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    Quite A Series Of Remarkable Things Happen In Quantum Theory
    By Hans van Leunen | August 30th 2010 05:43 PM | 24 comments | Print | E-mail | Track Comments
    About Hans

    I am a retired Physicist (born in 1941) with experience in chemistry, Fourier optics, image intensifiers, quantum logic, quantum physics, modular...

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    Wave functions and characteristic vectors. The first thing that draws attention is the concept of the wave function. It is a probability density function that is correlated with an observable. On its turn the observable corresponds to a continuous normal operator. The wave function corresponds with a single vector in Hilbert space and its values correspond to the inner product of this vector with the eigenvectors of the operator that belongs to the observable. The corresponding eigenvalues form the argument of the of the wave function. The function describes the probability of finding a given eigenvalue when a suitable measurement of the observable is performed. It is a kind of shadow image taken with long exposure of the internal movements of the observed item. For very small objects like atoms these movements are harmonic and repeat themselves after a certain period. The harmonic movements correspond to eigenfunctions of a different operator. In simple cases it can be shown that these eigenfunctions are eigenfunctions of Fourier transforms. It is possible to choose different observables as the base of a wave function. One choice is the position observable. Another choice is the momentum observable. Still other choices are the spin and the angular momentum. With each of these choices corresponds a different probability density vector. Each of these vectors characterizes an aspect of the observed object. None of these vectors offers a full representation of the object. According to the congruencies between quantum physics, quantum logic and the set of closed subspaces of a Hilbert space, a full representation of a physical object can only be given by a closed subspace of a Hilbert space. Since the vectors that correspond to different wave functions already differ, the subspace that represents the physical object will in general be multidimensional. With other words, it is false to interpret a wave function as a representative of a physical object. On the one hand the quantum physical item corresponds to a hierarchy of propositions about a physical item. On the other hand the top of the hierarchy of these propositions corresponds to a closed subspace of the Hilbert space. In general that is not a ray. It is not spanned by a single vector, but by a bunch of mutually independent vectors. Each of these mutually independent constituting vectors corresponds to an atomic proposition. Each of these constituting vectors can carry at least one property. This property is also an eigenvalue of a (normal) operator. Vectors that are linear combinations of the mutually independent constituting vectors may be eigenvector of other normal operators. Such an operator may also own a set of eigenvectors that together span the same subspace. In this way the subspace can support quite a series of properties that all belong to the considered physical item. It is interesting to consider observables that are each other’s canonical conjugate. Here the Heisenberg uncertainty relation plays its part. It already specifies a certain minimal area in phase space. This corresponds with a minimal dimensionality of the presentation of a physical item that is higher than unity. The wave function can be considered as a characteristic of a physical item. Another type of characteristic vector is used to indicate the precise location of a physical item in Hilbert space. A physical item possesses a series of different characteristic vectors. A characteristic vector can never on its own be considered as a full representation of a physical item. Further a characteristic vector cannot be interpreted as an eigenvector of all operators that act on the item’s representation. Static structure and dynamics Two important aspects of quantum physics are its static structure and its dynamics. The static structure is mainly set by its fundament on traditional quantum logic. For another part it is ruled by the Helmholtz decomposition theorem. Traditional quantum logic does not state anything about dynamics. It also does not state anything about influences that exist between propositions. These influences correspond to vector fields. The static decomposition of vector fields is treated by the Helmholtz decomposition theorem. Dynamics is correlated with the movement of the representations of physical items in Hilbert space. Unitary operators are usually considered as the actors that are capable of moving things around in Hilbert space. I will explain that this is a far too simple and in fact false interpretation of what is really happening. Let us first look at the operators that act in Hilbert space. There exist two pictures in quantum physics. In the first picture, the Heisenberg picture, the representation of the considered item is moved through Hilbert space. In this picture the eigenvectors of operators that represent observables stay fixed. The subspace representing the item moves over the forest of eigenvectors of operators that are selected as observables. In this way it dynamically meets eigenvalues of these operators that become its properties. It looks as if the definition of the subspace in terms of eigenvectors of operators becomes a dynamic function of the progression parameter t. In the second picture, the Schrödinger picture, the eigenvectors of the operators that represent observables move over the subspace that represents the considered item. It looks as if the operators become a dynamic function of the progression parameter. For the experience of the considered item it does not matter which of the two pictures governs its dynamics. However, these pictures are far too simplistic. Nature does not behave that way. Our observed space is curved. Further, its structure differs from the structure of the eigenspaces of the operators. Observed spacetime has a Lorentzian metric. Even when hyper complex numbers are used to define Hilbert spaces, the eigenspaces are Euclidean. Thus physicists must have played another trick. They use the results of differential geometry. However, curves and manifolds are not number fields or parts of number fields. Still, geodesic curves come very close to physics and geodesic equations have a great resemblance with equations of motions. But are the two things the same? Can you connect a moving subspace with a geodesic? Well if you restrict to one of its characteristic vectors you could. However, the move of the subspace occurs in Hilbert space and that space is not curved. The situation can only be explained when there exists something in Hilbert space that is curved. The curved subject is not constituted from the eigenvectors or eigenvalues of normal operators. These subjects are properly evenly distributed. There must be something else in Hilbert space and it must also be represented in both quantum logic and in quantum physics. The answer to this question is formed by fields. They are represented in quantum logic by influences between propositions. In quantum physics they are represented by physical fields. In Hilbert space the fields get a place in the arguments of the transformations that may push subspaces around. A unitary transformation cannot push a subspace around in Hilbert space. It is a far too simple mathematical tool for that purpose. However, the thing that can do it can be simulated by a trail of infinitesimal unitary transformations. Let us give that thing a name. Let us call it a redefiner. It redefines subspaces in terms of eigenvectors of linear operators. You cannot do realistic quantum physics without such a redefiner! The elements of the trail all differ from each other. They have different sets of eigenvectors. Otherwise the subspace cannot move. The reason is that a unitary transform cannot move its own eigenvectors. Thus, the redefiner moves its own eigenfunctions and it does not move other vectors. Further, the elements of the trail are only used locally. It means that these