MetaworldsTheory

Combining Bohm and Everett: Axiomatics for a Standalone Quantum Mechanics is blend of the Everettian and Bohmian interpretations of quantum mechanics, suggesting a new conceptualization known as the "Metaworld Theory".

In Everett's theory, there are some conceptual challenges and an ill-defined ontology, while Bohmian mechanics has a clear ontology but struggles with empty branches and indeterminate particle positions. These two theories seem to have complementary issues. Thus, the text proposes a merger of these two theories.

In the proposed metaworld theory, the universe consists of many point-like particles distributed across three-dimensional space. Every distinct configuration of these particles forms a distinct "world". All these worlds superpose to form a unified entity known as a "metaworld", within which all the worlds still exist. The continuum of worlds contained in the metaworld is described by a time-dependent universal wavefunction $\Psi_t$ such that:

$$ \mu_t (Q) = \int_{q \in Q} | \Psi_t (q) |^2 dq $$

This equation yields the volume of worlds whose configuration is within the set $Q \in \mathcal{Q}$, or the world volume of $Q$, at any given time $t$. Each point in the configuration space has a density at time $t$ given by:

$$ \rho_t (q) = | \Psi_t (q) |^2 $$

The wavefunction represents a physically existing field, and its absolute square denotes the density of this field, or a density of worlds. The text proposes a notion of "metamatter", which is a stuff made of a superposition of matter. The probability that a randomly chosen world has its configuration contained in a set $Q \subset \mathcal{Q}$ is given by:

$$ P_t (Q) = \frac{\mu_t (Q)}{\mu_t (\mathcal{Q})} = \frac{\int_{q \in Q} | \Psi_t (q) |^2 dq}{\int_{q} | \Psi_t (q) |^2 dq} $$

This probability density is given by:

$$ p_t (q) = \frac{| \Psi_t (q) |^2}{\int_{q} | \Psi_t (q) |^2 dq} $$

Each world follows a Bohmian trajectory, but unlike in Everett's theory, there is no branching of worlds. A clean axiomatic representation of the theory is also provided, which includes definitions for a "world" and a "metaworld", and several postulates.

This perspective challenges and extends traditional views by proposing a more integrated and dynamic relationship between particles and the wavefunction. Key points include:

1. Fluid Metaphor for the Wavefunction: The wavefunction is likened to a nonclassical compressible fluid flowing through configuration space, offering a visualization that contrasts with the guidance provided by the wavefunction in Bohmian mechanics. In this analogy, to say particles are "guided" by the wavefunction is inadequate; it is more accurate to say they are components of the wavefunction itself, moving with its flow.

2. Ontological Integration: Unlike in Bohmian mechanics, where particles and wavefunction are ontologically distinct, in this theory, particles are considered intrinsic parts of the wavefunction—or the metaworld. This integral view suggests that particles do not just follow the wavefunction but are part of its dynamical evolution.

3. Derivation of the Velocity Equation: The velocity equation, which in Bohmian mechanics needs to be postulated separately, is derived from existing postulates in this theory. This implies a foundational difference, providing a basis for describing the motion of particles (or worlds) without additional assumptions.

4. Existence of Continuous Trajectories: The theory posits the real existence of all worlds and their histories, represented by continuous trajectories in configuration space. This contrasts with the discrete projections of histories in the consistent histories approach and with the branching worlds of Everett's many-worlds interpretation. Here, trajectories do not split but remain continuous, ensuring a unique past and future for every world.

5. Deterministic Yet Subjectively Indeterministic Mechanics: While the dynamics of the wavefunction and, by extension, the universe, are deterministic, the theory accommodates the subjective experience of indeterminism through individual particle trajectories and probabilities. This bifurcation addresses the conceptual problem of the origin of probabilities in quantum mechanics, offering a deterministic framework for objective reality alongside a probabilistic interpretation for subjective observation.

In sum, this theory proposes a unified and deterministic description of quantum mechanics, where the wavefunction is the fundamental entity encoding all physical realities. It navigates the duality of determinism and indeterminism, offering a coherent framework that integrates the ontology of particles with the dynamical evolution of the wavefunction. This approach aims to reconcile conceptual challenges in quantum mechanics, such as the nature of probabilities and the wavefunction's role, by positing a fluid-like, interconnected ontology of particles and wavefunctions.