Relationships between QM and QFT Particles

[Buzz Bloom]

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]

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]

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]

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?