When Canonical Quantization Fails, Here is How to Fix It by John Rider Klauder

John Klauder

Please fire away!

John

Lawrence Crowell

I keep having trouble finding time for this. The one little point I can make is that it seem odd that one has (p,q| that acts on a state covector |ψ). I am not sure how to interpret (p,q|ψ). This little bit I am having a bit of difficulty understanding. (Note I use parentheses because carrot signs don't work in this format)

Cheers LC

John Klauder

I gather you are familiar with ((I have trouble with Psi) where say Q |x> = x |x> and int |x> that can be integrated or differentiated, etc. They have the property that the vectors |x> are mutually orthogonal in the form = delta(x - x') which is zero if x=/=x' yet integrate to 1. The important fact is they make a functional representation rather than an abstract vector like |u>..

But there can be other representations and that is what provides. The vectors |p,q> are continuously labeled which makes these representations only involve continuous functions unlike . They are NOT,mutually orthogonal, namely < p,q|p',q'> is often never zero and this inner product is normally bounded by one, but these vectors do lead to the important relation that int |p,q>

John Klauder

Somehow several symbols failed to service being posted. Terms like (x|u) and (p,q|u) (I now understand the () situation).

John Klauder

A new try to get my prior post clear

But there can be other representations and that is what (p,q|u) provides. The vectors |p,q) are continuously labeled which makes these representations only involve continuous functions unlike (x|u). They are NOT mutually orthogonal, namely (p,q|p',q') is often never zero and this inner product is normally bounded by one, but these vectors do lead to the important relation that int |p,q)(p,q| dp dq/2 pi = I the identity operator. In brief, (p,q|u) is just another representation of |u).

I hope this helps.

Lawrence Crowell

It is an odd notation. To say (p,q| is then a sort of "quantum and" that is a form of logical or. It is apparently a part of the argument for (в€‚П€/в€‚t)dt в†' . The total derivative

dП€ = [в€‚П€/в€‚t + (dq/dt)В·в€‡ П€]dt = (в€‚П€/в€‚t)dt + (i/Д§)pВ·dq П€.

If dП€/dt = 0 this gives the connection between the Lagrangian and the Schrodinger equation

iД§в€‚П€/в€‚t - HП€ в†' pВ·dq - H = L

The Lagrangian is in configuration variables, which is a way of casting physics according to one set of variables, or in the case of QM according to position variables and momentum expressed as operators.

So there seems to be something very subtle with the notation (p, q|. This appears to point to some subtle issue of quantum logic. I am still pondering this some for there seems to be something potentially subtle or deep going on here.

My next question will involve the Wheeler DeWitt equation.

Cheers LC

Lawrence Crowell

There is a problem with carrot signs, I meant to write: It is apparently a part of the argument for (в€‚П€/в€‚t)dt в†' pdq

John Klauder

I am unclear about your comments. Like (x| spans the Hilbert space means that if (x|u) = 0 for all x means that |u) = 0, there are other state families that do the same. (p,q| for all p and all q also span Hilbert space. This implies that if (p,q|u) = 0 for all p,q, then |u) = 0. An important property about Hilbert space representations is their inner product like (u|v) = int (u|x)(x|v) dx , where "int" means "integral". A very similar integral leads to (u|v) = int (u|p,q)(p,q|v) dp dq/2 pi h-bar. My previous relation like this one also had an h-bar, but units where h-bar = 1 were implicit. Thus these are two different functional representations, I.e., (x|v) and (p,q|v), that serve the same purpose.

The reduced action functional is given by

A = int (p(t),q(t)| [ i h-bar d/dt - £ ] |p(t),q(t)) dt = int { p(t) q-dot(t) - H(p(t),q(t)) } dt

where £ is the Hamiltonian operator. There is no need to introduce only position coordinates to build a traditional Lagrangian.

Georgina Woodward

Hi John, thank you for sharing your improvement, and for replies to comments that help explain why you have chosen to present it as an entry to the competition. As well as the function the improvement fulfills in improving the consistency the specific kinds of transformations (preventing the peculiar outcomes).

