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This is just a terminological question, not a question about reality or mathematics.

I often want to talk about state spaces in quantum field theory. For example the space of [all possible vector states in] a free scalar quantum field.

I have been told in a comment on my other question this this object is not called a "quantum field", because a "quantum field" is an operator field (or a space of operator fields). I know an operator is a kind of mapping, and takes an input. The entity I want to be able to talk about is not a mapping, it is like a vector (or it is a vector), it just exists and does not act on something else. What is the standard name for it?

Edits:

I hope this is quite a clear example: I may want to talk about the 'state of photons in the universe'. I have been told this cannot be called a quantum field, because the quantum field is an operator not a state. So I presume this cannot be called the photon field or similar? Obviously it is not a quantum field theory either because it is not a theory, it is physical. So I don't know what to call it. I have never seen a phrase like "state of photons" or "space of photon states" in use.

I think it is fair to say what I am looking for is a term that means "Hilbert space equipped with a quantum field theory interpretation" (or physical entity represented by it) based on the helpful comments and answers.

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    $\begingroup$ It's just a Hilbert space. Like in undergrad QM. $\endgroup$
    – knzhou
    Jul 29, 2018 at 10:49
  • $\begingroup$ @knzhou Thank you. So if I want to say "Hilbert space of a particle field" or "Hilbert space of a quantum field", there is no term for this? I just have to write that out in full? Also if I talk about an alternate theory where the state object does not actually have Hilbert structure, I will have to write "Object equivalent to the Hilbert space of the particle field if it were standard QFT"? $\endgroup$
    – user183966
    Jul 29, 2018 at 10:53
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    $\begingroup$ No offense, but I think you should learn standard quantum mechanics, out of a good undergraduate book, before trying to tackle QFT. You've asked a lot of QFT questions that indicate misconceptions in basic QM, and if you try to continue this way progress will be basically impossible. $\endgroup$
    – knzhou
    Jul 29, 2018 at 10:59
  • $\begingroup$ @knzhou I don't take any offence. I'm sorry for my badly founded questions. I have done a 4 year masters degree in maths & physics, specialising in particle theory, at a good UK university that specialises in particle theory, getting a fairly good grade. My problem is I guess that I don't know where my misconceptions lie. Honestly I suspect incorrect terminology is a large part of the confusion. $\endgroup$
    – user183966
    Jul 29, 2018 at 11:12
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    $\begingroup$ After I finished 4 year degree with several classes in quantum field theory, I realized I had misunderstood a lot of it, tossed out my notes, and started over from the basics. (By which I mean, I took the whole sequence_ again_ at another university.) The fact is that QFT is a big big jump up from undergrad physics, there are a lot of moving parts, and it's easy to walk away with almost no conceptual understanding despite having the ability to calculate cross sections. In particular, every conceptual error you had lurking from QM will come to haunt you tenfold when you try QFT. $\endgroup$
    – knzhou
    Jul 29, 2018 at 11:18

2 Answers 2

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The space of states of quantum field theory is a Hilbert space (or, if you want a space where every element is really a different state, the corresponding projective Hilbert space, since vectors that just differ by scalar multiplication represent the same state) just like in ordinary quantum mechanics.

Just like the classical observables of position and momentum get promoted to operators in quantum mechanics that act on such a space of states, quantum field theory promotes the classical fields of a field theory (e.g. electromagnetism) to operators acting on a space of states. The field is not the state, just like the position operator is not the state.

There is no such thing as a "space of states of a quantum field". The quantum field is the operator, not the state. There is a space of states of a quantum field theory, it is determined by all the fields of the theory and their interactions, not just by a single field, exactly how in non-field quantum mechanics there is no "space of states of position", just the single space of states both position and momentum act on.

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  • $\begingroup$ Thank you for trying to help me. I think this has made the terminology clearer, I have updated the question title to hopefully be more accurate. So when we say "field" we mean a value, like position, a "quantum field" is an operator/collection of operators, like like the position operator. So my question would be whether there is a standard name for a Hilbert space that has an interpretation as a state space for a quantum field. ie. a term that includes the Hilbert spaces of the electron quantum field theory, the photon quantum field theory, a scalar quantum field theory, etc. $\endgroup$
    – user183966
    Jul 29, 2018 at 11:27
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    $\begingroup$ you mean a projective Hilbert space? $\endgroup$ Jul 29, 2018 at 11:35
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    $\begingroup$ @user183966 I don't really understand what word you're looking for. What about "space of states" is insufficient? $\endgroup$
    – ACuriousMind
    Jul 29, 2018 at 11:42
  • $\begingroup$ @DavidBarMoshe Good point, included (although many do call the ordinary Hilbert space already the space of states). $\endgroup$
    – ACuriousMind
    Jul 29, 2018 at 11:43
  • $\begingroup$ @ACuriousMind "space of states" is completely correct, it's just not field theory specific. I have been using incorrect terminology for this concept, so I wanted to use correct terminology if it exists. Like I would rather not say "set" if I could say "magma" in abstract algebra. But it sounds like it doesn't. To distinguish a quantum field and a quantum field theory in your answer was already very helpful though. $\endgroup$
    – user183966
    Jul 29, 2018 at 11:55
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I think this is a good question because it goes to the heart of something rather confusing in the terminology: the fact that the words 'quantum field' are attached to an operator-valued quantity such as $\hat{\phi}$ and yet people also say that the vacuum state such as $|0\rangle$ or $|\Omega\rangle$ is a state 'of the field(s)'. But as answer from ACuriousMind points out, an operator is not itself a state, nor does it 'have a state', but rather it acts on states in some space (here, Hilbert space).

But now the question arises, what is a state such as $|\Omega\rangle$ a state of? The standard terminology is to say it is a state of a field, or of a collection of fields. So the standard terminology is inconsistent and confusing.

I think a good way around this is to refer to an operator such as $\hat{\phi}$ or $\hat{\bf E}$ not as 'the field' but as 'the field amplitude'. Then you can say things like 'the field amplitude acting on the field state' and this form of words maps unambiguously onto the mathematical quantity $$ \hat{\phi} | \psi \rangle. $$ (This could be compared to a quantity such as $\hat{p} |\psi\rangle$ in ordinary quantum theory, which can be put into words as 'the particle momentum acting on the particle state', only of course we normally say 'the momentum operator acting on the state').

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    $\begingroup$ I totally agree, and in fact there's another layer of confusion, which is that people often make no distinction between a quantum field and the particles it creates (e.g. they're denoted by the same letter). From my experience learning and teaching QFT, less than 10% of first-time students exit a QFT I course actually understanding the difference between a one-particle state, a field eigenstate, and a field operator. $\endgroup$
    – knzhou
    Nov 15, 2022 at 23:53
  • $\begingroup$ +1 If we're being consistent with the terminology, a "quantum field" should be the name of the physical system we're analysing, like "quantum harmonic oscillator" $\endgroup$
    – Ryder Rude
    Nov 16, 2022 at 6:32

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