Benefits of QM over QFT

[ydoaPs]

There are theories that we know are wrong, but we use them because they are good enough in certain (often most) circumstances and are easier to use. We know that GR is less wrong than Newtonian Mechanics, for instance, but we use Newtonian in most cases because the precision increase of going to GR is outweighed by the difficulty cost of going to GR in most cases.

 

 

On the smaller scale, we have QM vs QFT. They're both attempts to describe the nature of the fundamental constituents of matter and the ways in which they interact, but the latter is consistent with SR and and the former is not. Specifically, in single particle QM, there's a non-zero probability that a particle can travel outside of its light cone.

 

So, basically, why haven't we ditched QM altogether in favor of QFT? In what situations is it preferable use QM instead of QFT and why?

[studiot]

"Which is Professor Challenger's room?"

 

Down the corridor.

 

Down the corridor, on the left.

 

Down the corridor, third door on the left.

 

Down the corridor, room 15.

 

Each may be adequate or inadequate, depending upon circumstances.

[ajb]

Specifically, in single particle QM, there's a non-zero probability that a particle can travel outside of its light cone.

This would not be the case for relativistic quantum mechanics, though once you think about this you are better off thinking in terms of field theory anyway.

 

So, basically, why haven't we ditched QM altogether in favor of QFT? In what situations is it preferable use QM instead of QFT and why?

The reasons I see are as follows.

 

Mathematically standard non-relativistic quantum mechanics is well-posed. It reduces to studying operators on Hilbert spaces and there are no serious foundational questions. Relativistic quantum field theory is far less well-posed and lots of it need to be treated formally. That is we do not know if some parts of the theory mathematically exist, other than formally. This of course makes QFT very interesting.

 

From the physics point of view I think it is mostly a question of calculation. Quantum mechanics is easier to work with and is better suited to bound problems than the standard formulation of QFT. Moreover for a lot of systems relativistic corrections are going to be small, and could be ignored or calculated phenomenologically using perturbation theory.

[Sensei]

Where is solution to Schrodinger equations allowing to calculate Helium-4 spectral lines?

[ajb]

Hartree–Fock based methods can be applied to two electron systems quite accurately, others may be able to give more details here. The are other approximations that people use to get at the spectra of multi-electron atoms.

 

Anyway, you should not confuse the mathematical consistency of the framework with actually finding closed expressions for the things you want to calculate. You should view the two as related, but separate issues.

[ydoaPs]

Mathematically standard non-relativistic quantum mechanics is well-posed. It reduces to studying operators on Hilbert spaces and there are no serious foundational questions. Relativistic quantum field theory is far less well-posed and lots of it need to be treated formally. That is we do not know if some parts of the theory mathematically exist, other than formally. This of course makes QFT very interesting.

Can you go into more detail here? What doesn't exist mathematically in QFT? What's the difference between mathematical existence and formal existence?

[MigL]

I remember solving the Helium atom in 3rd or 4th year, using numerical methods.

Using Hollerith punch cards and a Burrows B6700 mainframe computer ( room sized ).

 

Only took about a week.

[Mordred]

Well considering how involved QM is involved in QFT, it's almost like describing the FLRW metric without relativity.

 

QFT is QM +relativity at its core, QM does an excellent job describing single probability interactions.

 

One example I can think of Is the double slit experiment. I for one don't recall ever seeing a QFT treatment to explain that. There probably is one. Just can't think of ever reading it lol

By the way this is an excellent fields resource book.

 

http://arxiv.org/abs/hepth/9912205: "Fields" - A free lengthy technical training manual on classical and quantum fields

[ajb]

Can you go into more detail here? What doesn't exist mathematically in QFT? What's the difference between mathematical existence and formal existence?

The biggest mathematical problem is that the path integral measure cannot be shown to exist, other than some very special cases. So the usual thing to do is just 'state that is does' and work with it. You know all the properties you want form it and so you just use it.

 

There are other questions about the convergence of the series expansions in perturbation theory and the mathematical status of renormalisation. Again, you can treat this rather formally and accept that your theory only works up to a certain energy scale. This is quite natural in physics, but mathematically not so great.