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October 23, 2019

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FQXi Essay Contest - Spring, 2017
[back]

TOPIC: When Canonical Quantization Fails, Here is How to Fix It by John Rider Klauder [refresh]

TOPIC: When Canonical Quantization Fails, Here is How to Fix It by John Rider Klauder [refresh]

Following Dirac [1], the rules of canonical quantization include classical and quantum contact transformations of classical and quan- tum phase space variables. While arbitrary classical canonical coor- dinate transformations exist that is not the case for some analogous quantum canonical coordinate transformations. This failure is due to the rigid connection of quantum variables arising by promoting the corresponding classical variable from a c-number to a q-number. A different relationship of c-numbers and q-numbers in the procedures of Enhanced Quantization [2] shows the compatibility of all quantum operators with all classical canonical coordinate transformations.

Employed as Member of Technical Staff, AT&T Bell Laboratories, for 35 years, and Professor, University of Florida, for 22 years. Head, Theoretical Physics Research Department, 1966-1967 and 1969-1971. Head, Solid State Spectroscopy Research Department, 1971-1976. National Science Foundation Physics Advisory Panel, 1972-1975. Consultant, Los Alamos National Laboratory, Theoretical Division, 1978-1989. Editor, Journal of Mathematical Physics, 1979-1985. Associate Secretary-General, Executive Council of International Union of Pure and Applied Physics, 1985-1990. Member Executive Committee of the International Association of Mathematical Physics, 1985-1991. President, International Association of Mathematical Physics, 1988-1991.

Hi John, Your paper is an exercise in pure math and to be very straight forward, I am wondering what is being presented as "fundamental" here? Can you help us out?

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The rules of quantization are fundamental. The usual rule of the relation of a classical/quantum connection fails sometimes. It’s replacement with a new rule performs much better. That is a fundamental change.

In this fundamental change in mathematical representation only or has the physical model changed or become more specific as to how the math comes out of a physical model? Or has the physical model remained unchanged with only the mathematical representation and notation changed? If only the mathematical notation changed, do you think that this math will bring about the answers bewildering physics such as why matter over anti-matter or dark energy, etc...

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John

Does this generalised canonical quantization procedure still guarantee the unitary property? That was the main strength of canonical quantization over the more manifestly covarient path-integral formulation; is this preserved?

Anton

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Does this generalised canonical quantization procedure still guarantee the unitary property? That was the main strength of canonical quantization over the more manifestly covarient path-integral formulation; is this preserved?

Anton

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Anton,

Then new rule preserves all unitary properties of the old way. What is different,is the choice of the Hamiltonian operator in relation to the classical Hamiltonian, when needed. This may effect properties of the path integral formulation. In short, that basic principles are preserved, just the choice of Hamiltonian operators may change.

John

Then new rule preserves all unitary properties of the old way. What is different,is the choice of the Hamiltonian operator in relation to the classical Hamiltonian, when needed. This may effect properties of the path integral formulation. In short, that basic principles are preserved, just the choice of Hamiltonian operators may change.

John

Scott

The change I propose is to the physical rule relating a classical and quantum system and not simply a change of the mathematical formalism. This change preserves the classical/quantum connection that works for some systems but provides a different result when the usual classical/quantum fails, e.g., when the classical system has a nonlinear interaction but it’s normal quantization has no nonlinear action. Such failures of the rule of quantization are unwelcome and the new rule provides an acceptable solution.

I do not expect that the change I propose will directly help in understanding the inequality you mention, but it could be part of the answer.

John

The change I propose is to the physical rule relating a classical and quantum system and not simply a change of the mathematical formalism. This change preserves the classical/quantum connection that works for some systems but provides a different result when the usual classical/quantum fails, e.g., when the classical system has a nonlinear interaction but it’s normal quantization has no nonlinear action. Such failures of the rule of quantization are unwelcome and the new rule provides an acceptable solution.

