Everett's work was debunked many time ago, both in physical and mathematical grounds. Alternative attempts presented in the last 50 years by Graham, DeWitt, Geroch, Deuthch,... have been debunked as well.
Not only Many-Worlds isn't a valid interpretation of QM, but its proponents disagree. E.g. Deuth's Many-Worlds is not Everett's Many-Worlds. Next link contains rebuttal of early Everett ideas
http://www.mat.univie.ac.at/~neum/physfaq/topics/manyworlds
"Quantum theory describes the evolution of a state vector in a complex Hilbert space, but we populate our theories with ideas like "spacetime", "particles," and "fields"." That is a very narrow conception of quantum theory because, for instance, the quantum state of an unstable system isn't given by any state vector in a Hilbert space. We populate our theories with concepts as particles because they are the basic building blocks of Nature.
Equation (1) is an approximation. It doesn't apply to system in mixed states, neither to irreversible phenomena. The notation used is also inadequate. If the state vector is only a function of time, then a partial derivative makes little sense; a total derivative would be used.
"The lesson we draw from this is that Nature at its most fundamental is simply described by a vector in Hilbert space". As mentioned above, this isn't true. What is more, even for those systems adequately described by ordinary state vector theory, it is possible to find alternative formulations without Hilbert spaces or vectors. A well-known example is the Wigner-Moyal formulation of quantum mechanics. Hilbert space and state vectors are replaced by non-commutative phase space and Wigner function W(p,q). One evident advantage of the Wigner-Moyal formulation is that is also works for quantum systems for which the Hilbert formulation doesn't work.
"Classical concepts must emerge from this structure in an appropriate limit". It has been rigorously demonstrated that classical systems aren't contained in a Hilbert space structure. And that is the reason why many physicists and mathematicians are working in extensions of quantum theory. The old theory is being extended at two levels: generalized spaces beyond Hilbert space, and nonlinear extensions of Schrödinger equation.
Eq (3) is only valid for quantum systems with discrete spectrum. The complete spectral decomposition of the Hamiltonian operator is given in the attachment.
"One might ask why, if the fundamental theory of everything is fixed by the spectrum of some Hamiltonian, we don't simply imagine writing the state of the universe in the energy eigenbasis, where its evolution is trivial?" Because we know a spectral decomposition of the Hamiltonian doesn't fix "everything".
"Consider the classical theory of N particles moving under the influence of some multi-particle potential in 3 dimensions of space. The corresponding phase space is 6N-dimensional, and we could simply think of the theory as that of one point moving in a 6N-dimensional structure. But by thinking of it as N particles moving in a 3-dimensional space of allowed particle positions, we gain enormous intuition; for example, it could become clear that particles in uence each other when they are nearby in space, which in turn suggests a natural way to coarse-grain the theory". This class of reasoning is what confused Boltzmann and a several generations of physicists. The classical state is given by a point in the 6N phase-space. A model of N particles moving in a 3-dimensional space (really 6-dimensional) fails to consider subtle elements of the full dynamics, such as the existence of long-range correlations. Coarse graining the the theory over distances larger than the range of the interactions will erase the correlations that are needed to drive systems to stable equilibrium.
Similar remarks about (8) and (9).
"This procedure is crucial to the Everettian program, where the interaction of systems with their environment leads to decoherence and branching of the wave function [...] The Born Rule for probabilities, p(i) = |Psi_i@|2, isn't assumed as part of the theory; it can be derived using techniques such as decision theory [12] or self-locating uncertainty [13]". The Born rule isn't compatible with the kind of unitary evolution this Essay is assuming. So it cannot be derived. At best, the rule can be introduced ad hoc as an additional postulate, but then we have the same dual-evolution inconsistency than in the traditional Copenhagen treatment.
"The former condition is ultimately cosmological -- the universe started in a low-entropy state, which we won't discuss here". But equation (1) conserves entropy, and cannot describe evolution to current state.
"The essential observation is that, if quantum behavior is distinguished from classical behavior by the presence of entanglement, classical behavior may be said to arise when entanglement is relatively unimportant". Another old argument has been refuted again and again in the literature. Eliminating the entanglement simply provides a mixture with only diagonal elements in the density matrix, but this is not the classical state; reason why the founding fathers of QM introduced an additional postulate to complement the postulate about Schrödinger evolutions.
Next section is based in many assumptions and unproven statements, some of them explicitly assumed "although it's unclear how to achieve this at this time".
WdW equation is incorrect. In fact one of his authors even renounced to it considering it "a very bad equation". Further attempts to extract a valid concept of time from it are condemned to failure. The problem is not the "clock ambiguity", the real problem is that time is an evolution parameter, not an observable one can get from a spectrum.
