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?

Edit: If the statement is true I believe a proof would go something like this:

  1. According to Schwartz' QFT book, the unitary operator $S$ satisfies the equation $iS'(t,t_0)=H(t)S(t,t_0)$. Hence we can get the Hamiltonian out of it as $$H(t)=iS'(t,t_0)S^{\dagger}(t,t_0).$$

  2. Duncan shows in his QFT book that the $h_{NM}$ above are the matrix elements of the Hamiltonian: $$h_{NM}(k_1,\dots, k_N,k_1',\dots k_M')=\langle k_1,\dots,k_N|H|k_1',\dots k_M'\rangle$$ so we know these functions in terms of $S$ $$h_{NM}(k_1,\dots, k_N,k_1',\dots k_M')=\langle k_1,\dots,k_N|iS'S^\dagger|k_1',\dots k_M'\rangle$$

  3. Next we should impose that $S$ satisfies clustering and see if we get the result out of that. If I remember correctly the definition is recursive, so we should expect an induction proof on $N,M$.

Now, am I on the right path or have I just stated nonsense? One thing that bothers me is that in the above approach we have time dependence of $S$, whereas I don't see time-dependence on Weinberg's/Duncan's approach.


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