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Foundations of Quantum Mechanics - Science topic

Principles and interpretations of Quantum Mechanics.
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Dear Sirs,
I did not find an answer to this question in Internet for both quasi-relativistic and relativistic case. I would be grateful if you give any article references.
As I think the answer may be yes due to the following simplest consideration. Suppose for simplicity we have a quasi relativistic particle, say electron or even W boson - carrier of weak interaction. Let us suppose we can approximately describe the particle state by Schrodinger equation for sufficiently low velocity of particle comparing to light velocity. A virtual particle has the following properties. An energy and momentum of virtual particle do not satisfy the well known relativistic energy-momentum relation E^2=m^2*c^4+p^2*c^2. It may be explained by that an energy and a momentum of the virtual particle can change their values according to the uncertainty relation for momentum and position and to the uncertainty relation for energy and time. Moreover because of the fact that the virtual particle energy value is limited by the uncertainty relation we can not observe the virtual particle in the experiment (experimental error will be more or equal to the virtual particle energy).
In the Everett's multi-worlds interpretation a wave function is not a probability, it is a real field existing at any time instant. Therefore wave function of wave packet of W boson really exists in the Universe. So real quasi relativistic W boson can be simultaneously located in many different space points, has simultaneously many different momentum and energy values. One sees that a difference between real W boson and virtual W boson is absent.
Is the above oversimplified consideration correct? Is it possible to make any conclusion for ultra relativistic virtual particle? I would be grateful to hear your advises.
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A virtual particle is a particle, whose energy-momentum relation doesn’t correspond to that of a real particle.
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I tried to publish a proof by which the Bohm interpretation of QM is problematic,
in a journal, and the editors claimed that they don't see a motivation for publishing my proof.
What you think? Is the correctness (or incorrectness) of Bohm's mechanics an issue enough relevant for the QM in order to justify investigation?
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To prove the mass is non-zero, you need to set a lower limit. The current upper limit is of the order of 10^-62 kg so your experiment would need to have an accuracy of ±10^-63 kg or better to reach the 5 sigma criterion for claiming a discovery. That should allow you to work out the experimental parameters and hence the cost of running it. Nobody will fund you if you cannot even tell them what it will cost.
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The special theory of relativity assumes space time is formed from fixed points with sticks and clocks to measure length and time respectively. The electromagnetic waves are transmitted at the speed of light through this space time. This classical space time does not explain the mysteries of quantum mechanics. Do you think that maybe there is more than one space time?
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Humans have two kinds of space-time observers: the chord (tonality) observer and the non-chord (atonality) observer. They observe two kinds of space-time: chord space-time and non-chordal (atonality) space-time. Space-time is two The second level of existence.
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Consider the polarization singlet of two photons 1 and 2
(1) |ψ> = (1/√2) ( |H>1 |H>2 + |V>1 |V>2 .
Let's represent the photon 2 in another base than { |H>, |V>}, e.g { |B>, |C>} the polarization B making an angle θ with H. So the wave-function (1) transforms into
(2) |ψ'> = (1/√2) [ |H>1 (|B>2 cosθ + |C>2 sinθ) + |V>1 (-|B>2 sin θ + |C>2 cos V)].
Assume that the experimenter Alice tests the photon 1 and finds the polarization H. What happens with the polarization with the photon 2?
Assume that the experimenter Bob tests the photon 2 and finds C. What happens with the polarization of the photon 1?
An additional question: what happens with the norm of the wave-function after one of the particles is tested? Does it remain equal to 1?
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To James H. Wilson,
Does your theory require global wavefunctions or states which are space- and time- independent, and as such can be be used anywhere, at any time and in any context?
There's a manuscript titled "Quantum Rayleigh annihilation of entangled photons" which is under consideration by Optics Letters and which can be found on this website. The correlation function which is alleged to be of a quantum nature can be derived without entangled states.
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There is an opinion that the wave-function represents the knowledge that we have about a quantum (microscopic) object. But if this object is, say, an electron, the wave-function is bent by an electric field.
In my modest opinion matter influences matter. I can't imagine how the wave-function could be influenced by fields if it were not matter too.
Has anybody another opinion?
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Nice discussion
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Bohm's mechanics considers the existence of a particle that triggers the detector. This particle is supposed to be guided by the wave-function, which is assumed to be a wave existing in reality.
My question is: which one between the two items, the particle and the wave, carries the properties of the respective type of particle (charge, mass, magnetic momentum, etc.)?
Specifically, how exactly is understood the guiding wave? Does it carry in each point and point all the above features? If not, how can it feel the presence of fields and be deflected by them?
Alternatively, is the particle the one which carries the physical properties? If the particle would be just a geometric point, how could it interact with the particles in the detector?
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The way I understand it, there is both wave and particle separately.
There is also postulated a force between the wave and particle, that keeps the particle close to the wave, so it finally rattles inside the wave. Dont think anyone has seen an "empty" wave yet, evidence is lacking.
When all is said and done the consequence of these assumptions is the same as in other (ie. Born interpretation) so I dont see much advantage in practice. It may be conceptually more clear to some this way.
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I am stuck between Quantum mechanics and General relativity. The mind consuming scientific humor ranging from continuous and deterministic to probabilistic seems with no end. I would appreciate anyone for the words which can help me understand at least a bit, with relevance.
Thank you,
Regards,
Ayaz
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I guess that the Scattering Theory always will be a trend in QM.
The experimental Neutron Diffraction field for example always is creating new tools where QM is widely used.
Although it is attached to a few experimental facilities around the world, still it is a trend.
We always see new discoveries using neutron diffraction in solid-state.
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  1. Is the GHZ argument more useful than BKS theorem or is only a misinterpretation of EPR argument?  
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Sorry but you didn’t understand neither the EPR argument nor Bell’s theorem.
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Imagine that we send the wave-packet of a neutron to sensitive scales. How much would weigh the wave-packet (after discarding the effect that impinging on the scales, the neutron transmits a certain linear momentum).
Imagine now that we split the neitron wave-packet into a three identical copies, by means of a beam-splitter (e.g. a crystal), and send only one of the copies to the scales. How much would weigh the copy?
I have some opinion but I want to see to which conclusion the discussion would lead.
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Dear Sofia,
It's not a matter of opinion. If you cannot find any means of empirically validating a concept or refuting it, then by definition, that concept is empirically vacuous.Science is not a democracy where one opinion is as good as another, but a dictatorship of the laboratory. Opinions don't count in science.
As to your other point: of course a detector could monitor momentum change. That is done literally billions of times in particle accelerators such as the Large Hadron Collider. As for detectors being `too big', that is surely an incorrect notion. All detectors are macroscopic devices designed to amplify greatly otherwise "small" changes.It's the only way we ever observe anything.
As for Bohmian mechanics and the original question being posed here, my reading is that Bohmian mechanics and your question share the same mind set regarding the nature of the wavefunction,, namely that a wavefunction has some sort of physical existence over and above that of the associated particles. Otherwise, the question of "weight" of a wavefunction would not arise. If that is not the case, then what exactly does your question mean?
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The hydrogen spectral lines are organized in various series. Lyman series are the lines corresponding to transitions targeting the ground state.
Most pictures dealing with hydrogen spectra and available in the web are recordings dealing with extraterrestrial hydrogen sitting in celestial entities. Otherwise they are illustrations obtained not from experimental recordings, but from the well known Rydberg formula.
Of interest for the undersigned are pictures of Lyman series as recorded in laboratory observations of hydrogen atoms, with the atoms sitting in the laboratory itself. Not extraterrestrial hydrogen, nor molecules H2, even if the molecules are sitting nearby.
Presumably such recordings would have required ultraviolet sensitive CCDs, UV photographic plates, or similars. Particularly relevant would be careful raw recordings of Lyman series that INCLUDE THE ALPHA-LINE at 1216 Å.
Experimental remarks about the Lyman alpha-line, difficulties to observe it ---if any---, line width, line broadening, etc., and difficult-to-explain anomalies, are of particular concern. So far Web searching has not been successful.
I would appreciate any link or suggestions as to how to obtain the pictures and experimentally based information of the kind explained above.
Most cordially,
Daniel Crespin
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Dear Daniel Crespin
The article “Anomalous Behavior of Atomic Hydrogen Interacting with Gold Clusters” contains information that might be useful for finding answers to your question....
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square of amplitude of quantum wave function refers to volumetric probability distribution of a quantum wave-particle. But what does the real and imaginary parts physically mean? or what does the phase angle physically infer?
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Dear Sumit Bhowmick, in addition to all the interesting answers posted here previously, probably you would like to look at Quantum Mechanics by Landau and Lifshits, Pergamon, 1965, chapt. on elastic collisions, discussion on pp. 512.
The imaginary part in the exponent determines the lifetime of the state. It is a resonance in a quasi discrete level, also that is the origin of the so-called quasiparticles with a quasi-stationary state.
Important to say that they are solutions of a Schrodinger equation with outgoing spherical waves at infinity, a more real physical system, than those that require the wave function to be finite at infinite.
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Due to the position-momentum uncertainty, it is impossible to measure the position of a microscopic particle exactly.
This means that it is impossible to measure the probability density, i.e. the square of the absolute value of the wave function, pointwisely, i.e. at any individual point, and by extension, the values of the wave function itself at any individual point of the physical space are irrelevant from the perspective of physics, they are, so to speak, “non-physical”.
Then, wouldn’t be more consistent, from the perspective of physics, to impose any mathematical condition on small regions of physical space instead of points?
Of course, if the wave function is to be continuous, then a condition imposed on a neighborhood of some point is translated to a condition on the point itself, but this is a consequence that follows from a mathematical property, it is not a physical requirement.
Besides, since the values of the wave function are not physically measurable, its continuity is not physically measurable either.
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Dear Spiros,
You ask: "Then, wouldn’t be more consistent, from the perspective of physics, to impose any mathematical condition on small regions of physical space instead of points?"
This is precisely what de Broglie proposed as QM was in process of being defined in the 1920's.
Actually, the Schrödinger wave function was meant to define a "resonance volume" within which the electron would be in axial resonance more.
One of the possible trajectories of the infinite set of the Feynman's path integral can even be established as completely electromagnetism compliant, despite his own opinion to the contrary:
Ref: Michaud, A. (2018). The Hydrogen Atom Fundamental Resonance States. Journal of Modern Physics, 9, 1052-1110. doi: 10.4236/jmp.2018.95067
Best Regards, André
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Do Einstein's Field Equations (EFE) allow a multitude of universes?
As far as I know, Everett proposed his interpretation two years after Einstein dead. But I think that Einstein should have known that such a proposal is about to be made. Does somebody know whether Einstein said something of it?
Another thing: do EFE allow pathologic points in the space-time, points at which the universe splits into two?
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“…As far as I know, there are some serious scientists that believe in it and there are others that think it is nonsense. Being a QM interpretation, that is no wierd.…..”
- as a QM interpretation that is simply nonsense, for rather evident reason
– even to create this Universe – more correct, though “lesser fundamental” – to create this Matter, it was necessary to find and to spend practically unbelievable portion of energy,
- when to create “Multiverse” , where are, as that is suggested in the interpretation, infinite “number” of Universes, it would be necessary to spend infinitely unbelievable energy.
Cheers
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Can someone suggest the steps to perform NTO (natural transition orbitals) analysis in Gaussian 09 and view it using GaussView 5?
I tried the following code after TDDFT on a molecule:
#Geom=AllCheck Chkbas Guess=(Read,Only) Density=(Check,Transition=1) Pop=SaveNTO
I opened the chk file using GaussView. It shows the normal HOMO and LUMO plots only - I don't see the hole and particle plots.
Is there a special procedure?
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Quantum entanglement experiments are normally carried out in the regime (hf>kT - where T is the temperature of the instrument) to minimise thermal noise, which means operating in the optical band, or in the lower frequency band (<6 THz) with cryogenically cooled detectors.
However, the omnipresent questions are whether in the millimetre wave band where hf<kT:
1) Could quantum entanglement be detected by novel systems in the at ambient temperature?
2) How easy might it be to generate entangled photons (there should be nothing intrinsically more difficult here than in the optical band - in fact it might be easier, as you get more photons for a given pump power)?
3) How common in nature might be the phenomenon of entanglement (this would be in the regimes where biological systems operate)?
Answers to 1) may lead to routes to answering 2) and 3).
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Dear Dimitry,
it may be possible to used the system proposed in:
to determine if entangled photons are generated by biological systems.
many thanks,
Neil
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Consider the wave-function representing single electrons
(1) α|1>a + β|1>b ,
with both |α|2 < 1 and |β|2 < 1. On the path of the wave-packet |a> is set a detector A.
The question is what causes the reaction of the detector, i.e. a recording or staying silent? A couple of possibilities are considered here:
1) The detector reacts only to the electron charge, the amplitude of probability α has no influence on the detector response.
2) The detector reacts with certainty to the electron charge, only when |α|2 = 1. Since |α|2 < 1, sometimes the sensitive material in the detector feels the charge, and sometimes nothing happens in the material.
3) It allways happens that a few atoms of the material feel the charge, and an entanglement appears involving them, e.g.
(2) α|1>a |1e>A1 |1e>A2 |1e>A3 . . . + β|1>b |10>A1 |10>A2 |10>A3 . . .
where |1e>Aj means that the atom no j is excited (eventually split into am ion-electron pair), and |10>Aj means that the atom no j is in the ground state.
But the continuation from the state (2) on, i.e. whether a (macroscopic) avalance would develop, depends on the intensity |α|2. Here is a substitute of the "collapse" postulate: since |α|2 < 1 the avalanche does not develop compulsorily. If |α|2 is great, the process intensifies often to an avalanche, but if |α|2 is small the avalanche happens rarely. How many times appears the avalanche is proportional to |α|2.
Which one of these possibilities seem the most plausible? Or, does somebody have another idea?
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Yes, Dinesh wants everything to be Classical Mechanics,
unfortunately he is very wrong.
classical Mechanics is identified with Newton and followers,
Lagrage and Hamilton. Special theory of relativity is also included.
Then Classical field theory covers Maxwell theory of Electromagnetism, and the General theory of relativity.
In summary almost everything that is non quantum is called
Classical.
The quantum markes a sharp break in methods and results.
In fact irreducible randomness is a characteristic, which
Dinesh would rather not admit.
Thermodynamics and Statistical Mechanics also admit randomness, although at not so fundamental level, unless it is quantum statistical mechanics.
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In his work "The Consistent Histories Approach to Quantum Mechanics" publish in the Stanford Encyclopedia of Philosophy, Griffiths claims that this approach overcomes the problem of the wave-function "collapse".
His suggestion is that in each trial of an experiment, a quantum system follows a "history" meaning a succession of states.
Here is an example: consider a Mach-Zehnder interferometer with an input beam-splitter BSi, and an output beam-splitter BSo, both transmitting and reflecting in equal proportion. The outputs of BSi are denoted b and c, and those of BSo, e and f. A single-particle wave-packet |a> impinging on BSi is split as follows
(1) |a> → (1/√2)( |c> + |d>).
Before impinging on BSo the wave-packets |c> and |d> have accumulates phases
(2) (1/√2)( |c> + |d>) → (1/√2)[exp(iϕc)|c> + exp(iϕd)|d>],
and BSo induces the transformation
(3) (1/√2)[exp(iϕc)|c> + exp(iϕd)|d>] → α|e> + β|f>,
where the amplitudes α and β depend on the phases ϕc and ϕd.
In his book "Consistent quantum theory" chapter 13, Griffiths indicates two possible histories:
(4.1) |a> → (1/√2)( |c> + |d>) → (1/√2)[exp(iϕc)|c> + exp(iϕd)|d>] → |e>,
(4.2) |a> → (1/√2)( |c> + |d>) → (1/√2)[exp(iϕc)|c> + exp(iϕd)|d>] → |f>,
the history (4.1) occurring with probability |α|2, and the history (4.2) with probability |β|2.
Does somebody understand in which way these histories avoid the collapse postulate?
