A Question about Lorenz invariance and cluster decomposition

[jordi]

From Weinberg, The Quantum Theory of Fields, Vol. 1, there is the statement that "the only way" to merge Lorentz invariance with the cluster decomposition property (a.k.a. locality) is through a field theory.

He uses this argument basically to justify that any quantum theory at low energies will be an (effective) field theory.

But this leaves out string theory: string theory is not a field theory, but it is thought as a candidate for a quantum theory. Is string theory not respecting Lorentz invariance and / or the cluster decomposition property?

Edit: from https://arxiv.org/pdf/hep-th/9702027.pdf , Weinberg states "(...) the whole formalism of fields, particles, and antiparticles seems to be an inevitable consequence of Lorentz invariance, quantum mechanics, and cluster decomposition, without any ancillary assumptions about locality or causality."

 

[stevendaryl]

From Weinberg, The Quantum Theory of Fields, Vol. 1, there is the statement that "the only way" to merge Lorentz invariance with the cluster decomposition property (a.k.a. locality) is through a field theory.

He uses this argument basically to justify that any quantum theory at low energies will be an (effective) field theory.

But this leaves out string theory: string theory is not a field theory, but it is thought as a candidate for a quantum theory. Is string theory not respecting Lorentz invariance and / or the cluster decomposition property?

Edit: from https://arxiv.org/pdf/hep-th/9702027.pdf , Weinberg states "(...) the whole formalism of fields, particles, and antiparticles seems to be an inevitable consequence of Lorentz invariance, quantum mechanics, and cluster decomposition, without any ancillary assumptions about locality or causality."

I don't see that string theory is a counter-example, because at low energies, string theory is equivalent to regular field theory. You don't see the strings at low energy.

 

[jordi]

I don't see that string theory is a counter-example, because at low energies, string theory is equivalent to regular field theory. You don't see the strings at low energy.

This is not the point. The argument of Weinberg is (should) be true irrespective of the energy.

 

[stevendaryl]

This is not the point. The argument of Weinberg is (should) be true irrespective of the energy.

So you think that the phrase "low energies" in "any quantum theory at low energies will be an (effective) field theory" was unnecessary?

 

[jordi]

So you think that the phrase "low energies" in "any quantum theory at low energies will be an (effective) field theory" was unnecessary?

No. What I am saying is he never states his general argument is valid only at low energy. He states that at any energy, the only way to reconcile relativistic invariance with the cluster decomposition property is through fields.

 

[stevendaryl]

No. What I am saying is he never states his general argument is valid only at low energy. He states that at any energy, the only way to reconcile relativistic invariance with the cluster decomposition property is through fields.

Well, every energy level is low-energy compared to higher energy levels.

I'm sort of serious. If the energies being considered are bounded above, then you can approximate the theory by an effective field theory that is good for that energy range, but might break down at higher energies.

 

[jordi]

Well, every energy level is low-energy compared to higher energy levels.

I'm sort of serious. If the energies being considered are bounded above, then you can approximate the theory by an effective field theory that is good for that energy range, but might break down at higher energies.

You are right. But your point does not address the goal of this post.

 

[kith]

But this leaves out string theory: string theory is not a field theory [...]

How about https://en.wikipedia.org/wiki/String_field_theory?

 

[jordi]

How about https://en.wikipedia.org/wiki/String_field_theory?

Maybe. But since string theory exists, it remains a puzzle regarding Weinberg's claim.

 

[stevendaryl]

You are right. But your point does not address the goal of this post.

Well, if Weinberg is saying that every relativistic quantum theory looks like quantum field theory in a certain range of energies, I don't see that string theory is a counterexample. It looks like quantum field theory in a certain of energies, as well.

The point (as I understand it, which is pretty shallow) is that at any energy range, a relativistic quantum theory will look like a quantum field theory. But you change the range, it might start looking like a DIFFERENT quantum field theory, with different particles (maybe), different coupling constants, etc.

