Prehistory and background:
The essence of two fundamental physical theories, quantum mechanics and general relativity is concentrated in their basic equations: The Schrödinger equation (SE) and the Einstein field equation (EFE). The mutual consistency of both theories is one of the biggest open problems in physics and its philosophy.
The mathematical equivalence of SE and EFE is provable under the axiom of choice. SE and EFE can be considered correspondingly as the nonstandard and standard interpretations (in the sense of Robinson’s nonstandard analysis) of one and the same mathematical structure. The physical sense of their equivalence consists in the equivalent transformation between a smooth continuum of inertial reference frames (as what space-time is considered in general relativity by pseudo-Riemannian space) and complementary pairs of qubits, the one member of which corresponds to the velocity (or dynamically, to the momentum) of a single reference frame, and the other to its position. The quantity of time is reduced to the well-ordering of those pairs for the axiom of choice. Thus, the axiom of choice is what reconciles the space-like time in general relativity to the arrow-like time in quantum mechanics and allows of their equating.
The axiom of choice, itself is interpreted physically as the choice of a single value after measurement by the measurement itself from the coherent whole of values before measurement. Indeed, the theorems of the absence of hidden variables in quantum mechanics exclude any well-ordering before measurement, while the results after measurement are always well-ordered according to the moments of registration time. The former can be mapped into the latter unambiguously only under the well-ordering principle equivalent to the axiom of choice.
The purely mathematical equivalence of SE and EFE might be as an occasional, eccentric, and curious, but meaningless coincidence (as the “channels on Mars”) as an expression of a certain deep ontological essence underlying both theories. The talk is intended to the research of the latter alternative and ontological conclusions implied by it.
Thesis:
The standard (EFE) and nonstandard (SE) interpretations can be embodied in an ontology equating the whole of the universe to the whole of a quantum. Then, EFE represents reality inside that whole, and SE outside it. The universe is situated within in a quantum.
A comment of the thesis:
Possible objections: (i) A quantum is the smallest, and the universe the biggest in our physics and its metaphysics. (ii) Both mass and energy of what is “alleged” to be inside of the universe, are about hundreds of exponents bigger than what “as if” is outside of a quantum. (iii) What is outside, a quantum, is inside, too, even as many, many, many quanta within the universe. (iv) The experimentally very well confirmed conceptions of the “Big Bang” and the expanding universe seem to be nonsense in the internality of a quantum.
A few arguments in favor of the thesis:
(1) That ontology is consistent to the mathematical equivalence of SE and EFE though it is not implied by their equivalence.
(2) Many examples of philosophical doctrines equating the biggest and the smallest (as Nicolas of Cusa) or introducing cyclicality (as Nietzsche) can be referred.
(3) Poincaré’s conjecture proved by Grigory Perelman implies for the topological equivalence (physically interpretable as equivalence in causality) of the usual three- dimensional Euclidean space and pseudo-Riemannian space of general relativity, the necessity of cyclic closure of the latter. This means right the equivalence of both biggest and smallest.
(4) Both mass and energy of a quantum cannot be directly compared with those of the universe because the physical dimension of a quantum (the Planck constant) is action, which depends of both energy (mass) and time. Time is right that physical quantity, which should not be compared directly between the non-standard interpretation outside of the whole and the standard interpretation inside of it for the time is continuous (smooth) inside and discrete outside in definition.
(5) A certain and unknown dimensionless fundamental constant is necessary to determine the smooth length of a unit of time inside per a discrete unite of time outside. Different conjectures are possible for it: e.g. that constant can be defined just postulating the equality of the actions of the universe and a quantum (the Planck constant) or it can be recognized as a certain one among the known fundamental physical constants (first of all, among the thermodynamic ones, such as Boltzmann’s).
(6) The totality such as the universe right being just ‘totality’ should contain itself in the final analysis for any externality of it contradicts to its definition. The transition from the externality to the internality of the universe implies the parallel one-to-many transformation: one single quantum “outside” into many, many, many quanta “inside”. The latter can be interpreted as “many worlds” or “many universes” as the different states in the whole of the universe (also in the thermodynamic sense of Gibbs). Any quantum “inside” generates a separable complex Hilbert space, which can be non-collinear to the others and thus, entangled to them. That general entanglement transferred from the non-standard to standard interpretation right represent gravity just as it is described by general relativity.
(7) The ontological borders of the universe as a quantum can be thought mathematically as those between finiteness and infinity. Indeed, the visible universe is necessary finite though exceptionally immense, while the nonstandard interpretation physically interpreted as quantum reality needs necessarily infinity. The conception of both beginning (the “Big Bang”) and expansion (the “expanding universe”) of a finiteness such as the universe within infinity such as a quantum is not nonsense.
(8) Skolem’s conception about the “relativity of ‘set’”, implying particularly one-to-one mappings of infinite sets into finite ones, can explain the transformation of the quantum “outside” into quanta “inside” as existing only “purely mathematically” because of the axiom of choice and representable constructively and physically only probabilistically.
