetime (as well as spacetimes with 1D and 2D spaces) , while currently QFTs fail on problems involving spacetime with 3D space. Is this correct?

Regards,
Buzz

#### atyy

I don't think this is mainly a question of mathematical rigor. The Wightman axioms are very physical.
If a reasonable physical QFT would exist, we should be able to write down its Hamiltonian.
What I meant was that there are QFTs in 2+1D that fulfill the Wightman axioms. From the informal point of view, physicists would understand these QFTs using a Fock space which is a "particle space", but from the rigourous point of view, the Hilbert space of these QFTs is not a Fock space in the physicists' sense.

#### naima

Gold Member
but from the rigourous point of view, the Hilbert space of these QFTs is not a Fock space in the physicists' sense.
What is missing for a rigorous Fock space in the cited models?

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