Quantum Logic

An original theoretical construction provides a solid foundation to support Afshar experiments http://en.wikipedia.org/wiki/Afshar_experiment that show the violation of Bohr Complementarity Principle in modern optical interactive experiments. http://www.quantiki.org/news/variant-simulation-system-using-quaternion-structures

How many electrons in (say) Uranium?-First of 3 notes We “know” the answer. It is 92 for Uranium. The question resolves itself into three parts: 1. What is the experimental basis for the conclusion and where are the electrons located? 2. Are the electrons permanent denizens of the atom? 3. There should be 92 protons in the nucleus. Where are these located? We shall be addressing these questions serially. The conclusion that I would be driving at is that in all these counts Quantum model for atomic structure fails to explain experimental observations. Skb. 1. Experimental basis for atomic number. The concept of atomic number as the number of extra-nuclear electrons depends on the work of Moseley in Rutherford’s laboratories almost a century ago. The theoretical basis is derived from the Bohr Theory of the hydrogen atom as modified and strengthened by the wave theory of electron. These are text-book matters. (Arthur Beiser, Concepts of Modern Physics, Tata-McGrawHill, 2004; W. E Burcham, Elements of Nuclear Physics; Longman, 1979) The Bohr model predicted that the square of number of electrons (Z 2) in any atom is proportional to the frequency of radiation emitted by it. A plot of Z2 against frequency should be a straight line. This was not the case for the Moseley X-ray frequencies. However a plot of (Z-1)2 against frequency was a straight line. So it was surmised that one electron from ‘1-s’ shell (closest to the nucleus) “fell” into the nucleus under the experimental conditions giving an effective atomic number of Z-1. I am not questioning the worth of Moseley experiments. I accept that it was of first grade quality in conception and execution. The young man’s life was unfortunately cut short by WW-I. His experiments were hailed as confirmation of the Bohr concept of the nucleus as point charge with an electron cloud around it. The nucleus as a heavy central unit was indicated by Rutherford’s work. But the “fall” of an electron into the nucleus was still a difficult proposition to justify. A clever method was soon found out to resolve this problem. If Z (instead of Z2 ) is plotted against square root (√ ) of frequency, a straight line was obtained. (cited in, Dyson 1990 N. A. Dyson, X-rays in atomic and nuclear physics, 2 Ed. Cambridge Univ. Press 1990.) This conclusion is questionable since it compresses the ordinates in a convenient manner. As I like to say, “An electron was falling into the nucleus till 1980; it no longer does so after the date.” There are other objections to the interpretation: 1. The location (disposition) of the extra-nuclear electrons had not been explained in a logical manner. 2. The electronic effects are dependent on the nature and number of substituents of the central atoms in binary compounds. They appear to vanish completely in some hexafluorides including that of Uranium. 3. A recent report in Scientific American India says that X-rays are produced when one unrolls scotch tape from a spool. Such low energy generation of x-rays would throw doubt on the high energy status of X-rays in the “Electromagnetic Spectrum” of radiation. In turn it shows that we understand very little about X-rays. We should not use such uncertain concepts to arrive at an important conclusion like atomic number. Moseley results still show that there is a gradation in the x-ray frequencies in the order predicted by mass numbers of the atoms. To that extent the experiments confirm the intuitive arrangement of elements in the Periodic table. But a quantitative interpretation is not tenable. The location of the electrons in the extra nuclear space: There are only three dimensions in Space, the x, y, z axes of co-ordinate geometry. Pauli’s Exclusion Principle would limit each dimension to two electrons. With two more electrons in first shell the extra nuclear space can logically accommodate only ten electrons or up to Neon. Beyond that only quantum specialists can understand. It is all imagination promoted by mathematics. We had seen the delusional element in the imaginary number of the wave function. The shapes of D-orbital sub-units and beyond are bizarre. This is not an emotional or aesthetic objection. The theory fails completely to account for the shapes of electron orbitals in real space. It also fails to explain the bonding patterns in several elements dealt with in the paper on unitary model and the critique. Skb. PS: A pdf version of this note is available on demand. In this version some symbols like the square may not be transmitted properly.