elements may be taken to be much more exotic subjects than just infinitesimal unitary transforms. They only need to act locally like infinitesimal unitary transforms. They might even accept higher dimensional hyper complex numbers as their local eigenvalue. For example the 2^n-ons of Warren Smith act locally as 2^m-ons in their lower 2^m dimensions. Why would you want higher dimensional eigenvalues? Well, they can store more properties! An 2^n-dimensional 2^n-on eigenvalue has place for 2^n real values. Isn’t that nice? The moving (≥ 3-dimensional) Frenet-Serre frame of a geodesic curve may be interpreted as the axes of a local hyper complex number field. That is a nice visualization of what is really happening. The conclusion of the above analysis is that the plain use of unitary transforms in quantum physics must be reconsidered! When you enter hyper complex Hilbert space you will encounter other features about which you never might have heard about. Hyper complex numbers perform transforms of their own. One very important number transform is the number waltz. In its simplest form it looks like: c=a•b/a. With real or complex numbers this always results in c=b. However, with hyper complex numbers that have dimension greater than two the manipulated subject b may get a turn. Not its real part. That part is not touched. The imaginary part of b gets a partial precession. When a unitary transform acts on an operator that represents an observable then the effect of the number waltz can easily get larger than the effect of the transfer of the eigenvector of the observable. Elsewhere it is shown that in this way the number waltz can form the reason of existence of special relativity. Thus, there is reason to take the number waltz serious! The number waltz may play a crucial role in the dynamics of quantum physics! If the given reason of existence of special relativity is correct, then physical dynamics cannot be understood properly without considering the influence of the number waltz. The number waltz, together with a particular choice of time step, introduces the Minkowski signature that forms the base of special relativity. Further the number waltz causes a clear split in quantum physics. It splits the manipulating part from the manipulated part. The game of the manipulating part takes place in Hilbert space. The game of the manipulated part takes place in a curved space that is characterized with a Minkowski signature and a Lorentzian metric. Reason for a universe wide progression parameter Unitary transforms are (continuous) normal operators and as such they possess an orthonormal set of eigenvectors that span the Hilbert space. The Hilbert space contains all subspaces that together represent the universe of all physical items. Thus, every unitary transformation, even the infinitesimal transformations that form elements of the trail that constitutes the redefiner, touches all items in universe. I consider this as a scaring idea, but it still might be right. It will mean that the redefiner has a universe wide scope. Each redefinition step will work universe wide. It means that the universe has a global clock. Before the redefiner takes a step the current status of the universe is governed by static laws such as quantum logic and the Helmholtz decomposition theorem. After taking the step these laws hold again but with different conditions. The step is a transfer from one static condition into another. Quantum logic rules how propositions about physical items relate to each other. The step rules how these propositions change. This change is NOT covered by quantum logic. The change of the propositions corresponds to a change of the subspaces of the Hilbert space that represent these propositions. The Helmholtz decomposition theorem states how vector fields can be split into a rotational and an irrotational part. The step rules how this partition changes. The change of the fields correlates to a change of the subspaces. In a static situation the fields compensate each other’s activity. A non-compensated activity results in a move of a subspace. That move is done during a step. The action of the fields is represented in the argument of the redefiner. The current infinitesimal unitary transform performs the step. As a reaction to the action the repositioned subspaces cause a reconfiguration of the fields. This mechanism governs the dynamics of quantum physics. It controls both the kinematics of the physical items as well as the dynamics of the fields. The Maxwell equations show how the electromagnetic fields changes. Other fields such as the gravitation field react similarly. Dennis Sciama gave an elucidating explanation of these effects in his “Origin of inertia”. It is interesting to note that the Helmholtz decomposition theorem is directly related to the fact that the Fourier transform of the vector field can be split into a longitudinal part and a transversal part. This again relates to the fact that the multidimensional Dirac delta function can be split into a longitudinal part and a transversal part. Thus, between steps these splits are reestablished as new status quos. Origins of dynamics Quantum physics has many aspects. Each of these aspects has its own origin of dynamics. These aspects have representations in four areas. The first area is nature. The second area is quantum physics. The third area is quantum logic and the fourth area is Hilbert space. For macroscopic movements the origin of dynamics can be found in the interaction between fields and the items that are subject of their actions. In nature fields cannot be sensed directly. They can only be noticed indirectly by their effect on items that exist in nature. In quantum physics fields have their own section of science in the form of quantum field theory. The fields influence physical items such as elementary particles and galaxies. Sometimes the interaction is very intense and the elementary items behave like waves of fields. In Hilbert space the fields have their place in the arguments of unitary operators. As stated above the operators are only used in their infinitesimal format. This restriction holds both in spatial as well as in “temporal=progressive” sense. The infinitesimal unitary operators act on closed subspaces. The fields are raised by physical items, which correspond to closed subspaces and to a certain kind of propositions. They act on other physical items/closed subspaces/propositions. In quantum logic the fields correspond to influences between propositions. They depend on how the proposition is redefined and they depend on the distance between the actuator and the influenced subject. In quantum logic, both are propositions. The redefinition concerns the way the enveloping proposition is constituted from atomic propositions. The distance in quantum logic corresponds to the distance in Hilbert space between characteristic vectors of the corresponding subspaces. These characteristic vectors must be of a suitable type. Apart from macroscopic movements there consist movements that occur inside physical items. Usually these movements are periodic. In well-formed physical items these movements are harmonic. Harmonic movements are directly related with eigenvectors of special operators. These operators have a discrete or mixed spectrum of eigenvalues. This indicates that internal movements are governed by processes that differ significantly from processes that govern macroscopic movements. The internal movements give rise to a next kind of dynamical behavior. That behavior concerns the creation and annihilation of items. These effects usually go together with a change of eigenvalue (property) of a related item that emits or absorbs the other item. So far we can see the following types of quantum physics (QP): • The stationary QP (Quantum logic; Helmholtz decomposition theorem) • The QP of the fields/influences (Quantum field theory) • The QP of the manipulators (redefiners) • The QP of the numbers (number waltz) • The macroscopic dynamic QP (operator driven / number driven) • The internal dynamic QP • The creation/annihilation QP Only the number driven macroscopic dynamic QP is governed by a Minkowski signature and must cope with a maximum speed of information transfer. More details can be found in http://www.scitech.nl/English/Science/Exampleproposition.pdf .

