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Corfu Summer Institute School and Workshops on Elementary Particle Physics and Gravity (CORFU)

Time, quantum entanglement and particle decay

J. A. Aguilar–Saavedra

Published Aug 24, 2026• 0 citations
Publisher

Abstract

We investigate the role of time ordering in entanglement experiments involving unstable particles, focusing on $\mu^+ \mu^-$ pairs produced in a maximally-entangled spin state. We analyse the correlations between measurements performed by two experimenters, Alice (who measures $\mu^-$ spin) and Bob (who measures $\mu^+$ decay products). Remarkably, the observed correlations persist irrespective of whether Bob's muon decays before or after Alice's spin measurement. We further discuss different interpretations of the same empirical results depending on the observer's reference frame. The findings reinforce the viewpoint that the Copenhagen interpretation of measurement is a mathematical tool rather than a literal account of physical reality.

DOI
10.22323/1.509.0060

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A sentence in the Search and Discovery story “Time-reversal asymmetry in particle physics has finally been clearly seen” (Physics Today, November 2012, page 16) has me somewhat puzzled. Bertram Schwarzschild writes, “If the first decay of a daughter reveals it to have been in a specific flavor or CP eigenstate, her still undecayed sister must—at that instant—be in the opposite state.” What is implicit in this sentence is that state-vector collapse takes place at the decay of the first B meson, so that the second meson is instantaneously projected onto the opposite state.I have two objections to this sentence. First, it is not relativistically invariant: Because the two events—decay of the first meson and “transformation” of the second—have spacelike separation, their time ordering is not defined. Second, the decay is governed by a relativistic version of the Schrödinger equation, which is reversible, whereas state-vector collapse is, by nature, irreversible.It is instructive to make a comparison with the analysis of a Bell-type experiment, in which a two-particle observable is measured via two detectors that are far apart, with each measuring the spin of one of the two particles. The quantum mechanical result is given by applying Born’s rule to the two-particle observable.The two experimentalists, Alice and Bob, then compare their results by exchanging them at a speed less than that of light, and they notice the results are correlated, in conformity with the theoretical predictions of nonlocal correlations. The same theoretical results may be derived from state-vector collapse: When Alice measures the first spin, Bob’s spin is projected instantaneously onto a well-defined polarization state; correlations may be easily computed and are, of course, in agreement with those derived from Born’s rule.State-vector collapse is strictly equivalent to Born’s rule in a nonrelativistic context, but I think it is safer to use Born’s rule when special relativity comes into play. Indeed, to the best of my knowledge, there is no satisfactory relativistic generalization of state-vector collapse, which is understandable since simultaneity is not defined for two spacelike separated events. Therefore, a full justification of the argument used in the Physics Today report should begin with state-vector collapse taking place in a well-defined reference frame at the instant when the decay products of the B mesons reach the detectors. Then, from the correlations observed between the results of the detectors, it should be possible to reconstruct the decays and show that they can be interpreted as an effective state-vector collapse.Such an analysis would likely vindicate that of the BaBar experiment, but as the use of Einstein-Podolsky-Rosen correlations goes far beyond the usual one, it should be fully justified.© 2013 American Institute of Physics.

