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It was proposed that the tensor product structure of the Hilbert space is uniquely determined by the Hamiltonian's spectrum, for most finite-dimensional cases satisfying certain conditions. I show that any such method would lead to…
In the many worlds community seems to exist a belief that the physics of a quantum theory is completely defined by it's Hamilton operator given in an abstract Hilbert space, especially that the position basis may be derived from it as…
Many-body Hilbert space is a functional vector space with the natural structure of an algebra, in which vector multiplication is ordinary multiplication of wave functions. This algebra is finite-dimensional, with exactly $N!^{d-1}$…
We argue in a model-independent way that the Hilbert space of quantum gravity is locally finite-dimensional. In other words, the density operator describing the state corresponding to a small region of space, when such a notion makes sense,…
Studies have shown that quantum states reside in a Hilbert space bundle. When a quantum system depends on continuous external parameters, these parameters define additional dimensions in the base space of the bundle. While much of the…
It is argued that the so-called holographic principle will obstruct attempts to produce physically realistic models for the unification of general relativity with quantum mechanics, unless determinism in the latter is restored. The notion…
For vector/AdS and dS holography we establish the structure of the emergent Hilbert space. This is done through implementation of finite $N$ trace relations on the infinite collective space. For fermionic theories a finite Hilbert space is…
Symmetries impose structure on the Hilbert space of a quantum mechanical model. The mathematical units of this structure are the irreducible representations of symmetry groups and I consider how they function as conceptual units of…
Generalized Fourier transformation between the position and the momentum representation of a quantum state is constructed in a coordinate independent way. The only ingredient of this construction is the symplectic (canonical) geometry of…
We construct two Hilbert spaces over the set of all metrics of arbitrary but fixed signature, defined on a manifold. Every state in one of the Hilbert spaces is built of an uncountable number of wave functions representing some elementary…
Space-time in quantum mechanics is about bridging Hilbert and configuration space. Thereby, an entirely new perspective is obtained by replacing the Newtonian space-time theater with the image of a presumably high-dimensional Hilbert space,…
We present a heuristic derivation of Born's rule and unitary transforms in Quantum Mechanics, from a simple set of axioms built upon a physical phenomenology of quantization. This approach naturally leads to the usual quantum formalism,…
Within the context of Newton's theory of gravitation, restricted to point-like test particles and central bodies, stable circular orbits in ordinary space are related to stable circular paths on a massless, unmovable, undeformable…
The quantum states or Hilbert spaces for the quantum field theory in de Sitter space-time are studied on ambient space formalism. In this formalism, the quantum states are only depended $(1)$ on the topological character of the de Sitter…
Studies have shown that the Hilbert spaces of non-Hermitian systems require nontrivial metrics. Here, we demonstrate how evolution dimensions, in addition to time, can emerge naturally from a geometric formalism. Specifically, in this…
Ergodicity breaking in isolated systems has emerged as an important frontier in the study of quantum many-body physics. While generic Hamiltonians are expected to obey the eigenstate thermalization hypothesis (ETH), recent studies on…
The holographic principle and the thermodynamics of de Sitter space suggest that the total number of fundamental degrees of freedom associated with any finite-volume region of space may be finite. The naive picture of a short distance…
In this paper, we study a possibility where gravity and time emerge from quantum matter. Within the Hilbert space of matter fields defined on a spatial manifold, we consider a sub-Hilbert space spanned by states which are parameterized by…
Hilbert spaces in theories of gravity are notoriously subtle due to the Hamiltonian constraints, particularly regarding the inner product. To demystify this subject, we review and extend a collection of ideas in canonical gravity, and…
So far, none of attempts to quantize gravity has led to a satisfactory model that not only describe gravity in the realm of a quantum world, but also its relation to elementary particles and other fundamental forces. Here, we outline the…