My understanding of the "particle" concept in Quantum Field Theory is that it describes something

  • infinite in extent in space (and also time?)
  • having no concept of trajectory (in absolutely any sense)
  • (possibly) effectively immutable or having no temporal extent, like mathematical concepts / global statements about a model / coordinates, or possibly a minimal existence, for example just binary existance or cardinality but certainly no properties at an individual particle level

In experimental physics, notably including areas dealing with the phenomena intimately tied to Q.F.T., there is a concept of a "particle" which follows conventional English usage, as nicely mentioned in an answer to this question.

  • localised in space
  • having a definite trajectory (under appropriate conditions)
  • having definite properties (up to some uncertainty) that vary with time (position, momentum, energy etc.)
  • observed in bubble tanks as a concrete example

I am not asking which of these definitions is "correct" usage of the word. I am also not trying to get a correct characterisation of either phenomenon; I'm sure I've misrepresented them in some ways, but that doesn't matter as long as you can understand what I'm referring to. What I'm interested in is do these correspond to the same thing.

I am aware of wave-particle duality in basic quantum mechanics but in this case it seems like people are simply talking about different things. Are these related the same way "fields" are in maths and physics (linguistically only)? Are they different perspectives of the same thing? Or are they maybe at opposite ends of a conceptual scale?



3 Answers 3


Your description of what a particle looks like in quantum field theory is not accurate, but you could be forgiven for getting that impression from a first course.

To recap, quantum field theory is a quantum mechanical theory describing an arbitrary number of particles. In order to start, we need to specify what the quantum states are. This is done by constructing the Fock space, which is roughly $$H = H_0 \oplus H_1 \oplus H_1^{\otimes 2} \oplus H_1^{\otimes 3} \oplus \ldots$$ where $H_0$ contains a single vacuum state, and $H_1$ contains the possible one-particle states, and $H_1^{\otimes 2}$ contains the possible two-particle states, and so on.

For concreteness, it's useful to pick a specific basis for the one-particle states. In the case of particle physics, we're often concerned with scattering experiments where particles come in from infinity with a very well-defined momentum, so we pick the momentum basis. In this basis, every particle is infinite in spatial extent. But in condensed matter physics, you can just stick wires into whatever solid you're studying, so you can do measurements in the position basis. Accordingly condensed matter field theory sometimes introduces $H_1$ in the position basis.

There's no fundamental difference, because we can always go back and forth by linear combinations. The other crucial thing is that, starting from momentum states, we can form finite wavepackets by superposing nearby momenta. Wavepackets can move, as shown here, and they have roughly defined trajectories. This is how we model the initial states in real scattering experiments, where our collider unfortunately has a finite size due to budget constraints.

To answer your other questions:

  • A particle can have plenty of other properties. For example, a particle can carry some spin or charge; this is all accounted for in $H_1$.
  • Particles evolve in time, even when they're in momentum eigenstates. For example, an unstable particle will decay. Or, if you have a state with an electron and positron in momentum eigenstates, they will annihilate.
  • In colloquial language the word 'particle' is reserved for states in $H_1$ that are reasonably localized in position, while 'wave-particle duality' is nothing more than the statement that some states in $H_1$ are localized and some are not. When doing quantum field theory, we just call everything in $H_1$ a particle; there is no need to split it up.
  • $\begingroup$ Thank you very much. To summarise your answer extremely crudely then, particles in QFT are rather similar to those in QM. I take it from this that there is nothing special then about the states on which a and a dagger operate and they themselves are not uniquely special particle creation operators. By the way, I have not seen it put that wave-particle duality is the term for fields having localised and non localised states before, I find that perspective very useful. $\endgroup$
    – user183966
    Commented Feb 23, 2018 at 23:30
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    $\begingroup$ @user183966 I'm glad it helped! To clarify more, QFT is not merely similar to quantum mechanics, it is a special case of quantum mechanics. It just happens to be such a complicated special case it deserves its own name. $\endgroup$
    – knzhou
    Commented Feb 23, 2018 at 23:32
  • $\begingroup$ @knzhou in what sense is QFT a special case of QM? Do you define QM to be $H\psi = i \partial_t \psi$, where $H$ has whatever definition and $\psi$ has whatever definition? $\endgroup$
    – Gonenc
    Commented Feb 26, 2018 at 23:26
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    $\begingroup$ @GonencMogol Yes, I define a QM theory to be one with a Hilbert space and an operator which generates time translations. $\endgroup$
    – knzhou
    Commented Feb 27, 2018 at 0:22

It is true that what we measure in particle detectors aren't the states you usually use to do computations in QFT. The place where your second, more intuitive notion of a localized particle pops up in the theory is when you define the S-matrix, which relates the in and out states of a scattering event. These are defined as well-separated wave-packets respectively far in the past and far in the future.

It turns out you can relate the two pictures -the theoretically convenient and the experimentally relevant- by means of the LSZ formula.

  • $\begingroup$ Thanks for the related concepts. Always helpful to have some leads for further research. $\endgroup$
    – user183966
    Commented Feb 23, 2018 at 23:43
  • $\begingroup$ You're welcome. One of the many places where you could do further reading on those concepts is in the chapters 4 and 7 of the Peskin & Schroeder book on QFT. $\endgroup$
    – Antoine
    Commented Feb 23, 2018 at 23:51

Your description of the notion of a particle in QFT is a bit off. What you're describing is a QFT particle in a momentum eigenstate. QFT can also describe particles with less degenerate wave-functions. These are more easily localized and correspond to what you label the experimental physics notion.

  • $\begingroup$ Thank you very much for the answer. I put the question in this way because every single description of QFT I have encountered seems to start with ladder operators, state that these add quanta of excitation to a field, and then define these quanta as particles. It isn't stated explicitly this is not the only instance of a particle, but if feels to me, perhaps wrongly, that this is hinted. Also I have also noted that often when people are asked questions about particles in QFT they say not to think like this. I guess you can't speak for them. $\endgroup$
    – user183966
    Commented Feb 23, 2018 at 22:53

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