    Comments

    vongehr
    "the wave function. It is a probability density function..."

    Only second sentence, already wrong. WF is NOT a PDF. After that, it goes steeply down hill. I applaud your trying to get people interested into QM, but it may be much better if you sleep somewhat more over your stuff before posting it, else it may just do the opposite of clarifying.
    fundamentally
    You are right. It is a probability amplitude rather than a probability density. Its squared modulus is the density. Further, I should have used "Hodge decomposition" rather than Helmholtz decomposition. The general focus of the article is to focus attention on the fact that much that is taken for granted in quantum physics appears to be questionable when you think deeper about it. But you are right. I am sloppy with some concepts. What interests me more is your opinion whether for example my criticism on the common use of unitary transforms is realistic and whether a thing like the redefiner is required to do proper quantum physics.
    If you think, think twice
    blue-green


    Normal
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    I could raise some quibbles with the above. For example,
    when you mention probability vectors, surely you meant quantum amplitude
    vectors. Probability is a real number between zero and one and not a
    vector.  To be fair, you did say“probability density vector”. then again, you used the word vector 40 times. Maybe you meant to say density matrix. (I see now that others have already posted about this).

    I think your point is that quantum mechanics and quantum
    field theory have become horribly complicated. A post-doc fellow is going to be
    buried in one, two or three corners of theory, while all of the other corners
    are bursting out with their own fancy mathematics

    There are some overarching principles to help one navigate
    the zoo. One application or another of a minimizing principle can always attain
    the dynamical equations. What is minimized is the Action and given the symbol S
    (not to be confused with the S that is used in other contexts for
    entropy).  Action has the same units as
    Planck’s constant and also the same units a pair of conjugate variables. The
    master equation for Action is the following: the variation of S = 0.   δS=0.

    It is a difficult formula to actually use, yet there it is.
    And yes, transient phenomena are not the same as near equilibrium steady-state phenomena

    Your essay reminds one of the difficult times in the late
    sixties and early seventies when the particle zoo had become a horrible looking
    mess. Quarks and a better understanding of gauge theories helped to bring some
    unity and order to the zoo.

    Although you mention quantum logic quite a few times in your
    essay and allude to information theory, your essay gives one no awareness of
    the reorganizing and simplification that is going on in work by people like
    Erik Verlinde. Our Hammock Physicist here has written a number of posts about
    this. I have tried to continue the conversation there at the following link.

    http://www.science20.com/hammock_physicist/it_bit_whole_shebang

    Some feedback would be appreciated. Will Verlinde’s approach
    make it into introductory courses to help make physics more intelligible?

