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I've been trying to understand the concept of locality in QFT, and I was reading this paper by Edward Witten, where he explains (on pg 13) that the state space cannot be factored into a tensor product of spaces corresponding to different regions in spacetime. If I'm understanding this correctly, this seems to suggest that we are unable to define a Hilbert space corresponding to the subset of the universe we are studying in a given example, unlike in ordinary quantum mechanics, as there is always entanglement between observables defined on our "system" and its "surroundings" (borrowing terminology from thermodynamics).

Is this an accurate statement? If so, then how are we able to use QFT to describe objects in the universe, without talking about the whole universe? I notice that we seem to have no trouble talking about interactions between particles in the LHC, without regard to what happens outside, seemingly using states in a space of the form $\mathcal{H}=\mathcal{H}_{\text{LHC}}\otimes\mathcal{H}_{\text{outside}}$—which would be prohibited, but I could be misunderstanding this. If what I described above is not accurate, then what did I get wrong, and what would a more accurate statement be?

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That is an accurate summary of the situation, yes: the Hilbert space cannot be split into a tensor product of system and environment, for the reasons that Witten explains. Briefly put, the reason that we can nevertheless use QFT to describe localized experiments is that QFTs satisfy the cluster decomposition principle. If we have an collection of incoming particles $\alpha$ and a collection of outgoing particles $\alpha'$ at the LHC and incoming and outgoing collections $\beta$ and $\beta'$ at the Tevatron, then cluster decomposition says that $$ \langle \alpha', \beta' | S | \alpha, \beta\rangle \to \langle \alpha' | S | \alpha\rangle\, \langle \beta' | S | \beta\rangle $$ in the limit where we treat the LHC and the Tevatron as infinitely distant. This means that the statistics observed at the LHC will be independent of whatever's going on at the Tevatron, even though we can't assign individual Hilbert spaces to the two colliders. Chapter 4 of Weinberg's The Quantum Theory of Fields and Chapter 6 of Duncan's The Conceptual Framework of Quantum Field Theory both discuss this issue in some more detail.

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  • $\begingroup$ So the circumstances of the LHC, the particles outside it, have small effect on the particles inside, eventhough particles "explore" all paths? $\endgroup$ May 17 at 9:54
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    $\begingroup$ @Felicia that's right, yeah. The effect is nonzero, but it gets very small quickly: the correlations die off exponentially for massive particles $\endgroup$ May 17 at 12:25
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How are we able to use quantum field theory to study systems?

The same question could also be asked about quantum mechanics, or even about classical mechanics.

For example, in the Newtonian mechanics of gravitation, every mass in the universe instantaneously influences any test mass you are studying. In principle, you should take into account every mass in the universe when you study the dynamics of a marble rolling down an incline plane. But, in practice you can ignore everything except the earth and the marble.

As another example, in the quantum mechanics of scattering, introductory treatments often describe the scattering of "plane waves." Plane waves have a wave function that looks like this: $$ Ae^{ikx}\;, $$ which extends over the entire universe and in fact has a probability of |A|^2 to be found anywhere in the universe (which is clearly not normalizable). In this case, the introductory treatments often elide the fact that the true scattering states need to be described by wave packets, which are localized wave functions with relatively precise positions and momenta. These localized wave still take on a value at every point in the universe, but they fall off to zero very quickly away from the locations of the particles being studied.

In quantum field theory, we demote the position from an operator to a parameter, and we promote the fields to operators. You might see the Dirac field operator describing the electron/positron field dynamics written like: $$ \hat\psi(x)\;, $$ where the field takes on a value at every point in the universe. But, as with the above examples, in any scattering experiment, such as at LHC, the field description of the colliding particles involves localization.

To bolster this argument with an authoritative reference, we can look at Schwartz's "Quantum Field Theory and The Standard Model" at page 97, where he states:

"Constructing local theories out of fields made from creation and annihilation operators guarantees cluster decomposition, as we have seen."

This means, for example, when you compute the scattering S-matrix you only need to consider "connected" diagrams (connecting your in and out scattering states), and you don't need to consider interference from diagrams related to scattering taking place on the other side of the country.

However, Schwartz does provide the caveat:

"It is also not clear how well cluster decomposition has been tested experimentally."

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  • $\begingroup$ I don't think your example with Newtonian gravity is really analogous to the situation with QFT, because in that case, you only need to know the value of the gravitational field around the object you're studying (which can be determined without reference to anything distant). In QFT, we don't have an analogous way of describing the state of some region. $\endgroup$
    – Sandejo
    May 19 at 4:44
  • $\begingroup$ It's not directly analogous, but it is related to the question of how can we study one subsystem without knowing everything about all other subsystems, even those very far away. $\endgroup$
    – hft
    May 19 at 4:56

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