I do not expect that the change I propose will directly help in understanding the inequality you mention, but it could be part of the answer.

John

Professor Klauder,

you have succeeded in getting me to follow that! I think I see your point. From the classical perspective, we might have an analytical observed momentum that falls one side or the other of a Planck value position on the real line, which might normalize (be made orthogonal) within a probable h-bar limit in real time. But in some cases, the convergence of two classical continuous functions will result in a real time window which would exceed that h-bar limit, and would nor renormalize. This of course is an analytical result, but must still be made true to the Planck value being the averaged least observable Action. And thus the metric for the Principle of Least Action being h-bar/2 in one dimension.

Did I get that correctly? Thank-you jrc

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you have succeeded in getting me to follow that! I think I see your point. From the classical perspective, we might have an analytical observed momentum that falls one side or the other of a Planck value position on the real line, which might normalize (be made orthogonal) within a probable h-bar limit in real time. But in some cases, the convergence of two classical continuous functions will result in a real time window which would exceed that h-bar limit, and would nor renormalize. This of course is an analytical result, but must still be made true to the Planck value being the averaged least observable Action. And thus the metric for the Principle of Least Action being h-bar/2 in one dimension.

Did I get that correctly? Thank-you jrc

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John

What you say has some merit but it is not connected with my essay. Let me say in simple terms the basic point I wish to make. A simple classical system involves two variables a position called q and a momentum p (normally a mass m times a velocity v, so p=m v), These variables are finite real numbers with no limitations. The Hamiltonian function H represents the energy of a system and is given as a function of p and q, e.g., H= p^2 +q^4, where p^2 = p squared, etc. One can change variables by a coordinate transformation, e,g, in 2 dimensions changing Cartesian coordinates x and y into polar variables r and u (u=the Greek letter theta) by letting x=r cos(u) and y= r sin(u), etc. A similar change can be made for p and q, e.g. b=p/q^2 and c=q^3/3, That leads to H having a different form, e.g., for our example H=b^2q^4+(3c)^{4/3}. Both of these forms are correct even though they are different. By choosing the right values for b and c the Hamiltonian H has the same numerical value as it did when we used p and q. Such equalities do NOT hold for quantum operators

such as P, Q, B, and C related to p, q, b, and c. One needs a rule to choose the right classical coordinates to promote to quantum operators. The present rule is what I want to change, i.e., what is the right quantum Hamiltonian operator for a given classical Hamiltonian. The current rule sometimes leads to nonsense. That is corrected with new rule. Briefly stated, my essay offers a new and much better way to.assign a Hamiltonian operator to a classical Hamiltonian.

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What you say has some merit but it is not connected with my essay. Let me say in simple terms the basic point I wish to make. A simple classical system involves two variables a position called q and a momentum p (normally a mass m times a velocity v, so p=m v), These variables are finite real numbers with no limitations. The Hamiltonian function H represents the energy of a system and is given as a function of p and q, e.g., H= p^2 +q^4, where p^2 = p squared, etc. One can change variables by a coordinate transformation, e,g, in 2 dimensions changing Cartesian coordinates x and y into polar variables r and u (u=the Greek letter theta) by letting x=r cos(u) and y= r sin(u), etc. A similar change can be made for p and q, e.g. b=p/q^2 and c=q^3/3, That leads to H having a different form, e.g., for our example H=b^2q^4+(3c)^{4/3}. Both of these forms are correct even though they are different. By choosing the right values for b and c the Hamiltonian H has the same numerical value as it did when we used p and q. Such equalities do NOT hold for quantum operators

such as P, Q, B, and C related to p, q, b, and c. One needs a rule to choose the right classical coordinates to promote to quantum operators. The present rule is what I want to change, i.e., what is the right quantum Hamiltonian operator for a given classical Hamiltonian. The current rule sometimes leads to nonsense. That is corrected with new rule. Briefly stated, my essay offers a new and much better way to.assign a Hamiltonian operator to a classical Hamiltonian.

