Particle-Wave duality and Hamilton-Jacobi equation

[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.

 

[Quantum River]

I was fascinated by the quaternion ring in my senior middle school, but I dropped it when I find it is even impossible to define simple analysis concepts such as limit, derivative, integral in the quaternion ring. In my opinion, our/mathematicians' understanding of such noncommutative things such as quaternion, matrix, operators is quite limited. Even the simple group of 2*2 matrix is not quite clear. Our limited understanding may be rooted in our naive understanding of more fundamental problems.

It is difficult to imagine a physical theory without derivative and integral, so I think a physical theory based on quaternion may not be so bright. But I still admit quaternion is fascinating.

 

[Anonym]

I find it is even impossible to define simple analysis concepts such as limit, derivative, integral in the quaternion ring.

Notice, that the quaternions are 4-dim quadratic normal division algebra with the signature {1,-1,-1,-1}. Enter Google, type quaternion analysis and you will get list of 330,000 publications. However, I have bad experience with the mathematicians. Almost always I find what I don’t need and don’t find what I need. Usually, I adopt the approach: do it by yourself. It is common practice in our room. Otherwise, why you need study that volume of math.? Indeed, you may be much more sophisticated and bring close to you Weyl and Minkowski. You should find your very specific point. Try to make it as simple as possible. Then apply all what you know. Don’t forget that “raffinert ist der Herr Gott, aber boshaft ist Er nicht".

 

[Anonym]

Quantum River, I would like to add something. We mentioned algebra and analysis as background. It is obvious, but I would like to emphasize it explicitly: you should speak fluently the theory of continuous groups.

I understand that you are a student. Most important: you should choose for you the teacher. Check your environment, if there is somebody in it that able to answer intelligently your questions. If not, go to Utrecht at Europe or Stony Brook,N.Y.

Dany.

 

[Quantum River]

Quantum River, I would like to add something. We mentioned algebra and analysis as background. It is obvious, but I would like to emphasize it explicitly: you should speak fluently the theory of continuous groups.

I understand that you are a student. Most important: you should choose for you the teacher. Check your environment, if there is somebody in it that able to answer intelligently your questions. If not, go to Utrecht at Europe or Stony Brook,N.Y.

Dany.

Anonym, thanks!
Now I am reading some papers on string theory and loop quantum gravity. It is more interesting and there is a lot of creative work to do.

I think duality is at the heart of physics. Besides the most important particle-wave duality, there are electromagnetic duality, T-duality, S-duality, Maldacena duality and much more. From a philosophical point, there are always two sides of a coin. There could be more than two sides, but the case of two sides is the most important one. Today particle-wave duality expresses the duality most clearly. But we may find a more fundamental duality which should encompass the particle-wave one as a special case. In mathematics, there is Langlands program. Fermat's last theorem is just a very small part of the program. I will not say more. Above duality, there must be the oneness, which is my personal faith. I have read William Hamilton's original paper On a general method of expressing the paths of light, and the planets, by the coefficients of a characteristic function. I think Hamilton still viewed the dynamics of the light as a path/orbit. So Hamilton didn't have a sense of duality. I'm not absolutely sure about this. Sometimes looking at a problem from a historical perspective is really interesting.

I am going to apply to a USA graduate school next year. Besides, why should I go to Utrecht?

Quantum River

 

[Anonym]

I am going to apply to a USA graduate school next year. Besides, why should I go to Utrecht?

QR, I did not say that you should. I only mentioned that it is a good place not only to study physics but also to study how excerpt the relevant point and not worry too much whether the suitable math tools already available.

 

[Anonym]

[QUOTE =Quantum River;1223047]Gross Nobel lecture on QFT and QCD.
http://nobelprize.org/nobel_prizes/p...ss-lecture.pdf [/QUOTE]

QR, thank you. It is amazing document and I enjoyed reading it. Using this example we may discuss better what are the interconnections between physics, mathematics and education.

1.Elementary Particles Physics ( Particle Physcis in the 1960’s):

“There was hardly any theory to speak of… Symmetries were all the rage… The biggest advance of the early 1960’s was the discovery of SU(3) by Gell-Mann and Y.Neeman… The most dramatic example of this is Gell-Mann and George Zweig’s hypothesis of quarks… Quarks clearly did not exist as real particles; therefore they were fictitious devices (see Gell-Mann above)… {interesting interpretation; notice, that M.Gell-Mann introduced quarks}… One was not allowed to believe in this reality… Weinberg was emphatic that this was of no interest since he did not believe anything about quarks {who is The Weinberg? just cloned Bohr?}.

To summarize:absence of the physical intuition, absence of understanding of the elementary particle physics, absence of understanding of group theory and Cartan classification.

2.Quantum Field Theory

“In my first course on QFT at Berkley in 1965, I was taught that Field Theory= Feynman Rules… My Ph.D. thesis was written under the supervision of Geoff Chew, the main guru of the bootstrap, on multi-body N/D equations… Nonetheless, until 1973 it was not thought proper to use field theory without apologies… Yang-Mills theory, which had appeared in the mid 1950’s was not taken seriously… I decided, quite deliberately, to prove that local field theory could not explain the experimental fact of scaling and thus was not an appropriate framework for the description of the strong interactions.

The Yang-Mills theory explain the origin of minimal coupling in Maxwell electrodynamics and covariant derivative in GR.

To summarize:absence of the physical intuition, absence of understanding of the differential geometry.

3.Asymptotic freedom

I would show that there existed no asymptotically free field theory… I knew that if the fields themselves had canonical dimensions, then for many theories this implied that the theory was trivial, i.e. free…The second part of the argument was to show that there were no asymptotically free theories at all.

To summarize:absence of the physical intuition, absence of understanding of the functional analysis.

Obviously, his ideas were interpretation dependent. However, he was successful in his “proves” and finally the coin falled inside his brain.

To summarize, on Hebrew we say: Lo mevin klum (American Education System outcome). By the way, during one of his lectures he said that he consider himself more clever then A. Einstein (Dubrovnik, June 1983).
For the nobel prize motivated kids it is the experimental confirmation that they have a chance.

Dany.

P.S. I did not read F.Wilczek lecture. However, “If two separated bodies, each by itself known maximally, enter a situation in which they influence each other, and separate again, then there occurs regularly that which I have just called entanglement of our knowledge of the two bodies.” (E. Schrödinger, “THE PRESENT SITUATION IN QUANTUM MECHANICS”, Paragraph10, Theory of Measurements,Part Two).