# Cluster decomposition and EPR correlations

#### metroplex021

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?

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#### metroplex021

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).

#### humanino

The cluster decomposition principle is an interpretation of the factorization of the S matric for separated reaction :
if $$S_{\alpha_1,\beta_1}$$ corresponds to the amplitude for $$\alpha_1\rightarrow\beta_1$$
and $$S_{\alpha_2,\beta_2}$$ corresponds to the amplitude for $$\alpha_2\rightarrow\beta_2$$
then $$S_{\alpha_1\alpha_2,\beta_1\beta_2}=S_{\alpha_1,\beta_1}S_{\alpha_2,\beta_2}$$
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.

#### Demystifier

2018 Award
I think that the cluster decomposition principle (CDP) IN THE FORM EXPRESSED BY WEINBERG is wrong.

Indeed, in Sec. 4.3 Weinberg explicitly says:
"It is one of the fundamental principles of physics ... that experiments that are sufficiently separated in space have unrelated results."
...
"... the cluster decomposition principle states that if multi-particle processes ... are studied in N very distant laboratories, then the S-matrix element for the overall process factorizes."

Clearly, these statements formulated as such are incompatible with EPR correlations, and are therefore wrong.

Yet, it does not mean that CDP cannot be reformulated in a more careful way, such that it becomes compatible with EPR correlations. After all, a form of the CDP principle can be derived from QFT, and QFT is compatible with EPR correlations. Probably, CDP can be viewed as a consequence of QFT & decoherence, but I am not aware of any detailed discussion of that idea.

2018 Award

#### Demystifier

2018 Award
The cluster decomposition principle is an interpretation of the factorization of the S matric for separated reaction :
if $$S_{\alpha_1,\beta_1}$$ corresponds to the amplitude for $$\alpha_1\rightarrow\beta_1$$
and $$S_{\alpha_2,\beta_2}$$ corresponds to the amplitude for $$\alpha_2\rightarrow\beta_2$$
then $$S_{\alpha_1\alpha_2,\beta_1\beta_2}=S_{\alpha_1,\beta_1}S_{\alpha_2,\beta_2}$$
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.
I would summarize and formalize it this way:
CDP says that if
1. the initial state (of spatially separated subsystems) can be factorized
and
2. the subsystems remain spatially separated all the time
then
the final state can also be factorized.

This is a correct form of CDP in QFT. But this is not the form explicitly stated by Weinberg.

#### humanino

This is a correct form of CDP in QFT. But this is not the form explicitly stated by Weinberg.
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.

#### Demystifier

2018 Award
Yes, that is essentially what I had in mind when I mentioned decoherence in post #5.

#### metroplex021

Thanks everyone for those replies. I'm glad it's not just me who thinks it's a bit odd. I'm going to keep looking for somewhere that Weinberg himself addresses (what seems to me) his strange wording of this principle. But any other refs would be really appreciated of course.

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