The concept of measurement in quantum mechanics is seen from a few philosophical and mathematical viewpoints further: quantum invariance, the foundation of mathematics and the axiom of choice, and quantum measure. Then a few general and more particular philosophical problems can be guided in a new direction. Those are:
- Quantum gravity: Can general relativity be considered as a theory of quantum gravity?
- The laws of conservation: Can quantum mechanics offer a more general law than that of conservation of energy (energy-momentum) so that to be consistent with general relativity where energy-momentum is conserved only locally?
- The principle of relativity: Can the Einstein relativity principle be generalized in a way to comprise quantum movements as well?
- Invariance to discreteness or continuity: Can quantum mechanics offer a more general viewpoint to unify continuous (or smooth) and discrete (quantum) motions?
- The alternatives of the “Big Bang”: Can the universe arise necessarily from nothing, in mathematical laws?
- Nothing and anything: Can nothing and anything have a common measure?
- Quantum information: Is quantum information that quantity both physical and mathematical one, which uses that measure?
- Mathematics and physics: Is there a smooth transition between them, in which a mathematical structure or its element like a number or a probability distribution can transform into a physical entity like a particle?
- The foundation of mathematics: Can set theory be “repaired” in a way to include directly quantum correlations or entanglement between the elements of a primary mathematical structure like ‘set’?
- The interpretation of Hilbert space: Can one think Hilbert space as that structure in the frameworks of the set-theory based mathematics, which is both the simplest and may involve quantum correlations or entanglement?
- Quantum measurement and the axiom of choice:
The theorem of Kochen – Specker (1968) and yet John von Neumann’s (1932) one before it require the absence of any hidden variables in any quantum state before measurement and thus they exclude any well-ordering in it. In fact any well-ordering before measurement is excluded by any theory utilizing differential equations to describe the states in the studied area. It can happen only if the axiom of choice is added externally for the smooth continuum necessary for any differential equation cannot be well-ordered otherwise than by it. However the difference between a point of the continuum and a corresponding point in a well-ordered subset of it can converge to zero. That way out is forbidden to quantum mechanics because of the Plank constant for it imposes a finite difference between the states before and after measurement. Furthermore any set of measured results is well-ordered, e.g. by the exact time of recording in a central computer.
If one combines the Kochen – Specker theorem with the well-ordering after measurement, the well-ordering theorem equivalent to the axiom of choice is unavoidable: So quantum measurement involves necessarily the axiom of choice. One can object that the sets before and after measurement are different so that the well-ordering theorem is irrelevant. In fact even then the mapping between a set, which cannot be well-ordered in principle, and that, which is always well-ordered, requires the axiom of choice for the Cartesian product between them to exist, of which the mapping is a subset. So the measurement of any quantum system or state designated shortly as ‘quantum measurement’ cannot be free of the axiom of choice.
Since the Kochen – Specker theorem does not admit the axiom of choice in any quantum system or state by itself, i.e. before measurement, and quantum measurement requires it as above, quantum mechanics has to be invariant to the axiom of choice: Any statement or equation in it has to be equally valid both if the axiom of choice is accepted or not.
If one combines the Kochen – Specker theorem with the well-ordering after measurement, the well-ordering theorem equivalent to the axiom of choice is unavoidable: So quantum measurement involves necessarily the axiom of choice. One can object that the sets before and after measurement are different so that the well-ordering theorem is irrelevant. In fact even then the mapping between a set, which cannot be well-ordered in principle, and that, which is always well-ordered, requires the axiom of choice for the Cartesian product between them to exist, of which the mapping is a subset. So the measurement of any quantum system or state designated shortly as ‘quantum measurement’ cannot be free of the axiom of choice.
Since the Kochen – Specker theorem does not admit the axiom of choice in any quantum system or state by itself, i.e. before measurement, and quantum measurement requires it as above, quantum mechanics has to be invariant to the axiom of choice: Any statement or equation in it has to be equally valid both if the axiom of choice is accepted or not.
