# B Quantum Theory and its Precision

#### Jimmy87

Hi,

I was watching a really interesting old video from Richard Feynman who was talking about how precise/accurate the predictions of quantum theory agree with observation. The number of decimal places it agrees he said is astounding. He likened it to measuring the circumference of the Earth to within the uncertainty of the width of a human hair. Could someone give me a few examples (or point me to some sources) of some common predictions of quantum theory and the subsequent accuracy of the observations?

Thanks.

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#### Stavros Kiri

It's particularly QED that is astonishingly precise. The term
quantum theory
("Quantum Theory") is a broad term. E.g. there is the "Old Quantum Theory", then Quantum Mechanics, Relativistic Quantum Mechanics, and eventually Quantum Field Theory (QFT ). Lots of times there are many confusions about those etc.

QED is really part of Quantum Field Theory, and part of the Standard Model.
[R. P. Feynman was one of the founders of QED, who also pointed out that accuracy etc.]
Thus perhaps you can specify more or limit down your request, to avoid confusions.
Could someone give me a few examples (or point me to some sources) of some common predictions of quantum theory and the subsequent accuracy of the observations?
(E.g. by no means QFT or QED is Quantum Mechanics [e.g. '2nd quantization' vs '1st quantization' etc. ...])

Within Quantum Theory (broadly), Relativistic Quantum Mechanics and the Dirac Equation is pretty accurate too.

#### Jimmy87

It's particularly QED that is astonishingly precise. The term

("Quantum Theory") is a broad term. E.g. there is the "Old Quantum Theory", then Quantum Mechanics, Relativistic Quantum Mechanics, and eventually Quantum Field Theory (QFT ). Lots of times there are many confusions about those etc.

QED is really part of Quantum Field Theory, and part of the Standard Model.
[R. P. Feynman was one of the founders of QED, who also pointed out that accuracy etc.]
Thus perhaps you can specify more or limit down your request, to avoid confusions.

(E.g. by no means QFT or QED is Quantum Mechanics [e.g. '2nd quantization' vs '1st quantization' etc. ...])

Within Quantum Theory (broadly), Relativistic Quantum Mechanics and the Dirac Equation is pretty accurate too.
Thanks. So this precise measurement of:

g/2 = 1.001 159 652 180 85 (76)

Is what they actually measured. What about what the theory predicted? Would it be the same? What I am trying to say is - if you did no experiment and just used calculations of QED would you get:

g/2 = 1.001 159 652 180 85 (76)

Then you actually measure it and get:

g/2 = 1.001 159 652 180 85 (76)

#### DrClaude

Mentor
Thanks. So this precise measurement of:

g/2 = 1.001 159 652 180 85 (76)

Is what they actually measured. What about what the theory predicted?
This is a bit complicated, as (to the best of my knowledge), there is no calculation that just spews out $g$ based on a purely ab initio calculation. Citing from the article where that measured value comes from [B. Odom et al., New Measurement of the Electron Magnetic Moment Using a One-Electron Quantum Cyclotron, Phys. Rev. Lett. 97, 030801 (2006)]
The most stringent test of QED (one of the most demanding comparisons of any calculation and experiment) continues to come from comparing measured and calculated g, the latter using an independently measured α as an input. The new g, compared to Eq. (6) with α(Cs) or α(Rb), gives a difference |δg/2| < 15 × 10-12.

#### Jimmy87

This is a bit complicated, as (to the best of my knowledge), there is no calculation that just spews out $g$ based on a purely ab initio calculation. Citing from the article where that measured value comes from [B. Odom et al., New Measurement of the Electron Magnetic Moment Using a One-Electron Quantum Cyclotron, Phys. Rev. Lett. 97, 030801 (2006)]
Thanks. Could someone give me an example of theory/equation vs measurement? Feynman was very clear that QM makes concrete predictions from its equations that agree with observation to an insane amount of precision. He said "You make a prediction, then do an experiment to confirm the prediction and the two agree almost perfectly". From what I see the wiki article is really interesting and informative but seems to demonstrate the ability of the equipment to make such precise measurements rather than confirming a prediction.

#### Vanadium 50

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[Moderator's note: Edited to remove reference to deleted post.]

The link DrClaude provided explains the measurement and how it was done, and Ref. [10] in that article is a summary of the theoretical calculations.

