E=hw vs [x,p]=ih: Quantum Postulates Compared

[arivero]

Old quantum postulate was E= h w, where w= 2 pi/ t. Nowadays, we start from the (afaik) equivalent postulate [x,p]=ih. I suposse this was originated by Dirac-Poisson methodology.

One wonders, Which are the merits and disadvantages of each postulate? For instance, the old one seems more physical, but I have never seen a quantum field theory formulated from the old principle. Also, it is very intriguing to see how to fit both postulates into the path integral formulation.

 

[dg]

Well how about the meaning of using the complex field?

Usually complex numbers simplify math but hide physical meaning, does this happen here too?

Dario

 

[arivero]

About complex numbers, I think here it is not the case. Complex numbers are typical of higher functional analysis, thus it is reasonable to work with them. Note that you can even find people liking to work with quaternions, which are complexer than complex.

It is possible to rid out of complex numbers by going to the pilot-wave formulation. If it is more physical or not, there is a lot of metaphysical discurrion about it.

Also, you could be thinking in other appeareance of complex numbers, namely the ones in Dirac equation, which a lot of people -the most famous being Hestenes- has show can be substituted by a Clifford algebra. Again, if a Clifford algebra is more physical, well, it is up to interpretation.

 

[arivero]

Canonical Quantization

In hte meanwhile, I have been readin Heisenberg books on quantum theory and I think I have got new views on the so-called Canonical Quantization. Most books take it as a blind imposing of [Q,P]=h where Q(x1,x2,x3,t) holds now infinite degrees of freedom, labeled by the index coordinates (x_i,t).

Instead of it, the original view is close to Fourier transform. In quantum mechanics a la Heisenber-Bohr-Jordan, one understands that a classical function q(t) has a Fourier transform, call it ^q(E), and that the original function q(t) can be composed from a sum of plane waves with coefficients in the q_E. Now, quantization is preached about this set of coefficients: there are to be defined for a matrix ^q_(E1,E2) and only for particular valid Energies. From this postulate it is derived that the [q,H] is q'(t) and than [q, m q'] = ih.

Then, the jump to field theory is to generalize this to a four-dimensional Fourier transformation, from Q(x_i,t) to ^Q(p_i,E) and then to preach a no commutative definion of this object, ^Q. The energy levels of the 0+1 dimensional case become particle numbers and so on. And again, the [Q,P] rule is a derived fact. This should to be stressed in the classroom, but it is not: in the usual presentation, Dirac approach is mixed with Canonical Quantization in an arbitrary manner.

Incidentally, a traditional drawback of quantum mechanics turns now to be a "feature". It is well known that the quantum uncertainty for energy-time is bad defined, because the time variable has so peculiar rule that there is not such a thing as a time operator. In canonical quantization of field theory this phaenomena will apply also to the rest of index of the Fourier transform, this is, to the x_i. So we do not expect in this formalism to be able to the get operators for the coordinates of the field. If you were happy about the situation with time in quantum mechanics, now you must be happy about the situation with space-time in QFT.

 

[dg]

Originally posted by arivero
About complex numbers, I think here it is not the case. Complex numbers are typical of higher functional analysis, thus it is reasonable to work with them. Note that you can even find people liking to work with quaternions, which are complexer than complex.

It is possible to rid out of complex numbers by going to the pilot-wave formulation. If it is more physical or not, there is a lot of metaphysical discurrion about it.

Also, you could be thinking in other appeareance of complex numbers, namely the ones in Dirac equation, which a lot of people -the most famous being Hestenes- has show can be substituted by a Clifford algebra. Again, if a Clifford algebra is more physical, well, it is up to interpretation.

Both of your replies are really interesting:

Let us start with the first, saying that higher functional analysis is based on complex field does not make it less abstract and anything more than a powerful MATHEMATICAL tool. The use of complex field in electrostatic and fluid mechanics problems are good examples: nobody would ever base its understanding (models) of those subjects on a complex formulation but formalism turn out to be largely simplified by introduction of complex REPRESENTATIONS. Do you have any example of physical meaning of a complex number?

