Last week I saw an excellent talk by philosopher Wayne Myrvold.

(Download MP4 video here.)

The topic was well-defined, and of reasonable scope. The theorem is easily and commonly misunderstood. And Wayne’s talk served to dissolve the confusion around it, by unpacking the theorem into a handful of pieces so that you could quickly see where the rub was. I would that all philosophy of physics were so well done.

Here are the key points as I saw them:

- The vacuum state in QFTs, even non-interacting ones, is entangled over arbitrary distances (albeit by exponentially small amounts). You can think of this as every two space-like separated regions of spacetime sharing extremely diluted Bell pairs.
- Likewise, by virtue of its non-local nature, the vacuum contains non-zero (but stupendously tiny) overlap with all localized states. If you were able to perform a “Taj-Mahal measurement” on a region
*R*, which ask the Yes-or-No question “Is there a Taj Mahal in*R*?”, you always have a non-zero (but stupendously tiny) chance of getting “Yes” and finding a Taj Mahal. - This non-locality arises directly from requiring the
*exact*spectral condition, i.e., that the Hamiltonian is bounded from below. This is because the spectral condition is a*global*statement about modes in spacetime. It asserts that allowed states have overlap only with the positive part of the mass shell. - This is very analogous to the way that analytic functions are determined by their behavior in an arbitrarily small open patch of the complex plane.
- This theorem says that some local operator, when acting on the vacuum, can produce the Taj-Mahal in a distant, space-like separated region of space-time. This is often mistakenly interpreted as it being possible to
*create*, in a causal sense, a Taj Mahal elsewhere. But in fact, the Taj Mahal is already “there”, deep in the belly of the vacuum, and the best you can do from your local region is make a measurement (essentially on the entangled Bell pairs) that has a (extraordinarily small likelihood) outcome that the Taj-Mahal will be found elsewhere. - Basically all of the key moving parts can be seen by looking at non-interacting QFTs (e.g., just the Klein-Gordan equation), Rindler wedges, and other simple idealizations.

In other words: The vacuum of a QFT is non-local (entangled) over arbitrary distances because of the spectral condition.

Things I still don’t understand:

- I believe any two regions can formally access an arbitrarily number of shared Bell pairs, but the energetic costs are extraordinary. However, I don’t know how to make this “scarce resource” picture precise. In particular, I’d like a better way to explain what’s fishy about this sort of thing.
- All of this depends on requiring that there is
*precisely zero*contribution from negative-energy modes, and then carefully analyzing exponentially tiny overlaps inherent in a non-local vacuum. But the justification I usually hear for the former are very weak, along the lines of “otherwise you’d have a Hamiltonian unbounded from below, which could set off a negative-energy cascade”. But a very*tiny*chance of this seems fine, for the same reason that we don’t worry about very tiny chances of spontaneous Taj Mahals. Is it obvious why we don’t go this route? Presumably there are deep considerations here with axiomatic QFT. - What is the precise connection (or “point of closest approach”) between this vacuum non-locality and the non-locality of analytic functions?

**Edit**: “Yet More Ado About Nothing: The Remarkable Relativistic Vacuum State” (arXiv:0802.1854) by Stephen J. Summers is a good place for further reading.

**Edit 2**: In a more recent blog post I constructed a state with the Reeh-Schlieder property in the traditional non-relativistic context (i.e., a finite number of countable-basis Hilbert spaces tensored together). This avoids relying on things like analyticity.

**Edit 3**: This recent Ed Witten lecture (video, notes) has the simplest derivation I’ve seen, and he interprets the result sensibly. As you can see, the derivation relies heavily on analyticity. (H/t Scott Aaronson.)

One statement that is pertinent to your third question (and that perhaps could be made precise) is that a Wightman field effectively projects each test function to its “analytic signal”, that is, that hatphi_f is equivalent to hatphi_{f_A}. There is a sense, in other words, in which a local test function is not local (though of course microcausality is satisfied, the two-point VEV is generally nonzero at space-like separation). Reeh-Schlieder might not apply to an interacting field if it’s not a Wightman field.

I find it quite helpful to formulate statements about QFT in terms of test functions insofar as using test functions can put us in a moderately clear relationship with signal analysis (where window functions are an almost equivalent concept, effectively of test functions used in convolution), and the raw data for which we use QFT models /is/ a set of signals.

What is f_A? I didn’t think there was a good notion of the “analytical part” of a function in the same sense in which there’s an imaginary part. Suppose, as an easy case, I had a non-analytic function f which is locally analytic in a finite number of distinct patches. Each patch encodes a different (and perhaps globally-defined) analytic function, so what’s the analytic part of f? More generally, how is this defined for a function f that’s nowhere analytic?

I was thinking in terms of something like https://en.wikipedia.org/wiki/Analytic_signal, though I now see f_a is used there for notation. I’ve worried at this overnight, because although there is a projection to positive frequency, there is also a projection to inside the light-cone, and I’m not clear how to talk precisely and briefly about the consequences of that, though of course Wightman field theory is all about such things (usually many pages of it). fmapsto f_a is manifestly nonlocal.

