What is the Principle of Cluster Decomposition in Quantum Field Theory?

In summary, the conversation touches on Weinberg's concept of Cluster Decomposition and its relation to quantum entanglement, with a focus on the correlation and dependence of in states and the issue of decoherence in practical scenarios. A previous discussion on the same topic is referenced, and a restatement of the cluster decomposition is suggested.
  • #1
fermi
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5
Recently a student brought the following to my attention from Weinberg’s Quantum Theory of Fields, Volume I, from page 177, which I must admit that it stumped me. Here Weinberg introduces the concept of Cluster Decomposition: “It is one of the fundamental principles of physics (indeed, of all science) that experiments that are sufficiently separated in space have unrelated results…” The quantum entanglement a la EPR paradox, however, tells us that the measurement of the polarization of a photon from a [itex]{\pi}^o[/itex] decay measured a year later will determine the polarization of the other photon (two light years away from this measurement.) This is because the decay photons from a [itex]{\pi}^o[/itex] are entangled forever, no matter how far apart they may be. This sort of thing was discussed and dissected endlessly in the past fifty years, but today we simply accept this non-local entanglement as part of Quantum Mechanics.

So back to Weinberg: what is he saying here then? If you were to restate his words a little more precisely, how would you restate the cluster decomposition?
 
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  • #2
Far be it from me to "restate Weinberg more precisely"! But what he's talking about is a situation in which all of the in states (α1, α2,...), (αjj+1,...) are known and independent. In your pion example the in states α1 and αj are correlated and dependent.
 
  • #3
fermi said:
The quantum entanglement a la EPR paradox, however, tells us that the measurement of the polarization of a photon from a [itex]{\pi}^o[/itex] decay measured a year later will determine the polarization of the other photon (two light years away from this measurement.) This is because the decay photons from a [itex]{\pi}^o[/itex] are entangled forever, no matter how far apart they may be.
This is only true if they your system of entangled particles is isolated. I think in most practical scenarios, interactions with the environment rapidly destroy this entanglement by decoherence.

/edit: deleted a scruffy analog
 
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  • #5
Demystifier said:
We already had a good discussion on it:
https://www.physicsforums.com/showthread.php?t=409861

In particular, for my restatement of the cluster decomposition see post #7.

Thanks Demystifier! I was not aware of the previous thread of a similar discussion. I agree with you that Weinberg's statement is a little careless (at best). Indeed your restatement in post#7 is almost exactly what I believe the right wording may be.
 
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  • #6
I'm glad that I've been helpfull.
 

Question 1: What is cluster decomposition in QFT?

Cluster decomposition is the principle in quantum field theory (QFT) that states that the expectation value of observables at spacelike separated points can be factorized into the product of expectation values at individual points. This means that the behavior of a quantum system at one point is independent of its behavior at another point, as long as the two points are sufficiently far apart.

Question 2: Why is cluster decomposition important in QFT?

Cluster decomposition is important because it allows us to understand the interactions between particles in a quantum system. By separating the system into individual points, we can study the behavior of each point independently and then combine them to understand the overall behavior of the system. This principle also helps us to calculate scattering amplitudes and other important quantities in QFT.

Question 3: How does cluster decomposition relate to the concept of spacetime locality?

Cluster decomposition is closely related to the concept of spacetime locality, which states that physical interactions can only occur between objects that are in close proximity to each other in spacetime. The principle of cluster decomposition shows that the expectation values of observables at spacelike separated points are independent of each other, which supports the idea of spacetime locality.

Question 4: Are there any limitations to cluster decomposition in QFT?

While cluster decomposition is a powerful principle in QFT, it does have some limitations. For example, it may not hold for theories with long-range interactions or for systems that are in a non-equilibrium state. Additionally, the principle only applies to expectation values of observables and not to individual measurements.

Question 5: How is cluster decomposition used in practical applications of QFT?

Cluster decomposition is used extensively in practical applications of QFT, such as in the calculation of scattering amplitudes and in the study of particle interactions. It also plays a crucial role in the development of new theories and in testing the validity of existing theories. Overall, cluster decomposition is an essential principle in QFT that helps us understand the behavior of quantum systems and make predictions about their interactions.

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