We combine the ideas of Dirac's orthonormal representation, Everett's relative state, and 't Hooft's ontological basis to define the notion of a world for quantum mechanics. Mathematically, for a quantum system $\mathcal{Q}$ with an associated Hilbert space $\mathbb{H},$ a world of $\mathcal{Q}$ is defined to be an orthonormal basis of $\mathbb{H}.$ The evolution of the system is governed by Schrödinger's equation for the worlds of it. An observable in a certain world is a self-adjoint operator diagonal under the corresponding basis. Moreover, a state is defined in an associated world but can be uniquely extended to the whole system as proved recently by Marcus, Spielman, and Srivastava. Although the states described by unit vectors in $\mathbb{H}$ may be determined in different worlds, there are the so-called topology-compact states which must be determined by the totality of a world. We can apply the Copenhagen interpretation to a world for regarding a quantum state as an external observation, and obtain the Born rule of random outcomes. Therefore, we present a mathematical formalism of quantum mechanics based on the notion of a world instead of a quantum state.We combine the ideas of Dirac's orthonormal representation, Everett's relative state, and 't Hooft's ontological basis to define the notion of a world for quantum mechanics. Mathematically, for a quantum system $\mathcal{Q}$ with an associated Hilbert space $\mathbb{H},$ a world of $\mathcal{Q}$ is defined to be an orthonormal basis of $\mathbb{H}.$ The evolution of the system is governed by Schrödinger's equation for the worlds of it. An observable in a certain world is a self-adjoint operator diagonal under the corresponding basis. Moreover, a state is defined in an associated world but can be uniquely extended to the whole system as proved recently by Marcus, Spielman, and Srivastava. Although the states described by unit vectors in $\mathbb{H}$ may be determined in different worlds, there are the so-called topology-compact states which must be determined by the totality of a world. We can apply the Copenhagen interpretation to a world for regarding a quantum state as an external observation, and obtain the Born rule of random outcomes. Therefore, we present a mathematical formalism of quantum mechanics based on the notion of a world instead of a quantum state.

arXiv:1504.08059Apr 30, 20150 citations

Quantum Mechanics and The Big World

van Jasper Wezel

Amsterdam University Press eBooksJan 1, 20076 citations

The Quantum Theory of Reproduction – How Unique is an Individual?

Zouhair Odeh Amarin

Our understanding of nature’s way is founded on quantum mechanics. In its existence of over 80 years, quantum theory has been describing the physical world. The attraction of studying quantum mechanics is the perception of the conceptual structure of nature. This is aided by the mathematical structure that exposes the internal logic of the subject by inventing a notation that embeds the philosophy of the question. To describe how unique each individual is. A calculation method was applied. The uniqueness of an individual is one in two nonillion, octillion, septillion, sextillion, quintillion, quadrillion, trillion, billion, million and thousand. Individuals are indefinitely unique.Our understanding of nature’s way is founded on quantum mechanics. In its existence of over 80 years, quantum theory has been describing the physical world. The attraction of studying quantum mechanics is the perception of the conceptual structure of nature. This is aided by the mathematical structure that exposes the internal logic of the subject by inventing a notation that embeds the philosophy of the question. To describe how unique each individual is. A calculation method was applied. The uniqueness of an individual is one in two nonillion, octillion, septillion, sextillion, quintillion, quadrillion, trillion, billion, million and thousand. Individuals are indefinitely unique.

IntechOpen eBooksJul 18, 20220 citations

Quantum mechanics and its philosophical implications

Tofigh Allahviranloo, Sovan Samanta, Prasenjit Mandal +1 more

Elsevier eBooksJan 1, 20260 citations

Can Many-Worlds Survive a Quantum Doomsday

Brett Maynard Bevers

A novel puzzle for the notion of probability in the Many-Worlds interpretation of quantum mechanics is presented. The puzzle makes use of a thought experiment that some have claimed would provide empirical support for many-worlds over alternatives. It is argued that, if the predictions of Many-Worlds do indeed differ from other inter-pretations as claimed, then Born’s rule must generally be invalid in Many-Worlds. It is shown that the thought experiment provides a counter example for recent decision-theoretic arguments that purport to establish Born’s rule. Finally, it is shown that the puzzle can be grounded in general considerations regarding the nature of prediction in Many-Worlds. The Many-Worlds interpretation (MW) of quantum mechanics is routinely mentioned alongside various hidden-variable, collapse and modal formulations as a proposed solution to the quantum measurement problem. The MW is unique among these alternatives, how-ever, for the fact that there is serious contention over what predictions the theory makes. Though Hugh Everett III is credited with first proposing a many-worlds interpretation,A novel puzzle for the notion of probability in the Many-Worlds interpretation of quantum mechanics is presented. The puzzle makes use of a thought experiment that some have claimed would provide empirical support for many-worlds over alternatives. It is argued that, if the predictions of Many-Worlds do indeed differ from other inter-pretations as claimed, then Born’s rule must generally be invalid in Many-Worlds. It is shown that the thought experiment provides a counter example for recent decision-theoretic arguments that purport to establish Born’s rule. Finally, it is shown that the puzzle can be grounded in general considerations regarding the nature of prediction in Many-Worlds. The Many-Worlds interpretation (MW) of quantum mechanics is routinely mentioned alongside various hidden-variable, collapse and modal formulations as a proposed solution to the quantum measurement problem. The MW is unique among these alternatives, how-ever, for the fact that there is serious contention over what predictions the theory makes. Though Hugh Everett III is credited with first proposing a many-worlds interpretation,

Jun 9, 20111 citation

\mathcal{Q}

with an associated Hilbert space

\mathbb{H},

a world of

\mathcal{Q}

is defined to be an orthonormal basis of

\mathbb{H}.

The evolution of the system is governed by Schrödinger's equation for the worlds of it. An observable in a certain world is a self-adjoint operator diagonal under the corresponding basis. Moreover, a state is defined in an associated world but can be uniquely extended to the whole system as proved recently by Marcus, Spielman, and Srivastava. Although the states described by unit vectors in

\mathbb{H}

may be determined in different worlds, there are the so-called topology-compact states which must be determined by the totality of a world. We can apply the Copenhagen interpretation to a world for regarding a quantum state as an external observation, and obtain the Born rule of random outcomes. Therefore, we present a mathematical formalism of quantum mechanics based on the notion of a world instead of a quantum state.