The correct transformation at BSo is (3), a unitary transformation, not (4.1) and not (4.2). Each one of the histories (4.1) and (4.2) involves a truncation of the wave-function at BSo. But this is exactly the mathematical expression of the collapse principle: truncation of the wave-function.
Hence my question: can somebody tell me how is it possible to claim that these histories avoid the collapse postulate?
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Nobody understands Griffiths. But if there is some need to explain, I believe that his view is that there could be multiplicity of projection operators of the state
\psi=1/SQRT(2) (|a> phase factor_a + |b> phase factor_b) on the final state but only two are consistent with observations. Still this explanation is based on assumption of zero coherence between channels |e> and |f>.
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Consider an experiment in which we prepare pairs of electrons. In each trial, one of the two electrons - let's name it the 'herald' - is sent to a detector C, and the other - let's name it 'signal' - to a detector D. The wave-function of the signal is therefore
(1) |ψ> = ψ(r) |1>,
i.e. in each trial of the experiment, when the detector C clicks, we know that a signal-electron is in the apparatus. Indeed, the detector D will report its detection.
Now, let's consider that the signal wave-packet is split into two copies which fly away from one another, one toward the detector DA, the other to the detector DB,
(2) |ψ> = 2ψA(r) |1>A + 2ψB(r) |1>B.
We know that the probability of getting a click in DA (DB) is ½, but in a given trial of the experiment we can't predict which one of DA and DB would click.
Then, let's ask ourselves what happens in a detector, for instance DA. The 'thing' that lands on the detector has all the properties of the type of particle named 'electron', i.e. mass, charge, spin, lepton number, etc. But, to the difference from the case in equation (1), the intensity of the wave-packet is now 1/2. It's not an 'entire' electron. Imagine that on a screen is projected a series of frames which interchange very quickly. The picture in the frame seems to be a table, but it is replaced very quickly by a blank frame, and so on. Then, can we say what we saw on the screen? A table, or blank?
The situation of the detector is quite analogous. So, will the detector report a detection, or will remain silent? What is your opinion?
For a deeper analysis see
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Dear Mazen,
You wanted me to reply to your question, but I have nothing to say.
"The particle didn't know all forces exist in space, but space itself know that, and know the particle itself, so when the particle appears at some point, the space (which is the second player that make the motion) can do (based on some internal mechanism) the sum-over-all-trajectories for this particle to give the particle (which is the first player that make the motion) the opportunity to exist in some specific points in space and time with different preferences (and this what I mean by "space gates")."
Exactly as you say that the space knows all sort of things, I can say that between my door and the door of my neighbor, exists a galaxy. You can say whatever you want, there is no limitation to that.
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NOTE: Please answer the question ONLY if you read what is the question.
Consider the well-known polarization singlet
(1) |S> = (1/√2) (|x>A |x>B + |y>A |y>B),
where as usually, the quantum object (Q.O.) A flies to Alice's lab and the Q.O. B flies to Bob's lab.
Consider that in each lab there is a polarization beam-splitter, PBSA, respectively PBSB, spliting the incomming beam in the base { |x>, |y>}. However, Bob has the option to input the two output beams to a second PBS - let's name it PBSC - which splits the input beams in the base { |d>, |a>} (d = diagonal direction, and a = the anti-diagonal, i.e. perpendicular on d).
(2) |x> → (1/√2) (|d> + |a>), |y> → (1/√2) (|d> - |a>).
The expression of the singlet wave-function becomes
(3) |S> = (1/2) {|x>A (|d>B + |a>B) + |y>A (|d>B - |a>B).
Assume now that Bob performs a test, with the detectors places on the outputs of PBSC, and gets the result, say, d. It is useful to write also the inverse of the transformation (2)
(4) |d> = (1/√2) (|x> + |y>), |a> = (1/√2) (|x> - |y>).
As one can see from the first equality in (4), to Bob's result |d>B contribute both beams |x>B and |y>B which exited PBSB and entered PBSC.
But, assume that while Bob does the test, Alice also performs a test, and gets, say, x. However, Alice has another story to say about what happened in the apparatus. She would claim that since she obtained the result x, in Bob's apparatus there was nothing on the output path y of PBSB. In consequence, she would claim that the beam |d>B recorded by Bob was just a component of |x>B as seen from the first relation in (2).
We do not know what is the wave-function, if it is a reality (ontic), or (epistemic) only represents what we know about the quantum object. But the quantum object travels in our apparatus, it has to be something real. Then, what is the truth about what was in Bob's setup? Was there, or wasn't, something on the output y of PBSB?
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Dear Sofia,
In this case, what is the meaning of wave-function?
if we maintain the probability meaning of the squared modulus of the wave-function, so can we found the particle it two locations at the same time?
This contradicts the energy conservation law, right?
or you need to change the meaning of wave-function?
With best regards.
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The classical limit of Feynman's path integral, gives us a partial view of how the 'collapse' process occurs. If the 'thing' that travels on all the paths between (t1, r1) and (t2, r2) increases in the number of components, or in mass - in short, becomes a classical object - we have destructive interference of all the paths, except in the vicinity of one of the paths. In this vicinity, the phases of the neighbor paths add up constructively. In this way, we get a classical trajectory. So, it's no 'collapse' of the wave-function, but destructive and constructive interference.
Indeed, when we perform the measurement of a quantum object, we do that with a macroscopic apparatus. For example, in an ionization chamber, the quantum object that enters the chambers produces a massive ionization, involving a huge number of particles.
Unfortunately, Feynman did not explain what happens when the wave-function has more than one wave-packet, i.e. how is picked one of the wave-packets. Thus, the non-determinism of the QM is, unfortunately, not explained by the Feynman's path integral.
Here came the GRW interpretation, and suggested a solution: supplementary terms in the Schrödinger equation. The parameters of these terms are so that as long as the quantum system contains only a small number of components, e.g. a small number of electrons/protons/atoms, the additional terms bring no significant change in the evolution of the wave-function. However, when the number of components becomes enough big that the object be macroscopic, the additional terms dominate in the Schrödinger equation and produce a random localization of the object.
G-C. Ghirardi and A. Bassi, "Dynamical reduction models", arXiv:quant-ph/0302164v2
The GRW interpretations has two big advantages: 1) it explains why the so-called 'collapse' occurs in the presence of macroscopic objects; 2) it shows that the density matrix of the macroscopic object has no extra-diagonal elements - i.e. represents a mixture of states not a quantum superposition.
It seems therefore that this interpretation comes in completion of Feynman's classical limit of the path integral.
Questions: a) why, in fact, should the Schrödinger equation be linear? b) any opinion about the GRW interpretation, any criticism?
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Yes, lambda is quite small. But at the end this is not really a problem. What one study nowadays is not GRW but the Continuous Spontaneous Localization (CSL) model. You can find a brief review of its theory in the review by Ghirardi and Bassi. This is the model one should actually apply since it holds also for undistinguishable particles, where GRW does not. The main difference is the scaling of the effective collapse rate which scales (roughly) with the square of the mass. This makes the collapse be effective also at the mesoscale level, where quantum features start to disappear. There is a big experimental effort in trying to test such a model. Please, take a look at my publications, where you can find the derivation of the latest experimental bounds: https://www.researchgate.net/profile/Matteo_Carlesso/research
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In his path-integral theory Feynman speaks of a particle that travels from a time-space point (t1, r1), to another time-space point (t2, r2). This particle travels on whatever possible trajectory between these two points, no matter how irregular is the trajectory. The trajectories are continuous, and their set visits in fact, between t1 and t2, all the points in the 3D space.
So far, so good. Though the abnormal fact in this story is that the particle does not travel one trajectory, after that another trajectory, and so on, but travels all these trajectories in parallel. That means, between t1 and t2, the particle goes simultaneously along all the trajectories. So at any given time t between t1 and t2, the particle is simultaneously in many points in the space.
Now, summing up the phases of all these trajectories Feynman obtains the path integral and also constructs the wave-function. However, the wave-function of a single particle is an eigenfunction of the operator number-of-particles, with eigenvalue 1. If the particle is simultaneously in many positions, we don't have a particle, but many particles.
How to explain this contradiction?
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Feynman makes an explicit comparison between his view of quantum mechanics and classical probability. Imagine a ball rolling down a Galton board, and moving randomly as it hits the various nails. If we do not observe the path taken by the particle, but merely its final position, then the probability of finding the particle in the position in which we observe it is simply the sum of the probabilities that the particle should have gone down any of the specific possible paths. Many paths are possible, and the probabilities of all are added up to obtain the complete probability. No multiple particles in that picture.
Now Feynman, as I understand him, says: ``Quantum mechanics is just the same, except you add up the amplitudes, which can be complex''. Now clearly, there is a lot that is not obvious in this substitution. However, it does not seem to me that, when we consider amplitudes, we are forced to view multiple coexisting particles for each different path. We did not do that in the case of probability, why do it with amplitudes?
Best wishes,
Francois
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Consider the simple wave-function describing single particles
(1) ψ(r, t) = 2L(r, t) + ψR(r, t)],
where the wave-packet ψL flies to the left of the preparation region, and ψR to the right. Since after some time t1 the two wave-packets are far from one another, their supports in space are disjoint. The continuity equation
(2) ∂|ψ(r, t)|2/∂t + ∇Φ(r, t) = 0
where Φ(r, t) is the density of current of probability, can be therefore written as
(3) ∂|ψL(r, t)|2/∂t + ∇ΦL(r, t) = -{∂|ψR(r, t)|2/∂t + ∇ΦR(r, t)}
because products as ψL(r, tR(r, t) and their derivatives, vanish. The current density is a functional of the functions ψL, ψR, and their derivatives, therefore can also be separated into ΦL and ΦL.
Let's further notice that when the position vector r sweeps the space on the left of the preparation region, the RHS of equation (3) vanishes. There remains
(4) ∂|ψL(r, t)|2/∂t + ∇ΦL(r, t) = 0.
Symmetrically, when r sweeps the space on the left of the preparation region, the LHS of equation (3) vanishes. There remains
(5) ∂|ψR(r, t)|2/∂t + ∇ΦR(r, t) = 0.
Imagine now that on the way of the wave-packet ψL is placed an absorber AL(ρ), where ρ defines the internal prameters of the absorber. The wave-packet ψL is splitted into an absorbed part and a part that passes unperturbed
(6) ψ(r, t) AL0(ρ) -> 2{ [e-γdψL(r, t) AL0(ρ) + (1- e-2γd)½AL1(ρ) + ψR(r, t) AL0(ρ) },
where the super-script 0 indicates the non-perturbed internal state of the absorber, 1 indicates its excited state, γ is the absorbing coefficient, and d is the absorber thickness. For d sufficiently big one will have total absorption of ψL,
(7) ψ(r, t) AL0(ρ) -> {AL1(ρ) + ψR(r, t) AL0(ρ) }.
Due to the presence of the absorber, the LHS of equation (5) should be multiplied by the factor AL0(ρ). But since AL0(ρ) ≠ 0, we can divide on both sides of the new equation by AL0(ρ), s.t. the original form of (5) returns. The meaning of this result is that the absorption of ψL does not imply the disappearence of ψR.
Now, let's replace the absorber with a detector. As long as the interaction with the detector proceedes inside the material of the detector, the analysis with the absorber remains valid (with the small difference that instead of absorption there may be inellastic scattering). Therefore, the equation (5) also remains valid and the conclusion that ψR is not affected.
The difficulty appears when the macroscopic circuitry surrounding the material in the detector, CL, enters into the play. Macroscopic objects cannot be in a superposition of the states as CL0 and CL1. So, we cannot have an equation similar to (7)
(8) ψ(r, t) AL0(ρ) CL0-> {AL1(ρ)CL1 + ψR(r, t) AL0(ρ)CL0 }.
However, when the circuitry clicks, what happens with the continuity equation (5)? For the collapse to be true, i.e. for ψR(r, t) to vanish suddenly, the derivative ∂|ψR(r, t)|2/∂t should be very big in absolute value for which the flux gradient should increase drastically in the outward direction from ψR. That doesn't mean that the wave-packet ψR disappears but that it disperses in space.
What is your opinion?
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Dear Sofia,
the continuity holds when the Schrödinger equation holds, i.e. when the time evolution is unitary. A measurement of an observable leads to a non-unitary change of the state vector of a system, but to the projection of the state onto some eigen-space of the observable under consideratuion. This means the continuity equation does not hold at the instance of measurement.
Best regards
Oliver
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In a course on quantum mechanics I took we were told that for the setting in which the gun shoots a particle with spin orientation +z and the analyzer is perpendicular to this orientation, 50% of the time the Stern-Gerlach apparatus will detect the particle coming out from the +x aperture, and 50% from the -x aperture.
I ran the PhET Stern-Gerlach simulator (https://phet.colorado.edu/sims/stern-gerlach/stern-gerlach_en.html) with one analyzer (magnet) and obtained the following results:
at 0 deg. angle: 100% from +x
at 90 deg. angle: 50% from +x, 50% from -x
at 180 deg. angle: 100% from -x
at 270 deg. angle: ca. 3% from -x, ca. 97% from +x
Is this last result an error of the simulator? Or how can it be explained?
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The e an m bosons make the 'field'. Thus, the 'field's is limited to v<<c. It therefore cannot propagate at v=c. The idea of an underlying non-local [qft] field requires it to be therefore infinite. If it is finite, it is therefore localized. The big bang is a lower limit, and by the most fundamental Theorem of the Limits at Infinty, defined our domain as finite. As such, no underlying non-local field is possible. Act is founded upon explaining clear observable by way of ungovernable (b-day and compactifued) dimensionalities.
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In quantum mechanics, the state space is a separable complex Hilbert space.
By definition, a Hilbert space is a complete inner product space.
The term complete means that any Cauchy sequence of elements (vectors) belonging to the Hilbert space converges to an element which also belongs to the space. In other words, completeness means that the limits of convergent sequences of elements belonging to the space are also elements of the space. Intuitively, we can say that Hilbert spaces have no “holes”.
If the state space is infinite-dimensional, we implicitly invoke its completeness every time we expand a state in terms of a complete set of eigenstates, such as the energy eigenstates or the eigenstates of another observable, since an infinite series of eigenstates is meant as the limit of the sequence of the respective partial sums when the number of terms tends to infinity. The sequence of partial sums is then a Cauchy sequence converging to the initial state, which must belong to the space.
Qualitatively, considering a convergent sequence of physical states, we expect that it converges to a physical state too, because it would be unphysical, by means of such a sequence, to end up at an unphysical state. For instance, assume that we perform a series of small changes to the state of a quantum system and suddenly we reach an unphysical state. This would be physically unacceptable. Thus, from a physical perspective, the completeness of the state space seems unavoidable.
However, looking in some of the so-called standard textbooks of quantum mechanics, particularly in Sakurai’s, Merzbacher’s, Gasiorowicz’s, and Griffiths’s, this essential property is either overlooked or just mentioned, and it is not highlighted properly.
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Mathematical subtleties that are of relevance in quantum theory are indeed discussed in many books, though they are not usually suggested as references to undergrads. For instance, see the two volumes by Galindo and Pascual. They discuss finer points such as the deficiency index, self-adjoint extension, etc, in addition to a neat introduction to Hilbert spaces. Another accessible introduction is the book by Capri. There are, of course, the well known mathematical treatises by Reed-Simon, Preguvecki, etc.
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In quantum mechanics, the state space is a separable complex Hilbert space.
A Hilbert space is separable if and only if it has a countable orthonormal basis [1, 2].
Why the quantum mechanical state space must be separable?
In [3], we read that separability is a mathematically convenient hypothesis, with the physical interpretation that countably many observations are enough to uniquely determine the state of a quantum system.
In Merzbacher’s quantum mechanics (3d ed.), page 185, we read that “The infinite-dimensional vector spaces that are important in quantum mechanics are analogous to finite-dimensional vector spaces and can be spanned by a countable basis. They are called separable Hilbert spaces.”
