Quantum mechanics as a measure theory: The theory of quantum measure

Quantum mechanics as a measure theory: The theory of quantum measure

Vasil Penchev, assoc. prof., DSc vasildinev@gmal.com , vaspench@abv.bg http://vasil7penchev.wordpress.com http://www.scribd.com/vasil7penchev CV: http://old-philosophy.issk-bas.org/CV/cvpdf/V.Penchev-CV-eng.pdf

Mechanical motions: • All motions should be smooth motions of particles in smooth trajectories according to classical mechanics as well as according to special and general relativity • Einstein’s principle of relativity (1916-1918) is the most general expression of that viewpoint on mechanical motion: All physical laws should be invariant to any smooth transformation (physically interpreted as the transformation between two reference frames moving to each other with an exactly defined speed which should be less than that of light)

The exception for the reference frame of light (electromagnetic radiation) The transformations between the reference frame of light (electromagnetic radiation) and another reference frame: either are not finite if the other reference frame is linked to a body with nonzero mass at rest or are identical if the other reference frame is linked to light (electromagnetic radiation)

Quantum-mechanical motions The quantum-mechanical motions include smooth, continuous or discrete motions in smooth, continuous or discrete trajectories for the Plank constant in quantum mechanics Einstein’s principle of relativity should be generalized so that to involve discrete transformations (physically interpretable as quantum leaps between two reference frames)

Mechanical motions and measures • The “naïve” viewpoint on the world shared by all people inexperienced in philosophy or science is that there are motions in the world, which is changed, but their measures or measuring units are unchangeable • However one can be tempted to postulate the opposite: the world is unchangeable, and the measures or measuring units are what are changed, or even to combine both viewpoints on the world: Both the world and its measures are changed and that joint change is visible as motions

Quantum mechanics as a measure theory • Furthermore that one can apply the new approach to quantum mechanics to generalized the notion of measure than that of motion as above • First the new approach is at least equivalent to the classical viewpoint on the world as external (or mathematically said, contrarvariant) motions.

In fact it turns out to be more general since the new measure resolves a series of other problems in philosophy

Problem statement: Mathematics arises as two fundamental kinds of a measure theory: Geometry studies how one can measure some continuous quantity like a length or an area or a volume Arithmetic investigates how one has to count, that is to measure some discrete quantity English (unlike Bulgarian) expresses very clearly the difference between an indefinite continuous and an indefinite discrete quantity:

Much or Many?

An indefinite continuous quantity will be called further a “much” or simply, a much, and correspondingly an indefinite discrete quantity a “many” – a many In terms of much and many, the problem statement would sound approximately so: Which is what is both much and many? Which or what is that measure which can measure equally well both a much and a many?

Infinity: Much or Many?

Thoralf Skolem (1922) introduced “the relativity of the notion of set” meaning that any infinite set can be represented as a countable set utilizing the axiom of choice or any “much” as a “many”: So infinity can unify the measurement of discrete and continuous quantities, which turn out of “both sides” of the axiom of choice Quantum measure should do the same about finite quantities. One will be able to see that the axiom of choice figures again in that case, too

The problem statement Quantum measure should be able to unify any much and any many as their joint measure In fact quantum mechanics is forced to resolve this general problem to unify discrete (quantum) and smooth (classical) motion. It introduces Hilbert space and thus implicitly quantum measure However quantum measure unifies not only the discrete and continuous, but also probability and quantity and even the real and virtual, quality and quantity, nothing and anything

The unit of quantum measure: a qubit The definition of a qubit (i.e. quantum bit) is: 1 đ?‘žđ?‘˘đ?‘?đ?‘–đ?‘Ą = Îą 0 + β 1 where 0 , 1 are any two orthogonal subspaces of Hilbert space, and Îą, β are any two complex numbers such that Îą 2 + β 2 = 1 In fact a qubit is isomorphic to a unit sphere (ball) with a point chosen on it Hilbert space can be represented by qubits as a tape (as in a Turing machine) of qubits (replacing the bits of a Turing tape)

Bit and Qubit Qubit can be thought also as a kind of generalization of the unit of information: 1 bit 1 bit is the choice between two disjunctive and equiprobable alternatives usually designated as “0” and “1” 1 qubit is the choice between an infinite number of disjunctive and equiprobable alternatives usually designated as the continuum of points of a unit sphere Thus as one bit is the unit of information as one qubit is the unit of quantum information

Information and quantum information The quantity of quantum information is a nontrivial generalization of that of information or entropy for the trivial generalization is impossible. Indeed: đ?‘?

đ??ź= đ?‘?

đ?‘„đ??ź = đ?‘

đ?‘„đ??ź = đ?‘

đ?‘

đ?‘“ đ?‘Ľ đ?‘™đ?‘œđ?‘”đ?‘Ž đ?‘” đ?‘Ľ đ?‘‘đ?‘Ľ

0(đ?‘Ľ)đ?‘™đ?‘œđ?‘”∞ ∞(đ?‘Ľ)đ?‘‘đ?‘Ľ =? ? ? đ?‘?

∞(đ?‘Ľ)đ?‘™đ?‘œđ?‘”∞ 0(đ?‘Ľ)đ?‘‘đ?‘Ľ =? ? ?

Wave function is the quantity of quantum information The next question could be: If the measure are qubits, how does the measured look like?