Jest to moja autorska praca. Szukam krytyki

the truth is not only thing that we found but is thing that there is

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Example proposition The set of quantum logical propositions is lattice isomorphic with the set of closed subspaces of an infinite dimensional separatable Hilbert space. Let us try to represent a quantum logical proposition in Hilbert space. The example is: All items in universe influence each other’s position. It can be answered with either yes or no. So, it is a valid quantum logical proposition. Proving ‘yes’ is cumbersome, but the ‘no’ is hardly less difficult. It requires finding an item that is not influenced by at least one of the other items. The first problem is to determine what in the representation stands for an item. The simplest solution is to attach a subspace of the Hilbert space to the item. The corresponding proposition can be phrased as: “This is the item”. Everything that can be attributed to the item can also be attributed to this subspace. Something either belongs to the subspace or it is outside that subspace. In this way the universe of items can be represented by a set of mutual orthogonal subspaces of the Hilbert space. Other subspaces do not represent an item. They can be considered as void. We will assume that on the average the ‘filled’ and the void subspaces are evenly distributed over a connected part of the Hilbert space. The Hilbert space can be specified by using a number field that allows the mutual orthogonalization and the closure of subspaces. The real’s, the complex numbers and the quaternions can perform that job. Horwitz showed that even the octonions with some trouble can achieve this (see: http://arxiv.org/abs/quant-ph/9602001). The fact that a number field is used for specifying linear combinations of Hilbert vectors does not mean that eigenvalues of operators must also be restricted to that same number field. In this sense a Hilbert space specified over the quaternions may allow eigenvalues of operators that are taken from the octonions or even higher 2n-ons (see http://www.math.temple.edu/~wds/homepage/nce2.pdf). The original proposition also speaks about the position of the item. The position must be related to something that is available in the Hilbert space. The Hilbert space is defined over a number field. Thus we might attach a number of this field (or a higher 2n-on) to the subspace that represents the item. The natural way of attaching numbers to subspaces of a Hilbert space is via the concept of eigenvalues of normal operators. The eigenvectors of normal operators span the whole Hilbert space. However, the Hilbert space has a countable dimension. It means that the eigenvalues may offer a dense coverage of the number field, but it is not a closed coverage. It does not include all limits of all convergent rows. Thus it is sensible to attach a tiny environment of the actual eigenvalue to each eigenvector. In this way the position is expressed in a tiny environment rather than in a single number. At least the position is represented by a single eigenvector and the whole number field is covered. The eigenvector represents an atomic proposition that represents the position attribute of the considered item. The eigenvector lies inside the subspace that represents the item. The atomic proposition states that the position of the item lies inside the environment that is represented by the eigenvector. The fact that the operator must be bounded in order to guarantee that its eigenvectors span the whole Hilbert space is not crucial to this approach. It is sufficient when all positions that are connected to the items stay in a finite sphere. The deliberations that the eigenvalues of operators need not stay in the number field that is used to specify the Hilbert space also hold for the position operator. In this way, the positions may be elements of a curved manifold. In this case we will call the position operator an extended position operator. The original proposition talks about influencing the position of the item. This implies that the position of the item changes due to the mentioned influence. Thus when the influence occurs, the eigenvector that represents the position of the item is exchanged against another eigenvector. That other eigenvector corresponds to another environment inside the eigenspace of the position operator. The new eigenvector takes the role of the old eigenvector and is the new characteristic for the item’s position. This replacement may take place inside the subspace, which represents the considered item, or the original eigenvector moves outside the subspace, while the new eigenvector moves in or stays in the subspace. In both cases the eigenvectors of the position operator move with respect to the subspace of the item. The movement is relative and takes place inside the Hilbert space. The original proposition claims that this movement is caused by other items. A unitary transform is capable to move a subspace