# From discrete to continuous - why quantum *fields*?

From what I understand, combining special relativity with quantum mechanics requires quantum field theory (QFT). Why fields?

I can see that for quantities already described by classical fields, such as the electromagnetic field, the quantum analogue will be some sort of field, but what is the motivation for describing particles in terms of excitations of underlying quantum fields?

### A possible line of reasoning

Perhaps we wish to describe the creation and annihilation of particles, and perhaps this can be done through introducing creation and annihilation operators that act on the vacuum state. Such operators should act at each point in spacetime since particles can be created at each and every point in spacetime. Furthermore, such particles should be able to propagate continuously through spacetime and so the object that describes them should "operate" at each point in spacetime. Hence we should consider a field of operators, a so called quantum field?!

• A good answer to this question should probably clarify whether or not there is a sensible notion of a relativistic quantum theory without fields. Note in particular that non-relativistic QFT is perfectly reasonable and in fact used all the time e.g. in condensed matter physics. – DanielSank Oct 19 '16 at 10:01
• Related: physics.stackexchange.com/q/122570/2451 and links therein. – Qmechanic Oct 19 '16 at 10:14
• @DanielSank Yes, if someone could provide an answer explaining the need for fields, as opposed to other approaches that would really help me. From the texts that I've read so far I don't feel I've really grasped the reason why. Does the need for fields simply arise from so-called "second quantisation", where one simply counts the number of indistinguishable particles in each quantum state, and considers particles as excitations of some ground state?! – user35305 Oct 19 '16 at 10:28

I think to get a real satisfying answer to this question, you should read parts of Weinberg, vol. I, especially chapter 4, where at the beginning he writes

[I]f we express the Hamiltonian as a sum of products of creation and annihilation operators, with suitable non-singular coefficients, then the $S$-matrix will automatically satisfy a crucial physical requirement, the cluster decomposition principle, which says in effect that distant experiments yield uncorrelated results. [...] In relativistic quantum theories, the cluster decomposition principle plays a crucial part in making field theory inevitable. There have been many attempts to formulate a relativistically invariant theory that would not be a local field theory, and it is indeed possible to construct theories that are not field theories and yet yield a Lorentz-invariant $S$-matrix for two-particle scattering, but such efforts have always run into trouble in sectors with more than two particles: either the three-particle $S$-matrix is not Lorentz-invariant, or else it violates the cluster decomposition principle.

The rest of chapter 4 is about supporting these statements and is of course far too long to repeat here.

Weinberg also talks about this in "What is quantum field theory, and what did we think it is?"

• So quantum fields are inevitable if one wishes to construct a quantum theory that satisfies the principles of special relativity (in particular Lorentz invariance) and the cluster decomposition principle, then?! I think what has confused me is the notes that I've read so fair (Srednicki, Zee and the like) do not discuss this in detail, and in fact proceed to constructing scalar quantum fields without much motivation for why one should use fields. – user35305 Oct 19 '16 at 10:59
• ...A scalar quantum field is then constructed as a solution to the Klein-Gordon equation, in which they first solve it classically via representing the classical field in terms of its Fourier transform and then subsequently.promoting the Fourier coefficients to operators – user35305 Oct 19 '16 at 11:01
• Yes, I think Weinberg is the only major textbook that gives this motivation in detail. In the linked talk, he says "When you teach any branch of physics you must motivate the formalism — it isn’t any good just to present the formalism and say that it agrees with experiment — you have to explain to the students why this the way the world is. [...] In the course of teaching quantum field theory, I developed a rationale for it, which very briefly is that it is the only way of satisfying the principles of Lorentz invariance plus quantum mechanics plus [the cluster decomposition] principle." – Robin Ekman Oct 19 '16 at 11:02
• He then goes on to say that, however, that his argument isn't entirely rigorous and that this probably isn't the final word: "Now, all of these caveats really work only against the idea that the final theory of nature is a quantum field theory. They leave open the view, which is in fact the point of view of my book, that although you can not argue that relativity plus quantum mechanics plus cluster decomposition necessarily leads only to quantum field theory, it is very likely that any quantum theory that at sufficiently low energy and large distances looks – Robin Ekman Oct 19 '16 at 11:06
• @RobinEkman that's a lot of comment. Stack Exchange comments are meant to be potentially short-lived notes. Could you edit the main post to contain the important information? – DanielSank Oct 19 '16 at 12:28

There is a reason other than the one you cited for picking fields. In QFT a particle is just a state in which the field has a particle number of 1 and n particles are just a field state in which the field has particle number n. In general, there is no such thing as the same particle over time, just changes in the patterns of particle numbers (and other field observables) in fields that propagate locally.

Suppose you have a relativistic quantum theory in which each particle was distinguishable from the others. The state of any such particle would in general spread outside the light cone. So if you measured the particle and found it in region x, it would not be found in region y that is spacelike separated from x. The physicists who found this out found it unacceptable and thought it might allow FTL communication. There are other problems solved by QFT: see Part III of 'Quantum field theory for the gifted amateur' by Lancaster and Blundell.