I do not know the pilot-wave formulation and if it has any drawback (if you could send me a good reference I will gladly learn more about it) but classical Schroedinger equation can be simply re-written as a system of two equations in the real field that have a much clearer PHYSICAL meaning: one is the continuity equation of probability, the other is a Hamilton-Jacobi equation containing an additional quantum potential energy.

You keep on saying that is "just" a question of "interpretation" but interpretation is not secondary to ability to perform calculation. Without a correct interpretation you can draw logical conclusion out of pure formalism but you will never develop an understanding of the underlying physical process. Without that a further development of the theory is strongly hindered. As an example consider how the Copernicus heliocentric model is just an "interpretation" of the same experimental data on which the geocentric model was based (nothing more than a simple change of reference frame if you want). BOTH WERE MAKING CORRECT PREDICTIONS (actually for a long time in absence of Kepler laws and perturbative corrections, geocentric model was largely superior), but the heliocentric model allowed to develop a much larger picture. Another good example is representation of SU(2) using unitary complex matrices or real O(3) matrices and inversion: the first is powerful mathematically but perform poorly in giving any geometric understanting, and viceversa for the latter.


For the the second reply you posted:
I think you have found something much closer to the kind of answers I am looking for. Is it the presence of infinite dimensions that demands for the introduction of commutation rules or equivalently for indetermination principles? Fourier transform properties are at the base of quantum theory (transformation from space to momentum representations) and this is the reason why a complex theory becomes so useful. They are also at the base of modern mathematical theory of measure and this is a strong indication that quantum theroy is something like classical mechanics with the addition of infinite dimension quantities, some theory of measure and... 'who-knows?' Another reason for using complex field is that it allows to represent a general real matrix using an hermitian one (antisimmetric part becomes the imaginary part). This is particularly true for field gradients and hessians by which almost all physics equations can be written.
I would really appreciate some reference in this case too ("Heisenbergs books..."). I agree that in the classroom the explanation usually falls into a mere presentation of hardly guessable postulates and the history of the development of quantum theory really helps.

I am also particularly interested in the last paragraph about the role of time in classical quantum theory: why a time operator cannot be defined along with the position operator? I think that a formal answer to this question could be remarkably insightful!

Good thoughts to you all, Dario

 

[arivero]

Hi Dario, uff, a lot of things you are mentioning! Let's try some links, at least :-)

complex field does not make it less abstract and anything more than a powerful MATHEMATICAL tool. Do you have any example of physical meaning of a complex number?

The closest thing to physical meaning could be a phase in a wave or a polatization. But Again, it is just the real "arg" of the complex number... Hmm what about x+ip? of course, just a pair of real numbers, but it has sense. In any case, I agree it is intriging: how it comes that x^2+1=0 does not have physical meaning if all the components of the expresion (the square, the unit, the zero, the addition) have?

I do not know the pilot-wave formulation and if it has any drawback (if you could send me a good reference I will gladly learn more about it) but classical Schroedinger equation can be simply re-written as a system of two equations in the real field that have a much clearer PHYSICAL meaning: one is the continuity equation of probability, the other is a Hamilton-Jacobi equation containing an additional quantum potential energy.

Well. basically that it the pilot wave interpretation! It was formalized by Bohm, check for authors in http://prola.aps.org/ if you have access. Incidentally, I got a link to an old paper from De Broglie, "Sur les equations et les conceptions generales de la mecanique ondulatorie", in www.numdam.org (it is free!). To be precise, http://www.numdam.org/item?id=BSMF_1930__58__1_0

You keep on saying that is "just" a question of "interpretation"

Hmm perhaps it is because I believe that a "real numbers" interpretation of quantum mechanics is just another geocentric view, not the heliocentric one. As discussing Eudoxus vs. Ticho Brahe. I agree there are probably a better interpretation, but my bet is to rethink the theory of calculus.