of the Hilbert space relative to a base of vectors such as the eigenvectors of a normal operator. It also may move all eigenvectors of a normal operator relative to a set of untouched vectors. Something that at least for a part acts locally like a unitary transform does the job. We will call this something a manipulator. With local we mean here the neighborhood of a Hilbert vector that lies inside the subspace of the Hilbert space that represents the considered item and that characterizes the item. Thus the influence from other items can be implemented by this manipulator. It means that at another location the manipulator may locally resemble a different unitary transformation. Not only the eigenvalues of the unitary transforms may differ. Even their eigenspaces may differ (the reference frames of their imaginary subspaces differ). The eigenvalues of the manipulator in the neighborhood of the item form a small environment. These environments can be considered to belong to a curved manifold. This manifold is a subset of a space that has a higher dimension than the number field over which the Hilbert space is specified. We will call that space the manipulator space. It is the eigenspace of the manipulator. Locally the eigenvalue of the unitary transformation resembles at least a part of the eigenvalue of the manipulator. For example the number field of the octonions contain several subfields that are quaternion fields. The ‘eigenvalues’ of the manipulator can be interpreted as locations in manipulator space. Part of the eigenvalue of the manipulator belongs to a number field that is equivalent to the number field, which is used to define the Hilbert space. Important for the manipulator space is that it allows the specification of a distance between two ‘eigenvalues’. That distance must be a non-negative real number. The eigenvalues of the manipulator represent the local influences. If several different local influences can be discerned, than all these influences must be present in the local eigenvalue. With other words if the Hilbert space is specified over the quaternions, then the eigenvalue of the manipulator is not a simple quaternion. It is more than that it must allows the storage of the attributes of all local influences. Let us summarize: Locally the ‘eigenvalues’ of the manipulator can be treated as elements of the number field that is used in the specification of the Hilbert space. In that role they form the eigenvalues of a unitary transformation. Globally they may be elements of a higher dimensional number field. As an example the local numbers may be quaternions and the global numbers, which are eigenvalues of the manipulator, are then the octonions or higher 2n-ons. The most central quaternion in this 2n-on specifies how the principle vector of the subspace moves around in Hilbert space. A normal quaternionic unitary transformation is an isomorphism between two Hilbert spaces. It conserves the inner product. There exists an adjoint unitary transformation that does exactly the inverse. When the target is the same as the source, then the unitary transformation is an automorphism. We only use the local activity of the unitary transform. The manipulator takes the global activity. Further it implements other influences than those that are implemented by the unitary transform. Let us now investigate what a unitary transform does. It takes a base of the Hilbert space and multiplies each of these vectors with a unitary number that is taken from the number field of the Hilbert space. (The one that is used to specify the Hilbert space.) The base is the set of eigenvectors of the unitary operator. The result is a new base. A normal operator does a similar thing, but its eigenvalues are in general not unitary. So the result of that action is not a new base. When a unitary transformation is applied to an arbitrary vector, then that vector is transferred into another vector. When a unitary operator U is applied to the eigenvector |q> of an operator Q with eigenvalue q, then the eigenvector is transferred into another vector |U q>. The expectation value for |QU q> is no longer q, but <q U|QU q> = <q|U†QU q> ≈ u*qu, where u is the expectation value <q|U q> of U for vector |q>. With real and complex numbers this just equals q, but for higher 2n-ons q can be affected by this transformation. However, its real part is not affected. I will call the product u*qu a number waltz, because when the action of U is stationary, then it causes a full rotation of part of the imaginary part of q. With the manipulator and with an extended position operator, locally at least the same thing will happen. The manipulator can do more. Here we will not further explore its capabilities. Each of the subspaces, which represent an item, contains at least one eigenvector of the manipulator, one of them acts as the principal vector for that item. This eigenvector corresponds to an eigenvalue of the manipulator. This eigenvalue corresponds to a location in manipulator space. That eigenvalue characterizes the item. When the representation of the item moves through Hilbert space, then the representing eigenvector is replaced by another eigenvector of the manipulator and the eigenvalue is replaced as well. The subsequent eigenvalues form a trail. This trail characterizes the life of the item. For that reason this number also represents the influence of all other items on the considered item. For that reason, the trail may be curved. Together the trail elements form a path on a curved manifold, which is a subset of the manipulator space. The steps from eigenvalue to eigenvalue can be identified by a trail progression parameter t. For the item this parameter takes the role of the manipulators time. This is not our usual notion of time but it is related with it. We will spend some remark on this at the end. The original proposition states that all items influence each other’s position. This includes that all items influence the considered item. It means that all items contribute to the action of the manipulator. The manipulator moves either the eigenvectors of the position operator or the subspace that represents the item. We are only interested in the relative movement, not in the absolute movement. It is not yet clear how the universe of items can influence the eigenvalue of the manipulator that is attached to the considered item. This can only be possible when the manipulator has a direct relation with the representation of each of the existing items. A unitary transformation is completely defined by its eigenvectors and the corresponding eigenvalues. The eigenvectors cover the whole Hilbert space. The eigenvalues are attached to the eigenvectors. In a similar way the manipulator may possess eigenvectors that cover the complete Hilbert space. It is sufficient that every subspace, which represents an item, contains an eigenvector of the manipulator that characterizes that item. This eigenvector corresponds to an eigenvalue that uniquely characterizes the current condition of the item. In this way the distance between items can be specified as the distance between the eigenvalues of the manipulator that characterize the items. The scalar vector product of the Hilbert space also allows the specification of a distance between vectors, but the new distance may characterize the strength of the influence. We could also choose the position of the items in order to specify the distance between items. However the position is affected by the number waltz. For that reason we prefer the distance provided by the eigenvalues of the manipulator. The influence may decrease with distance according to some function f(r) of the distance r. However the number of contributing items increases with the distance. The most probable result is that the strongest influence comes from the cooperative activity of the most distant items. Any variation of the influences of the distant items averages out. So there exists a spherically uniform background influence caused by the universe of items. Nothing occurs until the subspace that represents the item moves such that its characteristic value changes. What happens can be deduced from an equivalent of Denis Sciama’s analysis (see: http://arxiv.org/abs/physics/0609026v4.pdf) where the manipulator space is playing the role of space and the trail progression parameter plays the role of time. The investigation shows that a uniform speed causes a vector potential. Variation of the speed causes a field. Thus an influence in the form of a field goes together with an acceleration of the item. In this way the universe of items causes inertia. The analysis uses the fact that every item in universe causes an influence and that this influence reduces according to f = -k/r. (Compare this with Bertrand’s theorem in Wikipedia) When the number field of the quaternions is used for the specification of the Hilbert space, then the combination of the action of the manipulator and an observation of the position of the item causes a quaternion waltz. This quaternion waltz causes a partial rotation, which on its turn causes an acceleration. That acceleration generates an influence in the form of a field by the universe of states. The action of the field is represented in the argument of the eigenvalue of the manipulator. This completes our reasoning. This finding indicates that when Sciama’s analysis is correct the original proposition is false. The universe of items does not influence position. It counteracts acceleration of individual items. We may alter the original proposition. If Sciama’s analysis is correct, then the proposition All items in universe influence each other’s acceleration. Is true Final remark: The trail that is taken by the manipulator can be studied with the toolkit of differential geometry. The tantrix of that curve delivers the main part of the action. Only the imaginary part of this tantrix contributes, but the real part also influences the amount of rotation that the number waltz causes. The imaginary part of the tantrix can be related to the notion of spacetime. The spacetime step occurs perpendicular to the space step. The rectangular triangle can be closed with a step that we will call the coordinate time. The three steps define a Minkowski metric and as a consequence it raises special relativity. The curvature of the manipulator trail causes a curvature of the position trail. The observed position is also affected by the number waltz. The manipulator trail appears to be responsible for an equivalently curved influence of the field. The influence field can be interpreted as the gravitation field. This brings general relativity into the picture. As already said, this gravitation field is not the only influence that is supported by the manipulator. The electromagnetic field is another example. It has a shorter range than the gravitation field. There exists a list of fields with even shorter ranges. These are not treated here. A better readable and rigorously updated version of this text can be found on http://www.scitech.nl/MyENHomepage.htm

Rotations are fundamental to nature. The reason is the existence of the quaternion waltz. This is a transformation of the form ab/a. It only affects the imaginary part of the manipulated quaternion. Both the real and the imaginary part of the manipulator determine the effect of the transformation. The fact that this transform is fundamental to nature follows from quantum mechanics. A quantum logical proposition is represented by a closed subspace of a Hilbert space that is defined over a 2^n-on number field (See http://www.math.temple.edu/~wds/homepage/nce2.pdf). An atomic proposition is represented by a single Hilbert space vector |f>. After a while this vector is transformed into another vector |g> = |U f>. U is an unitary transformation. Let us chop the path from |f> to |g> into a trail of very small steps. Each step is controlled by a trail element, which is an (infinitesimal) unitary transformation U(t). With that transformation corresponds a local eigenvalue u that is a function of the trail progression parameter t. Let us assume that the eigenspace of U(t) is the quaternion number field. The eigenspaces of the trail elements need not be the same quaternionic space. Together they may form a curved manifold in an 2^n-on space (n > 2). An observable Q with eigenvalue (or expectation value) q for state |f> will be affected and gives a trail Q(t) = (U(t)^-1)QU(t) with eigenvalues (expectation values) q(t) = (1/u(t))qu(t). Here the quaternion waltz is introduced!. The trail concept has the advantage that locally everything is linear and quaternionic. Globally this need not be true. However, up to the 16-ons the number fields seems to be rather continuous. Thus these number fields might be “fit for physics”. I surpassed the complex numbers because the quaternion waltz has no effect in that field. By the way, the 16-ons are not sedenions. They are nicer! 8-ons are octonions. 4-ons are quaternions. 1-ons are the real numbers. If the above interpretation of unitary trails is correct, then the quaternion waltz is a universal law of nature! The quaternion waltz is the source of macroscopic dynamics. As a consequence space and time do NOT belong together in a natural way. They are not member of the same quaternionic (or 2^n-on) operator. The Hermitian part of Q is NOT affected and plays no part in (macroscopic) dynamics. It is neglected! Physicists tend to replace this part by the coordinate time, but that is an artificial construct. The trail progression parameter plays the role of proper time. Geometrically the proper time step relates o the action step along the tantrix of the trail. It is perpendicular to the q-step. The hypotenuse is the coordinate time step. In this way the action step plays the role of spacetime and the space gets a Minkowski metric or with curvature it gets a Lorentzian metric. Here relativity is born. The curved manifold is the source of general relativity. Bertrand’s theorem reveals how fields might intervene. Denis Sciama (http://physics.fullerton.edu/~jimw/general/inertia/index.htm) shows how inertia comes in. The universe of states causes inertia. A field is raised when a state accelerates. The usual critics that his approach is not valid because it uses instantaneous transfer of information are not applicable here because coordinate time need not be involved. This interpretation might throw new light on quantum mechanics and quantum field theory because it decouples the manipulation U(t) from the manipulated observation Q(t). It splits space and time. It brings nature back to a 2^n-on number theoretical problem. Minkowski metrics and Lorentzian metrics are artificial! Microscopic dynamics appears to be governed by a completely different mechanism. The harmonic movements of elementary particles inside a state indicate that the eigenfunctions of the manipulator (which is a 2^n-on Fourier transform) play an important role. Hans

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