Physics Today•Jul 31, 2013•0 citations
We investigate the role of time ordering in entanglement experiments involving unstable particles, focusing on μ+μ−\mu^+ \mu^-μ+μ− pairs produced in a maximally-entangled spin state. We analyse the correlations between measurements performed by two experimenters, Alice (who measures μ−\mu^-μ− spin) and Bob (who measures μ+\mu^+μ+ decay products). Remarkably, the observed correlations persist irrespective of whether Bob's muon decays before or after Alice's spin measurement. We further discuss different interpretations of the same empirical results depending on the observer's reference frame. The findings reinforce the viewpoint that the Copenhagen interpretation of measurement is a mathematical tool rather than a literal account of physical reality.
We consider a pair of unstable fermions in a spin-entangled state. After the decay of one fermion, a spin measurement is performed on the surviving partner, with a Stern-Gerlach experiment or similar. The measurement not only projects the spin of the surviving fermion, but is also physically equivalent to a spin projection for the decayed one — even when it no longer exists. This post-selection effect would be experimentally accessible using muon pairs in a maximally-entangled state, produced either in the decay of a scalar particle, or in e+e− collisions at wide angles.
We consider a pair of unstable fermions in a spin-entangled state. After the decay of one fermion, a spin measurement is performed on the surviving partner, with a Stern-Gerlach experiment or similar. The measurement not only projects the spin of the surviving fermion, but is also physically equivalent to a spin projection for the decayed one -- even when it no longer exists. This post-selection effect would be experimentally accessible using muon pairs in a maximally-entangled state, produced either in the decay of a scalar particle, or in e+e−e^+ e^-e+e− collisions at wide angles.
The role played by time in the quantum theory is still mysterious by many aspects. In particular it is not clear today whether the distribution of decay times of unstable particles could be described by a time operator (TO). As we shall discuss, different approaches to this problem (one could say interpretations) can be found in the literature on the subject. As we shall show, it is possible to conceive crucial experiments aimed at distinguishing the different approaches, by measuring with accuracy the statistical distribution of decay times of entangled particles. Such experiments can be realized in principle with entangled kaon pairs.
In 1935, Grete Hermann proposed a way to conciliate causality and quantum mechanics, detaching causality from determinism. In her view, a causal explanation for quantum phenomena can be inferred in a retrodictive way, after the corresponding measurements were performed, and verified via the obtained (non-deterministic) predictions for the future behavior of the system. Here we discuss the violation of Bell's inequalities, where pairs of entangled particles are measured by two parties named Alice and Bob in spacelike intervals, in view of retrodictive causality. We concluded that the causal order of the events is relative to the observer. For some observers, Alice's measurement occurs first and, combined with the physical laws that govern the production of the entangled state, can be retrodictively considered as an explanatory cause, though not a real cause, of Bob's observations. But for other observers, Bob's measurement occurs first and can be considered as an explanatory cause for Alice's observations. So, which event is the explanatory cause and which one is its consequence depends on the observer. But this odd conclusion does not conflict with Hermann's view, since for her the causal law is epistemic rather than ontological. We illustrate these ideas describing the quantum correlations between two spin-1/2 particles, when these particles are produced by the decay of a spin-0 particle, and the polarization quantum correlations of photons generated by parametric downconversion.
A sentence in the Search and Discovery story “Time-reversal asymmetry in particle physics has finally been clearly seen” (Physics Today, November 2012, page 16) has me somewhat puzzled. Bertram Schwarzschild writes, “If the first decay of a daughter reveals it to have been in a specific flavor or CP eigenstate, her still undecayed sister must—at that instant—be in the opposite state.” What is implicit in this sentence is that state-vector collapse takes place at the decay of the first B meson, so that the second meson is instantaneously projected onto the opposite state.I have two objections to this sentence. First, it is not relativistically invariant: Because the two events—decay of the first meson and “transformation” of the second—have spacelike separation, their time ordering is not defined. Second, the decay is governed by a relativistic version of the Schrödinger equation, which is reversible, whereas state-vector collapse is, by nature, irreversible.It is instructive to make a comparison with the analysis of a Bell-type experiment, in which a two-particle observable is measured via two detectors that are far apart, with each measuring the spin of one of the two particles. The quantum mechanical result is given by applying Born’s rule to the two-particle observable.The two experimentalists, Alice and Bob, then compare their results by exchanging them at a speed less than that of light, and they notice the results are correlated, in conformity with the theoretical predictions of nonlocal correlations. The same theoretical results may be derived from state-vector collapse: When Alice measures the first spin, Bob’s spin is projected instantaneously onto a well-defined polarization state; correlations may be easily computed and are, of course, in agreement with those derived from Born’s rule.State-vector collapse is strictly equivalent to Born’s rule in a nonrelativistic context, but I think it is safer to use Born’s rule when special relativity comes into play. Indeed, to the best of my knowledge, there is no satisfactory relativistic generalization of state-vector collapse, which is understandable since simultaneity is not defined for two spacelike separated events. Therefore, a full justification of the argument used in the Physics Today report should begin with state-vector collapse taking place in a well-defined reference frame at the instant when the decay products of the B mesons reach the detectors. Then, from the correlations observed between the results of the detectors, it should be possible to reconstruct the decays and show that they can be interpreted as an effective state-vector collapse.Such an analysis would likely vindicate that of the BaBar experiment, but as the use of Einstein-Podolsky-Rosen correlations goes far beyond the usual one, it should be fully justified.© 2013 American Institute of Physics.