There is no other quantum theory, besides many-worlds, that is
scientific, in the sense of providing a reductionist model of
reality, and free
of internal inconsistencies, that I am aware of. Briefly here are
the defects of the most popular alternatives:
1) Copenhagen Interpretation. Postulates that the observer obeys different physical laws than the non-observer, which is a return to vitalism. The definition of an observer varies from one adherent to another, if present at all. The status of the wavefunction is also ambiguous. If the wavefunction is real the theory is non-local (not fatal, but unpleasant). If the wavefunction is not real then the theory supplies no model of reality.
2) Hidden Variables [B]. Explicitly non-local. Bohm accepts that all the branches of the universal wavefunction exist. Like Everett Bohm held that the wavefunction is real complex-valued field which never collapses. In addition Bohm postulated that there were particles that move under the influence of a non-local "quantum- potential" derived from the wavefunction (in addition to the classical potentials which are already incorporated into the structure of the wavefunction). The action of the quantum- potential is such that the particles are affected by only one of the branches of the wavefunction.
The implicit, unstated assumption made by Bohm is that only the single branch of wavefunction associated with particles can contain self-aware observers, whereas Everett makes no such assumption. Most of Bohm's adherents do not seem to understand (or even be aware of) Everett's criticism, section VI , that the hidden- variable particles are not observable since the wavefunction alone is sufficient to account for all observations and hence a model of reality. The hidden variable particles can be discarded, along with the guiding quantum-potential, yielding a theory isomorphic to many-worlds, without affecting any experimental results.
[B] David J Bohm A suggested interpretation of the quantum theory in terms of "hidden variables" I and II Physical Review Vol 85 #2 166-193 (1952)
3) Quantum Logic. Undoubtedly the most extreme of all attempts to solve the QM measurement problem. Apart from abandoning one or other of the classical tenets of logic these theories are all unfinished (presumably because of internal inconsistencies). Also it is unclear how and why different types of logic apply on different scales.
4) Extended Probability [M]. A bold theory in which the concept of probability is "extended" to include complex values [Y]. Whilst quite daring, I am not sure if this is logically permissable, being in conflict with the relative frequency notion of probability, in which case it suffers from the same criticism as quantum logic. Also it is unclear, to me anyway, how the resultant notion of "complex probability" differs from the quantum "probability amplitude" and thus why we are justified in collapsing the complex- valued probability as if it were a classical, real-valued probability.
[M] W Muckenheim A review of extended probabilities Physics Reports Vol 133 339- (1986) [Y] Saul Youssef Quantum Mechanics as Complex Probability Theory hep-th 9307019
5) Transactional model [C]. Explicitly non-local. An imaginative theory, based on the Feynman-Wheeler absorber-emitter model of EM, in which advanced and retarded probability amplitudes combine into an atemporal "transaction" to form the Born probability density. It requires that the input and output states, as defined by an observer, act as emitters and absorbers respectively, but not any internal states (inside the "black box"), and, consequently, suffers from the familiar measurement problem of the Copenhagen interpretation.
If the internal states did act as emitters/absorbers then the wavefunction would collapse, for example, around one of the double slits (an internal state) in the double slit experiment, destroying the observed interference fringes. In transaction terminology a transaction would form between the first single slit and one of the double slits and another transaction would form between the same double slit and the point on the screen where the photon lands. This never observed.
[C] John G Cramer The transactional interpretation of quantum mechanics Reviews of Modern Physics Vol 58 #3 647-687 (1986)
6) Many-minds. Despite its superficial similarities with many-worlds this is actually a very unphysical, non-operational theory.
7) Non-linear theories in general. So far no non-linear theory
has any accepted experimental support, whereas many have failed
experiment. Many-worlds predicts
that non-linear theories will always fail experiment.