4
$\begingroup$

In Weinberg's The Quantum Theory of Fields Vol. 1, an argument is presented that the three postulates of

  1. Lorentz invariance
  2. quantum mechanics
  3. cluster decomposition principle

leads to quantum field theory as a unique description of nature.

In Weinberg's article What is Quantum Field Theory, and What Did We Think It Is?, he briefly reviews this argument, and then adds on top of p. 8

[an] objection to this not-yet-formulated theorem is that there’s already a counter example to it: string theory.

In string theory, we do not construct local operators as Fourier transforms of the creation/annihilation operators - instead, the creation/annihilation operators are packaged up in the string.

Later on p. 8 in the article, he writes

although you can not argue that relativity plus quantum mechanics plus cluster decomposition necessarily leads only to quantum field theory, it is very likely that any quantum theory that at sufficiently low energy and large distances looks Lorentz invariant and satisfies the cluster decomposition principle will also at sufficiently low energy look like a quantum field theory.

My question, then, is twofold:

1. What is the loophole in the argument presented in Weinberg's textbook that means string theory is a counterexample?

2. Why might this loophole not exist in the low energy limit?

$\endgroup$
4
  • 5
    $\begingroup$ String theory is clearly a counterexample because it fulfills these three conditions but is not a quantum field theory. Looking for a specific "loophole" would imply that Weinberg is trying to present a rigorous proof that is phrased at a level of specificity so that we can poke specific holes into it rather than a heuristic motivation of QFT, but the rest of your quotes ("not-yet-formulated theorem", what follows after "although you can not argue[...]") already makes clear that he doesn't think his argument rises to that level of rigor. $\endgroup$
    – ACuriousMind
    Feb 22, 2022 at 16:55
  • $\begingroup$ The point about low energy surely refers to the fact that low energy effective actions for string theory start to look familiar to a quantum field theorist again. An introduction to them is the second last chapter of Tong's notes. $\endgroup$ Feb 22, 2022 at 16:57
  • $\begingroup$ I think the question is perfectly meaningful. Weinberg's argument might not be rigorous but it's still an argument, and it makes sense to ask where specifically it fails. $\endgroup$
    – Javier
    Feb 22, 2022 at 17:35
  • 1
    $\begingroup$ ... e.g. in the same article (p.7), Weinberg says: "The bottom line is that quantum mechanics plus Lorentz invariance plus cluster decomposition implies quantum field theory. But there are caveats that have to be attached to this, and I can see David Gross in the front row anxious to take me by the throat over various gaps in what I have said, so I had better list these caveats quickly to save myself." :-) $\endgroup$
    – NickD
    Feb 22, 2022 at 18:36

1 Answer 1

2
$\begingroup$

Weinberg never claims that he is able to obtain "quantum field theory as a unique description of nature". In other words, he does not prove that in some sense Poincaré + Cluster Decomposition Principle (CDP) + Quantum Mechanics implies Quantum Field Theory. Rather, what he shows is the converse: that a QFT is a theory obeying all such constraints.

What is perhaps missing is to notice that there is, implied in all that Weinberg is doing, one criterium of simplicity. Weinberg writes down the constraints of Poincaré symmetry and CDP and looks for the simplest way that one could imagine solving such constraints.

Want one explicit example of that? Well, in page 144 Weinberg discusses Lorentz symmetry of the ${\cal S}$-matrix. Writing the ${\cal S}$-operator as $${\cal S}={\rm T}\exp\left(-i\int_{-\infty}^\infty dt\ V(t) \right)$$ one asks the form $V(t)$ must take so that ${\cal S}$ is manifestly Lorentz-invariant. Well, the simplest possible idea is to write $V(t)$ as one integral of a Lorentz scalar ${\cal H}(t,x)$ since then you have one integral over $d^4x$ of a scalar. Trying this out one further finds that ${\cal H}(x)$ must commute with ${\cal H}(x')$ when $x$ and $x'$ are either spacelike or lightlike separated.

Is it necessary that $V(t)$ be the integral of a density ${\cal H}(t,x)$ over space to have a Lorentz-invariant ${\cal S}$ operator? The answer is no, but Weinberg never claims that. What he claims is that this is sufficient. And I invite you to think for a moment: what else would you propose? In a sense, this is the simplest thing that would come to mind.

In a sense this kind of reasoning underlies all that Weinberg is doing from chapters 2 to 5. He is trying to motivate that the simplest and most natural way to satisfy all constraints coming from Poincaré symmetry and CDP is QFT.

He never claims it is the only way nor tries to formulate a precise statement of this proposal. String theory ends up being one counterexample that there are indeed relativistic theories obeying CDP which are not QFTs.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.