    Concerning unitary transformations: they hint at there being a "conservation of information"
    principle. I defer to Susskind on that for now. I have not seen the principle clearly stated or widely embraced. The old proverb is that an elephant never forgets.

    fundamentally
    I dislike quibbling. However, the text started with the name wave function. I indicated that this function can be related to a Hilbert vector via the inner products of that vector with the eigenvectors of a normal operator that represents an observable. The corresponding eigenvalues play the role of the function parameter. Thus the first mentioned vector represents the function. If the wave function represents a probability amplitude, then the corresponding vector does that as well. With another normal operator that same vector stands for a different function. In fact each Hilbert vector stands for a series of functions. It is easy to attach a given functionality that belongs to a function to a vector, but you must then state the relation between that vector and the function. One of the possible relations is the inner products with the eigenvectors of a normal operator. These eigenvectors span the Hilbert space and the eigenvalues can act as function parameters. Vectors can also relate to matrices. This is easily demonstrated with Dirac's bra-ket notation. In that way a density matrix can be constructed. But it is not what I meant. I did not touch minimized action S, but I mentioned action as the carrier of the influences of the fields in the argument of the operator that moves the representation of the considered physical item through Hilbert space. This move occurs along a trajectory that stays undisturbed by fields when it conforms to a geodesic. The corresponding geodesic equation presents similar conditions as the principle of minimal action. In his “The Principles of Quantum Mechanics” Dirac relates the unitary part of a normal operator to a representation ψ of a moving wave packet. In §31 of the book he writes that representation as ψ(qt) = A•exp(i•S/ħ) and via the Schrödinger equation he then arrives at the Hamilton-Jacobi equation. Thus, already Dirac used unitary operators to move representations of physical items through Hilbert space. There he was wrong in two respects. The wave function ψ cannot act as a full representation of the considered physical item. It is a characteristic rather than a representation. And it is only one out of a range of possible characteristics of that physical item. The full representation is given by a closed subspace and that subspace is multi-dimensional. Further, to my opinion that operator cannot be a unitary operator. Even when it is taken as a function of a progression parameter t it will fail the capability to move a closed subspace with dimension larger than one through Hilbert space. This is due to the fact that a unitary transform cannot move its own eigenfunctions and a multi-dimensional closed subspace contains at least one eigenfunction of a given unitary transform. The only thing that can move a multi-dimensional closed subspace is a construct that consists of a trail of infinitesimal unitary transforms, where each element of the trail has its own set of eigenvectors that differs from the set of eigenvectors of the previous trail element. Because it is such a strange construct, I have given it a special name. I call it a redefiner. Then that is what it does. After taking a step, it redefines the subspace in terms of the eigenvectors of the current trail element. The alternative is that each physical item is fully represented by a ray, which is a one dimensional subspace. Can you imagine a molecule or the sun being represented by a ray?
    If you think, think twice
    Bonny Bonobo alias Brat
    Interesting article Hans, though I think it would have been good to start the essay with an introduction that said something along the lines of ‘the general focus of this article is to focus attention on the fact that much that is taken for granted in quantum physics appears to be questionable, when you think deeper about it’, which is what you then added later in your comments. I have spent the last couple of months trying to understand relativity, the standard model and quantum physics and have had to admit defeat in this area, because I still can’t really understand it and I just haven’t got the time at present to keep giving it so much attention and time. It is the first subject that I haven’t been able to grasp after spending such a lot of time on it, but I haven’t given up completely yet, I hope to resume next year when I’ve finished my second degree or it may become a retirement hobby, if my brain is still up to it. One of my concerns regarding this field is that it has become mathematically overcomplicated and that just a few of the accepted hypotheses or beliefs are based upon false hypotheses or assumptions that somewhere down the track scientists have started to believe are true, as a result of a well known psychological phenomena called ‘groupthink’. The terminology involved is also huge and a challenge to the average person’s vocabulary. This article that you have written is a prime example of this. Another problem I think is that individuals specialise so much in specific areas of quantum physics that they lose sight of the big picture. Finally, I think that there is too much trust and reliance on computer simulations of what is happening in the detection processes, but that is just based on my own work experience and a gut feeling. In the meantime I keep plodding away reading every article that I can find, without being able to spend the time to decipher half of what I’m reading but just hoping I’m remembering enough of the gist of the article for it to be useful to me down the track or that some level of comprehension will fall into place eventually. It was interesting this week to find in Wikipedia that Tesla, who I have a lot of respect for as a creative genius, was critical of Einstein's relativity work, calling it:.. “...[a] magnificent mathematical garb which fascinates, dazzles and makes people blind to the underlying errors. The theory is like a beggar clothed in purple whom ignorant people take for a king ... its exponents are brilliant men but they are metaphysicists rather than scientists” And that Tesla also argued: “I hold that space cannot be curved, for the simple reason that it can have no properties. It might as well be said that God has properties. He has not, but only attributes and these are of our own making. Of properties we can only speak when dealing with matter filling the space. To say that in the presence of large bodies space becomes curved is equivalent to stating that something can act upon nothing. I, for one, refuse to subscribe to such a view.” Wiki states that Tesla also believed that much of Albert Einstein's relativity theory had already been proposed by Ruđer Bošković, stating in an unpublished interview: “...the relativity theory, by the way, is much older than its present proponents. It was advanced over 200 years ago by my illustrious countryman Ruđer Bošković, the great philosopher, who, not withstanding other and multifold obligations, wrote a thousand volumes of excellent literature on a vast variety of subjects. Bošković dealt with relativity, including the so-called time-space continuum.”
    My latest forum article 'Australian Researchers Discover Potential Blue Green Algae Cause & Treatment of Motor Neuron Disease (MND)&(ALS)' Parkinsons's and Alzheimer's can be found at http://www.science20.com/forums/medicine
    rholley
    Ruđer Bošković was certainly one of the most constructive of philosophers from the point of view of science.  It was discussion of his ideas among a group of friends that sparked in Michael Faraday the idea of fields.
    Robert H. Olley / Quondam Physics Department / University of Reading / England
    Bonny Bonobo alias Brat
    You can download for free or read online, Roger Boskovic's 'A Theory of Natural Philosophy' which is a very interesting book, at http://www.archive.org/stream/theoryofnaturalp00boscrich#page/n13/mode/2up
    My latest forum article 'Australian Researchers Discover Potential Blue Green Algae Cause & Treatment of Motor Neuron Disease (MND)&(ALS)' Parkinsons's and Alzheimer's can be found at http://www.science20.com/forums/medicine
    blue-green
    Normal 0