Is it the presence of infinite dimensions that demands for the introduction of commutation rules or equivalently for indetermination principles?

I believe not. It happens just in QM, and I can not see an infinite dimension involved directly. There are an infinite process, already in classical mechanics: the one of calculating the derivatives, for instance in F=m x''(t).

it allows to represent a general real matrix using an hermitian one

Indeed this is a key trick of the founding fathers.

I would really appreciate some reference in this case too ("Heisenbergs books...")

The online works of Heisenberg Born y Jordan are somehow dark. I have found more useful a final book, W. Heisenberg, {\it The Physical Principles of the Quantum Theory}, ed. Dover.

I am also particularly interested in the last paragraph about the role of time in classical quantum theory: why a time operator cannot be defined along with the position operator? I think that a formal answer to this question could be remarkably insightful!

This is, or should be, standard classroom lore. I doubt which is the proper form to present it. Most tipically it is just remarked that our operators are build from functions in phase espace (X,P), thus time does not appear. But there are for sure other approaches more insightful... I have read them, just I do not remember now.

 

[dg]

Hi Alejandro,

I have just had a look to your website, really interesting! This stuff about non-commutative geometry sounds pretty intriguing!
Now it is a little clearer why it feels like you are replying 'from quite a high chair': I guess you took much better classes than I did. ;)

Since it looks you have a pretty wide knowledge of math in contemporary physics, I wonder if you could help me clarifying other obscure points, that could actually deserve a new thread... Let us stick to this thread for now:

I would be really interested to see what you mean by a new calculus and how this would impact our current description of physical systems. I have read you can develope different calculus extension using these non-comm algebras but I do not have a clue about what it means.

I have included a point to point reply underneath but I am going to send you a more free-format one later.

The closest thing to physical meaning could be a phase in a wave or a polatization. But Again, it is just the real "arg" of the complex number... Hmm what about x+ip? of course, just a pair of real numbers, but it has sense. In any case, I agree it is intriging: how it comes that x^2+1=0 does not have physical meaning if all the components of the expresion (the square, the unit, the zero, the addition) have?

Ok at some point one might decide that the jump to complex number in QM is nothing worse than the jump from rational to real numbers (have you ever seen an instrument giving a real number as a measure?)
and keep going, but my feeling is still strongly in favor of complex as a math tool.
Let me add some more examples: you can treat a vector in the plane as a complex number and derive all kinematics (or theory of plane curves if you prefer) from one or from the other, but what a difference in the underlying physical insight. The imaginary unit is just a tool to represent an orthogonal projection to the real 1D space, to add a new dimension. The same happens in Minkowsky representation of Special Relativity: the addition of a temporal dimension can be performed by using an imaginary time component but a real representation introducing explicitly a non euclidean metric is much clearer as far as a geometric interpretation is concerned

Well. basically that it the pilot wave interpretation! It was formalized by Bohm, check for authors in http://prola.aps.org/ if you have access. Incidentally, I got a link to an old paper from De Broglie, "Sur les equations et les conceptions generales de la mecanique ondulatorie", in www.numdam.org (it is free!). To be precise, http://www.numdam.org/item?id=BSMF_1930__58__1_0

Thank you, hopefully I can still gain access to the aps website...

Hmm perhaps it is because I believe that a "real numbers" interpretation of quantum mechanics is just another geocentric view, not the heliocentric one. As discussing Eudoxus vs. Ticho Brahe. I agree there are probably a better interpretation, but my bet is to rethink the theory of calculus.

I agree that they can be just different views but, if using a Sun centered RF simplifies equations, still observations are performed in a geocentric RF. Eudoxus and Ticho were supporting non equivalent theories so I would not go there.

I believe not. It happens just in QM, and I can not see an infinite dimension involved directly. There are an infinite process, already in classical mechanics: the one of calculating the derivatives, for instance in F=m x''(t).