From a historical point of view, the two descriptions (or versions) of quantum mechanics that were initially developed in 1920’s, namely the Schrödinger’s wave mechanics and the Heisenberg’s matrix mechanics, were respectively based on the Hilbert spaces of square integral functions and square summable sequences of complex numbers, which are both separable, and physically equivalent (mathematically isomorphic). Thus, the invariant (or representation-free) description of quantum mechanics, through the abstract Hilbert space of Dirac kets, that was followed, had to be based on a separable Hilbert space too, otherwise it would not be equivalent to the two existing descriptions.
As it happens with the property of completeness [4], the property of separability of the quantum mechanical state space is also overlooked or mentioned very briefly in standard textbooks and the reader, especially the physics-oriented one, is left with the impression that it is rather a mathematical “decoration” of minor physical importance that can be forgotten.
From my own experience, it is also worth noting that the expression “a Hilbert space is separable if and only if it has a countable basis”, which is often given as definition of separability, is tricky and, to some extent, misleading. A reader with some background in functional analysis is rather easy to understand that, here, “it has a countable basis” actually means “ALL bases are countable”, as two basis sets are related by a one-to-one and onto mapping, thus they have the same cardinality, and then if one is countable, the other is countable too. But, a physics student may be confused and left with the impression that separable Hilbert spaces have also uncountable bases, which is the wrong picture, especially in connection with the uncountable (continuous) sets of the position and momentum eigenstates that although span the state space, they are not actually bases, because they are not belong to the state space, and this point is not highlighted in literature either.
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Of course not-only that it isn't correct to claim that it hasn't been taken into account.
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In standard quantum mechanics textbooks, the form of the momentum operator, in position space, is either given as definition, i.e. they write that the momentum operator is –id/dx (times the reduced Planck constant), or, in more advanced textbooks, like Landau & Lifshitz’s, it is derived as the generator of spatial translations.
I wonder if the form of the momentum operator, i.e. that it is a first-order differential operator, can be derived qualitatively, by means of physical arguments. In other words, does the slope of an arbitrary, i.e. non-stationary, wave function have a physical meaning?
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For people who insist on some definite definition, it is defined as the dual operator to x, that is whatever operator p
whose property is [x,p]=i hbar
In the previous message I showed how to recover m dx/dt
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I wasn't able to get von Neumann's book "Mathematical Foundations of Quantum Mechanics". But I saw many descriptions on his scheme of measurement, all of them saying the same things.
What I was interested in, was to see whether von Neumann claimed somewhere that after measuring a quantum system (with a macroscopic apparatus) and obtaining a result, say a, the rest of the wave-function disappears. As a simplest example, let the wave-function be α|a> + β|b>, and in one particular trial of the experiment one gets the result a. I saw nowhere a claim that von Neumann said that the part |b> of the wave-function disappears.
What I saw was the following claim: if we collect in a separate set A all the trials which produced the result a, the wave-function characterizing the quantum object in the set A is |a>.
I never saw a word about what happens with the part |b> of the wave-function in these trials. No assumption whether it disappears, or, alternatively, no opinion that we can say nothing about it. The fact that we collect the systems that responded a in a separate subset, does NOTHING to the part |b> if it survives in some way.
Did somebody see in von Neumann's work any opinion about the fate of the part |b>?
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Technical texts aren't scripture! Nor does the fact that anyone appeals to von Neumann-or any other ``famous'' name-imply anything about whether the statements made by the people-or by von Neumann or the authority appealed to-are correct, or not. That's why it's utterly futile to focus on the text and not the meaning in scientific issues.
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Please, I very need only in the number, not the references to book or common words.
Please, consider a photon as the quantum of electro-magnetic field, and thus, as carrier of magnet field. Thank you in advance.
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Is the photon motionless? Obviously not; it is moving at light speed. Then where 'its magnetic field' might be observed? One mile behind this photon, two miles? Stationary observer should experience time-varying field (electric and/or magnetic) or the energy flux in other words. If so, then the photon's energy should steadily decrease, thus its frequency should tend to null. Nothing like that is observed, although there are some hypotheses about 'tired light'. In conclusion: the amplitude of a single photon magnetic field is exactly zero.
Or in the other way: photon does not carry any electric charge and therefore produces no (time-varying!) electric field - so why it would be the source of magnetic field, which is uniquely related to electric field?
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This question is a reaction to the fact that some authors hold that the interaction between a microscopic object with a macroscopic object, leads to an entanglement between the states of the microscopic object and states of the macroscopic object. My opinion is that such an entanglement is impossible.
I recommend as auxiliary material the discussion
https://www.researchgate.net/post/What_is_the_quantum_structure_of_a_particle_detector_containing_a_gas_obeying_Maxwell-Boltzman_statistics
THE EXPERIMENT: From a pair of down-conversion photons, the signal photon illuminates the non-ballanced beam-splitter BS1 - see the attached figure. The idler photon is sent to a detector E (not shown) for heralding the presence of the signal photon in the apparatus. The signal photon exits BS1 as a superposition
(1) |1>st|1>a |0>b + ir|0>a |1>b , t2 + r2 = 1.
On each one of the paths is placed an absorbing detector, respectively A and B. The figure shows that the wave-packet |1>a reaches the detector A before |1>b reaches the detector B. Let |A0> ( |B0> ) be the non excited state of the detector A (B), and |Ae> ( |Be> ) the excited state after absorbing a photon.
Some physicists claim that the evolution of the signal photon through the detector A can be written as
(2) |A0> |1>s → (t|Ae> |0>b + ir|A0> |1>b) |0>a .
I claim that this expression is impossible, for a couple of reasons.
1) Are the states |A0> and |Ae> pure quantum states, or mixtures? I claim that a macroscopic object cannot have a pure quantum state, it can be in a mixture of pure states, all compatible with the macroscopic parameters. As supporting material see the discussion recommended above, and also the Feynman theory of path integral - the macroscopic limit.
2) In continuation, when the wave-packet |1>b meets the detector B, the state (2) should evolve into
(3) |A0> |B0> |1>s → (t |Ae> |B0> + ir |A0> |Be>) |0>a |0>b
= (t |Ae> |B0> + irt |Ae> |Be> - irt |Ae> |Be> + ir |A0> |Be>) |0>a |0>b
= [ t |Ae>( |B0> + ir |Be>) + ir|Be> ( |A0> - t |Ae>)].
That is similar with the following situation: if the cat A says "miaw" the cat B remains in the superposition ( |cat B dead> + ir |cat B alive>), and if the cat B says "miaw" the cat A remains in the superposition ( |cat A dead> - t |cat A alive>),
Did somebody see cats in such situations?
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The problem with this question is it depends on what you think the wave function represents. Mathematics simply relates symbols, but in physics, strictly speaking the symbols have to represent something in the world. In quantum mechanics the issue depends on what you think ψ represents. If all you do is consider it a mathematical process, then you write your formalism, and in the case of Sofia's question, your answer depends on what your formalism gives you.
I am sorry, Sofia, that I have not tried to give an answer because when push comes to shove, this problem has some similarity to the delayed quantum eraser experiment, and I have argued that there is an alternative possibility, and if the experiment were done properly, you would be able to distinguish them. What should happen depends very much on exactly what the down converter does with a polarised photon, and exactly what happens in the beam splitter. This can be approached by assertion, or one could do the experiment I want done. Most will wave their arms and say there is no need because they "know" what will happen, but if they were that confident, they should simply do it. (In the first step it involves getting the experiment to work as published, then blocking one of the streams of idlers going to the mixer and seeing the effect on the signal photons. The concept is you have to change the nature of what you KNOW gives the effect before you make your choice.)
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The Schrödinger self adjoint Hamiltonian operator H correctly predicts the stationary energies and stationary states of the bound electron in a hydrogen atom. To obtain such states and energies it suffices to calculate the eigenvalues and eigenfunctions of H. Since 1926 up to now, and for the foreseeable future of Physics, any theoretical description of the hydrogen atom has to assume this fact.
On the other hand the Schrödinger time dependent unitary evolution equation $\partial \Psi / \partial t = -iH(\Psi)$ is obviously mistaken. So much so that in order to explain transitions between stationary states the unitary law of movement has to be (momentarily?) suspended and then certain "intrinsically probabilistic quantum jumps" are supposed to rule over the process.
Transitions are physical phenomena that consist in the electron passing from an initial stationary state with an initial stationary energy, to another stationary state having a different stationary energy. Physically transitions always involve the respective emission/absorption of a photon. Whenever transitions occur the theoretical unitary evolution is violated.
It is absurd to accept as a law of nature an evolution equation that does not corresponds with the physical phenomena being considered. Electron transitions are not predicted, nor described by, nor deducible from the Schrödinger evolution equation. In fact Schrödinger evolution equation is physically useless. This is the reason for Schrödinger's "Diese verdammte quantenspringerei". Decades of belief in unitary evolution originated countless speculation, contradiction and confusion with enormous waste of human talent and time.
Assume then that physicists accept the mistaken nature of unitary evolution and proposes its replacement with a novel equation that a) is consistent with the predictive virtues of H b) deterministically describes transitions In principle a probability free, common sense, rational, deterministic, well constructed replacement of Quantism should be a welcome relief for physicists and chemists, and for philosophers of science as well. Then, among equations and theories currently accepted by mainstream Physics, which ones would be affected by the eventual replacement of unitary evolution? Here is a short list of prospective candidates that the reader can extend and refine Quantum chemistry Dirac equation Quantum field theories Quantum gravity Standard model Lists of physical theories are available at
For more on the inconsistencies of Quantism and details on a theory that could replace it see our Researchgate Contributions page
With most cordial regards, Daniel Crespin
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Dear Christian,
You wrote: "There seems nothing wrong with Schrödinger's equations. Only Bohr and Heisenberg did not like it. They preferred the mystery."
And so did most in the community almost from the start.
De Broglie and Schrödinger already considered transitions as "not instantaneous" even as Schrödinger introduced the wave equation, and this was practically 100 years ago, not a few decades ago, and this is what they were attempting to address, but could interest nobody else.
At the beginning of the 1950's, Schrödinger himself very publicly and futilely protested again about the very idea of instantaneous quantum jumps. Unfortunately, nobody paid attention.
If you had even glanced at the paper I referred Daniel to in the first answer in this thread, you would have found the direct quotes from him and the actual references where you can find them.
It is to this 100 years disconnect that I have been trying to draw attention.
At long last, some research seems to be resuming in the right direction.
Best Regards
André
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Is the canonical unit 2 standard probability simplex, the convex hull of the equilateral triangle in the three dimensional Cartesian plane whose vertices are (1,0,0) , (0,1,0) and (0,0,1) in euclidean coordinates, closed under all and only all convex combinations of  probability vector 
that is the set of all non negative triples/vectors of three real numbers that are non negative and sum to 1, ?
Do any  unit probability vectors, set of three non negative three numbers at each pt, if conceveid as a probability vector space,  go missing; for example <p1=0.3, p2=0.2, p3=0.5>may not be an element of the domain if the probability  simplex in barry-centric /probabilty coordinate s a function of p1, p2, p3 .
where y denotes p2, and z denotes p3,  is not  constructed appropriately?
and the pi entries of each vector,  p1, p2 p3 in <p1, p2,p3> p1+p2+p3=1 pi>=0
in the  x,y,z plane where x =m=1/3 for example, denotes the set of probability vectors whose first entry is 1/3 ie < p1=1/3, p2, p3> p2+p3=2/3 p1, p2, p3>=0; p1=1/3 +p2+p3=1?
p1=1/3, the coordinates value of all vectors whose first entry is x=p1=m =1/3 ie
Does using absolute barry-centric coordinates rule out this possibility? That vector going missing?
where <p1=0.3, p2=0.2, p3=0.5> is the vector located at p1, p2 ,p3 in absolute barycentric coordinates.
Given that its convex hull, - it is the smallest such set such that inscribed in the equilateral such that any subset of its not closed under all convex combinations of the vertices (I presume that this means all and only triples of non negative pi that sum to 1 are included, and because any subset may not include the vertices etc). so that the there are no vectors with negative entries
every go missing in the domain  in the , when its traditionally described in three coordinates, as the convex hull of three standard unit vectors  (1,0 0) (0,0,1 and (0,1,0), the equilateral triangle in Cartesian coordinates x,y,z in three dimensional euclidean spaces whose vertices are    Or can this only be guaranteed by representing in this fashion.
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Of course it is, its what it is by definition. I probably should have thought more about this, back when I originally posted. If it isnt, then nothing is. Its the "closed" convex hull of its vertices, or all points in [0,1]^3, that can be expressed by convex combinations of its vertices:, (1,0,0), (0,0,1), (0,1,0).
So that any vector (x,y,z), x+y+z=1 , or rather point, (x,y,z), in [0,1]^3, where x+y+z=1,and x >=0,y,>=0z>=0 can be simply expressed by x* (1,0,0)+(0,1,0)+z*(0,0,1)=(x,y,z), as closed under all convex combinations of the vertices convexity (of this form) simply means/ requires that its contains all points in [0,1]^3, that can be expressed by-negative c1, c2,c3 ; c1*(1,0,0)+c2*(0,1,0) +c3*(0,0,1) where c1+c2+c3 =1, it just is the set of 3 all and only non-negative coordinates, that sum to one . And clearly we can set c1=x, c2=y, and c3=y and these are non-negative and sum to one. So its all, and "only" probably triples (x,y,z),x+y+z=1 .
I should have thought this through.
The main questions are (1) and (2):
(1): does the canonical 2 probability simplex, contain the canonical 2 simplex of the sums at each point x+y, x+z, z+y, clearly at each point (x,y,z) in the canonical 2 probability simplex, x+y,in [0,1] x+z\in [0,1], z+yin [0,1] and their sum (x+y+x+z+z+y=2(x+y+z)=2*1=1, but does it contain at "every" point, every such convex combination of (x+y=l, x+z=g, z+y=h), 1>=h>=0,1>=g>=0, 1>=l>=0 & l+g, g+h, h+l in where h+g+l=2 at some point (x,y,z)?
Id say yes, because if there were a (l,g,h), l+g+h=2, 1>=h>=0,1>=g>=0, 1>=l>=0, but with no accompanying (x,y,z) x+y+z=1, (x,y,z)>=0, l= x+y, g=x+z, h=z+y
, g=x+h-y, y=x+h-g, so l=x+x+h-g, l=2x+h-: x=1/2*(l-h+g), which is uniquely determined where l-h+g<=l+g+h=2, as h, g, l>=0, and clearly x+y+z=1/2(2x+2y+2x)1/2(l+g+h)=1, the only way that point could not be in the simplex is if, for example x<0
which in this case of l+g<h, as h<=1, g<=1, l<l, as per the conditions above, that we have a contradiction as (l+g)+h=2 where, l+g<h gives (l+g)+(l+g)<(l+g)+h=2, to 2(l+g)<2, L+g<1 but h<=1, so 1+h<=1+1=2, so, 1+h<=2,
L+g<1 gives (l+g)+h< 1+h
and (l+g)+h<1+h<=2, L+g+h<2, ie l+g+h<1
1/2*(1-g+h)<=1/2*2=1
And moreover, (2):
(3)Does it contain all such h+g+l=1, where the l, g, h, are the sums of the first 2, the first and third, second and third coordinates, respectively of 3 "distinct" cartesian points in the simplex, (x1,y1,z1), (x2,y2,z2), and (x3, z3, y3), l=x1+y1, g=x2+z2, h= y3+z3, h+g+l=1, where none of the x1,x2,x3, x2,y2,z2, x3,z3, y3 are 0?.
Clearly for any m, in [0,1] there is a point in the simplex <x,y,z>, x=m.
So as x is an element of a point, in the simplex, then at those points, where the x coordinate is m, m=x>=0, and clearly x=m<=1.
As, 1>=m>=0 entails that 0<=1-m<=1, then (1-m) is in [0,1],. Then as, for any real in [0,1], there are pts in the simplex, where the x coordinate attains that real, then there is a pt, whose x coordinate is x1, where, (the x coordinate), assumes the value, x1=1-x=1-m.