    hoo boy … It did not think that I would read here in Science20.com that “[Dirac] was wrong in two respects” concerning unitary transformations …. and then a quotation that Einstein’s relativity “makes people blind to [its] underlying errors” … not that Helen is subscribing to it … yet gossip repeated becomes fact in some circles.

    An excellent primer on Special Relativity is Spacetime Physics by Wheeler and Taylor.
    It was originally a course for freshmen at Princeton.

    A classic on Quantum Physics is the same book by Dirac cited by Hans.

    For something easier and less confusing perhaps, yet still completely mainstream, you can try the link in my profile.

    Seriously Hans, a lot of operators in physics are infinitesimal operators. The transition from infinitesimal movements to macroscopic movements is called integration … on my island. Information is still preserved every step along the way, even if us mortals cannot recover the text of documents burned in a fireplace.

    To paraphrase from Wheeler …. a purrfectly classical black hole has no hair. A quantum mechanical black hole, on the other hand, sends out more hair than Tesla with his palms on a mountaintop Van de Graaff machine. In those tuffs of hair, one can in principle, reconstruct exactly how a black hole was formed (though it may take the life of the universe for one to do so). Microphysics is unitary, even at its extremes. It is reversible. At the macroscopic levels, course graining over hidden degrees of freedom causes irreversibility and a flow of time whose “arrow” (not a Hilbert space vector) points in the direction of increases of entropy. It is a more complicated than that perhaps, yet without keystone equations involving temperature, one can scarcely start her up and see if she runs.
    Hans, you need to get away from this "redefiner" stuff -- you're completely wrong about it. A unitary operator is the most general norm-preserving transformation in Hilbert space. While it does have eigenvectors, that fact does not make it any less general. True its eigenvectors are not "moved", but they are changed (multiplied by the eigenvalue - which is in the unitary case a phase) An analogy is three-dimensional rotations -- any three-dimensional rotation has an axis which is not "moved", but that does not make it less general. It just happens to be a theorem that three-dimensional rotations always have this property. Likewise its a theorem that the most general unitary operator can always be decomposed into actions on one- or two-dimensional eigenspaces. This is not a restriction.

    The "trail of infinitesimal unitary transforms" you talk about - a product of unitary transformations is again a unitary transformation. Imagine that! Your redefiner is just another unitary transformation, and yes it will have its own set of eigenvectors and eigenspaces all over again. The redefiner concept is completely redundant.

    fundamentally
    To Helen: I hoped that the title of the article would invite people to read it and think deeper about the subjects that are treated therein. To blue-green: I am an admirer of Paul Dirac. He wrote his book in a very turbulent episode of quantum physical theory development. About that time Birkhoff and von Neumann introduced their ideas about quantum logic. Dirac is a very pragmatic scientist and he picked ingredients from a variety of sources and mixed them in a very comprehensible theory. He united the visions of Heisenberg and Schrödinger and introduced his marvelous bra-ket methodology. Still I cannot avoid my impression that in the paragraphs 31 and 32 of his book he was provoking rather than putting an established theory. To me he was playing with unfinished thoughts. At least these paragraphs stimulated me to start thinking about trails of infinitesimal unitary transforms. This already happened during my studies in the sixties of the last century. At those times like Helen I studied all papers and books that became available to me. That included the papers of Jauch and Piron. These papers guided me to quaternionic Hilbert spaces. During my carrier I did not pursuit that direction. I started working on the development of high-tech imaging devices and later I landed doing software development. Only after my retirement I took again interest in research of fundamental physics. I restarted from my knowledge of quantum logic and quaternionic Hilbert spaces. I still belief that traditional quantum logic forms a solid fundament under quantum physics. It is not for nothing that most of quantum physics is done in a Hilbert space. It seems that many people have forgotten or do not know that quantum logic prescribes this habit. I am aware that infinitesimal operators, differential operators and integral operators are abundantly used in quantum physics. However that does not disprove my proposition that unitary operators are not fit to move representations of physical object through Hilbert space. If you disagree, then you must come with a more precise opposition or you should give a clear alternative. To Bill K: Let me state more precisely what I mean with the phrase “the eigenvectors of unitary operators are not moved”. I mean with this phrase that they are not moved beyond the realm of the subspace of which they currently are a member vector. So they cannot move that subspace. It does not matter whether they are multiplied with a number, even when that number is their own eigenvalue. “a product of unitary transformations is again a unitary transformation”. Remember that the elements of the trail all have different sets of eigenvectors. Because this makes the trail such a particular construct, I gave it a special name “redefiner”. I do not deny that you can interpret the combined effect of a part of the trail again as a unitary transform. I do it myself. However it is false to see the trail as a simple product of infinitesimal unitary transforms. So your comment does not take hold. I still belief in the necessity of my redefiner.
    If you think, think twice
    blue-green
    Is it your conviction Hans that even when recording devices are not present,
    real physical states carom off the planes and subspaces through which they
    would be confined by unitary transformations? What would be the experimental
    evidence for this? Chemical reactions?

