# I Particle number conservation and motivations for QFT

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1. Jul 13, 2016

### Frank Castle

I've read that one of the primary motivations for the need for QFT is that quantum mechanics cannot account for particle creation/annihilation, however special relativity "predicts" that such phenomena are possible (clearly they have been observed experimentally, but I'm going for a heuristic explanation for motivating QFT here). Hence, if one wishes to construct a fully consistent relativistic quantum theory, one must account for such phenomena, which naturally leads to the construction of QFT.

If this is correct, then there are a couple of things that I'm slightly unsure about.

Firstly, is the reason why quantum mechanics cannot treat scenarios in which there are a variable number of particles because of the the continuity equation: $$\frac{\partial}{\partial t}(\langle\psi\vert\psi\rangle) +\nabla\cdot\mathbb{j}=0$$ which implies conservation of particle number (I think it's correct to say that it is derived by assuming that no particles can be created or annihilated)?

Secondly, in what sense does special relativity "predict" that particle creation/annihilation should be possible? Is it implied by the energy dispersion relation $$E^2=m^2c^4+p^2c^2$$ so if the system has enough energy it can create a particle of mass $m$ from the vacuum?! Does it also come from Dirac's construction of a relativistic equation of motion (the Dirac equation) which predicts the existence of the positron which itself can annihilate with the electron?

2. Jul 13, 2016

### malawi_glenn

is a nice prelude to QFT

Furthermore, time and space are not treated on equal "footing" in non-rel QM. Time is not an operator in QM, it is just a label

If you give heuristically arguments for QFT, then the answers must be heuristic aswell :)

3. Jul 13, 2016

### Frank Castle

I get the reasoning from the point of treating space and time "on equal footing" a la relativity. What I don't fully understand is the reasoning from issues such as particle number (non) conservation, creation/annihilation of particles. (I don't mind the inclusion of mathematics in the explanation by the way )

4. Jul 13, 2016

### malawi_glenn

People tried to reformulate QM in terms of relativistic wave-equations, the Klein Equation and the Dirac Equation.

The paradox I linked to is related to the Klein Equation, which shows that probability is not conserved. Furthermore, you can examine the causality of the solutions to the Klein Equation and see that they do not respect causality (you have non-zero probability density outside the light-cone)

Here is a link to a "paper" I wrote many years ago in a course i took on relativistic quantum mechanics. It contains a mathematical treatment of the Klein Paradox
https://www.dropbox.com/s/ajoohwff2xvr8vf/RelativisticQM_handin.pdf?dl=0

The first chapters in Srednickis QFT book is also quite nice. Here you can find a draft of the book on his homepage http://web.physics.ucsb.edu/~mark/ms-qft-DRAFT.pdf

5. Jul 13, 2016

### Frank Castle

Ah, ok. Thanks for the references, I'll take a look and get back if I have further questions (if that's ok?!)

6. Jul 13, 2016

### malawi_glenn

Sure post here and more people can contribute and benefit :)

7. Jul 14, 2016

### Frank Castle

I have since had a chance to read through the first few chapters of Srednicki's QFT book, and whilst enlightening, it still doesn't provide a satisfactory explanation as to why QM+SR implies that particle number is not conserved (is this because of the mass-energy equivalence in SR and so if there is sufficient energy then particles can be created, and conversely, mass can be converted into energy implying particle destruction), and also why one can't use QM to describe systems in which there are a variable number of particles (is it because in QM we describe multi-particle states in terms of products of single-particle states and if the system has varying numbers of particles then it is not possible to do this?)

8. Jul 14, 2016

### hilbert2

Isn't it possible to have a nonrelativistic quantum field describing crystal vibrations, where phonon number isn't conserved?

9. Jul 14, 2016

### Frank Castle

From what I've read, there is no mechanism in standard non-relativistic quantum mechanics to deal with changes in the particle number. Equations in quantum mechanics are always written for fixed numbers of particles (Indeed, the continuity equation implies that particle number is conserved).

