William

If you want somebody to consider your question, you have to formulate it in simple sentences. Look at this sentence for example:

==> What this notion used by gleason (i presume not as von neumann was considering continuous type 1 geometries by that stage and i think abandoned hilbert space) or what his proof less reductive (he had a non-contextuality constraint, but i presume he must have had some other constraint that ie to the effect that probability is a monotonic function of amplitude, and that it must be a constant function (ie the same probability truth condition rule in all contexts).

Take in consideration that your English is not so good. If in addition you write so long sentences and with all sort of parentheses inside them, it is complicate even to read your text. 

Also, if you have a question about what von Neumann said and what Gleason said, don't mix what they actually said with what you presume that they meant.

I would warmly recommend you to reformulate your question.

Best wishes,

Sofia 

ive edited it somewhat-- i may still edit it somewhat more, the first part concerns the meaning of von neumanns statement and the second the assumptions behind gleasons theorem; the inter-relation that is alluded to is whether von neumanns view that probability could be derived from logic, would be or would not be in agreement with the proof given by gleason. Saunders (1985) appears to allude to the answer being yes. Redei ('on von neumans 1954 address on unsolved problems of mathematics) alludes to the answer being no

Hi William!

Let's take the issues one by one. Before going to von Neumann and Gleason let's see what is the problem with classical mechanics.

You say,

==> This is not true in classical mechanics. (which i may add neither gives any values to the probabilities, and even if it gives an axiomatization it hardly is one that falls out of logic or has a model theory from which its proven, except in the trivial sense of probability is measure theory)

The theory of probability is typical to the clasical statistical mechanics, i.e. not quantum and not even wave mechanics. Indeed, classical statistical mechanics uses positive probabilities smaller than 1, not complex amplitudes of probability.

Now, the theory of probability has its axioms. One doesn't prove axioms, they come rather from the everyday experience.

The concept of probability is a primary concept in the classical theory of probability, it is not derived from a more basic concept. In other theories, the probability is NOT a primary concept.

Best regards,

Sofia

John von Neumann worked closely with Garret Birkhoff, which was an expert in lattice theory. Together they discovered what in mathematical terms is known as an orthomodular lattice. It has nearly the same relational structure as the lattice that defines classical logic. This is probably the reason that they gave their discovery the name "quantum logic". On the other hand this is a strange decision, because in their introductory paper in 1936 they also showed that the set of closed subspaces of a separable Hilbert space has exactly the structure of an orthomodular lattice. The treacherous name-giving caused that many followers tried to interpret the elements of the orthomodular lattice as logical propositions. It would have been much better if the orthomodular lattice was given another interpretation.

Separable Hilbert spaces can only cope with number systems that are division rings. The most extensive division ring is formed by the quaternions. Quaternions are ideally suited for storing dynamic geometric data. These numbers will appear as eigenvalues of normal operators. Not all closed subspaces of a separable Hilbert spaces correspond to physical objects that are more than just dynamic geometric data. Thus what makes some of the closed subspaces of a separable Hilbert space so special that they figure as physical entities? This question was never solved in the first decades after the discovery of the orthomodular lattice and later the question was forgotten.

What if the orthomodular lattice is interpreted as part of a recipe for modular construction? With other words: what if the atoms of the orthomodular lattice correspond to elementary modules. Each elementary module is represented by a single Hilbert vector which is an eigenvector of a normal operator and the corresponding eigenvalue is a quaternion and that quaternion represents the current state of elementary module. However, if the real part of the quaternion represents progression and the imaginary part represents a spatial location, then the eigenvalue and the corresponding eigenvector can only represent the elementary module at a single progression instant. This does not correspond to a more persistent physical entity. Thus, the concept of elementary module must be extended to cover a sequence of eigenvalues and corresponding eigenvectors. If the real parts of the involved eigenvalues are properly ordered, then the elementary module hops through a path and after a while the landing locations of the hopping path have formed a location swarm. If the hopping dance is governed by a mechanism that ensures a coherent statistical behavior, then the statistical characteristics become properties of the elementary module. For example after a huge number of hops the statistical characteristics mature and the swarm can be described by a location density distribution. Depending on the controlling mechanism the swarm may become describable by a continuous location density distribution and that function may own a Fourier transform. Of course this is a very special condition, but if it happens, then the swarm as a whole owns a displacement generator and at first approximation the swarm can be considered to move as one unit. Thus in that case the elementary module becomes a physical entity that follows its own path, which differs from the hopping path of the point-like object. The swarm has its own kinematics.

The separable Hilbert space was introduced in the first decades of the twentieth century by David Hilbert and others. It immediately drew the attention of von Neumann. Von Neumann also experimented with projective geometries and for a small period he switched completely to this alternative representation. Later he turned back to Hilbert spaces.  Separable Hilbert spaces have the drawback that they can only support countable eigenspaces. They do not support continuum eigenspaces. Several scientists tried to expand the notion of the Hilbert space such that it also can store continuums. For example Paul Dirac introduced his famous bra-ket notation and the Dirac delta function that notation-wise could represent operators, which feature continuum eigenspaces. Mathematically, this was not a proper approach. Only after several decades Gelfand and others introduced the Gelfand triple, which represented a proper non-separable Hilbert space. It can be shown that every infinite dimensional separable Hilbert space owns a non-separable companion. It is possible to take the view that the non-separable Hilbert space embeds its separable companion. This enables inside the Hilbert space combination concepts, such as Gleason's theorem and the notion of the wave function. This is supported by an extension of Dirac's bra-ket notation, which is called the reverse bra-ket method.