Professor Klauder,

Thank-you very much for your time in reply, it's Christmas and I consider it a tribute that you would trouble to teach. That is truly a giving spirit.

Choosing a Hamiltonian might sound strange, as it would seem arbitrary. But not to my mind. Any physical experiment ultimately resolves to our limitation of being able to only know how a detector reacts, and we can...

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Thank-you very much for your time in reply, it's Christmas and I consider it a tribute that you would trouble to teach. That is truly a giving spirit.

Choosing a Hamiltonian might sound strange, as it would seem arbitrary. But not to my mind. Any physical experiment ultimately resolves to our limitation of being able to only know how a detector reacts, and we can...

view entire post

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I will comment more on your paper later. It is in many ways quite interesting. At this point I would tend to think that your quantization is based on conformal sympletic transformations.

Cheers LC

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Cheers LC

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LC

Please fire away!

John

Please fire away!

John

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

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Cheers LC

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LC

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>

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>

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

JK

JK

A new try to get my prior post clear

LC

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

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.

JK

LC

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

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.

JK

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 (∂ψ/∂t)dt → . The total derivative

dψ = [∂ψ/∂t + (dq/dt)·∇ ψ]dt = (∂ψ/∂t)dt + (i/ħ)p·dq ψ.

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

iħ∂ψ/∂t - Hψ → p·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

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dψ = [∂ψ/∂t + (dq/dt)·∇ ψ]dt = (∂ψ/∂t)dt + (i/ħ)p·dq ψ.

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

iħ∂ψ/∂t - Hψ → p·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

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There is a problem with carrot signs, I meant to write: It is apparently a part of the argument for (∂ψ/∂t)dt → pdq

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LC

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.

JK

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.

JK

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

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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

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GW

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.

JK

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.

JK

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

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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

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JF

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.

JK

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.

JK

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

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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

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JF

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.

JK

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.

JK

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

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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

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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...

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In response to a comment above, you note that "

"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...

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EK

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.

JK

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.

JK

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…

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

q^2=q1^2+q2^2+q3^2…

It's good to see you here John...

My experience, hearing your lecture at FFP15, was uplifting and enlightening. I felt like the top of my head popped open, at one point, and the heavens opened up so that new knowledge could pour in. I've had that kind of experience only from the best of the world's best scholars before, so I put you in the same category.

I would add that a visual metaphor was helpful for me to understand this work. An infinite-dimensional Hilbert space provides plenty of room for the Quantum action, and the Classical action can be seen as interactions on a specific restricted subset - a bounded 2-d surface. However; if we vary the characteristics of the restricted target space, we may derive an enhanced quantization rule that lets us obtain results even where that would not otherwise be possible.

I see this methodology as having great potential for studying the boundary between quantum and classical, automatic reduction vs. no-collapse models, and so on. This might even provide insights for the material covered in my essay (yet to post), which talks about thermodynamic models of gravitation, because I also deal with issues on the quantum-classical boundary.

All the Best,

Jonathan

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My experience, hearing your lecture at FFP15, was uplifting and enlightening. I felt like the top of my head popped open, at one point, and the heavens opened up so that new knowledge could pour in. I've had that kind of experience only from the best of the world's best scholars before, so I put you in the same category.

I would add that a visual metaphor was helpful for me to understand this work. An infinite-dimensional Hilbert space provides plenty of room for the Quantum action, and the Classical action can be seen as interactions on a specific restricted subset - a bounded 2-d surface. However; if we vary the characteristics of the restricted target space, we may derive an enhanced quantization rule that lets us obtain results even where that would not otherwise be possible.

I see this methodology as having great potential for studying the boundary between quantum and classical, automatic reduction vs. no-collapse models, and so on. This might even provide insights for the material covered in my essay (yet to post), which talks about thermodynamic models of gravitation, because I also deal with issues on the quantum-classical boundary.

