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Does anyone know of a good discussion of the compatibility (or otherwise) of Weinberg's 'cluster decomposition' principle with the fact that entangled states yield distant but correlated measurements?
Thanks for this - a useful link for future reference. But for now I'm looking specifically for discussion of CD and its relation to EPR-type correlations of distant measurement results, and to my knowledge, Weinberg doesn't discuss that here (which seems a strange oversight to me).Here is a link to the topic from Weinberg's 1995 book--found on internet:
http://books.google.com/books?id=3ws6RJzqisQC&pg=PA169&lpg=PA169&dq=weinberg+cluster+decomposition&source=bl&ots=Cu9rwmTFYG&sig=vbVZyOuLyWvlaw2mJ8nRZ4bGFCk&hl=en&ei=8vIhTIzOGMT6lwertPy-AQ&sa=X&oi=book_result&ct=result&resnum=6&ved=0CCwQ6AEwBQ#v=onepage&q=weinberg cluster decomposition&f=false
I would summarize and formalize it this way:The cluster decomposition principle is an interpretation of the factorization of the S matric for separated reaction :
if [tex]S_{\alpha_1,\beta_1}[/tex] corresponds to the amplitude for [tex]\alpha_1\rightarrow\beta_1[/tex]
and [tex]S_{\alpha_2,\beta_2}[/tex] corresponds to the amplitude for [tex]\alpha_2\rightarrow\beta_2[/tex]
then [tex]S_{\alpha_1\alpha_2,\beta_1\beta_2}=S_{\alpha_1,\beta_1}S_{\alpha_2,\beta_2}[/tex]
Each label indicates a specification for all particles in the initial (final) state, including momenta, spins, particle species, and anything else relevant to fully specify a particle state.
Now in an EPR-type experiment, we do not have independent reactions. We have only one final state which is not separable. Since we already cannot separate the state in QM, we have no reason to attempt to separate it in QFT and hope to get a sensible result. The problem is not with the cluster decomposition principle. The problem is with the thought experiment itself. It is generally necessary to assume that all relevant quantities are measured both in the prepared initial and the detected final state.
There are quite some caveats with scattering theory alone, and one must assume that all relevant indices are measured or otherwise summed over. That is assumed when we say that we prepare an initial state. It is not sufficient to use half of the final state of an EPR-type experiment, which would be inseparable from another half somewhere else. In this situation, once the initial state (half final state of an EPR exp.) has been measured it becomes separated and the CDP applies.This is a correct form of CDP in QFT. But this is not the form explicitly stated by Weinberg.