The conclusion is that the mechanisms, which control the hopping dances of the elementary modules are not part of the household  of the involved Hilbert spaces. Maybe this is the reason that they also are not part of quantum physical theories. Physical theories do not give any explanation of the coherence of the dynamical behavior of physical entities.

Personally, I believe that the mentioned mechanisms use a combination of a Poisson process and a binomial process, which is implemented by a three dimensional spatial spread function. It corresponds with my experiences as a developer of image intensifying devices. See what image intensifiers reveal.

http://www.e-physics.eu/#_What_image_intensifiers

http://vixra.org/abs/1605.0101

Hans,

You gave us a lesson worth of admiration about the birth of the quantum formalism. But, William asked "how logic alone compels me to say to events with equal amplitude have the same probability".

What is known to me is that, in the quantum world, the amplitude is a primary concept, not the probability. The latter is a derived concept from the former. The nature seems to operate directly with amplitudes, not with probabilities. 

This behavior is a consequence of the wave-like part of the quantum objects' behavior. As in the wave-mechanics, one first calculates the interference of all the contributing amplitudes, and from this result one calculates the intensity.

Similarly, in the quantum mechanics, for computing the probability of an event one first computes the amplitude of probability, which is a sum of all the contributions. From this one computes the probability: P = |ΣAi|2. One doesn't add probabilities for obtaining the total probability of an event, one adds the amplitudes.

I am aware that neither my comment is the formal mathematical answer that William asks for, I only said that the experiments confirm in full that the quantum behavior in what regards probabilities vs. amplitudes, is the same as in the wave mechanics.

Best regards

Sofia,

The confusions and misconceptions are due to the fact that physicists are reluctant to deliberate about what the wave function stands for. The wave function is not a measurable entity. It is a complex function of both space and time. Mathematically it is characterized as an amplitude probability distribution. In fact physically its squared modulus makes more sense, because that function can be interpreted as a location density distribution and in that interpretation it specifies the probability of the fact that if a corresponding observation is effectuated, then the probability that the location on which detection occurred conforms to the value of the location density distribution at the corresponding parameter value.

In combination with the Hilbert space as the applied storage medium the detected or detectable locations are precisely stored in the eigenspace of a normal operator. In this way the location distribution that underlies the wave function can be stored in a separable Hilbert space. As soon as an actual detection takes place, the process that generates the locations stops or proceeds in another mode. The consequence is that the location distribution looses its sense and together with it the wave function collapses.

This view corresponds to a picture in which a point-like particle hops around along a stochastic hopping path until it is captured and the hopping path is closed. The swarm of landing locations delivers the location density distribution.

Hans,

Which confusions and misconceptions? To endeavor to understand how the nature works is the heart of our research. If we wouldn't have done this along the centuries, we would still live in caves.

The dictum "shut up and calculate" is a declaration of incapacity. No doubt that we try to understand what is behind the formalism of QM.

But, not this is the issue here.

1) Of course the wave-function is measurable, however you have to use many identical quantum objects. By doing repeated measurements, and in different bases, you'll get amplitudes and phases.

Now, not even this is the issue. You say: "In fact physically its squared modulus makes more sense". FOR WHOM it makes more sense? The (trivial) formula 

(1) P = |ΣAi|2 

is known to you. All the experiments support this formula. So, can't you see that absolute square modulus of the wave-function is not a primary concept, but the contributing amplitides are a primary concept?

We can't be blind to experiments. The quantum objects are fragile, measuring them is not easy matter as in classical mechanics, the nature gives us only some hints. We have to catch those hints.

-------------------

You also say "This view corresponds to a picture in which a point-like particle hops around along a stochastic hopping path until it is captured and the hopping path is closed."

Let's first try to answer William's question, and after that go to other things. He asked where from comes the relation (1), and I say that it comes from wave-mechanics and the experiment shows that in QM it works. (You see, before explaining integral calculus we have to master well the simple algebraic calculus.)

Friendly regards,

Sofia

Simple answer to a complex question, the speed of light is unaffected, only its frequency ( red shift). The same would apply to gravitational waves.

The selection of a separable Hilbert space is a consequence of the orthomodular lattice structure of what von Neumann and Birkhoff called "quantum logic". That names giving was an unlucky choice, because the elements of that lattice are not logical propositions. The Hilbert space contains nothing that can guarantee the coherence of the dynamic processes that are described by QM. Stochastic mechanisms that are not part of the Hilbert space control this coherence. These mechanisms ensure that the location density distribution can be described by a continuous function, which relates to the wave function of the elementary object. Only a huge number of involved locations can implement sufficient continuity. The fact that the location swarm can be considered to move as one unit requires that the describing location density distribution owns a Fourier transform. These are very strong requirements that are far from straightforward and are completely due to the capabilities of the mentioned controlling mechanisms.

I consider it false to interpret the elements of the orthomodular lattice as logical propositions. There von Neumann and Birkhoff made an essential mistake. It is better to interpret these elements as (potential) construction modules and to interpret the atoms of the lattice as elementary modules, which are not constituted by combining lower level modules.