Last edited by a moderator:

#### bhobba

Mentor
For a specific example do some research on the Lamb Shift. It has a very interesting history. Modern measurements show astounding accuracy of experiment compared to theory:
https://journals.aps.org/pra/abstract/10.1103/PhysRevA.63.012505

The fine structure constant (as mentioned in another post) is also an excellent high precision test of QED theory vs measurement, but suffers from a complication associated with whats called the renormalisation group flow - it has different values depending on the energy scale used to probe it. That too has been confirmed experimentally. It is part of renormalization that Feynman called a dippy process, but Feynman was not the greatest at keeping up with the latest literature - despite doing some work on the eventual solution himself with Gell-Mann ie renormalisation group flow. His colleague Ken Wilson (I think he was at Cal-tech about that time) sorted it all out and there is actually nothing dippy about it:
http://math.ucr.edu/home/baez/renormalization.html

Thanks
Bill

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#### vanhees71

Science Advisor
Gold Member
It's particularly QED that is astonishingly precise. The term

("Quantum Theory") is a broad term. E.g. there is the "Old Quantum Theory", then Quantum Mechanics, Relativistic Quantum Mechanics, and eventually Quantum Field Theory (QFT ). Lots of times there are many confusions about those etc.
I'd however qualify that "Old Quantum Theory" is not really there anyore, and that's good. You can learn about it in a history-of-science study, which I strongly recommend to any serious student of physics since to fully understand the modern point of view of physics, particularly the development of the modern quite abstract quantum theory, it is very important to learn about how this modern view has developed from less comprehensive views, but it's not helpful for introducing you to modern physics. The same holds for what's called "Relativistic Quantum Mechanics", which is very cumbersome and problematic to understand, because it's "unnatural" to use it for relativistic particles to begin with. The reason is that "Relativistic Quantum Mechanics", which was the try to build a relativistic quantum theory in analogy to Schrödinger's wave mechanics, which deals with a fixed number of particles ("1st quantization formalism"). This works well in non-relativistic physics, but the very point of relativistic particle physics is that particle numbers are no longer conserved, i.e., given enough energy in collisions (which is what's particularly meant by high-energy/relativistic particle physics) you can always destroy particles and produce new ones, and this is most conveniently described by quantum field theory ("2nd quantiation"). In fact it was Jordan who has seen this early on already in 1926, when he insisted that also the electromagnetic field should be "quantized", i.e., made to "non-commuting matrices" in the matrix-mechanics representation of quantum theory developed by him together with Born, building on Heisenberg's heuristic Helgoland paper and then being finally formulated by Born, Jordan, and Heisenberg in the famous "Dreimännerarbeit" (Three-Men Paper, literally translated). At this time this idea was not appreciated by many other quantum physicists, perhaps mainly because of the unfamiliar matrix formalism and the difficulty to express continuous phenomena like scattering within this formalism. That's why many physicists preferred Schrödinger's wave mechanics, for which the scattering phenomena were a quite familiar thing, known from the scattering of light via classical Maxwell electromagnetics. That's why usually Dirac's work of 1928 on quantization of the radiation field is taken as the birth of quantum field theory rather than Jordan's pioneer work within matrix mechanics.

#### Vanadium 50

Staff Emeritus
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Education Advisor
The fine structure constant (as mentioned in another post) is also an excellent high precision test of QED theory vs measurement
How is this? What is the theoretical prediction of it?

#### Stavros Kiri

How is this? What is the theoretical prediction of it
Read carefully the wiki ref. given earlier (for how the conclusion is drawn).