Well the dimension of the space you use to represent operators is infinite.

Indeed this is a key trick of the founding fathers.

Well it took me a while to see it myself

The online works of Heisenberg Born y Jordan are somehow dark. I have found more useful a final book, W. Heisenberg, {\it The Physical Principles of the Quantum Theory}, ed. Dover.

Thanks for the reference, it should be easy to find...

This is, or should be, standard classroom lore. I doubt which is the proper form to present it. Most tipically it is just remarked that our operators are build from functions in phase espace (X,P), thus time does not appear. But there are for sure other approaches more insightful... I have read them, just I do not remember now.

This casts some light but I will try to come up at a later time with some more details

 

[arivero]

Hi dg,

It seems I waited a bit for your "later time" :-) Meanwhile, let me cover a couple of points

I would be really interested to see what you mean by a new calculus and how this would impact our current description of physical systems. I have read you can develope different calculus extension using these non-comm algebras but I do not have a clue about what it means.

It comes back from Heisenberg and, esp., Dirac, who noticed that the commutator was "the most general Leibnitz rule". In quantum mechanics, if you want to partial derivate an operator against a veriable, you take the commutator with the canonical conjugate one,
this is,
ih @A/@p = [A,X]
ih @A/@t = [A,H]
...
and so on.

This has evolved to a complete calculus in noncommutative algebras, which are to be understood in the same way that algebras of functions. An algebra of functions is commutative (with usual product) and it happens to be dual to the space where it is defined over.

A complet calculus, imho, should also incorporate notions from renormalization theory. Particularly the introduction and removal of cutoffs.

Some progress in this way has been done in the field of perturbative renormalization, which by now is completely incorporated in the theory of noncommutative differential geometry.

I believe not. It happens just in QM, and I can not see an infinite dimension involved directly. There are an infinite process, already in classical mechanics: the one of calculating the derivatives, for instance in F=m x''(t).

Well the dimension of the space you use to represent operators is infinite.

Ok. My guess is that this infinite is related to the "process". It does not matter very much if we are into a variational method or just a calculation of tangents (ie Hamilton or Newton), the point is that both methods imply a infinite process of approaching to some limit. The infinite dimension of Hilbert space could be just the container needed to codify this process.

 

[dg]

Hi Alejandro,

Thanks for your reply, but for now I would like to steer your attention to another thread. For once someone has posted something that looks sensible under the Theory Development section. This guy (getting a PhD in math) is trying to see what happens by describing a system in terms of space variables and proper time only and he claims that he can obtain some "quantistic results" just by doing that.

I would be really interested in reading your opinion about it.

The thread is https://www.physicsforums.com/showthread.php?threadid=1050&goto=newpost

Before closing let me also ask if we can find an operator T such that

ih @A/@E=[A,T]

where E stands for energy and if not why.

Hope to hear from you soon, Dario

 

[arivero]

?m going to read the thread.

As for the time operator, the honest answer is that I am not the one to explain why not. It is alwayd told about in the books of QM, but then randomly some paper appears on the contrary. When I was student I somehow got a sort of naive argument, as it was not easy to conceive time as a function of position and momenta. But I am not sure about the point so I am not going to pollute the net with adhoc-isms or dogmas :)

 

[planetology]

Originally posted by dg


... but classical Schroedinger equation can be simply re-written as a system of two equations in the real field that have a much clearer PHYSICAL meaning: one is the continuity equation of probability, the other is a Hamilton-Jacobi equation containing an additional quantum potential energy.

...

Another reason for using complex field is that it allows to represent a general real matrix using an hermitian one (antisimmetric part becomes the imaginary part).

These are both really intriguing statements. Could you give references?

I also remember someone once showing me a transformation that converts Maxwell's equations into complex form, so that they appeared as one equation perfectly analogous to the Schroedinger equation. I think this might help with the kind of physical insight you refer to, as it relates to the way our mathematical techniques sometimes hide the underlying physical properties. But now that I understand the significance of that, I can't find a reference that discusses it. If anyone knows one, please pass it along.