All the Best,

Jonathan

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JD

I support your view and applaude potential expansion of enhanced quantization.Thanks for your comments.

JK

I support your view and applaude potential expansion of enhanced quantization.Thanks for your comments.

JK

Then let me extemporize...

The reason I think your work represents an advancement in fundamental Physics is that it provides the missing piece to bring the search full-circle, and so could be a kind of holy grail for QM researchers. My thinking is that it is ill-advised to try to reduce the descriptive vocabulary for QM phenomena too far, because the core of what the subject is about resides in nature's optiony.

I had the privilege to correspond with Philip Pearle, when he first introduced statevector reduction theory, and I even have a typewritten manuscript of his 'gambler's ruin' paper. But I later became an advocate for no-collapse models, having studied and identifying with decoherence theory in its pure form, and had extended correspondence with H.D. Zeh and briefly with Joos.

I have more recently come to take seriously the idea that QM is more like a compass with a range of options of different formulations, depending on the kinds of information you have and are hoping to extract. This makes both automatic collapse as in CSL and no-collapse as in decoherence theory relevant to a full understanding of QM. All points on the compass are real, and only the pointer reveals the truth.

In this context; your enhanced quantization methodology is the missing link.

All the Best,

Jonathan

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The reason I think your work represents an advancement in fundamental Physics is that it provides the missing piece to bring the search full-circle, and so could be a kind of holy grail for QM researchers. My thinking is that it is ill-advised to try to reduce the descriptive vocabulary for QM phenomena too far, because the core of what the subject is about resides in nature's optiony.

I had the privilege to correspond with Philip Pearle, when he first introduced statevector reduction theory, and I even have a typewritten manuscript of his 'gambler's ruin' paper. But I later became an advocate for no-collapse models, having studied and identifying with decoherence theory in its pure form, and had extended correspondence with H.D. Zeh and briefly with Joos.

I have more recently come to take seriously the idea that QM is more like a compass with a range of options of different formulations, depending on the kinds of information you have and are hoping to extract. This makes both automatic collapse as in CSL and no-collapse as in decoherence theory relevant to a full understanding of QM. All points on the compass are real, and only the pointer reveals the truth.

In this context; your enhanced quantization methodology is the missing link.

All the Best,

Jonathan

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Hi John Rider Klauder

You said “This failure is due to the rigid connection of quantum variables arising by promoting the corresponding classical variable from a c-number to a q-number….When Canonical Quantization Fails, Here is How to Fix It ….wonderful solutions, you have very vast experience also sir John Rider Klauder…….wow……..….. very nice idea…. I highly appreciate...

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You said “This failure is due to the rigid connection of quantum variables arising by promoting the corresponding classical variable from a c-number to a q-number….When Canonical Quantization Fails, Here is How to Fix It ….wonderful solutions, you have very vast experience also sir John Rider Klauder…….wow……..….. very nice idea…. I highly appreciate...

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SNPG

You cite many conditions that you would like to impose on your analysis. I wish you good luck in finding one or more solutions that fit your constraints.

I will look at your essay.

JK

You cite many conditions that you would like to impose on your analysis. I wish you good luck in finding one or more solutions that fit your constraints.

I will look at your essay.

JK

Dear John Rider Klauder ,

Thank you very much for nice observation, that is one of the Dynamic Universe Model predictions came true

There were some recent experiments that showed velocities higher than that of light are achieved… in Europe as I remember. You can check in Wikipedia. Please see my paper on Velocities more than light for a more elaborate discussion ….

https://vaksdynamicuniversemodel.blogspot.com/

Waiting for more questions….

Best Regards

=snp

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Thank you very much for nice observation, that is one of the Dynamic Universe Model predictions came true

There were some recent experiments that showed velocities higher than that of light are achieved… in Europe as I remember. You can check in Wikipedia. Please see my paper on Velocities more than light for a more elaborate discussion ….

https://vaksdynamicuniversemodel.blogspot.com/

Waiting for more questions….

