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metroplex021
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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?
Salman2 said:Here is a link to the topic from Weinberg's 1995 book--found on internet:
http://books.google.com/books?id=3w...page&q=weinberg cluster decomposition&f=false
I would summarize and formalize it this way:humanino said: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.Demystifier said:This is a correct form of CDP in QFT. But this is not the form explicitly stated by Weinberg.
Cluster decomposition is a fundamental principle in quantum mechanics that states that the expectation value of a product of observables in a quantum system can be written as the sum of expectation values of individual observables. This means that the behavior of a quantum system can be understood by studying the behavior of its individual components.
Cluster decomposition is closely related to the concept of entanglement in quantum mechanics. Entanglement refers to the phenomenon where two or more particles become linked in such a way that the state of one particle cannot be described without considering the state of the other particle. Cluster decomposition allows us to understand how entanglement arises in quantum systems and how it affects the behavior of these systems.
EPR correlations, also known as Einstein-Podolsky-Rosen correlations, refer to the correlations between two distant particles that are entangled with each other. These correlations were first proposed by Albert Einstein, Boris Podolsky, and Nathan Rosen to demonstrate the incompleteness of quantum mechanics. They have since been experimentally verified and are now recognized as a fundamental aspect of quantum mechanics.
EPR correlations are closely related to cluster decomposition as they both involve the study of correlations between entangled particles. Cluster decomposition helps us understand the nature of these correlations and how they arise in quantum systems. It also allows us to make predictions about the behavior of entangled particles based on the properties of individual particles.
Yes, both cluster decomposition and EPR correlations have important applications in quantum communication. For example, cluster decomposition can be used to understand and improve the transmission of quantum information through noisy channels. EPR correlations, on the other hand, have been used to develop protocols for secure communication, such as quantum key distribution, which relies on the measurement of EPR correlations to ensure the security of transmitted information.