The question 'what is fundamental?' is very open. If founational is considered fundamental, and quantum physics to be the modelling of that foundation, then getting it to work consistently is fundamental to that specific area of physics attempting to model the fundamental foundation from which physics arises.

Section 1 were you talk about the desirability and problems of adding extra dimensions to phase space was interesting to me. Something for me to ponder. I wish I could understand more of the presentation but see why you have written it. Kind regards Georgina

John Klauder

The rules of canonical quantization fixed about 1930 are the foundation of quantization. One rule is postulated, namely the choice of Cartesian coordinates and momenta, is adopted as a necessary assumption. However, such phase space coordinates do not have a metric to choose Cartesian variables. The procedures under the heading of enhanced quantization provide a very different connection between classical and quantum variables and happily it identifies Cartesian variables to confirm their use in Quantization. The new connection permits arbitrary contact transformations which canonical quantization fails to achieve. An added bonus of the new approach not mentioned in my essay is the fact that certain models that fail to have successful canonical quantizations do have successful enhanced quantizations.

Georgina Woodward

Thank you for taking the time and trouble to reply. Georgina

Joe Fisher

Dear Professor John R. Klauder,

My research has concluded that Nature must have devised the only permanent real structure of the Universe obtainable for the real Universe existed for millions of years before man and his finite complex informational systems ever appeared on earth. The real physical Universe consists only of one single unified VISIBLE infinite surface occurring eternally in one single infinite dimension that am always illuminated mostly by finite non-surface light.

Joe Fisher, ORCID ID 0000-0003-3988-8687. Unaffiliated

John Klauder

Thanks for your views. I sympathize with the idea of a single universe that can survive the time and events there in. I would include what we can see and what we can not see. If I understand you correctly, I would have difficulty in admittting only a single dimension, however, because the naked eye as well as the many telescopes offer us a three dimensional view of objects within our Universe.

Joe Fisher

Dear Professor John R. Klauder,

Thank you for your courteous reply. Nobody has ever seen a ball. All observations and photographs of a ball will only ever show it to appear as a flat filled-in disk. That is why the sun, a full moon, and all of the planets always appear as flat filled in disks. Nobody has ever seen a cube. That is why photographs of the New York city skyline always appear to show squares, rectangles, and parallelograms. Nobody has ever seen a supposedly three-dimensional pyramid. Every photograph taken of the Pyramids at Giza only shows them to appear triangular.

There are no finite dimensions of length, width, depth and time. There only am one single infinite dimension.

Joe Fisher, Realist

[deleted]

Dear Professor John R. Klauder,

There are no finite dimensions of length, width, depth and time. There only am one single infinite dimension.

Joe Fisher, Realist

John Klauder

If we had only 2 dimensional pictures of a ball at a single moment of time I would support your view.

But for a ball or for a picture of you Joe we can feel the object and know it is really three dimensional. Photos at different times or even a movie are seen as 2dimensional but they may also show different photos as you, say, turn around. We accept that part of you is not visible after you turn around from a position where that part of you was visible. That is easy to understand if you are 3dimensional but not so clear if you accept only 2 dimensional views. Likewise we can see Jupiter and it's multiple moons as moving around that planet as photos and sometimes do not show some moons that were visible in other photos taken at different times. This sight led Galileo to suggest that the planets including the Earth rotate around the Sun.

I doubt that my words will change your views, but I am just citing additional data from which one can decide what 2 dimensional information observed over time is able to provide a reasonable interpretation of what may be seen.

Joe Fisher

Dear John Rider Klauder,

Please forget about finite dimensions. A picture of a ball, or a picture of me can be of infinite size. You can only touch the surface of a ball with the surface of your fingers, because I have concluded from my deep research that Nature must have devised the only permanent real structure of the Universe obtainable for the real Universe existed for millions of years before man and his finite complex informational systems ever appeared on earth. The real physical Universe consists only of one single unified VISIBLE infinite surface occurring eternally in one single infinite dimension that am always illuminated mostly by finite non-surface light.