    When you have a system tooling along in its subspaces … and then introduce
    an entirely new element into the system from outside, then yes, you have to
    redefine where you now are, because physics cannot predict when the long arm of
    a scientist or gremlin is going to drop something new (eye-of-newt) into a
    beaker.


    As Bohr indicated, when you change the experimental
    conditions -- the boundaries and/or framework -- you need to include these
    changes in your description. The redefinition of the apparatus works its way all
    the way down to the eigenstates. For Bohr, this was simply an extension of
    Einstein’s lesson concerning the dependence of variables on one’s frame of
    reference.
    fundamentally
    Instruments are not required. This is about mathematics. The fact that these mathematical tools are used in physics is in fact irrelevant. Hilbert spaces, their subspaces, their vectors, the unitary transforms are mathematical concepts. Traditional quantum logic is also a mathematical concept. It is an orthomodular lattice. Only its connection with quantum physics makes it a physical concept as well. Its congruence with the set of closed subspaces of a Hilbert space makes these items physical concepts. Birkhoff, von Neumann, Piron, Jauch and many others have established sufficient base for this connection. Their results are based on the outcome of experiments. It all happened between 1920 and 1970. It might be fruitfull when you study these ancient results. They are still valid! It must be stated that traditional quantum logic only relates to stationary quantum physics. There does not yet exist a logic that includes dynamics. It is this extension that I am searching.
    If you think, think twice
    blue-green


    Normal
    0


    As I already stated in my opening response: "transient phenomena are not the same as near equilibrium steady-state phenomena."


    Your approach is not yet comprehensible to me. I’ll ask again, what is its stance on conservation of information?

    The gee-whiz -- isn’t this remarkable -- method of introducing people to quantum
    physics can be counterproductive. A different civilization may feel that it is
    classical physics that is counter-intuitive.

    The Bell inequalities are proven using classical logic. One can tediously
    illustrate and prove them using Venn Diagrams. It’s all about Boolean logic and
    the classical assumption that things are either with you or not with you. The
    Biblical predecessor is that things are either with you or against you. The
    quantum-logical tolerance and acceptance of off-diagonal cross product terms is
    unthinkable.

    The difficulties and limitations of classical logic for many particle
    systems were sensed by the pioneers of quantum physics, including Einstein,
    long before Dirac and von Neumann. It was evident in the way one had to use the
    cross-product (tensor product) of state spaces H1 x H2 x H3 …. for many body
    systems and not a classical sum H1+H2+H3 …. The cross-products give one a
    higher dimensional arena and sophisticated ways for objects to be entangled
    that classical logic completely misses.

    A mnemonic for these cross-products is a Totem Pole or for that matter, any
    mythological beast that is an incongruous juxtaposition of seemingly
    irreconcilable objects. Such objects are rather common in non-western art and
    logic. The classical alternative is more like a painting of the Great Judgment;
    objects are sorted into separate sets and hierarchies; off-diagonal terms are
    not even considered.

    I realize that this may all sound far fetched. Nobody writes about quantum
    logic in this way. Or do they? When Bohr pleaded passionately about an “open
    world” tolerance and acceptance, was he not applying the quantum logic that had
    become natural to him?

    A more familiar issue is in the way Quantum Operators do not commute. Here
    too one can ask what is so sensational about that? One can even consider
    operators that fail the associative law. They exist algebraically in a higher
    dimensional analog of Hans’ waltzing quaternions. This was pointed out to Sir
    William Rowan Hamilton by a colleague long ago, and yet Hamilton kept toiling
    with his beloved quaternioins.

    There is nothing surprising or weird about a set of
    procedures being non-commutative. Suppose we start with a freshman student for
    which I’ll give the state vector |Fresh>.