<x1,y1,z1>, x1+y1+z1=1 so, y1+z1=1-x1=1-(1-m)=m, so x+z assumes the value m,
As the y and z coordinates can assume any value in [0,1] as well, then we have all non negative coordinate of three points, in [0,1]^3:, p1 =(x1,y1,z1), p2=(x2,y2,z2), and p3=(x3, z3, y3), where x1+y2+y3<=3 where x1,y2,y3, are all in [0,1], and thus, the subset, comprising all, sets of three points, p1, p2, p3 in [0,1]^3 where x1+y2+z3=2 , where x1,y2, z3 are non-negative and in [0,1,] are in the simplex.
As, at any pt, in the simplex, and at all pts , x+y+z=1, so: (x1+z1+y1)+)x2+y2+z2)+(x3+y3+z3)=3, and as x1+y2+z3=2 ,
so( x1+y1)+(y2+z2)+(z3+x3)= 3-(x1+y2+z3)= 1,
ie for any real in [0,1] x=m at(x,y,z) if and only there is a point (x1,y1,z1), x1+z1= m
(ie not on the edges of the simplex, where, aeprt from the vertices, only two of the coordinate of a point are non-zero and not 1 ?
At the edges we set y1=0, x1=x1+y1=0+x1= l,
x2=0, so as x2+y2+z2=1, x2+z2 in [0,1], so we set z2=g, z2+x2=z2+0=z2=h so,
z2+x2=g, and at the third point. Let z3=0, h=y3 so y3+z3=0+y3=y3=h
I suppose so, there was a counter-example there wou as if it did not there would have to be no point in the simplex (x,y,z), where 0<=y+z=l=1-g-h<=1 for some, (h,g, l), h+g+l=1 h>=0, g>=0, l>=0, where of course, this gives that 0=1-1<=1-(g+h)=l>=1-0=0 , It also has to express all such combinations because if did not for some l, g, h.
So as long as the the set of all probability triples, 2 canonical probability simplex , contain on its edges, (the 2 positive entries,), the set of call of all probability doubles, that is, the "canonical 1 probability simplex/ the set of 2 non-negative points in R^2 whose sum is 1 (all convex combinations of 2 non-negatives that sum to 1, then the it will contain some such three such edge points, meeting the above sum constraintm(x1,y1,z1), (x2,y2,z2), and (x3, z3, y3), l=x1+y1, g=x2+z2, h= y3+z3, h+g+l=1.
Where along the edges, the double in the canonical 1 probability simplex, which any triple on the edges of canonical 2 probability simplex, identifies, are those doubles, formed by the, 2 elements, of the 3 elements at that point (x,y,z), which happen to be positive (non-zero). SO, setting the first positive coordinate, in said triple (which will be x, or y) to be the x coordinate of the double, and the second positive entry (which will be z or y) to y, ie (x=1/6, y=0, z=5/6) goes to (x=1/6, y=5/6) which will be unique except at the vertices (where one can use both (1,0) or (0,1), and x+y=1 as the other point z will be zero, as the point was taken from the edge of the set of canonical 2 simplex (x,y, z), where precisely one of (x,y,z), is 0, (except at the vertices, themselves, where 2 of them are of course)
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Personally I don't find Objective-collapse-theory (QMSL), very appealing: Even though problems were resolved in the 1990's there are still inconsistencies in terms of diverging particle densities etc...
However, the Penrose interpretation, which is considered to be a QMSL variant, is a different story altogether: Penrose suggests that the collapse of the wave function is due to the energy differences between quantum states having reached a certain threshold. The limit being the Planck mass of the system/object at hand. In effect this still means that matter can exist in more than one place at one time. Nonetheless, a macroscopic system, like a human being, cannot exist in multiple places at once as the corresponding energy difference is too large to begin with. So a microscopic system on the other hand, like an electron, can exist in more than one location until its space-time curvature separation reaches the collapse threshold, which could be a thousands of years from the emergence of its superposition.
What are your thoughts and opinions on this?
  • Note: A macroscopic system, like a human being, could theoretically exist in a superimposed state for a very, very short period of time, at scales of Planck time or less [arXiv:1401.0176] ... So therefore it's not considered significant (if at all possible).
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Dear Daniel,
I think you may have forgotten about the Einstein–Smoluchowski Diffusion Equation, which is a differential equation for a classical probability wave in a diffusing system. Classically, I could label one of the particles, then throw it into a room and allow it to be pushed around randomly. I could then calculate the probability distribution at a subsequent time, then find that particle. The probability distribution immediately collapses. The probability distribution has no objective physical existence per se. Same thing in quantum theory, unless you have Bohmian-Vigier tendencies. Probability is a manifestation of context and ignorance, not a property of an object.
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The "collapse" postulate says that if part of the wave-function produces a click in a detector, the rest of the wave-function disappears. In the experiment described here, it is shown that no part of the wave-function disappears, namely, given a superposition of two wave-packets, while one wave-packet produces a click in a detector, the other wave-packet produces observable interference effects.
A quantum system is prepared in a state with maximum one particle, a photon, A;
(1) |ψ> = q{ |0>A + p( |1;a>A |0;b>A + eiθ |0;a>A |1;b>A ) },
see figure.
It is shown below that while the wave-packet |1;a>A
The wave-packet |1;b>A illuminates one side of the 50-50%beam-splitter BS, and on the other side lands a coherent beam
(2) |α> = N( |0>B + peiα|1>B + . . . ).
where N is the normalization factor. Thus, we have the total wave-function
(3) Φ = |α>|ψ> = Nq( |0>B + peiα|1>B + . . . ){ |0>A + p( |1;a>A |0;b>A + eiθ|0;a>A |1;b>A ) }.
At the beam-splitter the following transformations take place
(4) |1>B → (1/√2) ( |1;c> |0;d> + i|0;c> |1;d>);
(5) |1;b>A → (1/√2) (i|1;c> |0;d> + |0;c> |1;d>).
Introducing them in (3) one gets the following IMPLICATIONS:
(6) For θ = α - π/2, every click in the detector D is preceded by a the detection of the wave-packet |1>a in the detector U.
(7) For θ = α + π/2, every click in the detector C is preceded by a the detection of the wave-packet |1>a in the detector U.
Thus, one can see that by changing the phase θ, carried by the wave-packet |1;b>A , one can switch between a joint click in D and U, and a joint click in C and U.
CONTRADICTION: one can see in the figure that BS is more distant from the preparation region than the detector U. So, if the collapse hypothesis were correct, the tunning of θ would have no effect, since when the detector U clicks, the wave-packet |1;b>A would disappear instead of reaching the beam-splitter BS.
CONCLUSION: No part of the wave-function disappears - no collapse.
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Dear Tina Lindhard,
my viewpoint on Quantum Mechanics is basically expressed in this work:
You are not a physicist but if you raise questions I will search for giving an answer.
Regards
Daniele Sasso
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- A moment is the smallest difference between two states of the same matter in the space.
- Time is the continuous flowing of multiple consecutive moments.
I don't know if you could comment or not, but above is my proposition.
Comment définir le temps ?Aujourd'hui je vous propose une définition simple du temps. Dites-moi ce que vous en pensez !
- L'instant (le moment) c'est la plus petite différence entre deux états d'une même matière dans l'espace.
- Le temps c'est l'écoulement continue de plusieurs moments successifs.
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Time is an abstract concept that we experience. what we measure with clocks may not be the same as what we experience. There are many kinds of Time. Newton was the first physicist who used the symbol "t" for time in his equations to describe the physical phenomenon. This Newtonian Time was Absolute and was shown by Einstein that it is not measurable with our clocks. Our clocks measure Einsteinian Time. This time depends on the Frame of Reference as shown in the Special Theory on Relativity. The time measured by Galileo depended on Gravity and it is not the same as the Absolute Newtonian Time, since the pendulum moves at different speed depending on it's position. It moves slower on top of mount Everest than at the bottom. With General Theory on Relativity Einstein introduced another kind of time. This time is slower at the base of mount Everest than at the top. Neurologically there is the temporal lobe time. this is based on the way the temporal lobe arranges our experiences to form a narrative. there are disorders where this narrative is not exact and gives rise to phenomenon called de ja vu and jamais vu and amnesias. Genetically, there is another time which is based on the length of the telomeres on the chromosomes. they relate to the biological age of a cell or body which has limited correlation to the chronological age. this is seen in people whose physical health is much younger or much older than their chronological age. An example is progeria where the physical age of the person is highly accelerated. Therefore we cannot talk about a single entity called Time as it does not physically exist. thanks.
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Dear Syed
According to my mathematics called self-field theory (SFT) there may be two different methods of storing memories: one is based on electromagnetic (EM) fields while long term memories are stored as strong nuclear (SN) fields, probably within DNA. The conversion of short and long term memories presumably happens during 'deep' sleep when the two dimensional, EM fields are somehow converted into three-dimensional gluon encoded data within quarks. This is implied by the structure of the mathematics and its connections to particle physics.  
The mathematics 
If this hypothesis is correct the question is what happens to 'sound' in long term memories?
Could this be useful in your research into consciousness?
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With a SIM card and a code we have access to a huge amount of information which is not stored in the cell phone but in the Cloud, i.e. in big servers at another place in space. I claim the same for our memory. It is enough with a code in the brain but the memory of an event is not stored in brain. It does not have to be stored at all as all events past, present and future exists in spacetime, which is at least ontologically 4D but possible to extend to 6D see http://www.drpilotti.info/eng/sixdimensioinal-relativity.html
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What lies outside of the boundaries of space. I find the problem in this question is our knowledge, we have a prior belief that we are exist inside space, this belief that doesn’t based on any evidence, so I think before we asking, what lies outside of space? We must first ask, are the objects lying inside space or outside of it?
I think if objects are exist outside of space, it will suffer from superposition or uncertainty in position depending on the distance from space or how far is it from space.
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can i ask what is space?
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The standard QM offers no explanation for the collapse postulate.
By the Bohmian mechanics (BM) there is no collapse, there exists a particle following some trajectory, and and detector fires if hit by that particle. Therefore, there is no collapse. However, BM has big problems in what concerns the photon, for which no particle and no trajectory is predicted. Thus, in the case of photons, it is not clear which deterministic mechanism suggests BM instead of the collapse.
The Ghirardi-Rimini-Weber (GRW) theory says that the collapse occurs due to the localization of the wave-function at some point, decided upon by a stochastic potential added to the Schrodinger equation. The probability of localization is very high inside a detector, where the studied particle interacts with the molecules of the material and gathers around itself a bigger and bigger number of molecules. Thus, at some step there grows a macroscopic body which is "felt" by the detector circuitry.
Personally, I have a problem with the idea that the collapse occurs at the interaction of the quantum system with a classical detector. If the quantum superposition is broken at this step, how does it happen that the quantum correlations are not broken?
For instance in the spin singlet ( |↑>|↓> - |↓>|↑>) one gets in a Stern-Gerlach measurement with the two magnetic fields identically oriented, either |↑>|↓>, or |↓>|↑>. The quantum superposition is broken. But the quantum correlation is preserved. On never obtains, if the magnetic fields have the same orientation, |↑>|↑>, or |↓>|↓>.
WHY SO? Practically, what connection may be maintained between the macroscopic body appearing in one detector, and the macroscopic body appearing in another detector, far away from the former? Why the quantum correlation is not broken, as is broken the quantum superposition?
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Dear Sofia,
I have written about crises, about philosophy, about history of physics, about free will since these issues are directly relevant to your question. It is impossible to answer correctly on your question without an understanding of the Kant philosophy. Your question is VERY difficult without understanding of the Kant philosophy and the answer on this question will be simple and unambiguous if you, as Einstein, will saturate himself with the Kant philosophy. You write: “In the 21th century the result may be at all passed to a completely automatic process. But not this is the issue, whether the observer is human or a machine”. This opinion is typical for most modern scientists who sure that we can know the Nature as ’thing-in-itself’ and who don't make the distinction between ’res cogitans’ (thinking entities) and ’res extensa’ (extended entities). I draw your attention on the problem of free will in quantum mechanics since this problem is actual only if the observer is human since no machine can have free will. If the observer is a machine and the act of the observation in quantum mechanics is a real interaction of quantum system with a macroscopic body or apparatus, as you think, then the Nobel prize winner Gerard 't Hooft tries to solve a nonexistent problem of free will and numerous publications refuting realism (for example The BIG Bell Test Collaboration, Challenging local realism with human choices. Nature 557, 212216 (2018)) make no sense. I tried to draw the attention in my letter to an Associate Editor of Nature (see attached file) that the believers in quantum mechanics belong to different confessions: the believers of one confession refute realism whereas the believers of other confession don’t want to hear that quantum mechanics contradict realism. Your questions and remarks witness that you belong to the latter confession as well as most physicists of Soviet school. You are sure that quantum mechanics describes real processes. Therefore it's not obvious for you that the "collapse" of the wave-function occurs under influence of the mind of the observer. It was not obvious for most physicists but was obvious for Einstein, Schrodinger, Bell and other critics of quantum mechanics. It is obvious according to the Born interpretation of the wave-function. In order to understand the collapse of the wave-function occurs under influence of the mind of the observer it is needed to put the question: ”Why the realistic understanding of the wave function proposed by Schrodinger was rejected and Born’s interpretation was accepted?” The answer is obvious: ”We cannot think that a real density can change due to an observation whereas we know from our everyday experience that a probability of observation changes at first
observation”. For example, your friend takes away a one dice without looking from a box with one white dice and one black dice. You know before observation of the remaining dice that your friend will see the white dice with the probability 0.5. The probability will become 1 when you will see the black dice and 0 when you will see the white dice, regardless of distance between you and your friend. Thus, the wave function describes first of all the state of the mind of the observer, according to Born’s interpretation. We can think that only observer’s knowledge changes because of observation in the case of the dices. But the wave-particle duality observed for example in the doubleslit interference experiment cannot be described if only observer’s knowledge changes. Therefore Dirac postulated in 1930 ”that a measurement always causes the system to jump into an eigenstate of the dynamical variable that is being measured”. The wave function describes the states of both the mind of the observer and quantum system according to the Dirac jump or the collapse of the wave-function. This renouncement of the distinction between reality and our knowledge of reality creates the illusion that quantum mechanics can describe the wave-particle duality and other paradoxical quantum phenomena. But it is very strange that only few scientists understood that this renouncement of realism leads to the logical absurdity.
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In fact, I'm working on a thesis project on Quantum Information and precisely on quantum error correcting codes. I just started not long ago my research on the subject, and specifically how one can go from a classical signal to a quantum signal to describe the algorithms of error correction codes in physical channels.
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There is a lot of work going on here, it is a very active field of research. Also your question does not seem clear. Do you mean the uploading of classical information with a quantum oracle (alike Quantum RAM?)
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What experimental evidence (or any other) contradicts the use of non-unitary, non-Hermitian mathematics to represent pure quantum states? This question relates to pure states, not mixed states. Note that rational matrices have rational (real) eigenvalues.
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The mathematics used to describe quantum mechanics was chosen to correspond to the observed physical realities of experiment. The reason we use Hermitian operators is because they have real eigenvalues, corresponding to the fact that the numbers produced by measurements are real. While you could certainly claim to "measure" a complex or other sort of number, ultimately that could be seen as measuring a pair of real numbers (the real and imaginary parts), or four real numbers for a quaternion, so we don't lose any possibilities by requiring real numbers.
As a further example of this, measurements DID force us to use spinors rather than just the single complex wave function; again the predictions are collections of real numbers.
The need for unitarity is a consequence of our interpretation of quantum mechanics as probabilistic and evolving via the Schrodinger equation. The wave function is unchanged by an overall constant multiple. By interpreting the magnitude of the wave function to be one, we normalize probabilities in the usual way using the real part of this number. The phase that remains is arbitrary (*relative* phases are not, just an overall one).
