We propose a method for finding an initial state vector which by ordinary Hamiltonian time evolution follows a single branch of many-worlds quantum mechanics. The resulting deterministic system appears to exhibit random behavior as a result of the successive emergence over time of information present in the initial state but not previously observed.
We propose a method for finding an initial state vector which by ordinary Hamiltonian time evolution follows a single branch of many-worlds quantum mechanics. The resulting deterministic system appears to exhibit random behavior as a result of the successive emergence over time of information present in the initial state but not previously observed.
Beginning with the Everett-DeWitt many-worlds interpretation of quantum mechanics, there have been a series of proposals for how the state vector of a quantum system might be split at any instant into orthogonal branches, each of which exhibits approximately classical behavior. Here we propose a decomposition of a state vector into branches by finding the minimum of a measure of the net quantum complexity of the branch decomposition. We then propose a method for finding an ensemble of possible initial state vectors from which a randomly selected member, if evolved by ordinary Hamiltonian time evolution, will follow a single sequence of those branches of many-worlds quantum mechanics which persist through time. Macroscopic reality, we hypothesize, consists of an accumulating sequence of such persistent branching results. For any particular draw, the resulting deterministic system appears to exhibit random behavior as a result of the successive emergence over time of information present in the initial state but not previously observed.
According to quantum theory, pure physical states correspond to equivalence classes of state vectors, where any two members of one class differ by a complex factor. The point is that such a factor does not change the probability for the occurrence of any measurement result as computed within the formalism of quantum mechanics. In the formalism to be presented here, the state vector does not only determine the probabilities for the occurrence of measurement results, rather it determines the result itself. The information which guides the selection process is obtained from the state vector in a way that is not invariant against multiplication by a complex factor. It can be seen as a kind of global phase of the state vectors. This global phase then is a perfectly hidden variable. For this presentation of quantum mechanics we have a remarkable parallelism to classical statistical mechanics: If we would know exactly the initial state of a system, we could compute the outcomes for all experiments to be done later on this system. In the quantum case the initial state is to be represented by its state vector with specified phase, e.g. as a concrete complex-valued wave function.
The basic mathematical structure, QM-A, of the many worlds interpretation consists solely of the linear mathematics plus the Hilbert space properties of the state vectors. There is no collapse and there are no particles or hidden variables. It is remarkable that QM-A alone can account for all our observations except probability. There is no need for particles, hidden variables or collapse to explain perception of only one classical version of reality, the photoelectric effect, localized effects from a spread-out wave function in scattering and interference experiments, wave-particle duality, and so on. But probability cannot be defined within QM-A. Nevertheless, because of its astonishing success, it seems reasonable to require (1) that the mathematics of an interpretation be limited to the highly successful QM-A and (2) that "matter" be composed of state vectors alone. But the probability law requires, in essence, that one version of the observer be singled out. The most straightforward way to accomplish this under (1) and (2) is to assume there is an aspect of the observer-the Mind-which is outside the laws of quantum mechanics and perceives just one version of reality. Under that assumption, the probability law can then be derived. Thus we have an interpretation, QM-A plus "outside" observer, which explains all our perceptions.
Classical, i.e. deterministic theories underlying quantum mechanics are considered, and it is shown how an apparent quantum mechanical Hamiltonian can be defined in such theories, being the operator that generates evolution in time. It includes various types of interactions. An explanation must be found for the fact that, in the real world, this Hamiltonian is bounded from below. The mechanism that can produce exactly such a constraint is identified in this paper. It is the fact that not all classical data are registered in the quantum description. Large sets of values of these data are assumed to be indistinguishable, forming equivalence classes. It is argued that this should be attributed to information loss, such as what one might suspect to happen during the formation and annihilation of virtual black holes. The nature of the equivalence classes follows from the positivity of the Hamiltonian. Our world is assumed to consist of a very large number of subsystems that may be regarded as approximately independent, or weakly interacting with one another. As long as two (or more) sectors of our world are treated as being independent, they all must be demanded to be restricted to positive energy states only. What follows from these considerations is a unique definition of energy in the quantum system in terms of the periodicity of the limit cycles of the deterministic model.
The hidden variable theories are, more cumbersome and artificial—while the Copenhagen interpretation is hopelessly incomplete because of its a priori reliance on classical physics, as well as a philosophic monstrosity with a “reality” concept for the macroscopic world and denial of the same for microcosm. In recent work on the Everett (Many-Worlds) interpretation of quantum mechanics, it has increasingly been recognized that any version of the interpretation worth defending will be one in which the basic formalism of quantum mechanics is left unchanged. Properties such as the interpretation of wave-function as describing a multiverse of branching worlds, or the ascription of probabilities to the branching events, must be emergent from the unitary quantum mechanics rather than added explicitly to the mathematics. Deutsch’s strategy was to introduce a few assumptions from decision theory into the Everett interpretation and then to show how they work with quantum theory to create a natural value measure that is the same as the expectation value.
In quantum mechanics it is usual to represent physical reality as a vector in Hilbert space at a particular time, with evolution being governed by Schrodinger's equation which involves an externally imposed time parameter. This leads to difficulties if one wishes to regard the universe as a quantum mechanical system, because there should no time external to such a system. The approach in this paper is to represent the totality of reality as a vector in Hilbert space. The author shows how time evolution follows, where time now is defined in terms of the states of a quantum mechanical clock which is part of the system. Rather than the correlation between the clock states and the states of the rest of the system arising because both are governed by an imposed law involving an external time parameter, it is seen that this correlation is of the separation-independent Einstein-Podolsky-Rosen type. The total reality vector, which incorporates the whole history of the system, is shown to be a zero-energy eigenstate of the system Hamiltonian. He discusses systems of finite and infinite lifetime, and is able to answer the question: what was the state before the initial state? He concludes that the quantum mechanical system of this paper is a reasonable representation of the observed universe.