Interpretations of
Quantum Mechanics
Implications of
Quantum Mechanics

22. Mathematical Collapse Interpretations.

Summary
The GRW-Pearle mathematical collapse interpretation has severe drawbacks including non-linearity and correlations in fluctuations that extend over long ranges, and even across universes.

We have indicated in No Evidence for Collapse that there is currently no experimental evidence for collapse. In this section, because it is necessary to have some mechanism for singling out, we will consider the possibility of constructing a mathematical theory of collapse, with the presumption that it has characteristics which make it difficult to experimentally detect.

The primary (and just about the only) mathematical theory of collapse is the one proposed by Ghirardi, Rimini, Weber, and Pearle (Ref. 2). A random force is introduced into the time evolution of the wave function in such a way that, after a short period of time, a microsecond or less for systems with many particles, there is a collapse of the wave function down to just one version (without affecting the ‘shape’ of the wave function). The coefficients of all the other versions effectively shrink to zero, so those versions simply go out of existence. This idea is beautifully implemented in the Pearle model. But even though the mathematics is elegant, there are difficulties with the physical implications of the scheme (Ref. 2):
•There must be instantaneous coordination of billions of random events located far from each other (and for that reason, one is currently not able to make the theory relativistic).
•The coordination extends across versions, which is strictly forbidden in conventional quantum mechanics, because versions don’t communicate (principle [P5] in Separate Universes).
•The linearity of quantum mechanics must be abandoned (see Mathematical Collapse and Linearity), which runs counter to the group representational analysis in the section on Mass, Spin, and Charge, as well as to all the successes of quantum mechanics.
• Finally, the specific and very specialized form assumed (not derived) for the randomizing Hamiltonian just happens to give the |a(i)|2 probability law which agrees with the rest of quantum mechanics which goes against the reasoning implied by principle [P10] in The Probability Law.
Because there are no other serious candidates for a mathematical theory of collapse, and also because some or all of these problems would be expected to occur for any model, there is currently no reason to suppose that mathematically implemented collapse is the solution to the problem of perception of only one version of reality. (Actually the problem is not the perception of only one version (see Classical Perception); the problem is why the probability law holds. The GRW-P model does indeed give the probability law, but it has the problems just described.)
Thus we have the following principle:

[P21] There is currently no reason to be at all optimistic that there is a theory of collapse which augments standard quantum mechanics.
Penrose’s gravitational collapse. Penrose (Ref., Ref. 6) proposed the following (in connection with a theory of consciousness!): Suppose we again do the Stern-Gerlach experiment so there are two versions of the detecting apparatus and the observer. Since the arrangement of atoms is slightly different in the two versions, the definition of space and time—which depends on the distribution of matter—will be slightly different in the two versions. Penrose supposed that difference caused a ‘tear’ in the fabric of space-time. This tear would lead to a force that would pull the two versions together, thus causing a collapse to just one of them. But this inter-version force violates the principle that versions are in separate universes which cannot influence one another. So unless Penrose can show why the separate-universe principle doesn’t hold, I don’t believe his interpretation can be valid.  