If we were to transform a wave function by a non-unitary transformation, it is easy to show (try it!) that the Schrodinger equation does not preserve total probability. This, with the standard interpretation, means that there is not probability one that the particle is *somewhere*. The experimental observation you are asking for is that we do not see particles spontaneously vanish or come into being.
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The original question was wrong thus was completed rewritten.
Context:
Suppose I have a 3 d particle placed in a spherical box of infinite well from [-r,r] in all three directions, it's expectation value of position thus equaled to 0.
Thus its surface area equaled to $4 \pi r^2$
From quantum gravity, (existence) https://arxiv.org/pdf/gr-qc/9403008.pdf and (a fairly good numerical approximation) https://en.wikipedia.org/wiki/Planck_length we knew that space and time were quantized fractions. Suppose the minimum length equal to ds, then the maximum partition of the surface of the spherical ball equal to $N= 4 \pi r^2/ds^2$.
Which meant that, as the increase of r, the number of possible segment that our probe could be placed will increase.
In analogy, suppose I have a particle of spin 1/2. Where we place the particle at the center of the ball and measure its spin. Then the ball with larger r could have more "segment" area for observation.
Question 1
Was these analysis true? If not, why? Further what's its implication?
Question 2
Suppose I created a pair of such spin 1/2 particle entangled together. One placed in a ball of $r_a$ the other placed in a ball or $r_b$. If $r_b>r_a$, then our measurement could be more "precise" about the ball b than ball a.
In an imaginary extreme case where $N=4 \pi r_a^2/ds^2=2 $, measurement for ball a thus could only be up or down.
What's happened to the information here? Were they still consist?
Clarification:
1 in question 2, since it's an infinite well(although it was not possible in real), It did not had to be exact in the "center" of $(0,0,0)$. By the fact that the particle was not at the boundary, and, since it's spherical coordinates, by symmetry, position expectation value was at the center. In fact, it didn't even have to be at the center position. Wave was good. The encoding was based on the probability of $T_{funning}$ was selected such that it equaled to 0 or $<<1$. Thus could be ignored regard to numerical calculation.
2 the imaginary extrem was based on the fact that electron's classical readius was in e-16 and plunk length was in e-32 thus $N=2$ could not happen, but just to demonstrate the idea.
Feel free to ask question about the context.
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When the quantum gravity was assumed, it was essentially ignite the continuous of the space-time.
Especially, when taking the length comparable to $ds$(in this case, the plank length). Quote my professor: the superposition of states were destroyed. In the extreme case of $ds^3$, the wave function became a Dirac Delta function.
Thus, by taking the length comparable to $ds$, we essentially altered the particles and thus information was no longer valid, especially, the entanglement was altered.
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Is this meant as solitons? I think that I read that EM field does not have solitonslike solutions in most cases. But if they are solitons how is their ability accounted for to 'feel' entire space in almost 0 time (as is evident from Feynman trajectories approach)?
In a box the exitations imerge immediately and comprise the whole length of the box. So there is a probability to detect a photon far from the source. How is this to be consistent with the constancy of speed of light c?
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Dear Ilian,
Transient processes are not considered for quantum transitions. Photons creation or phonon it does not matter.
Quantum theory is a theory of metamorphosis. Each quantum process is metamorphosis of initial state to final state. Even if initial and final states are the same, the process is considered as a sort of metamorphosis.
If one consider field in each point as oscillator, these oscillators have to be coupled, because one CAN NOT initiate independent oscillations of each oscillator. If one consider each standing wave as oscillator, these oscillators ARE independent.
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i am very clear about momentum, spin, and polarization, performed on entangled particles are found to be correlated.
but i am not understanding ;
In what way 'position measurement' performed on entangled particles are found to be correlated?
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Yes. x is a dummy variable. If it represents the polarisation, it is exactly the same as the correlation of polarisation. But in this latter case, the polarisation is represented by only two eigenstates |0> and |1> instead of a continuum |x>. That doesn't change the principle.
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in relativistic quantum mechanics(quantum field theory) we take co-ordinate time as a time observable to bring space and time on equal footing ...why can't we take proper time as a time observable ? by this approach we may overcome the problem, called "renormalization."
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s h s> why can't we take proper time as a time observable?
The proper time of which objects, distributed throughout the universe? Given at space-time geometry defined by the line element
c22 = gμν(x) dxμdxν
it may be possible to choose a time coordinate such that g00 = 1, such that the proper time for objects at rest in these coordinates is equal to coordinate time. However, this would in most cases be a very bad idea, because then the other components of gμν become horribly ugly.
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So-called "Light with a twist in its tail" was described by Allen in 1992, and a fair sized movement has developed with applications. For an overview see Padgett and Allen 2000 http://people.physics.illinois.edu/Selvin/PRS/498IBR/Twist.pdf . Recent investigation both theoretical and experimental by Giovaninni et. al. in a paper auspiciously titled "Photons that travel in free space slower than the speed of light" and also Bereza and Hermosa "Subluminal group velocity and dispersion of Laguerre Gauss beams in free space" respectably published in Nature https://www.nature.com/articles/srep26842 argue the group velocity is less than c. See first attached figure from the 2000 overview with caption "helical wavefronts have wavevectors which spiral around the beam axis and give rise to an orbital angular momentum". (Note that Bereza and Hermosa report that the greater the apparent helicity, the greater the excess dispersion of the beam, which seems a clue that something is amiss.)
General Relativity assumes light travels in straight lines in local space. Photons can have spin, but not orbital angular momentum. If the group velocity is really less than c, then the light could be made to appear stationary or move backward by appropriate reference frame choice. This seems a little over the top. Is it possible what is really going on is more like the second figure, which I drew, titled "apparent" OAM? If so, how did the interpretation of this effect get so out of hand? If not, how have the stunning implications been overlooked?
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Here is a sort of "proof" that there is no photon-level quantum property corresponding to "OAM" which I developed in a series of interesting discussions (debates really) with Judy Kupferman:
We assume OAM is "not spin" but something else. We know at the beam level it is an angular momentum about the center of the beam. The question is whether it arises as a sum of photon quantum properties, or as a sum of external angular momenta components.
If you measure photons they will have either well defined position or momentum. The angular momentum, however, is a conjugate of orientation (angular position). Here is a list of conjugate pairs https://en.wikipedia.org/wiki/Conjugate_variables .
So one can precisely measure position of a photon in the beam without affecting angular momentum, since it is not paired with angular momentum.
Then the angular momentum, at each position, can be measured without scrambling the position measurements. How to do this experimentally I don’t really care, just that it is theoretically possible.
At that point, the linear momentum and angular orientation of the photon will be uncertain. They are obviously related to each other, and neither are of concern.
Notice I have not said what kind of angular momentum, other than “of the photon.” Doesn’t matter. Just measure its total angular momentum.
There are several ways to achieve non-zero angular momentum when summing the individual photon measurements:
  1. We will rule out having the measurements unevenly distributed about the beam center, because that is just external angular momentum, well understood and not OAM.
  2. I think we can rule out different magnitudes as a function of position, as that is just external angular momentum as well. So the magnitudes are either uniform or randomly or otherwise evenly and symmetrically distributed.
  3. That leaves only the vector direction of measurements. These are quantized, and we can take the beam direction as reference. They are either along the beam (+ or -) or transverse to it. The transverse angular momenta don’t matter for our purposes, so throw them out. The + and – amount to magnitude differences, and as noted in 2 will have no relevance unless distributed unevenly which amounts to external angular momentum.
Well, those three cases are all that there are, and there is no room in them for a net non-zero sum over the beam, except for external angular momentum. Therefore, a hypothetical property “orbital angular momentum” which is different than ordinary quantum angular momentum, aka spin, is some form of external angular momentum.
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Energy transitions are classified according to spectral series such as Balmer, Lyman, Paschen, etc., which assume a single transition between two non-contiguous levels (except the first transition). What the question really means is, can cascading transitions from one level to another, emitting a photon at each contiguous level, occur or have been observed?
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As we all know, the majority of the softwares used in condensed matter physics are based on DFT (such as VASP, CASTEP, ......).But they can only handle the problems at zero temperature, so is there some ab-initio software which could solve the Schrodinger Equation at finite temperature?
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structural relaxation in DMFT has been implemented, and is freely available here:
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A non-absorbing detector is supposed, at least in theory, to report that a particle passes through it, though, the particle is allowed to exit the detector, s.t. we can do additional tests on it. No doubt, the non-absorbing detector "collapses" the wave-function, but additional detectors in continuation, may tell us in which state the 1st detector left the particle, which we can only guess we use absorbing detectors.
Now, does somebody know, do we have such detectors "on shelf", i.e. in practice?
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Dear @MUNISH KUMAR,
I see on your profile page that you are an expert in radiation detection. I am no experimenter, so, I'd like to ask you some questions.
Please see, you speak in your comment of gamma detection. Well, if the efficiency is poor with gamma, what about alpha detection? A strongly charged particle may leave part of its energy in a medium, and get out of the detector through a back window. Then, can we have non-absorbing alpha detectors with not so low efficiency?
I am aware that inside the medium in the detector the alpha may be scattered, but since it is a charged particle, can we place near the back window collimators consisting in electric fields?
I am also aware that the alpha may attach one or two electrons, s.t. a further detection becomes less probable. Finally, as far as I know about cross sections, th more rapid is the alpha, the cross section of interaction with the medium is smaller. But my question is not if we can achieve very high efficiency, it is about non-negligible efficiency.
With best regards,
Sofia
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For understanding my question I invite everybody to read the example.
In the Bohmian mechanics (BM) the velocity formula gives an infinite value to the velocity of the Bohmian particle at points where the wave-function vanishes, but the gradient doesn't vanish.
Do we have any proof that this is wrong, i.e. that attributing superluminal velocity to a particle is wrong? Could it be that this feature of the BM is a flaw, and implies that BM is wrong?
EXAMPLE:
In a Hanbury-Brown and Twiss (HB&T) type experiment with a pair of identical photons, one passing through a slit A and the other passing through a slit B - see attach - the wave-function of the pair looks as follows:
Ψ(r1, r2) = 2 {exp[ik|r1 - rA| exp(ik|r2 - rB|) + exp(ik|r1 - rB|) exp(ik|r2 - rA|) }
= 2 exp[iθ(r1, r2)] cos[ϕ(r1, r2)]
where r1 , r2, denore the positions of the photons, and rA, rB, the positions of the slits.
ϕ(r1, r2) = (κ/2) { (|r1 - rA| + |r2 - rB|) - (|r1 - rB|) + |r2 - rA|) }.
If ϕ is an odd integer multiple of π/2, the wave-function vanishes. Assume that so happens for the pair of points P'1 and P'2. If one places detectors at these two points and, say, the detector at P'1 makes a detection, the detector at P'2 remains silent. By the BM, the particle moving towards P'2 jumps over this point with an infinite velocity, and this is why it cannot be detected.
The problem is that the presence of the detector at P'2 reduces the probability Prob(P'1, Q2) of joint detection at P'1 and Q2, where Q2 is any point below P'2. This probability shows no more interference effect, it is given only by the cross waves from A to Q2 and from B to P'1.
Obviously, the detector at P'1 although cannot detect the particle passing through it, but yes stops something. The probability of joint detection in P'1 and Q2 decreases due to the detector at P'2 not only by the disappearence iof the nterference, but also below the sum of the isolated probabilities of detection from the crossed rays and detection from the direct rays.
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Dear Daniel,
Thanks for trying to help. Please see, my question is related to an article. The example there is of Hanbury-Brown and Twiss type, as I wrote in the question. So, semiconductors, or Bohm-Aharonove, are not connected with it. But, I wrote you a message with details, please look at it.