 

[king vitamin]

From Weinberg, The Quantum Theory of Fields, Vol. 1, there is the statement that "the only way" to merge Lorentz invariance with the cluster decomposition property (a.k.a. locality) is through a field theory.

Is that an actual Weinberg quote? I couldn't find "only way" in his book, and indeed I found the following lecture by him where he points out that string theory is a counterexample: https://arxiv.org/pdf/hep-th/9702027.pdf

He does mention a "folk theorem" equivalent to what is being said above - it appears that all such theories will at least look like QFTs at long enough distances.

 

[Haelfix]

From Weinberg, The Quantum Theory of Fields, Vol. 1, there is the statement that "the only way" to merge Lorentz invariance with the cluster decomposition property (a.k.a. locality) is through a field theory.

He uses this argument basically to justify that any quantum theory at low energies will be an (effective) field theory.

But this leaves out string theory: string theory is not a field theory, but it is thought as a candidate for a quantum theory. Is string theory not respecting Lorentz invariance and / or the cluster decomposition property?

Edit: from https://arxiv.org/pdf/hep-th/9702027.pdf , Weinberg states "(...) the whole formalism of fields, particles, and antiparticles seems to be an inevitable consequence of Lorentz invariance, quantum mechanics, and cluster decomposition, without any ancillary assumptions about locality or causality."

Can you provide a quote for the second paragraph. QED is valid at low energies, but it is presumably not an effective field theory (depending on what you mean). Also its worth pulling out specific quotes here, b/c he is very careful about what he says, and if memory serves in volume 1, he was talking about the necessity of the field formalism in constructing relativistic lagrangians that satisfy Poincare invariance and the C.D.P.

Perturbative String theory (and string field theory) incidentally, can be thought of like a sort of field theory with an infinite amount of fields all constrained by a tower of extra 'gauge' symmetries. In any event, it satisfies clustering and lorentz invariance (essentially by construction).

 

[jordi]

Is that an actual Weinberg quote? I couldn't find "only way" in his book, and indeed I found the following lecture by him where he points out that string theory is a counterexample: https://arxiv.org/pdf/hep-th/9702027.pdf

He does mention a "folk theorem" equivalent to what is being said above - it appears that all such theories will at least look like QFTs at long enough distances.

That's it, then. The paragraph about the folk theorem explains that string theory is a counterexample of his "theorem", but he sees his "theorem" as only applying to low energies. My question is then answered. I believe it is better explained in the paper than in the book.

 

[jordi]

Can you provide a quote for the second paragraph. QED is valid at low energies, but it is presumably not an effective field theory (depending on what you mean). Also its worth pulling out specific quotes here, b/c he is very careful about what he says, and if memory serves in volume 1, he was talking about the necessity of the field formalism in constructing relativistic lagrangians that satisfy Poincare invariance and the C.D.P.

Perturbative String theory (and string field theory) incidentally, can be thought of like a sort of field theory with an infinite amount of fields all constrained by a tower of extra 'gauge' symmetries. In any event, it satisfies clustering and lorentz invariance (essentially by construction).

From the paper quoted above:

"(...) 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. Picking up a phrase from Arthur Wightman, I’ll call this a folk theorem. At any rate, this folk theorem is satisfied by string theory, and we don’t know of any counterexamples. This leads us to the idea of effective field theories. When you use quantum field theory to study low-energy phenomena, then according to the folk theorem you’re not really making any assumption that could be wrong, unless of course Lorentz invariance or quantum mechanics or cluster decomposition is wrong, provided you don’t say specifically what the Lagrangian is. As long as you let it be the most general possible Lagrangian consistent with the symmetries of the theory, you’re simply writing down the most general theory you could possibly write down. This point of view has been used in the last fifteen years or so to justify the use of effective field theories, not just in the tree approximation where they had been used for some time earlier, but also including loop diagrams "

 

[mitchell porter]

Cluster decomposition in string theory was also discussed here.

 

[MathematicalPhysicist]

Has anyone gone through all of Weinberg's 3 QFT books?
Just wonder.
It reminds me of principia mathematica of Russell though not as thick.