Joe Fisher, ORCID ID 0000-0003-3988-8687. Unaffiliated

Edwin Klingman

Dear John Rider Klauder,

In response to a comment above, you note that "the current rule sometimes leads to nonsense." and elsewhere,

"The usual rule of the relation of a classical/quantum connection fails sometimes."

I confess not to have followed your complete argument, however it seems to derive from Dirac's interpretation of all contact transformations in which the classical transformation has a role to play in the quantum theory.

The classical variable is either random or varies with respect to something. If it varies with respect to something, say z, then we typically form an 'operator' of the form O ~ d/dz, and the formalism that results (incorporating h) is a quantum theory. It is difficult to see how the quantum operator could possibly be more fundamental than the classical variable from which it is derived, and, in fact, the correspondence principle insists that the quantum operator equations must be derived from the appropriate classical Hamiltonian. This seems compatible with your statement that

"The Hamiltonian operator is the same function of P and Q as the classical Hamiltonian is of p and q ... [plus o(h)]..

I will not repeat your logic but will say that I agree with your conclusion that

"The special role played by Cartesian coordinates in canonical quantization is essential."

Your goal, as I understand it, is "an alternative quantization procedure relating classical and quantum variables to each other", and your derivation of equation (5) seems to yield the correct classical action function without the need to modify the quantum operators at all.

I did not follow your switch to affine variables, but, to return to your point that "the current rule sometimes leads to nonsense", I would note that Steven Kauffmann (viXra:1707.0116) has shown that the Dirac equation was not derived from a classical Hamiltonian, but was instead influenced by 'space-time symmetry' considerations, and, accordingly, leads to some nonsense results, such as particle speed ~1.7c. Additionally, the result of Dirac's equation is a spin that differs from Pauli's eigenvalue spin; in fact Dirac does not have a spin eigenvalue equation, and such is obtained only after a Foldy-Wouthuysen transformation smears a spin over an extended region (viXra:1411.0096 ).

If you're wondering where this extended comment is leading, it is the fact that Dirac based his equation on space-time symmetry considerations and my current essay treats the development of space-time symmetry concept. I hope you will read my essay and comment on it.

Thanks for a very interesting essay,

My best regards

Edwin Eugene Klingman

John Klauder

The lack of all canonical transformations finding a place within conventional canonical quantization is a minor failure suitable for such an essay. But the real failure, which is not discussed in my essay, is illustrated by the following property. Namely, in canonical quantization the classical Hamiltonian H(p,q) is promoted to a quantum Hamiltonian £(P,Q) which, in the classical limit (h =0) recovers the original classical Hamiltonian H(p,q). I now offer an example in which this criterion fails. Let the classical Hamiltonian be given by (using p= p1, p2, p3,..., q=q1,q2, q3,..., p^2=p1^2+p^2+p^3 + ...,

and q^2=q1^2+q^2+q^2+..., where the sums run to infinity). We choose

H(p,q) = a p^2+ b q^2 + c(q^2)^2

where a, b and c are positive real constants. The quantum Hamiltonian for such a model is given by

£(P,Q) = a P^2 + b Q^2 + c(Q^2)^2 + ...

Where Q and P represent a string of conventional canonical quantum operators. Now we ask what is the classical limit of this proposed quantum Hamiltonian and the answer has the form

H(p,q) = a p^2. + b' q^2

NAMELY THE QUARTIC INTERACTION TERM HAS VANISHED ANd A NEW COEFFICIENT b'

HAS BEEN INTRODUCED. This is what I mean that conventional canonical quantization 9has

FAILED. There are many examples that fail in this manner. I also add the fact that our new procedures in the essay do NOT fail and lead to a satisfactory solution for such problems. I only illustrated the problems with canonical transformations because proving the failure as illustrated here was too complex for an essay.

John Klauder

Some errors in the equations of the preceding comment. I put the correct equations below

p^2= p1^2+p2^2+p3^2 ...

q^2=q1^2+q2^2+q3^2...

« Previous Page Next Page »