Best Regards

=snp

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JD

I do not recognize CSL. Please explain. I like your compass picture, but I imagine that different directions reveal alternative formulations that all lead to the same conclusion. I assume your compass includes the Copenhagen view. I find that Copenhagen views can correctly deal with many problems but not all problems that arise in quantizing a given classical model. In a.response above I gave a classical interacting model that becomes non-interacting when quantized. This same model finds a good interacting quantum result using the enhanced quantization rules. I like to say that

CQ (canonical quantization) is contained in EQ (enhanced quantization) I agree that EQ offers a kind of closure to CQ but surprisingly EQ can provide solutions that CQ cannot. One of these examples deals with covariant scalar fields with a quartic interaction in five and more space time dimensions for which CQ leads to a trivial (= free) solution while EQ leads to a satisfactory non-trivial solution.

JK

I do not recognize CSL. Please explain. I like your compass picture, but I imagine that different directions reveal alternative formulations that all lead to the same conclusion. I assume your compass includes the Copenhagen view. I find that Copenhagen views can correctly deal with many problems but not all problems that arise in quantizing a given classical model. In a.response above I gave a classical interacting model that becomes non-interacting when quantized. This same model finds a good interacting quantum result using the enhanced quantization rules. I like to say that

CQ (canonical quantization) is contained in EQ (enhanced quantization) I agree that EQ offers a kind of closure to CQ but surprisingly EQ can provide solutions that CQ cannot. One of these examples deals with covariant scalar fields with a quartic interaction in five and more space time dimensions for which CQ leads to a trivial (= free) solution while EQ leads to a satisfactory non-trivial solution.

JK

JD

Let me propose a different story. For canonical quantization there is a building with a round central open space on the ground floor which has many doors that open to different formulations of CQ. However, we now can see there also is a first floor above with many doors that lead to different formulations of EQ.

JK

Let me propose a different story. For canonical quantization there is a building with a round central open space on the ground floor which has many doors that open to different formulations of CQ. However, we now can see there also is a first floor above with many doors that lead to different formulations of EQ.

JK

I am happy to be having this conversation...

I like your alternative ending to the CQ story. CSL stands for Continuous Spontaneous Localization, and it is a logical endpoint of statevector reduction theory, by combining automatic reduction with gravity (i.e. - reduction by gravitation). I do agree that Copenhagen deserves a place on the compass, with the antipode being the notion that the fundamentals of QM and how a result is arrived at having equal relevance to 'shut up and calculate.'

Beyond this; the idea of reduction being induced by gravitation may be seen to imply entropy or an entropic force associated with gravity. I presented a poster at FFP10 championing the idea of a common basis for thermodynamic entropy and non-local effects in QM, extending the ideas of spreading and sharing of energy as an entropy metaphor to include information.

I based these assumptions largely on what I was learning about decoherence theory, but it turned out the pure formulation was ill-equipped to deal with dissipative phenomena or processes. On the other hand; I later learned Sandu Popescu was formulating a very similar picture in the same timeframe (2009) so I was actually more on-target than I imagined at the time.

More later,

Jonathan

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I like your alternative ending to the CQ story. CSL stands for Continuous Spontaneous Localization, and it is a logical endpoint of statevector reduction theory, by combining automatic reduction with gravity (i.e. - reduction by gravitation). I do agree that Copenhagen deserves a place on the compass, with the antipode being the notion that the fundamentals of QM and how a result is arrived at having equal relevance to 'shut up and calculate.'

Beyond this; the idea of reduction being induced by gravitation may be seen to imply entropy or an entropic force associated with gravity. I presented a poster at FFP10 championing the idea of a common basis for thermodynamic entropy and non-local effects in QM, extending the ideas of spreading and sharing of energy as an entropy metaphor to include information.