- Quantum measurement and quantum invariance:
This unique relation of quantum mechanics to the axiom of choice is due to the true fundament of it, wave-particle duality and thus, to involving quantum leaps in mechanical motion. One can speak of quantum invariance underlying the above noticed invariance to the axiom of choice for quantum measurement. The wave-particle duality has another side, which can be designated as the discrete-smooth duality of mechanical motion, since there is an exact correspondence of a wave function to a quantum leap (a discrete motion) as well as of a particle to a smooth trajectory (a world line). Thus that discrete-smooth duality is a kind of generalization as to Einstein’s (1918) principle of relativity, which requires the invariance of all physical laws only to smooth relative motions (reference frames), so that it comprises already also quantum leaps.
That new and more general invariance is quantum invariance: It means that all physical laws have to be invariant to any, both discrete and continuous (smooth), transformation between two or more reference frames. The generalization pioneers the simplest pathway between general relativity and quantum mechanics, i.e. that of quantum gravity.
Any discrete motion does not allow defining a finite value of relative speed. If yet it is defined, this excludes to determine the distance of the leap. Both complementary restrictions constitute together the Heisenberg principle of uncertainty. This uncertainty is a new and unique physical variable, a free variable unboundable in principle, by a natural law such as uncertainty. Its physical dimension is action: The physical quantity of action is exceptional and singular since it is as the dimension of that unboundable free variable (in quantum mechanics) as the dimension of the corresponding bound variable (in general relativity) of the same name and dimension. Consequently what is conserved passing from each to the other theory is action in a rather extraordinary way of conservation: A dimensionless physical quantity like entropy is transformed in its reciprocal also dimensionless physical quantity like information and vice versa. Philosophically said the disorder of entropy is transformed in the order of information and vice versa conserving action as in each of both theories as between them: The theory of quantum gravity turns out to be the theory of quantum information.
Furthermore the mutual transformation between entropy and information or between disorder and order in the framework of conservation of action describes well quantum measurement and quantum invariance, and even the invariance to the axiom of choice: Indeed quantum measurement transforms an initial fundamental disorder of coherent state into the order of a well-ordering of measured results, and quantum invariance means the equivalence of the disorder of probable values in a quantum leap and of the determined order of a smooth trajectory.
That new and more general invariance is quantum invariance: It means that all physical laws have to be invariant to any, both discrete and continuous (smooth), transformation between two or more reference frames. The generalization pioneers the simplest pathway between general relativity and quantum mechanics, i.e. that of quantum gravity.
Any discrete motion does not allow defining a finite value of relative speed. If yet it is defined, this excludes to determine the distance of the leap. Both complementary restrictions constitute together the Heisenberg principle of uncertainty. This uncertainty is a new and unique physical variable, a free variable unboundable in principle, by a natural law such as uncertainty. Its physical dimension is action: The physical quantity of action is exceptional and singular since it is as the dimension of that unboundable free variable (in quantum mechanics) as the dimension of the corresponding bound variable (in general relativity) of the same name and dimension. Consequently what is conserved passing from each to the other theory is action in a rather extraordinary way of conservation: A dimensionless physical quantity like entropy is transformed in its reciprocal also dimensionless physical quantity like information and vice versa. Philosophically said the disorder of entropy is transformed in the order of information and vice versa conserving action as in each of both theories as between them: The theory of quantum gravity turns out to be the theory of quantum information.
Furthermore the mutual transformation between entropy and information or between disorder and order in the framework of conservation of action describes well quantum measurement and quantum invariance, and even the invariance to the axiom of choice: Indeed quantum measurement transforms an initial fundamental disorder of coherent state into the order of a well-ordering of measured results, and quantum invariance means the equivalence of the disorder of probable values in a quantum leap and of the determined order of a smooth trajectory.