A propos of your second point, if one takes a signal analysis PoV one possibility that is probably discountenanced from an axiomatic PoV is to consider transforms that introduce negative frequencies. In particular, the free EM field can be projected into left and right helicity components, so for a 2-form EM quantum field and a 2-form test function f, hat F_f=hat L_f+hat R_f, one can consider the field X_f=hat L_f+hat R_{f^-}, where f^-(x)=f(-x), reversing the frequency of the right helicity components. With this, one finds that [X_f,X_g]=0 — in other words, that X_f is a classical stochastic field — and also that =. fmapsto f^- is a nonlocal transform and is not translation invariant, but nonetheless a vacuum sector for X_f and hat F_f can both be defined in Poincar’e invariant ways, and they are isomorphic as Hilbert spaces (obviously X_f and hat F_f generate different algebras acting on the same Hilbert space, but {X_f+the vacuum projection operator} generates the same algebra as {hat F_f+the vacuum projection operator}).

X_f has quite a few failings, which can be partly summarized by saying that it does not satisfy the correspondence principle relative to non-stochastic classical EM, but one might wonder whether absolute insistence on positive frequency is too inhibiting. One could say, indeed, that the insistence that we use the analytic signal is what causes measurements to be noncommutative. You’re right that the reason given for not allowing negative frequencies is that “otherwise you’d have a Hamiltonian unbounded from below, which could set off a negative-energy cascade”, but I think in a signal analysis PoV that’s not quite as absolutely off-putting.

I hope the LaTeX is not too unreadable. Same for the concept.

Thanks very much for this comment Peter. I have a few elementary questions:

(1) When you say that [X_f,X_g]=0, am I correct in thinking that (a) this applies to a

allrelativistic quantum fields when f and g are test (“window”?) functions with space-like separated regions of compact support, but (b) the thing that makes X aclassicalfield is that this commutation relation holds for arbitrary f and g?(2) Why do you put hats on L and F but not X? Is this just to signal that X is a classical stochastic field (though still formally an operator as defined), or is it because, by virtue of its definition, X is actually

nota linear operator?(3) Does this depend importantly on using spin-1 massless E&M with different helicity states, or can X_f be defined for massive/massless scalar fields? If not, what is the physical significance of this?

(1) (a)yes, (b)yes. (2)yes, I intended it to signal that X is a classical stochastic field, though I doubt now that it was a good choice, because X_f /is/ an operator. X:fmapsto X_f /is/ a complex linear map, as is fmapsto f^-; for constant complex scalar c, X_{cf}=cX_f. (3) One can do the same with a quantized complex scalar field (one has to not introduce extra DoFs), as in my EPL, 87 (2009) 31002, doi: 10.1209/0295-5075/87/31002 (pre-print arXiv:0905.1263v2 [quant-ph]), though not all of the text that surrounds the math in that paper is good. Losing the Correspondence Principle as a guide in some ways gives too much rope to play with, particularly the worrisome fact that energy no longer corresponds to the generator of time-like translations.

I would happily use the term “window function” instead of “test function”, because it puts us rather firmly into signal analysis language, but one doesn’t see test functions used in convolution, AFAIK ever (a small difference, perhaps, but it feels like one sign of a significant cultural shift, from engineering to serious analysis).

I know almost nothing about axiomatic QFT, but I have a feeling that something very similar to this should also be true if you “slightly relax” the spectrum condition. Think for example a simple scalar phi^4 theory with a small negative coupling g0 theory and just change the sign of the coupling. In this case I think it works because you do a perturbative expansion around a theory that actually satisfies the spectrum condition (free field theory).

Thanks this is useful. Can you say more about why g<0 violates the spectrum condition? I'm out of my depth here, so it's not obvious to me.

My intuition is that you've relaxed the spectrum condition, but not in a way that actually reduces the non-locality of the vacuum. In other words, the spectrum condition is a sufficient but not necessary condition for non-locality; relaxing it only reduces non-locality if you relax it in the right way.

Now I also watched the talk, and I really liked it, thank you for the recommendation.

Now for your questions: The explicit way to see it is to write down the Hamiltonian with the field variables, it will have a term

int d^3 x g phi(x)^4, if g<0 this will make the Hamiltonian unbounded from below. This is basically the same thing as the Hamiltonian of a simple anharmonic oscillator being unbounded from below if we have a -x^4 term in the Hamiltonian.

Of course, the spectrum condition is a sufficient for the non-locality of the vacuum, and that does not mean it is necessary. Unfortunately, I don't know if it can be relaxed in a way that you still get some kind of sensible theory, but local correlations, but I doubt it.

For some reason, my answer was marked as spam. Can you do something about that?

Yea, sorry about that. Not sure why you tripped the Disqus filter. I’ve whitelisted you.