# What is a fundamental particle according to QFT?

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1. Sep 19, 2015

### ajv

In quantum field theory, a fundamental particle is an excitation in the underlying field, but what does that mean? Do fundamental particles have any physical existence according to QFT?

2. Sep 19, 2015

Staff Emeritus
Of course they have a physical existence. You're made of them. We can count them. We can move them around. What more do you want?

3. Sep 19, 2015

### ajv

Ok, so according to QFT all fundamental particles are quanta of the underlying field. What is a "quanta"? Is it energy?

4. Sep 19, 2015

### Staff: Mentor

QFT is more advanced than QM. QM cant really be expressed except using math, even though many popularisations attempt it with varying degrees of success, and also a lot of misconceptions. QFT is much much worse. As one mentor here says everything you have read about QFT outside a QFT textbook is likely wrong - and that even includes well respected non QFT textbooks - it really is that bad at the lay level.

That said, and running into that issue, its an extension of the harmonic oscillator of QM:
https://en.wikipedia.org/wiki/Quantum_harmonic_oscillator

Mathematically it can be described using creation and annihilation operators as well as the number operator. The interesting thing here is that its actually possible to formulate QM in terms of creation and annihilation operators - it one of a number of different formulations of QM:
http://susanka.org/HSforQM/[Styer]_Nine_Formulations_of_Quantum_Mechanics.pdf

Now heuristically here is what's going on. You write down the equations of a quantum field analogously to a classical field. You do a trick called a Fourier transform and low and behold the Fourier components are mathematically the same as the harmonic oscillator, hence you have creation and annihilation operators. Mathematically its exactly the same as the creation and annihilation formulation of QM. That's how particles come about in QFT.

As you can see its not trivial, but really its the best I can do to explain this advanced area in basic terms.

If you want to study the detail recently some good books have started to appear that can be tackled, admittedly with effort, after a basic course in QM

Sussliknds book is good enough preparation:
https://www.amazon.com/Quantum-Mechanics-The-Theoretical-Minimum/dp/0465062903

While doable, be warned it will take time and attention to detail. But at the end you will have an understanding way beyond the lay level.

Thanks
Bill

Last edited: Sep 21, 2015
5. Sep 20, 2015

### ajv

Pretend we have a super powerful microscope that could see infinitely small things.
If we zoomed in to an electron with an infinite magnification, according to QFT, what would we see? Would we see absolutely nothing?

6. Sep 20, 2015

### Staff: Mentor

According to QM such does not exist so its a meaningless question like what would happen if an immovable object meets one that's unstoppable. Specifically you run into the uncertainty principle. Heuristically if you observe a quantum particle at a specific position it has a totally unknown momentum and will scoot off elsewhere.

In QFT the very existence of particles is nebulous - we don't even know the number of particles there is little alone the characteristics of a single one:
https://en.wikipedia.org/wiki/Fock_space

Like I said its way way beyond lay level explanations like you are trying to pin it down to.

Thanks
Bill

Last edited: Sep 20, 2015
7. Sep 20, 2015

### vanhees71

In relativistic quantum theory, it's even worse than in nonrelativistic. Trying to pin down the position of a particle needs a lot of energy to confine it in some small spatial volume. The smaller you want this volume, i.e., the more accurate you try to determine ("prepare") the particle's position, the more energy you need, but then in relativistic quantum theory it turns out that there are no models for interacting particles with conserved particle numbers but only certain charge-like quantities (like electric charge or baryon number). So nothing prevents reactions of creating and/or destroying particles. E.g., if you want to determine the position of an electron precisely you could shoot a photon on and look at its scattering. Now to resolve the electron's position you need very energetic photons, and if the photon's energy exceeds the energy $2 m_e c^2 \simeq 1 \; \text{MeV}$ it can create another electron and a positron. Then you don't gain more precision in the determination of the original electron's position.

For a more detailed treatment of this quite simple argument, which can be made semi-quantitative, using the Heisenberg-Robertson uncertainty relation and is due to Landau and Peierls (1931; the paper is in German), see Landau, Lifshitz, vol. 4. The original paper is

L. D. Landau, R. Peierls, Erweiterung des Unbestimmtheitsprinzips für die relativistische Quantentheorie, Zeitschr. f. Physik 69, 56 (1931)
http://dx.doi.org/10.1007/BF01391513

8. Sep 21, 2015

### stevendaryl

Staff Emeritus
9. Sep 21, 2015

### atyy

Your original question asked about "existence" which is a tricky question, since in the standard interpretation quantum theory is not about what "is" but what we can say about "nature".