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Is there exist at least 9 distinct a bi-j-ective, diffeomorphic& homoeomorphic. analytic  functions F(x,y), dom i of two variables in the x,y, cartesian plane
F(x,y); dom F:[0,1]\times[0,srt(3)/2],\toΔ^2, unit 2 probability simplex
CO-DOM(F)=IM(F)=delta_2: {p(i)=<p_1,p_2,p_3>_i; |\forall p(i)\in [0,1]^3, s.t: p1_i+p2_i+p3_i=1, 1>=p_1i,x_p2i,x_3_i>=0\in [0,1]\forall (x1,x2,x3)\in [0,1]
F(0,0)=(1,0,0)
F(1,0)=(0,1,0)
F(1/2, sqrt(3)/2)=(0,0,1)
F(x=1/2,y=sqrt(3)/6))=(1/3,1/3,1/3)_(1/2, sqrt(3)/6)   =
The inverse function being
i=(x,y)=F-1(<p1i,p2i,p3i>_i=(x,y))= <x=[2p2+p3+1]/2,y= p3 *[sqrt(3)/2]]
F(x=1/2,y=sqrt(3)/6))=(1/3,1/3,1/3)_(1/2, sqrt(3)/6) ;
Incidentally it also has to accomodate the 8 element boolean algebra of events on each pt as well  <p1,p2,p3>, pi1+p2+p3=1 pi>=0 omega=1, emptyset =0,
PR(A or B)=PR(A)+PR(B)= p1+p2 >=0
and <p1+p2, p2+p3, p3+p4)
ie F(x,y)= <p1,p2,p3>,\to <1,0, p1, p2, p3, p1+p2, p2+p3, p3+p1}
and as every element of the simplex must be present at very least six times,
 It actually must consist of at least six identical simplexes, that where the euclidean
F(x,y, {1.2,3.4.5.6})=Δ^2\cup_i=1-6, these can be the same simplex but with the order, of each pi in each <p1, p2, p3> interchanged
<Omega={A,B,C}, F= {{Omega},{ emptyset}, {A}, {B}, {C}, {A V B}, {AVC}, {B VD)
PR(A)=p1
PR(B)=p2 PR(C)=p3
PR(A or B)=PR(A)+PR(B)= p1+p2 >=0
PR(A or C)=PR(A)+PR(C)=p2+p3
PR(B or C)=PR(B)+PR(C)=p1+p3
where in addition there is a further triangle probability function that is ranked by this chances. The triangle frame function G(x,y,1,6)={{1,0, g1,g2,g3, g1+g2, g2+g3, g1+g3}, 1<=gi, g1+g2, g2+g3, g3+g1>=0, g1+g2 +g3=1;
such that for F(x,y,{i,6]  on the same coordinates, \forall Ei\in(A, B, C,A or B, A or C, B or C)
{{1,0, g1,g2,g3, g1+g2, g2+g3, g1+g3}= [G(x,y,{1,,,6,1), G(x,y,{1,,,6},2), G(x,y,{1,6},3), G(x,y,{1,,,6},4), G(x,y,{1,,,6},5),..... G(x,y,{1,,,6},8),]
; G(x,y,{1,,6},1)=1, G(x,y,{1,,,6},2)=0
1>G(x,y,{1,,6},3)=g1,>0 in the interior
1>G(x,y,{1,,6},4)=g2 >0,
1>G(x,y,{1,,6},5)=g3>0
1>G(x,y,{1,,6},5)=g1+g2>0, >{g1,g2}
1>G(x,y,{1,,6},7)=g2>0
1>G(x,y,{1,,6},8)=g1+g3>0, g1+g3>g3, g1+g3>g1
1>G(x,y,{1,,6},7)=g2+g3>0, g2+g3> (g2, g3)
g1+g2+g3=1
1>G(x,y,{1,,6},5)=g1+g2>0
\forall i in [i in 6} G(0,0,i,)=(0,0,1),
G(0,0,i\in [1,6},3)=G(0,0,i\in [1,6},, 1),=G(0,0,i\in [1,6},5)=G(0,0,i\in [1,6},8}=1
G(0,0,i\in [1,6},2)=G(0,0,i, 4),=G(0,0,i\in [1,6},{7}=,G(0,0,i\in [1,6},{8})=0
\forall i in [i in 6} G(1,0)=(0,1,0)
G(1/2, sqrt(3)/2)=(0,0,1)=
G(x=1/2,y=sqrt(3)/6, {1,,,6}))=(1/3,1/3,1/3)=(1,0,1/3.1/3.1/3. 2/3,2/3.2/3}
\forall {x,(y),l {i....6},\forall j \in {1,,,8}, G(x,y,i,j)=gj>1/3, iff F(x,y,i,j)=pj>1/3,
\forall {x,(y),l {i....6},\forall j \in {1,,,8}, G(x,y,i,j)=gj>1/3, iff F(x,y,i,j)=pj>1/3,
\forall {x,(y),l {i....6},\forall j \in {1,,,8}, G(x,y,i,j)=gj=1/2, iff F(x,y,i,j)=pj=1/2,
\forall {x,(y)}\forall,l {i....6},\forall j \in {1,,,8}, F(x,y,i,j)=pt<|=|>F(x1,y,1i1,t)=pt[\forany {x1,(y1)}\foranyl,l1 {i....6},forany t \in {1,,,8},  iff G(x,y,i,j)=gj|<|=|> G(x1,y1,i1,t)=gt,  ,
\forall {x,(y)}\forall,l {i....6},\forall t \in {1,,,8}, \forall {x1,(y1)}\forall,i1 {i....6},\forall t_1 \in {1,,,8},  such that
F(x,y,i,,t1)+F(x1,y1,i1,t2)=p_t(y,x,i)+p_t1(x1,x1,i1)<|=|> F(x2,y2,i2, t3)+F(x3,y3,i3,t4) =p_t3(x2,x2,i2)+ p_t4(x3,x3,i3)\forany{x2,(y2),i3},(x3,y3,i3}dom(F) such that, forany (t2,t3) \in {1,,,8},  where t2 @,sigma F(x2,y2,i3_, t3 in sigma @,F( x3,y3, t3) iff
G(x,y,i,,t1)+G(x1,y1,i1,t2)=g_t1(y,x,i)+g_t2(x1,x1,i1)<|=|> G(x2,y2,i2,j,t3)+ G(x3,y3,i, t4)=g_t3(y2,x2,i2)+g_t4(x3,x3,i3)
where
, iff F(x,y,i,j)=pj=1/2,
\forall {x,(y),l {i....6},\forall j \in {1,,,8}, 0<G(x,y,i,j)=gj<1/3, iff 0<F(x,y,i,j)=pj<1/3,
\forall {x,(y),l {i....6},\forall j \in {1,,,8},2/3 <G(x,y,i,j)=gj>1/3, iff 2/3>F(x,y,i,j)=pj>1/3,
\forall {x,(y),l {i....6},\forall j \in {1,,,8}, G(x,y,i,j)=gj=2/3 iff F(x,y,i,j)=pj=2/3}
\forall {x,(y),l {i....6},\forall j \in {1,,,8}, 1>G(x,y,i,j)=gj>2/3 iff 1>F(x,y,i,j)>2/3
\forall {x,(y),l {i....6},\forall j \in {1,,,8}, 1>G(x,y,i,j)=gj>0iff 1>F(x,y,i,j)>0
\forall {x,(y),l {i....6},\forall j \in {1,,,8}, 1>G(x,y,i,j)=gj= 0 iff F(x,y,i,j)=0
\forall {x,(y),l {i....6},\forall j \in {1,,,8}, 1>G(x,y,i,j)=gj= 1 iff F(x,y,i,j)=1
forall
_(1/2, sqrt(3)/6)   =
{{1,0, p1,p2,p3, p1+p2, p2+p3, p1+p3}
 \forall Et\in(A, B, C,A or B, A or C, B or C), \forall Ej\in(A, B, C,A or B, A or C, B or C)
ie for t\in {1,......8), j in {1,,,,,8}
G(,x,y,{i,,,,6},t,) @ x,y,i>G(x,y,{1,,,6},j)  or G(,x,y,{i,,,,6},t,)=G(x,y,{1,,,6,j,) or G(,x,y,{i,,,,6},t,) @ x,y,i<G(x,y,{1,,,6},j) 
G(,x,y,{i,,,,6},t,)_t @ x,y,i>G(x,y,{1,,,6},j) iff F((x,y, {i,6},t)>F(x,y,{i,,6},j)= PR(F(x,y,{i,,6})>PR(x,y,{i,,,6}
G(Ei)=G(Ej) iff P(Ei)>PR(Ej) ie g1=g2 iff p1=p2,
g1+g2= g3 iff p3=p1+p2,
G(Ei<G(Ej) iff P(Ei)>PR(Ej)
G(0,0)=(1,0,0)
G(1,0)=(0,1,0)
G(1/2, sqrt(3)/2)=(0,0,1)=
F(x=1/2,y=sqrt(3)/6))=(1/3,1/3,1/3)_(1/2, sqrt(3)/6)   =
where on each vector, it is subject to the same constraints F-1{x,y,i} G(A)+G(B)+G(C)=1, G(A v B)=G(A)+G(B), G(sigma)=1, G(emptyset)=0 etc whenver G-1(x,y, i \in {1,,6})=F-1(x,y,{i,6}
for all of the 8 elements in the sigma algebra of each of the uncountably many vectors in the interior of each of the six simplexes of uncountably many vectors
  and all elements F_i in the algebra of said vector in each in simplex,  except omega, 0, G(
Such in addition every element of Δ^1, the unit one probability simplex, set of all two non non negative numbers which sum to one, are present and within the image of the function; described  by triples like (0, p, 1-p) on the edges of the triangle in cartesian coordinates
to, the unit 2 probability simplex
consisting of every triple of three real non-negative numbers, which sum to 1. Is the equilateral triangle, ternary plot representation using cartesian coordinates over a euclidean triangle bi-jective and convex hull. Do terms  p[probability triples go missing. 
I have been told that in the iso-celes representations (ie the marshak and machina triangle) that certain triple or convex combinations of three non -negative  values that sum to one are not present.
Simply said, does there exist a bijective, homeomorphic (and analytic) function F(x,y)of two variables x,y, from the x-y plane to to the probability 2- simplex; delta2   where delta2,  the set ofi each and every triple of three non negative numbers which sum to one <p1, p2, p3> 1>p1, p2 p3>=0; p1+p2+p3=1 
F(x,y)=<p1, p2, p3>  where F maps each (x,y) in dom(F) subset R^2 to one and only to element of the probability simplex delta2  subset (R>=o)^3; and where the inverse function,  F-1 maps each and every element of delta 2
<p1, p2, p3>;p1+p2+p3  P1. p2. p3. >=0 , that is in the ENTIRE probability simplex, delta 2  uniquely to every element of the dom(F), the prescribed Cartesian plane.
Apparently one generally has to use a euclidean triangle,  with side lengths of one in Cartesian coordinates, often an altitude of one however is used as well according to the book attached attached, last attachment p 169. 
(which suggests that certain elements of the simplex will go missing there will be no pt in Dom (F), such F(x,y)=<p1,p2, p3> for some <p1,p2,p3> in S the probability simplex
is in-vertible and has a unique inverse, such that there exists no <p1, p2, p3> in the simplex such that there is no element (xi,yi)of dom(F) such that F(xi,yi)= <p1, p2, p3> in
F(x,y), that is continuous and analytic
map to every vector in the simplex, ie there exists no set of three non negative three numbers p1, p2 p3 where p1+p2 +p3=1 such that
ie for each of the nine F,
Where CODOM(F)=IM(F)=delta_2: {p(i)=<p_1i,p_2i,p_3i>_i; |\forall p(i)\in [0,1]^3, s.t: p1_i+p2_i+p3_i=1, 1>=p_1i,x_p2i,x_3_i>=0}
where p(i( described a triple and i whose  Cartesian index is i= (x,y), ie F(x,y)=p(i)<p_1i,p_2i,p_3i>_i).
and
,
CO-DOM(F)=IM(F)=delta_2: {p(i)=<p_1,p_2,p_3>_i; |\forall p(i)\in [0,1]^3, s.t: p1_i+p2_i+p3_i=1, 1>=p_1i,x_p2i,x_3_i>=0\in [0,1]\forall (x1,x2,x3)\in [0,1]
F(0,0)=(1,0,0)
F(1,0)=(0,1,0)
F(1/2, sqrt(3)/2)=(0,0,1)
(with a continuous inverse)he car-tesian plane, incribed within an equilateral triangle to the delta 2,
coDOM(F)=IM(F)=delta_2: {p(i)=<p_1,p_2,p_3>_i; |\forall p(i)\in [0,1]^3, s.t: p1_i+p2_i+p3_i=1, 1>=p_1i,x_p2i,x_3_i>=0\in [0,1]\forall (x1,x2,x3)\in [0,1]
where, no triple goes missing, and where delta 1, the unit 1 probability simplex subspace, (the set of all 2=real non negative numbers probability doubles which sum to one, described as triples with a single zero entry),
 delta _1 subset IM(F)=codom(F)=Dela_2
and each probability value in [0,1] , that is each and every real number in [0,1] occurs infinitely times many for each of the p1-i, p2_2, p3_3 , on some such vector,
1. and one each degenerate double
2, And which contains, as a proper subset, the unit 1 probability simplex, delta 1 (set of all probability doubles)contained within the IM(F)=dom(F)=delta2; in the form of a set of degenerate triples, delta_1*, subset delta 2=IM(F)=codom(F) ( the subset of vectors in the unit 2 (triple) probability simplex  with one and only one, 0, entry),
ie <0.6, 0.4,0>, <0, 0.6, 0.4>
3. where for each pi /evctor (degenerate triple) in the degenerate subset  delta 1 * of delta  2; the map delta1*=delta 1 is the identity (that is no double goes missing). The unit 2 simplex (set of all real non negative triples= im(F) must contain along the edges of the equilateral, every element of delta 1, every set of two number which sum 1).
4. And where among-st these degenerate vectors in delta 1* (the doubles inscribed as triples with a single 0), (not the vertices), must contain, for  each, and every of the two convex, combinations, or  positive real numbers in the unit 1  probability simplex (those which sum to one) at each of them, at least three times, such that every real number value p in (0,1), such that :
p1 +1-p =1,  occurs  at least six times, among-st six distinct degenerate, double vectors <p1, p2,. p3}{i\in {1...6}
 that for all p in (0,1)and there exists six distinct degenerate triple vectors mapped to six distinct points in the plane
5. in addition  in must contain  the unit 2 simplex  as the sum of the entries in each triple.
ie among-st the triples <p1, p2, p3> in IM (F) it must be that \for reals, r, in (0,1) and for each possible value of p1+p2 assumes that value , infinitinely many times,
p2+p3 on a distinct vector assumes that value r in (0,1) infinitely many times.
p1+p3 must assume that value r in (0,1) infinitely many times, , Corresponding there must not exist some real value in (0,1) such that one of p1, p2 p3 assumes that value e, and moreover, not infintiely times, and but no sum value on some vector  (ie element of F(p1+p2, p2+p3, p3+p1) that also assumes that value and infinitely times. The entire  unit interval of values must for each such rin [0,1]and for each of three distinct sums in Fsimplex must be contained and assumed individuallyt infinitely many
6, Finally for every element of a 2or more distinct vectors such that the two elements  (p1, p2 , p3) sumto one
such that for any given p1\in vector 1 , p2\in vector 2 ,p1+p2=1
for any given p1 in vector 1 p3 in vector 2, p1+p3=1
for any given p1 in vector 1 p1in vector 2, p1+p1=1
such that for any given p2\in vector 1 , p2\in vector 2 ,p2+p2=1
for any given p2 in vector 1 p3 in vector 2, p2+p3=1 ,
for any given p3 in vector 1 p3 in vector 2, p3+p3=1
6, Finally for every 3 elements (,p1 p2 , p3, p1, +p2, p3+p1, p2+ p3)  in 2 or 3or more distinct vectors v1, v2, v3, such that the three elements in  the three distinct vectors,in  sum to  any of these numbers \forall n \in {\forall n\in {1....48}n/28,1, 4/3, 1.25, 1.5, 4/3. 1.75, 1.85, 2}, or such that  or any 2 elements (p1, p2 , p3, p1 +p2 p1+p3, p2+p3) in common vector v1 and another for all possible other elements in (p1, p2 , p3, p1 +p2 p1+p3, p2+p3) in distinct vector v3 sum to those values these must be present
Moreover it must also at least extend to any given 4/5/6/7/8 /9/10distinct  elements of (p1, p2 , p3, p1 +p2 p1+p3, p2+p3) that sum to for all possible combinations of being in up to 10 distinct vectors, 9 distinct vectors two elements in common,,,,,,,,,,,,,,,,, such that for any such one of all such combinations there must be uncountable many versions of each element of p1, p2 , p3, p1 +p2 p1+p3, p2+p3) in each combination individually for each of the above sum values & for each of the above cardinalities of entries,.
More over the entire simplex must be present for each of foralll of the ten possible combinations  or sum  term number values,10
and for all r each different combinations of elements that sum to that those values in (p1, p2 p3, p1+p2.....)
and for each of the distinct number of distinct vectors that could  be present in that sum upl to ten
and, that could sum to each of all those approximately 70 distinct values. that could to those values, and for each of the different number of terms in each sum,. the entire simplex must be present and every such value in [0,1[ must be assumed individually for every term in every
(1)for each of the term length sum (any given 2 that sum to one, any given three which sum to one, any given four that sum one m any given five which sum to one, any three which sum to 2, any given four which sum to 2 any five which sum to 2, any given four which sum to three, any five which sum to 3
(2)for all of the 70 or so , values mentioned d
(3)for all number of distinct vectors of which those terms are in that sum are associated
(4) for all 6 distinct terms types of  in the sum p2 p3, p1+p2.....) 
forall n \in {\forall n\in {1....48}n/28,1/2, 2/3, 1,1.125, 4/3, 1.25, 1.5, 5/3. 1.75, 1.877,11/6, 2,2.25, 2.33, 2.5, 2.66, 2.75, 3, 3.25, 3.33 ,3.5,3.666.3.75, 4, 3.333, 4.5, 4.666, 5, 5.5, 6, 6.5, 7 7.333, 7.5, 7,666, 8,  }
IE there vcant be any GAPS
  elements  elements such that in  four or five distinct vectors  with no elments in common
, three or four distinct vectors, two elements in a common vector,the other three/2 being in distinct vectors when there are fouir elements
three distinct vectors, with two elements in two  common vectors, or three elements in one vector common vector,  and two and the other two in either one or two common vectors and the other elements or in 2 elements in one common vector and one in a   common vector, 2  distinct vectors with 2  elements common to each of either one or /two of the vectors, 2 vectors with  3 elements in one vectors and 2 in the other
n a distinct vectors,
 such p1 vector 1 +p2 in vector 2 +p3 in vector 3=1
such p1 vector 1 +p3 in vector 2 +p2 in vector 3=1
such p1 vector 1 +p2 in vector 2 +p2 in vector 3=1
 such p1 vector 1 +p3 in vector 2 +p2 in vector 3=1
such p1 vector 1 +p1 in vector 1 +p1 in vector 2
such p1 vector 1 +p1 in vector 1 +p3 in vector 2
such p1 vector 1 +p1 in vector 1 +p2 in vector 2
such p1 vector 1 +p1 in vector 1 +p2 in vector 3
such p1 vector 1 +p1 in vector 2 +p2 in vector 3
such p1 vector 1 +p1 in vector 3 +p2 in vector 3
such p1 vector 1 +p3 in vector 1 +p2 in vector 3
such p1 vector 1 +p2 in vector 1 +p2 in vector 3
such p1 vector 1 +p3 in vector 1 +p2 in vector 3=1
 such p1 vector 1 +p2 in vector 2 +p2 in vector 3=1
 such p1 vector 1 +p2 in vector 3 +p3 in vector 3=1
 such p1 vector 1 +p1 in vector 2 +p1 in vector 3=1
 such p1 vector 1 +p1 in vector 2 +p1 in vector 3=1
 such p2 vector 1 +p2 in vector 2 +p2 in vector 3=1
 such p3 vector 1 +p3 in vector 2 +p3 in vector 3=1
 such p3 vector 1 +p3 in vector 1 +p3 in vector 3=1
 such p3 vector 1 +p3 in vector 2 +p3 in vector 2=1
 such p3 vector 1 +p3 in vector 3 +p3 in vector 3=1
 such p3 vector 1 +p3 in vector 2 +p3 in vector 2=1
 such p3 vector 1 +p3 in vector 3 +p3 in vector 3=1
 such p2 vector 1 +p1 in vector 2 +p3 in vector 3=1
 such p2 vector 1 +p1 in vector 2 +p1 in vector 3=1
 such p2 vector 1 +p3 in vector 2 +p2 in vector 3=1
 such p2 vector 1 +p2 in vector 2 +p2 in vector 2=1
 such p1 vector 1 +p2 in vector 1 +p2 in vector 2=1
 such p1 vector 1 +p2 in vector 1 +p3 in vector 3=1
 such p1 vector 1 +p2 in vector 1 +p3 in vector 3=1
 such p1 vector 1 +p2 in vector 1 +p3 in vector 3=1
uch that for any given p1\in vector 1 , p2\in vector 2 ,p1+p2=1
for any given p1 in vector 1 p3 in vector 2, p1+p3=1
for any given p1 in vector 1 p1in vector 2, p1+p1=1
such that for any given p2\in vector 1 , p2\in vector 2 ,p2+p2=1
for any given p2 in vector 1 p3 in vector 2, p2+p3=1 ,
for any given p3 in vector 1 p3 in vector 2, p3+p3=1
for any given p1 in vector 1 p1+p2 in vector 2 p1+(p1+p2)=1
for any given p1 in vector 1 p1+p3 in vector 2 p1+(p1+p3)=1
for any given p1 in vector 1 p2+p3 in vector 2 p1+(p2+p3)=1
for any given p2 in vector 1 p1+p2 in vector 2; p2+(p1+p2)=1
for any given p2 in vector 1 p1+p3 in vector 2; p2+(p1+p3)=1
for any given p2 in vector 1 p2+p3 in vector 2 p2+(p2+p3)=1
for any given p3 in vector 1 p2+p3 in vector 2 p3+(p2+p3)=1
for any given p3 in vector 1 p2+p3 in vector 2 p3+(p2+p3)=1
for any given p3 in vector 1 p2+p3 in vector 2 p3+(p2+p3)=1
for any given p1+p2 in vector 1 p1+p2 in vector 2  p1+p2)+(p1+p2)=1
for any given p1+p2 in vector 1 p1+p3 in vector 2 p1+p2)+(p1+p3)=1
for any given p1+p2 in vector 1 p2 +p3in vector 2( p1+p2)+(p2+p3)=1
for any given p2+p3 in vector 1 p3 +p1in vector 2 (p2+p3)+(p3+p2)=1
for any given p2+p3 in vector 1 p2 +p3in vector 2
for any given p1+p3 in vector 1 p3+p1in vector 2
for any given p2+p1 in vector 1 p3 +p2in vector 2
for any given p2+p1 in vector 1 p3 in vector 2
such that for any given p2\in vector 1 , p1\in vector 2 ,p1+p2=1
for any given p1 in vector 1 p2 in vector 2, p1+p3=1
for any given p1 in vector 1 p3in vector 2, p1+p1=1
plus each of the
such that for any given p3\in vector 1 , p2\in vector 2 ,p1+p2=1
for any given p3 in vector 1 p1 in vector 2, p1+p3=1
for any given p3 in vector 1 p3n vector 2, p1+p1=1
for all of the 36 or distinct sombination such that p1 in one vector and one of (p1, p2, p3,p1+p2 p2+p3, p3+p1) on a disitnct vector sum to one , for all reals in [0,1]each of these combination must obtain  infinitely many times, each oif these combinations must be surjective/ bijective with regard to the unit 1 probability simplex, and for each such combination listed, each of the two terms must assume individually ,for each  of the uncountably many real values i m [0,1],, uncountably many times.
for each of the way, and each of the 36 values in each of  that one the 36 and obtain infinitely many time, assume each value within in [0,1] infinitely many time,, and
 In additions
p2\in vector 2 , p3\in vector 2
, p3 p1
, p3+p1=1, p2+p1=1, p3+p2=1,
p2+p3=1
where these are entries on distinct vectors each of these entries must contain infinitely many distinct probability
to either 1, 2, 3, or the
Moreover, it must that both horizontally and vertically, the entire set of element in the sum to 2 non negative simplex
Where CODOM(F)=IM(F)=delta 1, it must be such that the for bijective function G which for each element of Im (F) , G: maps <p1,p2, p3> \in IM (F)=delta 2 ,to G(<p1+p2, p2+p3, p3+p1)>
, ie F(x,y)=<p1,p2, p3) then G[(x,y))=G(F-1(p1,p2, p3)])=<p1+p2, p2+p3, p1+p3>
such that G is also a bi-jective and analytic diffeomorphism onto
delta_2: {p(i)=<p_1i,p_2i,p_3i>_i; |\forall p(i)\in [0,1]^3, s.t: p1_i+p2_i+p3_i=2, 1>=p_1i,x_p2i,x_3_i>=0}, the set of all three real number that sum to 2.
ie dom (G)=dom (F) and and thus for any<p1, p2 p3,> domain we compute F-1(,p1, p2, p3) to get the cartesian coordinates of that vector and feed them into G, where G computes the probabilities of the disjunctive events
(unit 2 probability simplex)
that delta_2: {p(i)=<p_1i,p_2i,p_3i>_i; |\forall p(i)\in [0,1]^3, s.t: p1_i+p2_i+p3_i=2, 1>=p_1i,x_p2i,x_3_i>=0}
in these sums in this sense.
Moreover, it also be the case, that there must exist
, p2+p3, p1+p3
see (4) below
i=(x,y), and i(2)=(x2,y2); . i(3)=(x3, y3), i(4)=(x4, y4), i(5)=(x5, y5), i(6)=(x6,y6);
(x6,y6)\neq (x5, y5)\neq (x4, y4)\neq(x3,y3)\neq(x2,y2)\neq (x,y)
where
F(x2, y2)= p(i(2)=<p_1(i2)=p, p_2(i2)=0, p_3_(i2)=1-p>{i2}; p1+p3=1; 0>(p3, p1)<1,    p2=0, p1=p
F(x3, y3)= p(i(3)=<p_1(i3)=p, p_2(i3)=1-p, p_3_(i3)=0>^{i3}; 0>(p1, p2) <1,  p1+p2=1, p3=0, p1=p
F(x4, y4)= p(i(4)=<p_1(i4)=0, p_2(i4)=p, p_3_(i4)=1-p>{i4); 0>(p3, p2) <1,  p3+p2=1, p1=0, p2=p
F(x5, y5)= p(i(5)=<p_1(i5)=1-p, p_2(i5)=p, p_3_(i5)=0>^{i5)  0>(p1, p2) <1,  p1+p2=1, p3=0, p2=p
F(x6, y6)= p(i(6)=<p_1(i6)=1-p, p_2(i6)=0, p_3_(i6)=p>{i6) 0>(p1, p3) <1,  p1+p3=1, p2=0, p3=p
F(x, y)= p(i)=<p_1(i)=0, p_2(i)=1-p, p_3_(i)=p>{i);
0>(p3, p2) <1;  p3+p2=1, p1=0, p3=p
where for all i\in {i,i(1)...i(5)} and p_t1(m)+p_t2(m) +pt3(m)1 etc
,p1(i)+p2(i) +p3(i)=1
p1(i2)+p2(i2) +p3(i2)=1,
such that p_j_i(2)\in p(i(2); p_j_i(2)\=p
\oplus p2 oplus p3 =0, and
 
where  ONE and only of p1, p2, p3 =0, where  that precise values occurs at least twice in the first entry of two distinct vectors,
<0.6, 0.4, 0>
<0.6, 0, 0.4>, at least twice in the second entry, p2, here p2=0.6
<0, 0.6, 0.4)
<0.4, 0.6, 0)
and in the third entry p3, p3=0.6; at least twice
<0.4,6, 0.6)
<0, 0.4, 6)
. In other words \forall (p1)\in (0,1) and for all p2=in \in (0,1), amd for all  p3 \in (0,1), where p2+p3=1, there exists two distinct vectors (if only in name) such that <p1=0, p2, p3=1-p2), and < p1=0, p2=1-p3, p3>
there exists
\forall p\in [0,1] ;  0<p1, p2<1; <p, p2=1-p>
possible degenerate triple combinations, for each degenerate convex combination in for all real positive p ;  0<p<1 p, 1-p
<0.6, 0.4, 0> <0.4, 0.6, 0> <0.6, 0.4, 0>, <0, 0.6, 0.4)
[delta_1* ]subset delta2=IM(F)=codom(F)= {p(i)=<p_1,p_2,p_3>_i; |\forall p(i)\in [0,1]^3, s.t: p1_i+p2_i+p3_i=1, 1>=p_1i,x_p2i,x_3_i>=0, & \exists in p(i), one and only one(p2_i, p1_i, p3_i)=0; where the other two entries  \neq 0, s.y1< p1 p2>0}
for all convex combinations in delta 1 , all possible probability doubles,  tuples element of [0,1]^2, of non negative real numbers which sum to 1
where map G(delta1*)=delta1 is the identity
or some subset of I^2\subset R^2 to the unit probability simplex, (the triangle simplex of all triples of non-negative numbers <=1,  which sum to one.?
Are such functions convex, that those which use absolute bary-centric coordinates over the probability simplex, when defined over an equilateral triangle with unit length in the Cartesian plane.
seehttps://en.wikipedia.org/wiki/Affine_space#Affine_coordinates
I presume that such function are hardly homogeneous in that infinitely many possitive triples will not be present?
F:I^2, to {<x_1,x_2,x_3>_m; x1+x2+x3=1, (x_1,x_2,x_3)\in [0,1]\forall (x1,x2,x3)\in [0,1]}
from the set of all or triples {<x_1,x_2,x_3>_m; x1+x2+x3=1, (x_1,x_2,x_3)\in [0,1]\forall (x1,x2,x3)\in [0,1]} to a unique index m,\in I^2, a real interval in the cartesian plane?
dom(F):[0,1]\times[0,srt(3)/2]
IM(F)=F: {p(i)=<p_1,p_2,p_3>_i; |\forall p(i)\in [0,1]^3, s.t: p1_i+p2_i+p3_i=1, 1>=p_1i,x_p2i,x_3_i>=0\in [0,1]\forall (x1,x2,x3)\in [0,1]}
where
F(0,0)=(1,0,0)
F(1,0)=(0,1,0)
F(1/2, sqrt(3)/2)=(0,0,1)
F(x=1/2,y=sqrt(3)/6))=(1/3,1/3,1/3)_(1/2, sqrt(3)/6)   =
ie x=2p2+p3]/2,= [2 times 1/3 +1/3]/=1/2
y=srt(3)/2-sqrt(3)/2p1-sqrt(3)/2 times p2= sqrt(3)/2*(p3)= sqrt(3)/2*(1/3)=sqrt(3)/6
i=(x,y)=F-1(<p1i,p2i,p3i>_i=(x,y))= <x=[2p2+p3+1]/2,y= p3 *[sqrt(3)/2]]
, ill have to check the properties there a lot of other roles that it has to fulfill then just this.
Where in addition, every no value of x1+x2, x2+x3, x3+x1 can be missing these must assume each value in [0,1], prefererably infinitely many if positive and <1, and cannot assume a value, that is not assumed by one of the x1,x2, x3, somewhere in the structure,
Preferably this must property contain the unit 1 simplex, as a function x1, x2+x3, where every convex combination which sum to one, of two values must be assumed,  by x1, 1-x, on distinct vectors <x1,x2,x3>m
x2, 1-x2=x1+x3, <x1,x2,x3>m_1 ,m_1\neq m
x3,1-x3 =x2+x3=<x1,x2,x3_m2, m2\neq m1\neq m,
There also cannot be any mismatched between element of the domain on distinct vectors, ie diagonal or vertical sums, where any two of them sum to one, any three of that sum to 1, or 2,
<0.6, 0.25, 0,15>
   <0.4, 0.32, 0.28>
<0.26, 0.4, 0.34>
<0.3   0.4 , 0.3>
<0.26, 0.38, 0.36>   <0.35, 0.35, 0.3>, <0.26,0.4, 28>
I presume if its convex it would contain the doubly stochastic matrices or the permutation matrics
,
<0.7, 0.25, 0,15>
   <0.8, 0.32, 0.28>
   <0.5, 0.32, 0.28> must be a vectors <0.3=1-0.7x1, 0.5=1-x2=0,5, x1=1-0.8=0.2)
 and conversely for <x1,x2,x3> there must be triad of three distinct vectors, such that one elements =1-x1, , another =1-x2, and another =1-x3
<x1,x2,x3, >
<y1,y2,y3>, where one of y1,y2,y3, = 1-x1, one of z1,z2,z3, =1-x2,
<z1,z2,z3>
there must be distinct vectors such that <x1,x2,x3>, x1=25+0.32,
as well for x2, x3, x1+x2, x3+x2, x1+x3,
and which sum to 0.15+0.28,  and common vectors, where all of the six events,  or rather 12 events ,   whose collective sum <= 1, lie on a common vector as atomic events <x1,x2,x3> or disjunctive events <x1+x2,x2+x3, x3+x1>
ie a vectors <0.4, 0.25, 0.35>, and one where <x1=0.4, x2, x3> where x1+x2=0.6
< 0.4, 15, 0.45>, <0.6, 0.28,0.12>
<x1,x2,x3> where x1+x2=0.4
, any 'two sums';,  three, sum  of two elements, to one, or any three sum sum to one
, or three elements of distinct vectors which sum to 2,
and the set of three non positive numbers which sum to 2, as s
Where m denote a Cartesian pair of points in the x,y, plane which uniquely denotes a specific vector, built over an equilateral triangle  in Cartesian coordinates,
and with unit length in side,in Cartesian coordinates
-where this is distance in Cartesian coordinates (x,y) in euclidean norm  of each Cartesian coordinate  probability vectors vertex = F-1(1,0,0),F-1(0,1,0),F-1(0,0,1)=1, where the respective euclidean  norm in probability coordinates is  clearly sqrt (2)  sqrt sqrt(1-0)^2+ (1-0)^2+(0-0)^2)in
sqrt (2) in probability coordinates, in 2- norm,1, in 1 norm,
distance from the triangle center/circum-centre and vertexes, (the Cartesian coordinates of the mid spaced probability vector (1/3, 1,3, 1.3)whose distance from each vertex as a euclidean norm in probability =2/3=equal to the 1-norm distance between value in the triple and its relevant vertex)=2/3, where the overall 1-norm difference in probability between the cent-roid and each vertex =4/3,
=  1/sqrt (3), in Cartesian coordinates in euclidean norm,
all three altitudes=medians (distance from each vertex in Cartesian coordinates, to center of each opposing side of the triangle)-in 2-norm again=
=,sqrt(3)/2 
the probability vertices whose Im(F)=(1,0,0),(0,1,0),(0,0,1) whose untoward cartesian coordinates are given above.
and the three, apothem (2-norm distance from the cir-cum center, the Cartesian coordinates (1/2,sqrt(3)/6) of the centroids/probability vector, (1/3,1.3, 1.3),)  and the Cartesian coordinates of the  mid point of each side of the equilateral triangle,
= 1/(2 * sqrt(3))
where the Centro id (1/3,1/3,1/3 )is the vector in n simplex, entries are just the average , n-pt average of the unit (the only vector with three values precisely the same, and the sums precisely the same 2,3,2.3,2,3)
mid probability vector all entries 1/n here 1/3, whose Cartesian coordinate are the circumcentre of the triangle, the point were all three medians cross, (ie the Cartesian point equidistant from each vertex.
that being the circumcentre , (1/2, sqrt(3)/6)  =in 2-norm of the  (the pt whose probability coordinates are just the average for an n simplex of a unit vector (1/n, 1/n,1.n)
 denoting the distance between the cen are
F-1(1/n, 1/n, 1/n) here F-1(1/3, 1.3, 1.3)
probability/bary-centric coordinates, with side lengths, between the vertics of sqrt (2)
length  sqrt(2) in probability coodinates (euclidean norm ) ie sqrt ([1-0]^2+[0-1)^2+ [0,-0,]^2)=sqrt(2), and have side lengths =1 in cartesian coordinates,  with distance from the centre 1/sqrt (3),  andall median/altitudes /angular bisectors/perpendicular bisectors=sqrt(3)/2, area=sqrt(3)/4 and apothem=1/2sqrt(3)
F(x,y)=<x-1,x_2,x_3>m=(x,y)
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Dear Professor Mateljevic,
When you say that the unit disk is a real analytic bijection to R2, and has a bijective analytic function rto R^2, does it have one ; is it b-ijective to the simplex; are all pts in (p1, p2, p3)  in IM(F) of the unit disk function.  Do any go missing. I presumed that there were certain issues, with the surface area of the sphere
F: dom (unit disk) to . I was wondering if a quadratic function of the northern surface of the unit sphere, would be bijective as defined by F:[0, 2pi times [0, pi]in R^2\to < [p1=sinpi* cosphi]^2, p2=[sin pi * sin phi]^2, p3=cos^2phi> in R^3where  p1+p2+p3=1, and should be non-negative due to the squares.
Whilst the triangular functions appear to be the other way around and maximize surface area
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What is the name for the identities (2) and (3) in the functional analysis literature F-1(is the inverse function?
where (1) F:[0,1] to [0,1] and F is strictly monotonic increasing
where F(0)=0, F(1/2)=1/2 and F(1)=1,
2)\forall (x)\in (F) F(1-x)+F(x)=1
(3)\forall (p)in codom(F)F-1(1-p) +F-1(p)=1; these are the equality bi-conditional (expressed in (2) and (3) , the equality cases of (4), bi-conditional, because it applies to the inverse function so they are expressed as
forall (x, x1)\in dom(F)=[0,1],[x+y =1]  (iff) [F(x)+F(y)=1]
forall (p, p1), in IM(F)\subseteq[0,1];[p+p =1] iff [ F-1(p1) +F-1(p)=1], F-1 is the inverse function and thus F-1(p) F-1(1-p) are elements of dom(F)=[0,1]
x+y=1 iff F(x)+F(y)=1
see, the attached paper 'order indifference and rank dependent probabilities around page 392, its the biconditional form of segal calls a symmetric probability transformation function
I presume that if in addition F satisfies (4)\forall x in [0,1]=dom(F); F(1/2x)=1/2F(x)
That such a function will be identity function, as F(x)=x for all dyadic rationals and some rationals and F is strictly monotone increasing and agrees with F(x) over a dense set note that
given midpoint convexity at 1 and 0
I presume that if in addition
@1\forall x in [0,1]=dom(F) F(1/2x)<=1/2F(x)
\@0 forall x in [0,1]  F(1/2+x/2)<=1/2+F(x)/2
That these equations collapse into equal-ties  
.F(1/2x)=1/2F(x)
F(1/2+x/2)=1/2+F(x)/2
given the symmetry equation;(2) F(1-x)+F(x)=1, andd(1) F:[0,1]to [0,1] and F(0)=0 (which gives F(1/2)=1/2, F(1)=1, follows from F(0)=0,), it follows then F(x)=x for all dyadics  rationals in [0,1]
where F(1)=1 and F(0)= and F strictly monotonic increasing as above
and some rationals and F becomes odd at all dyadic points in [0,1]
n
I am not sure if (3) is required but then given F is strictly monotone increasing (it should be applied by injectivity and (2) . In any case I presume F would collapse into F(x)=x.
What is the general form of a function that merely satisfies F(1)=1 F(0)=0 F(1/2)=1/2 and is strictly mo-none increasing and continuous and satisfies the inequalities?
@1\forall x in [0,1]=dom(F) F(1/2x)<=1/2F(x)
\@0 forall x in [0,1]=dom(F)  F(1/2+x/2)<=1/2+F(x)/2
 The function also satisfies
(4)\forall x,y\in dom (F) x+y>1 \leftrightarrow F(x) +F(y)>1,
\forall x,y\in dom (F) x+y<1 \leftrightarrow F(x) +F(y)<1,
\forall p,p2\in codom(F) p+p1>1 \leftrightarrow F-1(p) +F(p2)>1
\forall p,p2\in codom(F) p+p1<1 \leftrightarrow F-1(p) +F(p2)<1
]
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if u introduce new variable s=t-0.5 and new function q(s)=f(s)-0.5 then the equation is rewritten as q(s)=q(-s), i.e. it is just a condition of antisymmetry.
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How are you progressing ?
How have you selected your respondents?
All the best from Copenhagen, Bo
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I don't take part in medical projects! S.L.
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Is there a distinction between strong or complete qualitative probability orders which are considered to be strong representation or total probability relations neither of which involve in-com parables events, use the stronger form of Scott axiom (not cases of weak, partial or intermediate agreements) and both of  whose representation is considered 'strong '
of type (1)P>=Q iff P(x)>= P(y)
versus type
(2) x=y iff P(x)=P(x)
y>x iff P(y)>Pr(x) and
y<x  iff P(y)<Pr(x)
The last link speaks about by worry about some total orders that only use
totallity A<=B or B<=A without a trichotomy https://antimeta.wordpress.com/category
/probability/page/3/
where they refer to:
f≥g, but we don’t know whether f>g or f≈g. S
However, as it stands, this dominance principle leaves some preference relations among actions underspecified. That is, if f and g are actions such that f strictly dominates g in some states, but they have the same (or equipreferable) outcomes in the others, then we know that f≥g, but we don’t know whether f>g or f≈g. So the axioms for a partial ordering on the outcomes, together with the dominance principle, don’t suffice to uniquely specify an induced partial ordering on the acti
.
 The both uses a total order  over
totality
A <=B or B >=A
l definition of equality and anti-symmetry,  A=B iff A<=B and B>=A
A<= B iff [A< B or A=B] iff not A>B
A>=B iff [A>B or A=B]iff not A<B
where A>B equiv B<A,and
A>=B  equiv B<=A iff (A<B)
where = is an equivalence relation, symmetric, transitive and reflexive
<=.=> are reflexive transitive, negative transitive,complementary and total
, whilst <, > are irreflexive and ass-ymetric,
transitive
A<B , B<C implies A>C
A<B B=C implies A>C
A<B, A<=B implies A>C
and negatively transitive
and complementary
A>B iff ~A<~B
<|=|>, are mutually exclusive.
and where equality s, is an equivalence class not denoting identity or in-comparability but generally equality in rank (in probability) whilst the second kind uses negatively transitive weakly connected strict weak orders,r <|=|>,
weak connected-ness not  (A=B) then A<B or A> B
whilst the second kind uses both trichotomous strongly connected strict total orders,  for <|=|>,.
(2) trichotomoy A<B or A=B or A>B are made explicit,  where the relations are mutually exclusive and exhaustive in (2(
(3) strong connectected. not  (A=B) iff A<B or A> B, and
and satisfy the axioms of A>= emptyset, \Omega > emptyset , \Omega >=  A
scotts conditions and the separability and archimedean axioms and monotone continuity if required
In the first kind <= |>= is primitive which makes me suspect, whilst in the second <|=|> are primitive.
Please see the attached document.And whether trich-otomoy fails
in the first type, which appears a bit fuzzier yet totality holds in both case A>=B or B<=B  where
What is unclear is whether there is any canonical meaning to weak orders (as opposed total pre-orders, or strict weak orders) .
In the context of qualitative probability this is sometimes seen as synonymous with a complete or total order. , as opposed to a partial order which allows for incomparable s, its generally a partial order, which allows for comparable equalities but between non identical events usually put in the same equivalence class (ie A is as probable as B, when A=B, as opposed, one and the same event, or 'who knows/ for in-comparability) Fihsburn hints at a second distinction where A may not be as likely as B, and it must be the case
not A>B and not  A< B  yet not A=B is possible in the second yet
A>= B or A<=B must hold
which appears to say that you can quasi -compare the events (can say that A less then or more probable,  than B ,but not which of the two A<B, A=B, , that is which relation it specifically stands in
but yet one cannot say that A>B  or A<B
)
and satisfy definitions
and A<=B iff A<B or A=B iff B>=A, iff ~A>=~B, where this mutually exclusive to A<B equiv ~B>~A
A>=B iff A>B or A<=B
iff iff B>=A where this mutually exclusive to A>B equiv ~B<~A
and both (1) and (2) using as a total ordering over >= |<=
(1)totalityA<= B or B<=A
(2)equality in rank and anti-symmetric biconditional  A=B iff A<=B and B>=A where = is an equivalence relation, symmetric, transitive and reflexive
(2) A<=B iff A<B or A=B, A>=B iff A>B or A<=B
(3) and satisfy the criterion that >|<|>=|<=,  are
complementary, A>B iff ~B<~A
transitive and negatively transitive,
where A<B iff B<A and where , =, <|> are mutually exclusive,
The difference between the two seem to be whether A>=B and A<= B is equivalent to A=B; or where in the first kind, it counts as strongly respresenting the structure even if A>=B comes out A>B because one could not specify whether A>B or A=B yet you could compare them in the sense that under <= one can say that its either less or equal in probability or more than or equal, but not precisely which of the two it is.
either some weakening of anti-symmetry of the both and the fact that the first kind use
whilst the less ambiguous orders trich-otomous orders use not  (A=B) iff A<B or A> B; generally trichotomy is not considered, when it comes to using  satisfying scotts axiom , in its strongest sense, for strict aggreement
and I am wondering whether the trich-otomous forms which appear to be required for real valued or strictly increasing probability functions are slightly stronger, when it comes to dense order but require a stronger form of scotts axiom, that involves <. > and not just <=.
but where in (1) these <=|>= relation is primitive and trich-otomoy is not explicit, nor is strong connected-ness whilst in (2)A neq B iff A>B or A<B
>|=|< is primitive and both
(1) totality A<= B or B<=A
(2) A<B or A=B or A>B are made explicit,  where the relations are mutually exclusive and exhaustive in (2(
and (2) trichotomy hold and are modelled as strict total trichotomous orders,
as opposed to a weakly connected strict weak order, with an associated total pre-order, or what may be a total order,
, or at least are made explicit.  I get the impression that the first kind as deccribed by FIshburn 1970 considers a weird relation that does not involve incomparables, and is consided total but A>=B and B<=A but one cannot that A is as likely as B, or that its fuzzy in the sense
that one can say that B is either less than or equal in probability to A, or conversely, but if B<= A one cannot /need not  whether say A=B or A<B,
not A=B] iff A<B or A>B
and strongly connected in the second.
where A=B iff A<=B and B>=A in both cases
where <= is transitive , negative transitive, complementary, total, and reflexive
A>=B or B<=A
are considered complete
and
y
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You are way too dispersed.
Try to undestand well the more basic stuff, one topic at a time.
Then move on.
Do not start from Gleason or weak measurements  for eample.
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What is general, is meant by the orthogonality relation x⊥y 
in the functional equation: orthogonal additivity below (1)
(1)∀(x,y)∈(dom(F)∩[x⊥y]):F(x+y)=F(x)+F(y)
Particularly as used in the following two references listed below (by De Zelah and Ratz, in the context of quantum spin half born rule derivations
What is general, is meant by the orthogonality relation x⊥y 
denotes x is orthogonal to y.
.
(1)∀(x,y)∈(dom(F)∩[x⊥y]):F(x+y)=F(x)+F(y)
 
See Rätz, Jürg, On orthogonally additive mappings, Aequationes Math. 28, 35-49 (1985). ZBL0569.39006.
2.See 'Comment on `Gleason-type theorem including Qubits and pro-jective measurements: the Born rule beyond quantum physics' ", by Michael J. W. Hall ">https://www.researchgate.net/profile/Francisco_De_Zela
where x⊥y
denote: -'logical/set theoretical Disjointedness' of events, on the same basis.That is mutual exclusivity? Or - or geometrical orthogonality events . That is, on perpendicular/possibly non-commuting vectors/bases in spin 12 system? (for example, spin up x direction versus spin up ydirection)? - OR something else?
That is, within QM, in the Hilbert inner product metric, what does it mean for the frame function two events to be explicitly allowed to add as per x⊥y
, in functional equation, Orthogonal additivity.
.
Does z⊥m
denote events mean events whose amplitude modul-i squared (P(A),PR(B))=(z,m)respectively and which lie on the same basis/vector in a spin system, for example
:
A spin up on basis in x direction,P(A)=||amplitude(spin up_x)||2=z
and¬A spin down on basis in x direction;P(¬A)=||amplitude(spin down_x)||2=m
 
2.Versus Non commuting bases:
A spin up in x direction
andA spin down in y direction
 
Corresponding to the logical and geometric notions roughly, respectively
How does one distinguish this, from events that are non-commuting from events& orthogonal events in the geometric sense from events, that are orthogonal in the logical sense and disjoint events on the same basis?
Is there a distinct operator, inner product in quantum mechanics (which equals zero) which determines when the frame function probabilities of events can explicitly be assumed to add?
That is disjoint, in the sense of **Kolmogorov that is disjoint or mutually exclusive, or its analog in quantum mechanics as in (1)?
(1)F(X∪Y)=F(X)+F(Y)where in (1);X∩Y=∅,∧X,Y∈Ω,,X,Y are mutually exclusive and lie on the same basis
(1a)P(A∪AC)=1,A∪AC=Ω=⊤∧A∪AC=∅
(1.b)∀(Ai)∈Ω;where,Ai∈F,the singletons, atoms, of which are in the algebra of events F;P(∪n=|Ω|i=1Ai)=[∑i=1n=|Ω|P(Ai)]=1
where, in 1(b)[∪n=|Ω|i=1Ai]=⊤∧∀(j);j≠i⟺Ai∩Aj=∅
 
This being opposed to: when the events on distinct vectors, that are geometrically orthogonal or non commuting/complementary bases/distinct bases, which are not explicitly specified to add, they may happen to, as was derived for n≥3
, where this is really a form of the much stronger global ,Cauchy additivity equation (2) which relate to unique-ness , when one has (1), in addition ? (2)(2)∀(x,y)∈dom(unitsphere);F(x+y)=F(x)+F(y)
where x+y,,x,y, can be translated as ||x+y||2,||x|2,||y||2 Of these two notions, to which does orthogonal additivity relate to, is probabilism (or quantum probabilism) with some further richness properties that make it closer to (2)in a restricted form
That is the sense of being on bases that are at right angles or being at 90
degrees from one another, not simultaneously measurable such as (A),(B):
as
(A)spin-up at angle y with||spin−up−y||2=P(spin up, measured@ angle,y)12
(B)spin-up at angle x with||spin−upxy||2=P(spin up, measured@ angle,x)=12
 
Both seem to use the Hilbert space inner product? I presume one is for bases, and another is for events.
Does this relate to when the events commute, are on vectors at right angles from each other, or rather do commute and lie on the same basis. If the former is just standard probabilism.
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Answer
An arbitrary wave functionin Hilbert space may be written as
psi) = sum over i of i)(i  psi) = sum over i of P(i) psi)
where P(i)=i)(i  is a projector to the state i)
Each P(i) squared is equal to itself, The product of any two is zero for different índices.
As a ressult of a measurment, one of the P(i) has eigenvalue 1, all the rest zero, and it is
determined that the particle is in state i, mutually exclusive to all other meauement posible results.
There is then a kind of additivity, and mutually exclusive posible results
I =sum over all i of P(i)
is the unit operator.
Im using ) and ( for the ket an bra rspectively in Dirac language.
Read Diracs Q. M.
as for orthogonality it is in (i,j) = 1 if i=j and 0 if i not equal to j
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In the case of diffraction of photons by an edge it can be shown simply by means of classical relativistic mechanics that the assumption of quantized angular momentum leads to a diffraction pattern which looks very similar to what is observd in experiments.