# Is quantum field theory a field theory of quantum mechanics or a quantum theory of fields?

Quantum field theory can describe and extend phenomena of classical fields, such as electromagnetism. I had assumed for a long time that it was itself a "field theory", by which I mean it is a set of rules for the evolution of 'states' with the states being fields in the physics sense, ie. some function from a point in space to a value in a domain that is fixed for the theory. However the Fock space formalism seems to strongly suggest against this, with states being expressed as amplitudes relating to configurations of particles (in position representation). Particle configurations being nonlocal (and this being the essence of entanglement as I understand) it seems very distant from any description as a field in the sense I mean above.

In case what I want is not precise enough or clear enough from my description above to answer, I would like to know if quantum field theory can admit a description such that:

• The theory has a 'state' which is some kind of data for each point in a region of space of interest
• The phenomena at a point can be calculated from only the state data at that point (up to best knowable probabilities etc.)
• The evolution of state data at a point over time can be calculated from only state data in the spacetime neighborhood of that point, or where the point is near the boundary of the region of interest, with some extra boundary information
• If we choose to reconsider a scenario with an expanded region of interest, identical results are achieved as with the previous smaller region, and all the data within that region is the same, with all the information required for these calculations that was previously provided by the boundary information of the smaller region now being provided by the state data at the points where the boundary previously was. ie. there is no requirement for new information within the old region to represent additional (ie nonlocal) interactions now we have a larger region.

Note: Because there appears to have been some confusion; when I write 'state' I don't mean a quantum state ie. a ket, I mean a state in the general sense.

• Qmechanic don't you like my English spelling of "neighbourhood" :..( – user183966 Jul 27 '18 at 16:47
• "I thought we were the quantum theory of fields?" "Field theory of quantum mechanics!" "Whatever happened to the quantum theory of fields?" "He's over there." ... "Splitter!" – user93146 Jul 27 '18 at 16:51
• Apologies to Qmechanic it was in fact Daniel Sank who 'standardised' my spelling of neighbourhood ;) – user183966 Jul 27 '18 at 17:11
• It sounds like what you're looking for is the Algebraic QFT framework — also called the "local QFT" formalism. I'm far from an expert in it, but at least that will give you something to Google; you could probably do worse than reading this review of QFT in curved spacetimes, where the particle interpretation is a lot less natural. – Michael Seifert Jul 27 '18 at 17:34
• @MichaelSeifert Thank you for the references. Although sadly I had seen that page and despite already having a taster of category theory did not understand much of it. – user183966 Jul 27 '18 at 17:42

Is quantum field theory a field theory of quantum mechanics or a quantum theory of fields?

Yes.

The theory has a 'state'...

Yes, quantum field theory has states in the quantum mechanical sense. Sometimes this is a little bit glossed over (or just implied), for example, if there is a ground state expectation value and the ground state is the vacuum (no particle state). But, yes, there are states. There are excited states above the ground state, etc.

The phenomena at a point can be calculated from only the state data at that point (up to best knowable probabilities etc.)

This is not true even in "normal" quantum mechanics. You also need the hermitian operators that correspond to the measurable/observable physical quantity.

It's also not clear what you mean by "phenomena at a point..." and "state data at that point..."

This might also help aid your understanding of quantum field theory: In "normal" quantum mechanics the position (of a single particle) is an operator and the time is a simple parameter. Attempts were made (historically because of the influence of special relativity) to "promote" the time to an operator and so be able to treat it similarly to position. This did not work (for a variety of reasons). So, instead of promoting time to and operator, position was demoted to a simple parameter. The position and time parameters were used as the arguments of fields, and the field values are operators. This is quantum field theory; both position and time are simple parameters, but the fields are promoted to operators. I.e., we are looking at operator-valued fields. This did work, and in fact, it is a very useful way to look at multi-particle systems.

• Thank you for your answer. Unfortunately I don't think I've made my question clear enough. I am aware that quantum mechanics has a 'state' in a quantum mechanical (and therefore time-local) sense. What I would like to know is whether that state in QFT can be expressed as as a "field in the physics sense" ie a function of position, which is what I was trying to make more formal with my descriptions of phenomena and state data at each point. – user183966 Jul 27 '18 at 16:52
• Yes, if you have a multi-particle state (ket, e.g., $|\Psi>$) and, for example, you don't mind projecting out the N-particle component of it, you can take the inner product of the state with eigenstates of the position operator $|x_1> x_2>...|x_N>$ and you will get the "usual" N-particle wave function $\Psi(x_1, x_2,...,x_N)$ – hft Jul 27 '18 at 16:56
• Regarding "This is not true even in "normal" quantum mechanics.", for traditional quantum mechanics I would say that it is true for position observation. I believe it is also true for time evolution as this can be expressed as a differential ie. local equation. For momentum it is not true, and that probably requires some more clarity in my question, maybe to define 'position local observables' using wave packets or something similar which I don't think I am sophisticated enough to do at the moment sadly. – user183966 Jul 27 '18 at 17:00
• perhaps another way to answer my question then is whether it is possible to reformulate the N-particle wavefunction Ψ(x1, x2 ,..., xN) as a field Ψ(x, y, z) – user183966 Jul 27 '18 at 17:07
• No. You can reformulate N-particle quantum mechanics in terms of a functional (not function) of a function with fewer arguments. For example, the N-particle wave function is considered to be a functional $\Psi[\rho]$ of the density $\rho(x,y,z)$ in density functional theory (developed by Walter Kohn et al.) – hft Jul 27 '18 at 17:29

Following E. Fermi, I can say that QFT is a QM description of specific (compound, complex, whatever) systems in terms of quasi-particle occupation numbers of the system.

• ...which RPF vulgarized as "QFT is an infinite bunch of quantum oscillators, packaged right". – Cosmas Zachos Jul 27 '18 at 21:59