- Quantum measurement and the foundation of mathematics:
The mutual transformation of order and disorder as the invariance to the axiom of choice deserves to be independently described for it can refer to the foundation of mathematics and thus to a new understanding how mathematics and physics are connected in reality and in quantum measurement. In fact the invariance to the axiom of choice is well known in the foundation of mathematics for a long time as the so-called paradox of Skolem (1922). He introduced the “relativity of the notion of set” meaning that any infinite set can be enumerated by the axiom of choice or even to be interpreted as a finite one. So a bridge exists from quantum measurement to a new, quantum foundation of mathematics. Its main idea is to borrow from nature the way how mathematics is founded in it generalizing the well-ordered set of natural numbers to that of qubits, which is equivalent to Hilbert space:
In other words, the idea is the founding set of natural numbers, which is always countable and thus it is noninvariant to the axiom of choice, to be replaced by the simplest one, which is invariant to the axiom of choice being uncountable and countable as a quantum coherent state before and after measurement. Such a one is Hilbert space, or the set of all well-ordered series of qubits. A point in it (or a sequence of qubits, or a wave function) can represent equally well both a coherent state before measurement and its corresponding statistical ensemble after measurement. That point is invariant to the transformation between entropy (E) and information (I) if and only if the definition of entropy and information is modified in a way to be invariant to the reciprocal transformation of their variables: . The Shannon type definition is not invariant so. However, the definition of physical quantity or observable in quantum mechanics by a selfadjoint operator is invariant just so. Consequently the latter is to be accepted as the relevant definition of information at least as to the quantum foundation of mathematics.
That mathematics founded in this way cannot involve undecidable statements since decidability can be generalized as the invariance of entropy and information to the reciprocal transformation of their variables as above. Furthermore any statement Gödel codable can be coded as a series of qubits and thus undecidable statements cannot exist in that mathematics. Furthermore it cannot be divided from physics in the bridge of quantum mechanics and thus from reality. The conception of quantum measurement serves as the base of total decidability. It leads to some kind of quantum Pythagoreanism.
In other words, the idea is the founding set of natural numbers, which is always countable and thus it is noninvariant to the axiom of choice, to be replaced by the simplest one, which is invariant to the axiom of choice being uncountable and countable as a quantum coherent state before and after measurement. Such a one is Hilbert space, or the set of all well-ordered series of qubits. A point in it (or a sequence of qubits, or a wave function) can represent equally well both a coherent state before measurement and its corresponding statistical ensemble after measurement. That point is invariant to the transformation between entropy (E) and information (I) if and only if the definition of entropy and information is modified in a way to be invariant to the reciprocal transformation of their variables: . The Shannon type definition is not invariant so. However, the definition of physical quantity or observable in quantum mechanics by a selfadjoint operator is invariant just so. Consequently the latter is to be accepted as the relevant definition of information at least as to the quantum foundation of mathematics.
That mathematics founded in this way cannot involve undecidable statements since decidability can be generalized as the invariance of entropy and information to the reciprocal transformation of their variables as above. Furthermore any statement Gödel codable can be coded as a series of qubits and thus undecidable statements cannot exist in that mathematics. Furthermore it cannot be divided from physics in the bridge of quantum mechanics and thus from reality. The conception of quantum measurement serves as the base of total decidability. It leads to some kind of quantum Pythagoreanism.
- Quantum measurement and quantum measure:
The unit of quantum measurement is specific, quite different from the classical one even as the mathematical notion of measure: It is not a Lebesgue or Borel measure. This is quantum measure, which is complete as the former and universal as the latter. It is neither arbitrarily dimensional nor one-dimensional. It is three-dimensional. Its universal unit is a qubit isomorphic to a unit ball. This is the unit of quantum information: So quantum measure serves as the quantity of quantum information of any measured. Indeed it is really universal. It can measure equally well both the disorder of entropy and the order of information, both anything, which exists being actual, and nothing, which does not exist, being virtual, only possible and probable. Consequently, quantum measure can describe the “genesis from nothing” as a process, of course not as a process in time, because that is the genesis of time: For example, the “Big Bang” is not more than a visualization of quantum measure onto the “screen” of the usual understanding of measure.
No comments:
Post a Comment