    Let B stand for Bathing, U stand for the operation of putting on one’s
    Underwear and let P stand for putting on one’s Pants. The sequence of operations
    PUB acting on |Fresh> will put the our freshman in a prepared state PUB|Fresh>
    for performing a wide range of tasks, whereas a change in the order of PUB to
    BUP, for example, is going to be … disastrous.
    fundamentally
    What has all of this to do with the simple fact that a plain unitary operator cannot move a closed Hilbert subspace? I only used traditional quantum logic because it has a clear relation to Hilbert space. The quantum logical propositions form an orthomodular lattice. The set of closed subspaces of an infinite dimensional separatable Hilbert space possesses the same orthomodular lattice structure. Thus quantum logical propositions and closed Hilbert subspaces can represent each other. Now take the proposition "this is physical item X" and take the complete hierarchy of propositions that relate to this proposition, including the atomic propositions that state the properties of that item. Let this hierarchy be represented by the corresponding hierarchy of Hilbert subspaces. This is a stationary picture of what can be stated about this quantum physical item. (It is still pure lattice theory, where things are given names.) Now try to introduce dynamics into this picture. It means that atomic propositions in the mentioned hierarchy are replaced by others. Now translate this to the Hilbert space and tell me what you get. Rays that include eigenvectors of observables represent the atomic propositions and some of them are moved out of the covering subspace. At the same time rays containing other eigenvectors enter that covering subspace. What thing does this replacement? I called it a “redefiner”. Either you deny traditional quantum logic and its relation to Hilbert space or you must accept the picture painted above. If you deny, then you deny the work of those who laid the foundation of today’s quantum physics. All stuff that you stated does not disprove my point.
    If you think, think twice
    blue-green
    Hans, you claim “that traditional quantum logic only relates to stationary quantum physics. There does not yet exist a logic that includes dynamics. It is this extension that I am searching.”

    That’s a tough fish of bones to swallow, even for a cat

    Why would quantum electrodynamics have dynamics in its name
    if it did not include dynamics?

    Newtonian mechanics includes Newtonian dynamics using
    classical logic.

    Quantum mechanics includes Newtonian mechanics. It covers it
    completely and extends it using traditional quantum logic. It has all of
    Newton’s dynamics in it and more. So how can you say that is not a logic that
    includes dynamics?

    Can someone else help me here?

    ((I only have a couple of days left here before I head north to remote areas of British Columbia. I presume the Hammock Physicist is already on a sabbatical of sorts.))
    fundamentally
    Quantum logic does not contain any axioms that concern dynamic features. In a similar way Helmholtz decomposition theorem and Hodge decomposition theorem only concern stationary vector fields. There are groups working on extending quantum logic such that it includes dynamic elements. One of these groups works on "Logic of Quantum Actions". They introduce operational elements that can best be compared to unitary transformations. To my opinion this falls short of the real source of dynamics. Just like dynamics couples electrostatics with magnetostatics resulting in the well-known Maxwell equations, will dynamics also couple the fields with the movement of closed Hilbert subspaces. The fact that the Hilbert space that relates to traditional quantum logic has an infinite but still countable dimension, does not rule out the applicability of a rigged Hilbert space. It is required to handle Fourier transforms, Dirac delta functions and other kinds of distributions. However, these concepts must be seen as limit cases rather than as parts of the model. A rigged Hilbert space is not a real Hilbert space. Only the name is reused. The seperability of the Hilbert space means that for all operators the sets of eigenvalues are countable. With other words, our observable spaces are granular! In rigged Hilbert spaces there exist differential operators. In Hilbert space only the corresponding difference operators exist. If nature works with a Hilbert space rather than a rigged Hilbert space, then our world is granular rather than continuous. There exist many indications in that direction. LQG aims in that direction. Even the rigged Hilbert space has to cope with minimal sizes of phase spaces. The Heisenberg uncertainty principle, which is a corollary of the properties of Fourier transforms shows this. Birkhoff, von Neumann and others concluded that traditional quantum logic guides the way how quantum physics must be done. This guides to the separable Hilbert space rather than to the continuous rigged Hilbert space. In this sense a stepper like the "redefiner" is not such an odd concept.
    If you think, think twice
    blue-green


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    On 9/1/2010 I asked the following question:

    “Is it your conviction Hans that even when recording devices are not present, real physical states carom off the planes and subspaces through which theywould be confined by unitary transformations?”

    To which he replied:

    “Instruments are not required. This is about mathematics ..."

    In my inquiry, I was checking to see if Hans' “dynamics” has anything to do with the so-called measurement problem. He signals that it does not.

    To understand you Hans, I have to start from something familiar. Since I originally specialized in General Relativity and the singularity theorems of Penrose and Hawking, I have long been familiar with Penrose.

    Readers here may be aware that Penrose puts all of the operators of dynamical quantum physics into two distinct classes. One of them he calls U for unitary. The other class he calls R for reduction. This latter class has to do with the measurement problem and the projection of states into subspaces. Both classes have dynamics; maybe not dynamics as you imagine dynamics to be, Hans. You seem to be on a solitary trip.
    fundamentally
    Either a "measurement" is a kind of mathematical concept, or it has to do with experiments, where (rather large) classical objects influence (very small) quantum physical items and return a measured value. It means that the experiment may completely destroy the original state of the quantum physical item until it reaches a well-defined new state. In the picture that I painted above, the experiment squeezes the Hilbert subspace that represents the considered item. If you then redo the measurement, the representation is still in the same format and will repeat the result. This is what happens when (rather large) classical objects act on (very small) quantum physical items. It does not say anything on what happens when two very small quantum physical items influence each other. This is a more common interaction than the interaction between a classical object and a quantum physical item. The small quantum physical items hardly influence each other’s state. Still there is some influence. If you confine to measurements, then you miss a large part of quantum physical behavior. In nature a full de-coherence as it occurs during a measurement of position is an exception rather than a common case. Thus, indeed it is sensible to also investigate what moves the tiny objects. As I indicated this is NOT described by plain unitary transforms. The situation is much more complicated. Sir Penrose wrote his book for the masses. Not for solitaries like me.
    If you think, think twice
    blue-green
    For the tiniest possible steps to nudge a "system" from one level to another (even if it is just a single pure state particle or wave) we already have the annihilation and creation operators. What more dynamics can you envision Hans?