10. Jul 14, 2016

### hilbert2

^ Yes, we can't define a wave function unless we know how many variables that function depends on, and the number of variables in 3D space is three times the number of particles (ignoring spin variables). But I still don't think that you necessarily need a relativistic system to have a non-constant number of quasiparticles such as phonons.

http://eduardo.physics.illinois.edu/phys582/582-chapter6.pdf

11. Jul 14, 2016

### Frank Castle

Is the point here though that one has to deviate from standard quantum mechanics (in which products of single-particle Hilbert spaces are used to construct multi-particle wavefunctions) and introduce the notion of a Fock space to account for the changing number of particles?

12. Jul 14, 2016

### hilbert2

^ Yes, you need the Fock space to do that.

13. Jul 14, 2016

### Staff: Mentor

Quantum field theories allow for a variable number of particles which makes them a good starting point for a relativistic theory, but they don't have to be relativistic. There's a good example, the field-theoretical treatment of an array of coupled harmonic oscillators, towards the beginning of Lancaster's "Quantum field theory for the gifted amateur". You may find that this book is a better starting point than Srednicki.

14. Jul 14, 2016

### Frank Castle

Thanks for the recommendation, I shall take a look.

Does one need field theory, whether relativistic or not, then to describe cases in which the particle number varies?

Is the point also that in single-particle quantum mechanics is not compatible with relativity since it violates causality (there is a small, but non-zero amplitude for the particle to exist outside its forward light-cone)?! (Also, the energy-uncertainty relation along with the relativistic energy-dispersion relation implies that pair production can occur even in the case where there was originally one particle present)

Last edited: Jul 14, 2016
15. Jul 14, 2016

### vanhees71

Of course, you can as well formulate nonrelativistic QFT, and that's what condensed-matter physicists do nowadays all the time. Phonons are the vibrations of the crystal lattice, and in the QFT description they appear as particle-like excitations. That's why they are also called quasiparticles. There are tons of interesting quasiparticles in condensed-matter theory, including very exotic stuff like magnetic monopoles, Weyl fermions, etc.

16. Jul 14, 2016

### stevendaryl

Staff Emeritus
I am not sure if I understand, at a deep level, why relativity would imply the possibility of particle non-conservation, but treating space and time on an equal footing intuitively makes it compelling to me.

The following diagrams aren't seriously meant to be Feynman diagrams--they're just visual illustrations of processes involving particles. On the picture on the left, we have a pretty unremarkable process: An electron (marked by the solid line) is hit by a photon (marked by the wavy line) and is deflected. In the diagram, time is vertical and the one spatial dimension is the horizontal axis.

Now the diagram on the right is the same process, "rotated" in spacetime. In this diagram, the same process is described by: Originally, (in the bottom of the diagram), there is only a photon. At some point, the photon produces an electron-positron pair. So pair production and deflection of an electron by a photon are closely related processes. (In actuality, these simple processes have zero amplitude, because there is no way to get energy/momentum to balance for a single photon to split into an electron/positron pair.)

17. Jul 14, 2016

### stevendaryl

Staff Emeritus
I think that there is a sense in which many-particle quantum mechanics simply is field theory, if you drop the restriction on particle conservation.

18. Jul 14, 2016

### Frank Castle

So is the point that, in spacetime it is quite natural, in some sense, for particle production to occur and such a phenomenon cannot be described in standard quantum mechanics?

Is this purely because one introduces the occupation number representation of quantum states and also the notion of a Fock space to describe a varying number of particles?

19. Jul 18, 2016

### malawi_glenn

20. Jul 18, 2016

### Staff: Mentor

Combining QM and relativity inevitably leads to particle number not being fixed.

See section 8.3 of the following book where its carefully explained:
https://www.dur.ac.uk/physics/qftgabook/

Once that is recognized you are naturally led to QM formulation F of the equivalent formulations of QM: