Particle-Wave duality and Hamilton-Jacobi equation

[Anonym]

how about talking QM, ok ? Let's go back to the electron's position stuff.

Sorry, Marlon. It is just personal feature. I am not able to talk when I have no idea what I am talking about. You invite me to discuss the relativistic QM, not it's non-relativistic limit. If and when I will have an answer, I will talk.However, I am optimist. " Raffinert ist der Herr Gott, aber boshaft ist Er nicht"

 

[marlon]

Sorry, Marlon. It is just personal feature. I am not able to talk when I have no idea what I am talking about. You invite me to discuss the relativistic QM, not it's non-relativistic limit. If and when I will have an answer, I will talk.However, I am optimist. " Raffinert ist der Herr Gott, aber boshaft ist Er nicht"

It's not going to be THAT easy. I asked you a specific question in post 38 concerning your claim "that a particle is always accompanied by a wave in QED". Why won't you answer that question.

Also, i explained to you why we cannot be talking about stuff like "a photon's size" because it violates the ONLY existing QM formalism.

marlon

 

[vanesch]

What I have in mind is entire reformulation of the classical physics in terms of functional analysis. Then it will be a functional.

In post #23, I was just making the objection that quantum river seemed to imply that in classical physics, particles are guided by a wave (I took it: a field in real space, of the F(x,y,z,t) kind) which was given by the function S in the H-J equation. This is only possible in the case of a single particle, because only there, S corresponds potentially to a function of the kind S(x,y,z,t) which can be a genuine field in space.
The S in the H-J equation *can* be seen as a "field", but over configuration space, in which the entire world is just one point, and then this field guides the "world point" in configuration space.

 

[Anonym]

The S in the H-J equation *can* be seen as a "field", but over configuration space, in which the entire world is just one point, and then this field guides the "world point" in configuration space.

That sounds similar to what we are doing when we introduce Fock space and second quantization. What do you think?

 

[vanesch]

That sounds similar to what we are doing when we introduce Fock space and second quantization. What do you think?

Of course, that's a basic postulate of quantum theory: superposition ! A quantum state being a superposition of all non-quantum states (points in configuration space), this automatically defines a "field" (wavefunction) over the previous configuration space, where the "field values" are nothing else but the complex weights in the superposition. The configuration space in the non-quantum version becomes the index space of a basis for quantum state space (= hilbert space). That's exactly what happens in the Fock space description: to each individual non-quantum E-M configuration corresponds a basis vector in Fock space - although these are not the usual basis vectors in Fock space, which is usually spanned with the eigenvectors of the (E,P) observables, and not the EM configuration states.

 

[Anonym]

Of course, that's a basic postulate of quantum theory: superposition ! A quantum state being a superposition of all non-quantum states (points in configuration space), this automatically defines a "field" (wavefunction) over the previous configuration space, where the "field values" are nothing else but the complex weights in the superposition. The configuration space in the non-quantum version becomes the index space of a basis for quantum state space (= hilbert space). That's exactly what happens in the Fock space description: to each individual non-quantum E-M configuration corresponds a basis vector in Fock space - although these are not the usual basis vectors in Fock space, which is usually spanned with the eigenvectors of the (E,P) observables, and not the EM configuration states.

First of all: thank you. But what you mean "these are not the usual basis vectors in Fock space, which is usually spanned with the eigenvectors of the (E,P) observables, and not the EM configuration states"?

 

[Anonym]

It's not going to be THAT easy. I asked you a specific question in post 38 concerning your claim "that a particle is always accompanied by a wave in QED". Why won't you answer that question.

Marlon, come on! Renormalization.

 

[Anonym]

If someone talks about the 'size of [insert favorite item here]', then one must always look at how that [insert favorite item here] is defined. If someone thinks there is a size of [insert favorite item here], then I'd like to hear how such a thing is measured, especially when in a standard formalism, we simplify our theory by adopting a point particle. We still see no deviation between that simplification and what we observe YET.

I did mistake above (#29, corrected). Everything consistent with delta x=0. Perhaps,it may be used as definition what point-like object (particle) means in wave mechanics. Bounded fermion is extended object. Free moving fermion/boson is point-like. If so, the gluons are not point-like. Excuse me that I am lazy. What we see in deep-inelastic? Only quarks, right? Why we do not see gluons? May be they are “blurred” over entire volume of the hadron?

 

[marlon]

Marlon, come on! Renormalization.

Sorry, that was post 30...Anyhow, no i don't get it...renormalization ? Besides, i asked for an equation of that wave that accompanies a particle in QED. Why won't you give me that equation ? Answering with renormalization is a bit hollow and basically useless.

marlon

 

[marlon]

If so, the gluons are not point-like. Excuse me that I am lazy. What we see in deep-inelastic? Only quarks, right? Why we do not see gluons? May be they are “blurred” over entire volume of the hadron?

But a gluon is defined based upon the colour quantumnumber. Why are you then talking about the gluon's "position" in a hadron ? Don't you see the flaw in that ? Same goes for what you say about photon size BTW.

marlon

 

[lightarrow]

I'm not sure how this has degenerated into the issue of a discussion in this forum.

Remember, your initial claim made it appear to be that there is a difference between "quantum physics" and "quantum formalism", the latter being the phrase I had used (and is often used in other instances and books). I continue to ask for examples to support your claim. Somehow, you have managed to change this into an issue of what is being discussed on here.

It is in what you write after:
"If someone talks about the 'size of [insert favorite item here]', then one must always look at how that [insert favorite item here] is defined. If someone thinks there is a size of an 'electron', then I'd like to hear how such a thing is measured, especially when in a standard formalism, we simplify our theory by adopting a point particle. We still see no deviation between that simplification and what we observe YET. Thus, anyone claiming that an electron has a "size" will have to redefine all the properties of an electron and how such a quantity should be measured."

If in theory is adopted a point particle for the electron, then the value of it's size is defined: it is zero. But, at the same time, the electron's size is not defined in QM formalism, so how can we know its value?. It's this that I don't understand.
As you always point, and I thank you for this, we must always give precise definitions of concepts, especially in quantum physics. Are we well aware (I'm not) of what are the consequences of assuming the electron as point like?
I know this has worked well, up to now, however I don't believe the electron is point like; at the same time, I still don't know if and how would be possible to create a definition for an electron's size.

 

[ZapperZ]

If in theory is adopted a point particle for the electron, then the value of it's size is defined: it is zero. But, at the same time, the electron's size is not defined in QM formalism, so how can we know its value?. It's this that I don't understand.

But you seem to have missed the point of that reply from me that you quoted. It is a matter of CONVENIENCE that we select these things to be "point particle" and that so far, this convenience WORK. However, nowhere in the formulation is the property of the width of it is DEFINED!

Secondly, we are now playing with semantics. You are adopting the "x=0 is not the same as x is undefined" concept. While I agree that technically this is correct, it doesn't fit into this scenario. Why? Because if you want to be absolutely technical about it, setting something to have a width of zero means that that something isn't there! We're not talking infinitesimal here, we're talking about ZERO. So that object isn't there! Yet, how could it have a mass, charge, and can collide with other particles and make itself known? So herein lies the contradiction.

So by calling an electron to be "point particle", we IGNORE the issue of its width till we know more to be able to address it. Till then, the concept of an electron being "point" particle works in the majority of the situation we deal with. But by doing that, does that mean the width of an electron is well-defined? Nope! Just because I can set the geometry of a cow on Alpha Centauri to be a sphere doesn't mean that it is. That is why you don't see anywhere in the Particle Data book the radius of an electron being set to zero.

But where in here is the example of "quantum physics" different than "quantum formalism"?

Zz.

 

[marlon]

If in theory is adopted a point particle for the electron, then the value of it's size is defined: it is zero. But, at the same time, the electron's size is not defined in QM formalism, so how can we know its value?. It's this that I don't understand.

First of all, a point does not have a size equal to zero. "Nothing" has a size which equals zero. You should have written "the size of a point particle has zero dimensions"

Now, a photon is a point particle in the energy base. Just like in a spatial coordinate base (like the Euclidean frame of reference) where the points denote coordinate values, the points of the energy base denote energy values. A photon is nothing more than such an energy value and thus a point in the energy base. It is with respect to this base that a photon is a point particle with zero dimensional size (which is NOT the same as saying that the size is zero). Again, a photon is NOT a point particle in a spatial base so it is NOT defined as an object with finite spatial boundaries.

marlon

 

[lightarrow]

..the DEFINITION of a photon. A size is a spatial coordinates thing whuile a photon is a point particle in an energy base.

Can you explain me better the exact definition of photon? I have never found it; the only one I have found is "quantum of energy in the electromagnetic field".

 

[marlon]

Can you explain me better the exact definition of photon? I have never found it; the only one I have found is "quantum of energy in the electromagnetic field".

Not a quantum of energy in the electromagnetic field but a quantum of energy ASSOCIATED to the electromagnetic field. The "association" is ofcourse the quantisation of the EM waves. That is all

marlon

 

[Anonym]

Lightarrow:”Are we well aware (I'm not) of what are the consequences of assuming the electron as point like?”

You should not required to assume anything. This is firmly established experimental evidence (as well as all fundamental fermions and bosons).
The consequence is that QT exactly as Classical Physics are elementary particles physics.

Lightarrow:”But, at the same time, the electron's size is not defined in QM formalism”

The size is well defined notion in the standard formulation of QM: dispersion, delta x. There is no room in the QM formalism for other definition. In the non-relativistic QM it is adequate (see for example the calculation of the first Bohr radius for H-atom). The point is whether or not delta x=0 is consistent with the high energy experimental evidence (relativistic QM).

Lightarrow:”however I don't believe the electron is point like”

Physics in general and PF in particular have nothing to do with religion.

ZapperZ:” setting something to have a width of zero means that that something isn't there! We're not talking infinitesimal here, we're talking about ZERO. So that object isn't there!”

ZapperZ, come on! You give classical example of the circular argument. In addition, your statement is in contradiction with everything that is known in Classical Physics. And a cow on Alpha Centauri also have nothing to do with that.

 

[Anonym]

Besides, i asked for an equation of that wave that accompanies a particle in QED. Why won't you give me that equation ? Answering with renormalization is a bit hollow and basically useless.

I do not know what you have in mind. To avoid any misinterpretation, if you mean physics/0504008, I have no doubt that the eqs. are adequate. In order to convince yourself you just substitute the presented solutions into the suggested eqs. It is a matter of few minutes. In addition, I can’t imagine more simple form of the solutions. Obviously, the presented set is the complete orthonormal set of the positive energy solutions. No sea and Zitterbewegung are required, they are the artificial consequences of the Dirac formulation. However, I do not understand the content of the obtained result. I hope that if I will find the basis in the real Hilbert space, I will understand better what I am doing.

 

[ZapperZ]

ZapperZ:” setting something to have a width of zero means that that something isn't there! We're not talking infinitesimal here, we're talking about ZERO. So that object isn't there!”

ZapperZ, come on! You give classical example of the circular argument. In addition, your statement is in contradiction with everything that is known in Classical Physics. And a cow on Alpha Centauri also have nothing to do with that.

You still don't know how to use the quote function?

The argument WAS intentionally circular to illustrate the absurdity in stating that a zero width is a valid and KNOWN size of an electron! You just stated MY point!

Zz.

 

[vanesch]

First of all: thank you. But what you mean "these are not the usual basis vectors in Fock space, which is usually spanned with the eigenvectors of the (E,P) observables, and not the EM configuration states"?

For fields, this is a kind of tricky issue. Compare it to the NR single-particle quantum system. The configuration space of a featureless point particle is simply R^3 (generalized coordinates x,y,z). The hilbert space of this single-particle quantum system is then, by definition, spanned by basis vectors which are indexed upon the configuration space: each |x,y,z> is a (generalized) basis vector spanning the Hilbert space. But the "fock space basis" equivalent of this Hilbert space is spanned by the common eigenstates of the {E,Px,Py,Pz} set of operators, which turn out to be non-degenerate in this case and correspond to the |E,px,py,pz> vectors, with E = 1/(2m) (px^2 + py^2 + pz^2).
Of course, in this example, they are simply related by a Fourier transform.

For the EM case it is more tricky. The Fock space basis is again given by the eigenvectors of {E,Px,Py,Pz}, but this time, there is a lot of degeneracy, and it turns out that a tidy way of writing down all these eigenvectors is by using finite n-tuples of natural numbers, which correspond to "photon states" with n different modes. The relationship to a classical EM *configuration* is not so evident, and I'm kinda struggling myself to get a clear view on this. Mind you that a classical *mode* is not a configuration, but an entire solution to the classical EM field equation ; one should picture a classical configuration rather as a totally arbitrary vector potential distribution *at a single instant*. (if I understand this well, one shouldn't even take A but a related quantity). Note that a configuration doesn't determine the energy (we also need the derivatives of the potential for that) for instance. To each such configuration must correspond then also a basis vector, which should be able to be expressed as a function of the Fock elements.

This is (partially) treated in Mandel and Wolf, section 10.4 - but I have to say that this is some time ago that I studied this, and I remember vaguely not having understood everything clearly myself. But the idea goes more or less as follows: to each classical EM mode is assigned a harmonic oscillator, which has a single q and a single p variable assigned to it. The independent quantization of each of these harmonic oscillators corresponds to the canonical quantization procedure for the EM field and leads to the Fock basis. But the set of all these q variables of all these harmonic oscillators is nothing else but a set of generalized coordinates of the configuration space of the EM field. We can write the inproduct between a specific Fock basis state and a configuration eigenstate, using the position representation for the energy eigenstates of harmonic oscillators.

 

[Anonym]

You still don't know how to use the quote function?

I already learned “the quote” but still have problems with “the multi quote”. Any way I prepare post outside the PF and should keep quote text in front of my eyes. Then copy-paste do a job in one step. In addition, “each one has his small shortcomings”

The argument WAS intentionally circular to illustrate the absurdity in stating that a zero width is a valid and KNOWN size of an electron! You just stated MY point!

You may check that I consistently say that argue with the experimentalist is waste of time.

 

[Anonym]

I'm kinda struggling myself to get a clear view on this.

Thank you. I have Mandel and Wolf. Please give me ref. to your papers and others that you consider relevant.

 

[ZapperZ]

You may check that I consistently say that argue with the experimentalist is waste of time.

Well, that's funny. I know of a theorist here in our division that said the same thing about OTHER theorists.

Besides, if you truly believe that's the case, when why did you engage in an "argument" with me in the first place?

Zz.

 

[vanesch]

Thank you. I have Mandel and Wolf. Please give me ref. to your papers and others that you consider relevant.

Well, as I said, it is worked out in Mandel and Wolf, section 10.4: the so-called q-basis is close to what I called the "configuration basis".

(and BTW, I'm also an experimentalist )

 

[Anonym]

Well, that's funny. I know of a theorist here in our division that said the same thing about OTHER theorists.

“Each one has his small shortcomings”.

Besides, if you truly believe that's the case, when why did you engage in an "argument" with me in the first place?.

My intention was to contribute something that maintain “duality” or “complimentarity”. All that remind me old Jewish story. Three man came to Rabbi, two of them in sharp disagreement with each other. The first presented his point of view. Rabbi said:”you are right.” The second presented his point of view. Rabbi said:”you are right”. The third said :”Rabbi, it is impossible”. Rabbi said:”you are also right” (human realization of the linear superposition).

You said:This is a point, but I know it down to that bound.
I said: This is a point, but I do not know what internal angular momentum of the point means.
Marlon said: This is not a point, since the dispersion of dimensionality not =0.
And Vanesch said: you have some souvenirs of that chapter.

 

[ZapperZ]

You said:This is a point, but I know it down to that bound.

Really? All of the stuff where I said that the width of an electron really is not defined you just ignored?

Then my version of the story is:

You said "I only read what I care to".

Zz.

 

[vanesch]

And Vanesch said: you have some souvenirs of that chapter.

What I never figured out to my satisfaction, is what exactly constitutes the configuration space of the classical EM field. For instance, I don't know if giving simply the E-field at a single instant, but not the B-field, qualifies. In my bones, I feel that it should, somehow. But the q-variables in M&W are not exactly this. They are the "position" variables of the (classical) harmonic oscillators associated with each individual mode, but these mix E and B fields.

 

[lightarrow]

Originally Posted by lightarrow
If in theory is adopted a point particle for the electron, then the value of it's size is defined: it is zero. But, at the same time, the electron's size is not defined in QM formalism, so how can we know its value?. It's this that I don't understand.

First of all, a point does not have a size equal to zero. "Nothing" has a size which equals zero. You should have written "the size of a point particle has zero dimensions"

Now, a photon is a point particle in the energy base. Just like in a spatial coordinate base (like the Euclidean frame of reference) where the points denote coordinate values, the points of the energy base denote energy values. A photon is nothing more than such an energy value and thus a point in the energy base. It is with respect to this base that a photon is a point particle with zero dimensional size (which is NOT the same as saying that the size is zero). Again, a photon is NOT a point particle in a spatial base so it is NOT defined as an object with finite spatial boundaries.

marlon

Thank you for this answer, however I was talking about an electron, not about a photon.

 

[lightarrow]

Not a quantum of energy in the electromagnetic field but a quantum of energy ASSOCIATED to the electromagnetic field. The "association" is ofcourse the quantisation of the EM waves. That is all
marlon

I know nothing about QED. What does "quantisation of the EM waves" mean? I believed it meant "quantisation of the EM waves's energy".

 

[lightarrow]

But where in here is the example of "quantum physics" different than "quantum formalism"?

Since, for you, quantum physics = quantum formalism, they are not different by definition. But I have never seen this statement before, so, maybe, someone should have specified it in the books, or at university.

But you seem to have missed the point of that reply from me that you quoted. It is a matter of CONVENIENCE that we select these things to be "point particle" and that so far, this convenience WORK. However, nowhere in the formulation is the property of the width of it is DEFINED!

If it is for convenience, and not because of QM formalism, then everybody could use his own "convenient" value, for example 10^-34m (just a number as others).

Secondly, we are now playing with semantics.

If we say that an electron is a point particle, we are making a physical statement, not semantic. If the correct word is not "size" but "dimension" it doesn't change anything to me, we are talking about a physical concept, that is "the electron's dimension". If this concept is not defined, then it's wrong to say "the electron is pointlike".

 

[ZapperZ]

Since, for you, quantum physics = quantum formalism, they are not different by definition. But I have never seen this statement before, so, maybe, someone should have specified it in the books, or at university.
If it is for convenience, and not because of QM formalism, then everybody could use his own "convenient" value, for example 10^-34m (just a number as others).If we say that an electron is a point particle, we are making a physical statement, not semantic. If the correct word is not "size" but "dimension" it doesn't change anything to me, we are talking about a physical concept, that is "the electron's dimension". If this concept is not defined, then it's wrong to say "the electron is pointlike".

I believe you have misunderstood completely what "quantum formalism" is. If you look at the general form of Schrodinger Equation, nowhere in there is the size or width of ANYTHING is included. The theory wasn't made for a specific system or particle. Whether an electron has a width or not is NOT part of the quantum formalism, even though one might USE it to either interpret its value, or even get it from First Principle. One certainly can arrive at values such as the electronic gyromagnetic ratio via QED, but the value itself is NOT part of a quantum theory.

So again, I really don't see what the setting of the value of the width of anything has anything to do with quantum formalism.

Zz.

 

[Anonym]

Vanesch:” I don't have the book handy, but have some souvenirs of that chapter”
“What I never figured out to my satisfaction, is what exactly constitutes the configuration space of the classical EM field. For instance, I don't know if giving simply the E-field at a single instant, but not the B-field, qualifies. In my bones, I feel that it should, somehow. But the q-variables in M&W are not exactly this. They are the "position" variables of the (classical) harmonic oscillators associated with each individual mode, but these mix E and B fields.”

I usually find “souvenirs” in each “chapter”. Relativistic QM included. To be able to continue our discussion of the classical EM will take for me a while. Thanks again.

Lightarrow:” Quote:
Originally Posted by ZapperZ
But where in here is the example of "quantum physics" different than "quantum formalism"?

Since, for you, quantum physics = quantum formalism, they are not different by definition. But I have never seen this statement before, so, maybe, someone should have specified it in the books, or at university.

Quote:
But you seem to have missed the point of that reply from me that you quoted. It is a matter of CONVENIENCE that we select these things to be "point particle" and that so far, this convenience WORK. However, nowhere in the formulation is the property of the width of it is DEFINED!

If it is for convenience, and not because of QM formalism, then everybody could use his own "convenient" value, for example 10^-34m (just a number as others).

Quote:
Secondly, we are now playing with semantics.

If we say that an electron is a point particle, we are making a physical statement, not semantic. If the correct word is not "size" but "dimension" it doesn't change anything to me, we are talking about a physical concept, that is "the electron's dimension". "

We are talking about “the electron’s size”. The difference is not semantic. They are different operators. The dimension is property of the classical space-time continuum and =4. Marlon statement “You should have written "the size of a point particle has zero dimensions"” is simply absurd (I do not know from where the quotation brought). Except that, I agree with everything you said.

Lightarrow:” Considering that I don't want to discuss what "a particle is always accompanied by a wave and vice versa" means, can you show me where exactly Quantum River wrote that QM formalism have that?
Or, discussing about quantum physics = only discussing about QM formalism, for you?”

Lightarrow:”I still don't know if and how would be possible to create a definition for an electron's size.”

You refuse to follow the standard rules of the scientific development. The notion of size was introduced by Egyptians, perhaps 7000 years ago. You take two points and stretch a cord. Then you ask what is an angle (firstly what is 90deg angle. It lead to the phenomenological result: 3^2+4^2=5^2 and to the corresponding theoretical generalization). It provide foundation for Euclidian geometry. That provide foundation for the mathematical formalism of metric spaces.
That provide foundation for Newtonian formalism. Later you introduce the communication problems. That provide the foundation of special and general relativity based on non-Euclidian geometry. And so on. In the non-relativistic QM it is well defined notion: the eigenvalues of the self-adjoint operator called dispersion of a position are measurable quantities. The mathematical formalism of relativistic QM still open problem. You can discuss, but can’t require to give you definition of the size there. For sure, the answer to a question what is the size of an electron, quark or gluon a posteriori will be consistent with the Egyptians.

ZapperZ:” Really? All of the stuff where I said that the width of an electron really is not defined you just ignored?Then my version of the story is:
You said "I only read what I care to".

Completely wrong with certainty since you are now in my reference frame and I easily can compare. I consider the C.E. Shannon theory of communication adequate. To the best of my knowledge and understanding it require the final bandwidth in any practical realization. Your statements were not communicated to me since they are outside of my bandwidth. This does not mean that they are wrong. My intuition said to me that all fundamental elementary particles, fermions and bosons, are points physically as well as mathematically down to the range where Minkowski metric remains the well defined notion.

 

[marlon]

Thank you for this answer, however I was talking about an electron, not about a photon.

Same counts for the electron, except the stuff on the energy base ofcourse.

marlon

 

[Anonym]

Quantum River:”“So how many mathematics do a theoretician need to do Quantum physics?”

It seems that nobody intend to add something to our discussion. So, let me write the conclusions.

It concludes also my current active participation in PF for a while. It is obvious what I intend to do next: to establish the connection between classical and quantum electrodynamics. I usually start drawing the circle with the infinite radius. Then the problem will be there. Then I try to shrink it to the reasonable size keeping the central point inside. Then I start to work.

I began my PF with the discussion of the statistical interpretation in “the wave packet description” session. It became clear that the Born statistical interpretation is unavoidable and fundamental. The problem only with the interpretations of that interpretation. The discussions were around double-slit experiment and the meaning of self-interference. We continue that discussion in the “single particle interference” session. Does not a matter whether we discuss electrons or photons. The point is that the notion of simultaneous is different in CM vs QM. In QM simultaneous is dispersion of time, size of the time if you like. In CM the simultaneous means instant, delta t=0. It is clear that the notion of time is the special relativity all about and thus classical as well as quantum electrodynamics. The HJ formulation already “know” special relativity, look to the Poisson brackets.

All our session “Particle-Wave duality and Hamilton-Jacobi equation” is the discussion of what we don’t know. Lightarrow point of view is that the size is a matter. Vanesch and I say that the size is not a matter (here only, not in every problem). The matter is the number of particles and formalism of non self-adjoint operators. Quantum River ask what the relevant math one should learn. That I do not know. Nobody ask what is the mathematical content of P.A.M. Dirac q-number algebra. That I know: it is well known and widely used by mathematicians Cayley-Dickson process.

Let us return to the physics. If the matter is the particle number operator, who is his canonically conjugate observable? Phase? Certaintly not. Phase difference? Certaintly yes. Notice that HJ formulation is all about phase. But what about number of particles operator? Vanesch said : you have some souvenirs in Ch. 10 of M&W (and Ch. 11,12 also). I do not need more than that. The size of the problem is now point-like.

By the way about the double-slit. If it is the relativistic QM or Classical Electrodynamics problem and if the number of particles is not conserved quantity, then self-interference or Born statistics are “dual” expressions of the linear superposition and the statistics emerges naturally exactly in the same manner as in the classical statmech. Statistical ensemble is not prepared on Alpha Centauri (L.E. Ballentine) . It prepared by macroscopic device called lossless beamsplitter and thus the particle don’t do statistic with itself and do not interfere with itself.

Is here somebody that did not understand?!

ZapperZ, we are not in disagreement. It is impossible. I am educated on M.Gell-Mann words (approx. quotation): “One “ugly” experiment destroy thousand “beautiful” theories”. In my own words:

Physics are an empirical science,in practice this means that the only perfect mathematical frameworks will survive.

That is the content of my story. Thanks to everybody.

Daniel Gleekstein.

 

[Quantum River]

Anonym, I have read quant-ph/0606121 and found several problems.
What is the "Real Hilbert Space"? You gave some description about it. But there is no clear definition. What does "tr" mean? "tr" can operate on a complex number (such as the complex number i=sqrt(-1)) and also a matrix. So what is "tr"? Do you mean the physical state of a system in classical mechanics could be described by a special Hilbert vector? The CM is embedded in the QM because the physical state (a vector in the special Hilbert space) in CM is embedded in the bigger quantum Hilbert space and the total set of the physical state in CM forms a subspace in the quantum Hilbert space . I could misunderstand your meaning. If I understand quant-ph/0606121's meaning correctly, I think this view is alluring, but there are lots of problems.
When I asked how many mathematics we need, I am confused by the mathematics in the Gutzwiller trace formula, which is a formula connects CM and QM. Mathematicians may look at a problem from an algebra perspective or a geometry perspective, but only few and few mathematicians could look at a problem from a physics perspective. Actually I am not just asking a mathematics problem, because we do need to know lots of mathematics to understand the problem, the connections between CM and QM.
Sorry for replying so late. I am preparing for the examinations recently.

 

[Anonym]

What is the "Real Hilbert Space"? You gave some description about it. But there is no clear definition. What does "tr" mean? "tr" can operate on a complex number (such as the complex number i=sqrt(-1)) and also a matrix. So what is "tr"?

Welcome to the club! I am glad to meet you here! As introduction let me present my discussion with CarlB from “Good Introductory Books” session:

CarlB:” He (R.P. Feynman) describes complex numbers as arrows, where one adds up a bunch of arrows, and then takes the squared magnitude. But he really does this in a way that is to be understood by EVERYONE.”

My answer:I do not understand that and never did. This is irrelevant for the physics abstraction made by stupid mathematicians. Instead, they were required to understand that the new mathematical object enter into the Game (matrices) and that the proper generalization of usual multiplication rule is required. Then one will have x^2>0 as needed for the measure and the measurement theory (metric space:geometry).

I consider the above point crucial for understanding of the Quantum Theory and why it is formulated in terms of wave packets
exp(i*phi)=cos(phi)+i*sin(phi):
two component wave packet and/or two-level QM system.

Do you mean the physical state of a system in classical mechanics could be described by a special Hilbert vector? The CM is embedded in the QM because the physical state (a vector in the special Hilbert space) in CM is embedded in the bigger quantum Hilbert space and the total set of the physical state in CM forms a subspace in the quantum Hilbert space .

You use a circular argument. This is The Error made by J. von Neumann in the formulation of the Theory of Measurements. If so, then the state of the QM system (system under test)+measurement apparatus should be represented by the Kronecker product of the subsystem states. But Quantum world is not a Classical world (W. Heisenberg UR’s; E. Schrödinger cat). They are different worlds connected through the act of measurement that express itself by the collapse of wave packet. The measurement instruments belong to the Classical world, therefore the collapse should be a natural feature of the Classical Physics formalism. The proper generalization (wave mechanics) of the non-relativistic limit (Newtonian mechanics) is presented in quant-ph/0606121. No doubt that the relativistic extensions of J.C. Maxwell electrodynamics and A. Einstein GR may be achieved within the suggested frameworks.

What is the "Real Hilbert Space"? You gave some description about it. But there is no clear definition. What does "tr" mean? "tr" can operate on a complex number (such as the complex number i=sqrt(-1)) and also a matrix. So what is "tr"?”

To the best of my knowledge, the real and quaternionic “Hilbert” spaces were originally considered by J. von Neumann, E.P. Wigner, P. Jordan and G. Birkhoff. They disregarded octonions since they considered octonions as nonassociative algebra. It is another error, but not essential. The Cayley numbers are only alternative algebra, it is not a true nonassociative algebra. My guess that the physical motivation was dictated by the structure of the Dirac equation and the Majorana-Weyl description of neutrino. Notice, that the real quaternions are the number system; the complex quaternions are ordinary 8-dim C3 Clifford algebra among the infinite examples of other algebras. P.A.M. Dirac tried to reduce the algebraic dimension of his eq. of motion during all his life. Also notice, that the q-number (operators) Dirac algebra is Cayley-Dickson process.

The definition of tr presented in quant-ph/0606121 is the only correct and mathematically rigorous definition of trace operation existed (see literature below).

You enter the room in the mathematical physics called the axiomatic foundation of the physical theory. You should realize that the volume of the mathematical knowledge required is several orders of magnitude more than the average physicist. In addition you are required not to lose your physical intuition (just every day sense of reality). If you are ready for that, the good point to start in algebra is:

1.B.L. Van Der Waerden, “Algebra”
2.R. Bellman, “Introduction to Matrix Analysis”
3.R.D. Schafer, “An Introduction to Nonassociative Algebras”.

Dany.