    The whole notion of “dynamics” can be a chimera. If you flow along with the particles … if your coordinate system itself is tied to the particles themselves …. as one would do with 19th century Lagrangian coordinates …. then what changes may be an average distance from one molecule (or galaxy) to another. However, in the overall scheme of things, the total energy, momentum, angular momentum, charge, and more exotic flavors never change. Not even the information. I don't believe in evolution. Everything is as sophisticated as it has ever been.

    It is a matter of taste and efficiency as to whether one considers "dynamics" to be an active rotation (for example) or a passive one. When it comes to down to what is measurable, it makes no difference if you want view things from on or off the merry-go-round. There is a coordinate
    transformation from one frame of reference to another. This is why “dynamics” is a bugaboo. Einstein's equivalence principle taught us that before anyone here was born.

    Instead of imagining dynamics, one can simply say the Universe IS.
    Neither stance is more fundamental. It is one of those duality thingies. Ladies' choice.
    fundamentally
    Creation and annihilation concerns mostly the change of discrete levels that are eigenvalues of operators that work in an harmonic mode. Other cases are creation of particles from energetic bosons and the reverse, which is the combination of particles resulting in an energetic boson. This has nothing to do with the macroscopic movement that I was telling about. The change of the "coordinates" in the Lagrangian goes together with a change in the action S, which has a different background and is more related to the physical fields that constitute this action. Any conserved quantity is directly related to the corresponding independence of the Lagrangian or as you wish of the action S with respect to the corresponding canonical conjugate observable. This is the sense of Noether's theorem. I am puzzled by the statements that you put here and I cannot find the goal that you want to achieve with it. In any case, none of your statements does disprove anything that I stated.
    If you think, think twice
    blue-green
    Normal 0

    I don’t know if anyone is reading this blog or can follow it. I still have the feeling that you are  making up theory where none needs to be made.

    I mentioned the annihilation and creation operators because these basic raising and lowering operators include the smallest changes that the quantum of action will allow.

    You wrote that you are interested “when two very small quantum physical items influence each other … The small quantum physical items hardly influence each other’s state. Still there is some influence … it is sensible to also investigate what moves the tiny objects. ... this is NOT described by plain unitary transforms. The situation is much more complicated.”

    What would Feynman say to that?

     

    Methinks you have chosen a pretty lonely path for yourself.
    fundamentally
    My honest advice to you is to read the thin and readable book of Dirac about the principles of quantum mechanics. He builds a comprehesible theory from scratch. In this way you may get a consistent picture of quantum physics. May be you will then understand something of my approach. And if you get the chance, also read some papers about the contributions of Birkhoff and von Neuman. E.g. “Quantum Theory: von Neumann” vs. Dirac; http://www.illc.uva.nl/~seop/entries/qt-nvd/. then you might understand some of my motivations.
    If you think, think twice
    blue-green


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    Thanks for the link. I'll see if I can print it and read it on my trip.

    A funny thing happened on the way to the forum.

    You have not given us Hans, a real physical situation in which your more complicated theory can calculate something at a higher degree of precision than can quantum mechanics. You have not even shown that you can do calculations with your theory.

    If your theory is better at describing an array of microphenomena, then it should be able to account for something that standard quantum mechanics misses. Good luck with that.

    In my posts I have indicated that quantum mechanics is sufficient and complete enough. No additional layers of complexity are needed, no additional hidden variables and no transformations unforeseen by Dirac and his students Penrose and Feynman. If it ain’t broke, why fix it.

    An excellent thin book on the merits and meaning of Dirac’s bra kets is Primer of Quantum Physics by Marvin Chester; he had Feynman as a teacher. It needs to be supplemented with a standard presentation of experiments.

    Currently, I am reviewing statistical physics because Erik Verlinde’s approach offers a simplification of what we already know.  Occam’s razor still rules in the sciences.  Theology is different. 

    I’ll leave with a word from Feynman in his Lectures, Volume I

    “ … although most problems are more difficult in quantum mechanics than classical mechanics, problems in statistical mechanics are much easier in quantum theory.”

    We accepted the complications of General Relativity because it allowed us to calculate at a higher degree of precision. Much later (long after Tesla), we learned how to make the theory simple and natural in our understanding of it. It took gifted teachers like John Wheeler to do so; he practically created a new generation of physicists conversant in Einstein's lessons.

    In the same vein Hans, you need to simplify and show where you can do more accurate calculations than can current theory.
    fundamentally
    If you take that paper with you, then also take Dennis Sciama's paper on the origin of inertia. See: http://www.adsabs.harvard.edu/abs/1953MNRAS.113...34S. Compare this with with what you can find about Helmholtz decomposition theorem. This gives deep insight in what fields do to physical dynamics. "In the same vein Hans, you need to simplify and show where you can do more accurate calculations than can current theory." The editor of this site does not allow a very scientific treatment in formulas. So, if you want more, you have to look at my website http://www.scitech.nl/English/Science/Exampleproposition.pdf .
    If you think, think twice