Renninger negative-result experiment

It is sometimes noted that the time of the decay of the nucleus cannot be controlled, and that the finite half-life invalidates the result. This objection can be dispelled by sizing the hemispheres appropriately with regards to the half-life of the nucleus. The radii are chosen so that the more distant hemisphere is much farther away than the half-life of the decaying nucleus, times the flight-time of the alpha ray.

To lend concreteness to the example, assume that the half-life of the decaying nucleus is 0.01 microsecond (most elementary particle decay half-lives are much shorter; most nuclear decay half-lives are much longer; some atomic electromagnetic excitations have a half-life about this long). If one were to wait 0.4 microseconds, then the probability that the particle will have decayed will be ${\displaystyle 1-2^{-40}\simeq 1-10^{-12}}$; that is, the probability will be very very close to one. The outer hemisphere is then placed at (speed of light) times (0.4 microseconds) away: that is, at about 120 meters away. The inner hemisphere is taken to be much closer, say at 1 meter.

If, after (for example) 0.3 microseconds, one has not seen the decay product on the inner, closer, hemisphere, one can conclude that the particle has decayed with almost absolute certainty, but is still in-flight to the outer hemisphere. The paradox then concerns the correct description of the wave function in such a scenario.

Another common objection states that the decay particle was always travelling in a straight line, and that only the probability of the distribution is spherical. This, however, is a mis-interpretation of the Mott problem, and is false. The wave function was truly spherical, and is not the incoherent superposition (mixed state) of a large number of plane waves. The distinction between mixed and pure states is illustrated more clearly in a different context, in the debate comparing the ideas behind local-hidden variables and their refutation by means of the Bell inequalities.

A true quantum-mechanical wave would diffract from the inner hemisphere, leaving a diffraction pattern to be observed on the outer hemisphere. This is not really an objection, but rather an affirmation that a partial collapse of the wave function has occurred. If a diffraction pattern were not observed, one would be forced to conclude that the particle had collapsed down to a ray, and stayed that way, as it passed the inner hemisphere; this is clearly at odds with standard quantum mechanics. Diffraction from the inner hemisphere is expected.

In this objection, it is noted that in real life, a decay product is either spin-1/2 (a fermion) or a photon (spin-1). This is taken to mean that the decay is not truly sphere symmetric, but rather has some other distribution, such as a p-wave. However, on closer examination, one sees this has no bearing on the spherical symmetry of the wave-function. Even if the initial state could be polarized; for example, by placing it in a magnetic field, the non-spherical decay pattern is still properly described by quantum mechanics.

The above formulation is inherently phrased in a non-relativistic language; and it is noted that elementary particles have relativistic decay products. This objection only serves to confuse the issue. The experiment can be reformulated so that the decay product is slow-moving. At any rate, special relativity is not in conflict with quantum mechanics.

This objection states that in real life, particle detectors are imperfect, and sometimes neither the detectors on the one hemisphere, nor the other, will go off. This argument only serves to confuse the issue, and has no bearing on the fundamental nature of the wave-function.