The essence of two fundamental physical theories, quantum mechanics and general relativity is concentrated in their basic equations: The Schrödinger equation (SE) and the Einstein field equation (EFE). The mutual consistency of both theories is one of the biggest open problems in physics and its philosophy.
The mathematical equivalence of SE and EFE is provable under the axiom of choice. SE and EFE can be considered correspondingly as the nonstandard and standard interpretations (in the sense of Robinson’s nonstandard analysis) of one and the same mathematical structure. The physical sense of their equivalence consists in the equivalent transformation between a smooth continuum of inertial reference frames (as what space-time is considered in general relativity by pseudo-Riemannian space) and complementary pairs of qubits, the one member of which corresponds to the velocity (or dynamically, to the momentum) of a single reference frame, and the other to its position. The quantity of time is reduced to the well-ordering of those pairs for the axiom of choice. Thus, the axiom of choice is what reconciles the space-like time in general relativity to the arrow-like time in quantum mechanics and allows of their equating.
The axiom of choice, itself is interpreted physically as the choice of a single value after measurement by the measurement itself from the coherent whole of values before measurement. Indeed, the theorems of the absence of hidden variables in quantum mechanics exclude any well-ordering before measurement, while the results after measurement are always well-ordered according to the moments of registration time. The former can be mapped into the latter unambiguously only under the well-ordering principle equivalent to the axiom of choice.
The purely mathematical equivalence of SE and EFE might be as an occasional, eccentric, and curious, but meaningless coincidence (as the “channels on Mars”) as an expression of a certain deep ontological essence underlying both theories. The talk is intended to the research of the latter alternative and ontological conclusions implied by it.
Thesis:
The standard (EFE) and nonstandard (SE) interpretations can be embodied in an ontology equating the whole of the universe to the whole of a quantum. Then, EFE represents reality inside that whole, and SE outside it. The universe is situated within in a quantum.
A comment of the thesis:
Possible objections: (i) A quantum is the smallest, and the universe the biggest in our physics and its metaphysics. (ii) Both mass and energy of what is “alleged” to be inside of the universe, are about hundreds of exponents bigger than what “as if” is outside of a quantum. (iii) What is outside, a quantum, is inside, too, even as many, many, many quanta within the universe. (iv) The experimentally very well confirmed conceptions of the “Big Bang” and the expanding universe seem to be nonsense in the internality of a quantum.
A few arguments in favor of the thesis:
(1) That ontology is consistent to the mathematical equivalence of SE and EFE though it is not implied by their equivalence.
(2) Many examples of philosophical doctrines equating the biggest and the smallest (as Nicolas of Cusa) or introducing cyclicality (as Nietzsche) can be referred.
(3) Poincaré’s conjecture proved by Grigory Perelman implies for the topological equivalence (physically interpretable as equivalence in causality) of the usual three- dimensional Euclidean space and pseudo-Riemannian space of general relativity, the necessity of cyclic closure of the latter. This means right the equivalence of both biggest and smallest.
(4) Both mass and energy of a quantum cannot be directly compared with those of the universe because the physical dimension of a quantum (the Planck constant) is action, which depends of both energy (mass) and time. Time is right that physical quantity, which should not be compared directly between the non-standard interpretation outside of the whole and the standard interpretation inside of it for the time is continuous (smooth) inside and discrete outside in definition.
(5) A certain and unknown dimensionless fundamental constant is necessary to determine the smooth length of a unit of time inside per a discrete unite of time outside. Different conjectures are possible for it: e.g. that constant can be defined just postulating the equality of the actions of the universe and a quantum (the Planck constant) or it can be recognized as a certain one among the known fundamental physical constants (first of all, among the thermodynamic ones, such as Boltzmann’s).
(6) The totality such as the universe right being just ‘totality’ should contain itself in the final analysis for any externality of it contradicts to its definition. The transition from the externality to the internality of the universe implies the parallel one-to-many transformation: one single quantum “outside” into many, many, many quanta “inside”. The latter can be interpreted as “many worlds” or “many universes” as the different states in the whole of the universe (also in the thermodynamic sense of Gibbs). Any quantum “inside” generates a separable complex Hilbert space, which can be non-collinear to the others and thus, entangled to them. That general entanglement transferred from the non-standard to standard interpretation right represent gravity just as it is described by general relativity.
(7) The ontological borders of the universe as a quantum can be thought mathematically as those between finiteness and infinity. Indeed, the visible universe is necessary finite though exceptionally immense, while the nonstandard interpretation physically interpreted as quantum reality needs necessarily infinity. The conception of both beginning (the “Big Bang”) and expansion (the “expanding universe”) of a finiteness such as the universe within infinity such as a quantum is not nonsense.
(8) Skolem’s conception about the “relativity of ‘set’”, implying particularly one-to-one mappings of infinite sets into finite ones, can explain the transformation of the quantum “outside” into quanta “inside” as existing only “purely mathematically” because of the axiom of choice and representable constructively and physically only probabilistically.
The presentation also as a PDF or a video; furthermore as slides @ EasyChair
A related post abouth the link of the Schrödinger equation and the Einstein field equation at Bloogger ot at WordPress
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