I based these assumptions largely on what I was learning about decoherence theory, but it turned out the pure formulation was ill-equipped to deal with dissipative phenomena or processes. On the other hand; I later learned Sandu Popescu was formulating a very similar picture in the same timeframe (2009) so I was actually more on-target than I imagined at the time.

More later,

Jonathan

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Dear John,

Thank you for this essay, I was pleased to read it! I didn't know about enhanced quantization so far, but it seems to me that it fixes some problems already, and that even more significant results will probably follow. It seems more natural than the canonical quantization, and much better connected with the classical systems that are quantized. Maybe some of the developments will shed more light not only to QFT, but also to the problem of emergence of a classical world from the quantum one, and to the problems related to quantum gravity. I will follow with interest your work.

Best wishes,

Cristi

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Thank you for this essay, I was pleased to read it! I didn't know about enhanced quantization so far, but it seems to me that it fixes some problems already, and that even more significant results will probably follow. It seems more natural than the canonical quantization, and much better connected with the classical systems that are quantized. Maybe some of the developments will shed more light not only to QFT, but also to the problem of emergence of a classical world from the quantum one, and to the problems related to quantum gravity. I will follow with interest your work.

Best wishes,

Cristi

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CS

Thanks for the nice words, and they came on my birthday as well! Yes, I believe that Enhanced Quantization is a novel way to connect quantum and classical systems and removes some fuzzy parts of canonical quantization. There is a book entitled Enhanced Quantization published by World Scientific in 2015 that covers many of its benefits.

JK

Thanks for the nice words, and they came on my birthday as well! Yes, I believe that Enhanced Quantization is a novel way to connect quantum and classical systems and removes some fuzzy parts of canonical quantization. There is a book entitled Enhanced Quantization published by World Scientific in 2015 that covers many of its benefits.

JK

Dear John,

You Essay is impressive. I came to it based on a suggestion of a friend of mine. I did not know your approach of Enhanced Quantization, but it seems consistent. Your Essay is well written and seems mathematically correct. Maybe I will try to use it in a quantum gravity framework in the future.

In any case, it has been a pleasant reading for me. You deserves the highest score.

Maybe an expert of quantum theory like you could be interested in my Essay, where I discuss a Bohr-like approach to the “gravitational atom” with... Albert Einstein!

Congrats again and good luck in the Contest.

Cheers, Ch.

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You Essay is impressive. I came to it based on a suggestion of a friend of mine. I did not know your approach of Enhanced Quantization, but it seems consistent. Your Essay is well written and seems mathematically correct. Maybe I will try to use it in a quantum gravity framework in the future.

In any case, it has been a pleasant reading for me. You deserves the highest score.

Maybe an expert of quantum theory like you could be interested in my Essay, where I discuss a Bohr-like approach to the “gravitational atom” with... Albert Einstein!

Congrats again and good luck in the Contest.

Cheers, Ch.

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I must commend you John,

You have given us a powerful tool and insights that will fuel the progress of fundamental Physics. So I must give you high marks on both the idea presented and its clarity of presentation. I admire that you are able to incorporate mathematical expressions into your sentences so seamlessly, and that every time you leave a reader with questions about what the Math means you follow up in the next sentence or two with an explanation of the how or why. It gets pretty Math-intensive in places, but the clarity is nearly transparent.

I could not give you full credit for weaving your subject into the theme of the contest - in terms of showing why this new approach advances fundamental Physics in your essay. But after seeing your talk at FFP15, that piece is obvious. I'm glad to see Cristi Stoica concurs that you not only expand your core topic, but give us insights "also to the problem of emergence of a classical world from the quantum one, and to the problems related to quantum gravity." I am so happy you clearly explain why "the special role played by Cartesian coordinates in canonical quantization is essential," which I had not understood before Orihuela, and how varying the target surface leads to enhanced quantization.