Since quantum measure is a tape (vector) of qubits, it represents an orthonormal basis of Hilbert space, and the measured by it is a point in it, i.e. a wave function Consequently any physical quantity is a change of wave function: The world is only a collection of changes of quantum information, nothing else

Quantum information and quantum quantity (observable) The definition of quantity (observable) in quantum mechanics as a selfadjoint operator adds a condition: The change of quantum information has to be invariant between conjugates cutting down half the variables of classical mechanics Furthermore the change of quantum information has to be equal to that of quantum entropy: Observable in quantum mechanics equates order and disorder: It has an equal value for the change of either of them

The universal substance of quantum information If qubit can be the joint measure of the discrete and continuous, of the real and possible, of probability and quantity, of disorder and order, of quality and quantity, and even of nothing and anything, quantum information is the universal substance of the world: Anything in its fundament is quantum information, and furthermore even nothing is quantum information, too So quantum mechanics resolving a scientific problem has been forced to resolve the most fundamental philosophical problem of what underlies the world

All physical processes are the computations of a quantum computer Hilbert space has an equivalent of a quantum Turing machine, in the infinite tape of which all bits are replaced by qubits: Any state of that quantum computer is a wave function Any computation defined as the transition from a state to another is a selfadjoint operator in Hilbert space and thus an observable (a quantity in quantum mechanics) Consequently all physics for its substance of quantum information can be interpreted as the computations of the quantum computer of the universe and as mathematics

Quantum measure is nonlocal Given the axiom of choice, any qubit is equivalent to all Hilbert space (for the so-called Banach - Tarski paradox) Not given the axiom of choice, any qubit cannot be equivalent to all Hilbert space, and quantum measure is local Since the axiom of choice is both valid and not valid to quantum measure, it manages to measure both locally and nonlocally (globally), i.e. it is the joint measure of a whole and of any element in it

The Aharonov – Bohm effect explained by quantum measure Quantum measure claims to represent a common viewpoint to nothing and anything: If so then it has to offer an explanation for ABE since it shows how nothing (absolute vacuum) affects a thing like an electromagnetically charged particle Of course it can be explained classically and locally involving a new physical quantity like electromagnetic potential as a definite integral of electromagnetic field (more exactly of electric field E and magnetic induction B). Its integration range can be a finite or infinite space-time region

The case of a finite integration range Then the electromagnetic field in this range only converges to zero (which is experimentally equivalent to zero) and the electromagnetic potential can have some finite value over the area of the integration range So there is a thing (namely, the electromagnetic potential), and not nothing, which can affect on the charged particle passing through that finite domain Then the explanation of the Aharanov – Bohm effect is local

The case of an infinite integration range Then the explanation of ABE is nonlocal since it supposes some nonzero physical influence from beyond the light cone admissible for the phenomena of entanglement In that case the value of electromagnetic field can be exactly zero everywhere in the integration range and nevertheless the electromagnetic potential can have some finite value originating from infinity So again there is a thing (namely, the electromagnetic potential), and not nothing, which can affect on the charged particle passing through that domain

Objections: The main objection to that kind of explanations is ... Occam’s razor: A new entity (electromagnetic potential) is involved ad hoc to be explained an experimental effect and then it is confirmed by this effect, which is a vicious circle One does so to be conserved at any price the prejudice that “there is not anything from nothing” Now we will try to abandon this prejudice to see what will happen ... 

The first step: space-time as a cause Three steps are necessary where the first one can conserve the prejudice yet as long as space-time is accepted as a matter with nonzero energy: Indeed quantum mechanics allows of ascribing an energy to any space-time region so:

đ?&#x;? đ??Ąđ?’„ đ?‘Ź = đ?’‰đ??‚ = đ?’‰ = đ?’• đ??ˆ đ?‘Ź – energy, đ?’‰ – the Planck constant, đ??‚ – frequency, đ?’• – time, đ?’„ – the light speed in vacuum, đ??ˆ – the radius of a space-time region

The interpretation Please notice that the energy of space-time should decrease with increasing the size of the space-time region The moving particle comprises a larger and larger space-time region with a smaller and smaller energy đ?‘Ź(đ??ˆ) as above: This is equivalent to the action of a force on the particle for the conservation of energy: đ?’…đ?‘Ź đ?‘­=− đ?’…đ??ˆ

Why be this force electromagnetic so that to affect only on electromagnetically charged particles, though?

The explanation follows:

The second step: Newton space measured by quantum measure Minkowski space can be obtained if one measures Newton’s absolute space by quantum measure: Indeed Minkowski space represents an expanding ball, which fills gradually the usual three-dimensional Euclidean space. The speed of this expansion is that of light in vacuum It has not one center for Lorentz invariance: Any point in Minkowski space can be accepted as that center Since the expansion is smooth, it can be represented by a countable set of balls or qubits

Space-time as an electromagnetic field If space-time is Minkowski space as special relativity implies, it can be interpreted as an electromagnetic field: Consequently it can affects only particles electromagnetically charged, and the Aharonov – Bohm effect is what describes that physical influence originating from the space-time itself That explanation does not involve a new entity, which Occam’s razor will cut off, but links two old and independently well-confirmed ones: space-time and electromagnetic field

The third step: The transformation of a “much” into space-time However one has to distinguish space-time as an electromagnetic field from any other electromagnetic field already within it: Space-time appears measuring any “much” by quantum measure and thus transforming it into a well-ordered “many” of space-time: The further the well-ordering has gone, the less the energy of the corresponding well-ordered region of space-time is: Consequently the energy will appear by itself or “from nothing” as a “by-product” along space-time

Conclusions: Quantum mechanics is well representable as a theory of a new kind of measure: quantum measure.

It can be interpreted as a covariant counterpart of its usual consideration The unit of quantum measure is a qubit and thus all physical quantities share a common fundament, that of quantum information, and all physical processes can be interpreted as computational ones That interpretation of quantum mechanics is fruitful and heuristic: This can be visualized by an explanation of the Aharonov – Bohm effect