In Weinberg's "The Quantum Theory of Fields, Vol. 1", Section 4.4, page 182, the author says:

We now ask, what sort of Hamiltonian will yield an $S$-matrix that satisfies the cluster decomposition principle? It is here that the formalism of creation and annihilation operators come into its own. The answer is contained in the theorem that the $S$-matrix satisfies the cluster decomposition principle if (and as far as I know, only if) the Hamiltonian can be expressed as in

$$H=\sum_{N=0}^\infty\sum_{M=0}^\infty \int dq_1'\cdots dq_N' dq_1\cdots dq_M a^\dagger(q_1')\cdots a^\dagger(q_N')a(q_M)\cdots a(q_1)h_{NM}(q',q)$$

with coefficient functions $h_{NM}$ that contain just a single three-dimensional momentum-conservation delta function (returning here here briefly to a more explicit notation):

$$h_{NM}(\mathbf{p}_1'\sigma_1'n_1',\cdots \mathbf{p}_N'\sigma_N'n_N',\mathbf{p}_1\sigma_1n_1,\cdots,\mathbf{p}_M\sigma_Mn_M)=\delta^{(3)}(\mathbf{p}_1'+\cdots+\mathbf{p}_N'-\mathbf{p}_1-\cdots -\mathbf{p}_M)\tilde{h}_{NM}(\mathbf{p}_1'\sigma_1'n_1',\cdots \mathbf{p}_N'\sigma_N'n_N',\mathbf{p}_1\sigma_1n_1,\cdots,\mathbf{p}_M\sigma_Mn_M)$$

where $\tilde{h}_{NM}$ contains no delta function factors.

Now, Weinberg says that as far as he knows the converse holds: if the $S$-matrix satisfies the cluster decomposition principle the interaction has this form.

My question here is, out of curiosity, is it known today a proof of this statement? Or Weinberg's belief was wrong and indeed it has been found $S$-matrices satisfying the cluster decomposition principle without the interaction having this form?


This answer raises more questions, which perhaps others can clarify.

The cluster decomposition (CD) condition can be formulated as a smoothness condition on the momentum-dependence of the $S$-matrix, as in Weinberg's "The Quantum Theory of Fields, Vol. 1", end of section 4.3. As Weinberg says, loosely speaking, CD means the only $\delta$-function singularities in the connected part of the $S$-matrix must be those associated with conservation of momentum. Using this formulation of the CD condition, I think the $S$-matrix of string theory is generally believed to satisfy CD. (I think that's the case even though Cluster decomposition in string theory seems ambivalent.)

Meanwhile, I think the $S$-matrix of string theory is thought not to arise as the $S$-matrix associated to a Hamiltonian of the form given by Weinberg in the original question, i.e. $$H=\sum_{N=0}^\infty\sum_{M=0}^\infty \int dq_1'\cdots dq_N' dq_1\cdots dq_M a^\dagger(q_1')\cdots a^\dagger(q_N')a(q_M)\cdots a(q_1)h_{NM}(q',q),$$ with the smoothness condition on $h_{NM}$. (How does one show this?)

In that case, string theory would provide a counterexample to the claim that any $S$-matrix satisfying CD must arise from a Hamiltonian of the given form. Perhaps this is what Weinberg means in his talk "What is Quantum Field Theory, and What Did We Think It Is", where he says

The bottom line is that quantum mechanics plus Lorentz invariance plus cluster decomposition implies quantum field theory... A much more serious objection to this not-yet-formulated theorem is that there’s already a counter example to it: string theory.

However, when Weinberg cites string theory as a counterexample, maybe he only means it as a counterexample to the stronger claim that any theory satisfying CD arises from a local Hamiltonian density constructed using field operators (i.e. is a field theory), which is perhaps stronger than requiring it arise from a Hamiltonian constructed from ladder operators as above.

Finally, even if string theory were a counterexample, Weinberg suggests a modified claim that any Poincare-invariant $S$-matrix satisfying CD must resemble a field theory at low energies, as does string theory.


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