JC

 

[dg]

I think for the first statement you can look into Messiah. I have a doc that summarizes all the passages if you want... actually I will go ahead and attach it (it will take some time before they filter it... and show it at the bottom of this reply)

For the second one I remember figuring it out by reading a book by Postnikov, Lectures in Geometry, II semester o something like that.

Thanks for the idea about Maxwell equation in Schroedinger form... if you find a reference I will be really interested!

 

[marcus]

fascinating story about Isaac Barrow
at Alejandro's website---audacious
and potent high-fantasy style. glad
you posted the link.

John Aubrey (fl. 1626-1697) included
Isaac Barrow (1630-1677) in his
collection of short biographies called
"Aubrey's Brief Lives".

Barrow taught Newton math at Cambridge
and resigned his math professorship in favor of
Newton in 1669, moving up the ladder
to become Master of Trinity in 1672.
Aubrey's two final paragraphs:

[[He was a strong man, but pale as the Candle he studied by.
His pill (an opiate, possibly Matthews his pil) which he was wont
to take in Turkey, which was won't to doe him good, but he
took it preposterously at Mr. Wilson's, the Saddlers, neer
Suffolk House, where he was won't to lye and where he dyed,
and 'twas the cause of his death.

As he laye unravelling in the agonie of death, the Standers-by
could hear him say softly, *I have seen the Glories of the world.*]]

Someone on this forum recently declared that Stephen Hawking should be ejected from the window of a tall building, for reasons which seemed adequate at the time, and I have to point out that Hawking occupies the Lucasian Chair at Cambridge and that Isaac Barrow was the first Lucasian, Newton the second, and so on down to Stephen Hawking. Also that Barrow said Glories of the *world* meaning the universe. And that Aubrey could write better, even in a hurry, than people do now, and used words like unravelling and preposterously.
So here is this story about Barrow by arivero at his website
http://dftuz.unizar.es/~rivero/research/0001033.pdf
In this story it says somewhere that one tries to understand nature out of a sense of honor, so you might want to read it.
It is wacko too and thus deserving of serious attention.

 

[damgo]

Another reason for using complex field is that it allows to represent a general real matrix using an hermitian one (antisimmetric part becomes the imaginary part).

This is a fairly basic theorem in linear algebra: for a real matrix M, let MA=1/2 (M+M_tp) and MB=1/2 (M-M_tp). Then the matrix MA+i*MB has Hermitian conjugate MA-i*MB_tp = MA-(i/2)*(M_tp-M) = MA+i*MB

So MA+i*MB is Hermitian. (tp=transpose)

A related neat fact is Stone's Theorem -- if you have any continious single-parameter group of unitary operators -- like the time evolution operators T(t) -- there is always a unique Hermition operator H with T(t)=exp(-iHt), ie you can get the Schrödinger equation by just considering states as rays in Hilbert space and making reasonable assumptions about the nature of the time development operator T(t).

IIRC you can find the proof in Reed&Simon among other places.

 

[arivero]

I am ashamed. Just I hope that the adjetive "wacko" refers to
the narrator in the text and not to myself :) Thanks in any
case.

Well, I'll add some comments in the math forum:

https://www.physicsforums.com/showthread.php?s=&threadid=1606

(to fight against thread pollution, you see).

 

[marcus]

Originally posted by arivero
I am ashamed. Just I hope that the adjetive "wacko" refers to
the narrator in the text and not to myself :) Thanks in any
case.

Well, I'll add some comments in the math forum:

https://www.physicsforums.com/showthread.php?s=&threadid=1606

(to fight against thread pollution, you see).

I totally do not understand your saying ashamed. No occasion for that! On the contrary it is a fine, interesting and orginal piece you wrote.

I should have used a smilie :) with "wacko". I meant it as an ironical compliment.

Approve of the move. stay on topic (hermitian) and avoid
thread pollution!

rgards