I am of the opinion that yours is the correct methodology, but I have doubts about the approach taken by Carroll and Singh - which lacks a well-defined target space entirely. If you have read their essay; I'd like to know your opinion.

All the Best,

Jonathan

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You have given us a powerful tool and insights that will fuel the progress of fundamental Physics. So I must give you high marks on both the idea presented and its clarity of presentation. I admire that you are able to incorporate mathematical expressions into your sentences so seamlessly, and that every time you leave a reader with questions about what the Math means you follow up in the next sentence or two with an explanation of the how or why. It gets pretty Math-intensive in places, but the clarity is nearly transparent.

I could not give you full credit for weaving your subject into the theme of the contest - in terms of showing why this new approach advances fundamental Physics in your essay. But after seeing your talk at FFP15, that piece is obvious. I'm glad to see Cristi Stoica concurs that you not only expand your core topic, but give us insights "also to the problem of emergence of a classical world from the quantum one, and to the problems related to quantum gravity." I am so happy you clearly explain why "the special role played by Cartesian coordinates in canonical quantization is essential," which I had not understood before Orihuela, and how varying the target surface leads to enhanced quantization.

I am of the opinion that yours is the correct methodology, but I have doubts about the approach taken by Carroll and Singh - which lacks a well-defined target space entirely. If you have read their essay; I'd like to know your opinion.

All the Best,

Jonathan

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JK

That looks quite brilliant, but the details were a little beyond me as I'm no mathematician (though scored top on Wigner with red/green lined socks in 2015!)

I hope you can help. It looks to me as if you have an algorithmic solution to non-linear classical 'quantum' interactions. That's of great interest as my essay describes a full ontology (and scaled up experiment) achieving...

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That looks quite brilliant, but the details were a little beyond me as I'm no mathematician (though scored top on Wigner with red/green lined socks in 2015!)

I hope you can help. It looks to me as if you have an algorithmic solution to non-linear classical 'quantum' interactions. That's of great interest as my essay describes a full ontology (and scaled up experiment) achieving...

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John,

Thanks for your nice comments on mine. As my conclusions were a logical consequence of the classical reproduction of the Cos[su]2 curve can you identify what you thought was missing from the mechanism, or logically 'wrong' in the conclusions?

The finding is very important if correct, though I know varies a little from your prior views, but I suspect I may not have described the ontological sequence in a way to allow it to be kept all in mind at once.

If not I need to identify any error you saw.

Many thanks

Peter

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Thanks for your nice comments on mine. As my conclusions were a logical consequence of the classical reproduction of the Cos[su]2 curve can you identify what you thought was missing from the mechanism, or logically 'wrong' in the conclusions?

The finding is very important if correct, though I know varies a little from your prior views, but I suspect I may not have described the ontological sequence in a way to allow it to be kept all in mind at once.

If not I need to identify any error you saw.

Many thanks

Peter

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John,.. .. (Thanks for your reply red the Dirac Eq. this copied from mine);

I really appreciate your reply. I agree, in fact more than*'a stretch'!* such a *"new way of seeing things will involve an ***imaginative leap that will astonish us**. In any case it seems that the quantum mechanical description will be superseded." JB p.27.

It followed from Majorana, (e it's...

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I really appreciate your reply. I agree, in fact more than

It followed from Majorana, (e it's...

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John, ..& consistent with optical science, i.e. this, just passed to me by Marianne M.

B Jack et al. Phys. Rev. A 81, 043844 30 April 2010

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B Jack et al. Phys. Rev. A 81, 043844 30 April 2010

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John

WOW! & check out the "Poincare sphere; --..complex superposition of two orthogonal polarisation states"

It's been there all along and ignored!

https://books.google.co.uk/books?isbn=1107006341 /link]

Peter

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WOW! & check out the "Poincare sphere; --..complex superposition of two orthogonal polarisation states"

It's been there all along and ignored!

https://books.google.co.uk/books?isbn=1107006341 /link]

Peter

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