However, this question about the powerful microscope is much better, since it asks about measurement results. In our current understanding, quantum electrodynamics is generally believed to be not a fundamental theory, because of the "Landau pole". This belief is not entirely justified, so something like "asymptotic safety" may be possible.

10. Sep 21, 2015

### Lord Crc

FWIW it was and is Amazon for me, both text and link. Maybe time todo some scans?

11. Sep 23, 2015

### stevendaryl

Staff Emeritus
Very weird. I just did a malware scan, and it didn't find anything. But apparently there is something running on my browser that substitutes URLs to give preference to Walmart over Amazon. The book is available in both places.

12. Sep 23, 2015

### vanhees71

I've the strange behavior of my chrome browser, that sometimes I can't get to arxiv.org. It works always for de.arxiv.org and also always with firefox and konqueror; very strange too!

13. Sep 24, 2015

### fxdung

Elementary particles are ''wave packet'' of field or quantum of field,so they must have the size(the size of packet).But we know that elementary particles are point partcles.Why they have zero size?

14. Sep 24, 2015

### Staff: Mentor

That does not follow.

Thanks
Bill

15. Sep 24, 2015

### wotanub

I have a different answer than what has been offered so far. I think that explaining what a particle is a very subtle question. When I'm asked I usually say something like it is an excitation of the quantum field as you said in the OP, but if you have a more mathematical background, I'd say a particles are irreducible representations of the Poincaré group, à la Winger.

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

I don't know how helpful this link is but take a look, and if you know some algebra, I'd take a look at the book Relativistic Quantum Physics by Ohlsson. He walks the reader through Winger's classification early on in the book.

16. Sep 25, 2015

### fxdung

Is there any concept relationship between quantum field and wave function of particle?

Last edited: Sep 25, 2015
17. Sep 25, 2015

### DrDu

As far as I understood, fundamental particles are rather an auxiliary construction than a fundamental physical entity. The point is that all we observe are interacting particles and these have virtually nothing in common with hypothetical free non-interacting particles. We do not even know whether a perturbational description of the interacting reality starting from some non-interacting particles is unique in the sense that we can also do it in terms of another set of non-interacting particles. All we can say is that there are some quantized charges, like electric, weak or strong.

18. Sep 25, 2015

### vanhees71

This is a subtle question. First of all one has to emphasize that the short answer to the question in the subject line is that the theoretical definition of an elementary particle is that it can be described by a local microcausal quantum field theory that induces an irreducible representation of the proper orthochronous Poincare group (see also #15).

This is a strict definition from a theoretical point of view, but physically it's quite useless. From an empirical point of view, using the abstract definition, it depends on the collision energies of the particles you look at. At low energies, e.g., a proton simply looks as if it were an elementary particle. At very low energies, in atomic physics, it's even accurate enough for a quite good description to just treat it non-relativistic and treat it as a point-particle electric Coulomb field, leading to the binding of an electron to form a hydrogen atom. At higher energies, you'll resolve a bit more of the structure of the proton and realize that it is not elementary, but that it has a non-trivial charge distribution. Then you introduce form factors. At very high energies ("deep inelastic electron scattering"), you won't find this accurate enough anymore, but you find socalled scaling relations, leading to the conclusion that a proton consists of three quarks. With today's awailable energies in particle colliders, there's no hint yet that the quarks are not elementary. So we assume that they are, but it's not said that one day one finds that also the quarks are composed of other constituents, yet not known.

Now to your question about the wave function. A wave-function description is always applicable, when you deal with a problem of fixed particle number, i.e., when the interactions involved do not create or destroy particles. This is the case for low energies, and then you can often treat the problem in the non-relativistic approximation of a potential problem like in atomic physics. Of course, you can make stystematic improvements of the non-relativistic approximation to take into account relativistic effects as perturbations and also radiative corrections, leading to phenomena like the Lamb shift of spectral lines, which is one of the most accurate results in particle physics (QED).

In the relativistic realm, i.e., at high energies, the description of particles with wave functions is quite difficult and not very consistent, because one can create and destroy particles, e.g., an electron and a positron can annihilate to two photons or an electron scattering at a nucleus irradiates a photon (bremsstrahlung) or additional electron-positron pairs (pair creation), etc. This is why the most natural description of relativistic quantum theory is relativistic quantum field theory, and here the local microcausal ones provide the successful models.

This brings me back to the beginning of this posting: Now you have a very clear recipe to guess particle models. You start from Poincare invariance and look for the irreducible ray representations (which all are induced by unitary representations of the covering group) and pick out the local and microcausal models which have a ground state (i.e., the Hamiltonian is bounded from below). Than you take into account additional symmetries (e.g., the approximate chiral symmetry to describe hadrons) and so on.