#### Jimmy87

I'd however qualify that "Old Quantum Theory" is not really there anyore, and that's good. You can learn about it in a history-of-science study, which I strongly recommend to any serious student of physics since to fully understand the modern point of view of physics, particularly the development of the modern quite abstract quantum theory, it is very important to learn about how this modern view has developed from less comprehensive views, but it's not helpful for introducing you to modern physics. The same holds for what's called "Relativistic Quantum Mechanics", which is very cumbersome and problematic to understand, because it's "unnatural" to use it for relativistic particles to begin with. The reason is that "Relativistic Quantum Mechanics", which was the try to build a relativistic quantum theory in analogy to Schrödinger's wave mechanics, which deals with a fixed number of particles ("1st quantization formalism"). This works well in non-relativistic physics, but the very point of relativistic particle physics is that particle numbers are no longer conserved, i.e., given enough energy in collisions (which is what's particularly meant by high-energy/relativistic particle physics) you can always destroy particles and produce new ones, and this is most conveniently described by quantum field theory ("2nd quantiation"). In fact it was Jordan who has seen this early on already in 1926, when he insisted that also the electromagnetic field should be "quantized", i.e., made to "non-commuting matrices" in the matrix-mechanics representation of quantum theory developed by him together with Born, building on Heisenberg's heuristic Helgoland paper and then being finally formulated by Born, Jordan, and Heisenberg in the famous "Dreimännerarbeit" (Three-Men Paper, literally translated). At this time this idea was not appreciated by many other quantum physicists, perhaps mainly because of the unfamiliar matrix formalism and the difficulty to express continuous phenomena like scattering within this formalism. That's why many physicists preferred Schrödinger's wave mechanics, for which the scattering phenomena were a quite familiar thing, known from the scattering of light via classical Maxwell electromagnetics. That's why usually Dirac's work of 1928 on quantization of the radiation field is taken as the birth of quantum field theory rather than Jordan's pioneer work within matrix mechanics.
Thanks for all this info. So is relativistic QM an older and now outdated theory? I thought most of modern QM was based on SR? So do QED & QFT not use any relativity?

#### Jimmy87

This is a bit complicated, as (to the best of my knowledge), there is no calculation that just spews out $g$ based on a purely ab initio calculation. Citing from the article where that measured value comes from [B. Odom et al., New Measurement of the Electron Magnetic Moment Using a One-Electron Quantum Cyclotron, Phys. Rev. Lett. 97, 030801 (2006)]
Thanks. Is this more a measurre of accuracy or precision or both? When we talk about these terms in class we are told that if an expeirment agrees with prediction then it is accurate. If there is a very small uncertainty then it is precise. For example, measuring acceleration due to gravity of 9.81 + 5m/s2 is accurate but not precise whereas a result of 9.81 + 0.01 m/s2 is accurate and precise. So the QED result of 10-12 sounds like an accuracy measurement because it is the different between prediction and observation. What would the precision/uncertainty of the QED experiment be? If this 10-12 is the uncertainty then how can you determine the accuracy?

#### Vanadium 50

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Education Advisor
Read carefully the wiki ref. given earlier (for how the conclusion is drawn).
Not there. The wiki shows several different measurements, converted to measurements of alpha so they can be compared, but they are not a comparison between theory and experiment. There is, as far as I know, no ab initio calculation of alpha.

#### vanhees71

Science Advisor
Gold Member
Thanks for all this info. So is relativistic QM an older and now outdated theory? I thought most of modern QM was based on SR? So do QED & QFT not use any relativity?
No! There's a subtle difference between "Relativistic Quantum Mechanics", which is outdated and, in my opinion, should not be taught in the standard physics courses anymore. It's more or less defined by the content of volume 1 of Bjorken&Drell's classic.

Then there is "Relativistic Quantum (Field) Theory", which is the modern exposition of the subject and basis of the Standard Model of elementary particles, which is such a successful theory that it withstands all tries of the particle experimentalists to find "physics beyond the Standard Model". There are some 2-$3\sigma$ signals of deviations always present, but so far all struggles of the experimentalists to make the hint or evidence a discovery, i.e., reaching $5\sigma$ confidence level, lead to another triumph of the SM. So it's really fair to say that the SM is among the most successful theories ever discovered, as incomplete as it is (since it doesn't incorporate gravity) and a bit unsatisfactory for some physicists (hierarchy/fine-tuning problem etc). That's by the way is treated in Bjorken's and Drell's other classic "Relativistic Quantum Field Theory", which is also quite old but in many parts still among the best introductions. Particularly you find a very good treatment of the LSZ reduction formalism, which always keeps me puzzling as often as I try to explain it to somebody. On the downside is that the part on renormalization theory is outdated, because it was written before the final answer by BPHZ. Another one is that it was written before the importance non-Abelian gauge theories and the Higgs mechanism has been fully recognized.

For a modern introduction to the subject my favorite is

M. D. Schwartz, Quantum field theory and the Standard Model, Cambridge University Press, Cambridge, New York, 2014.

The ultimate books are

S. Weinberg, Quantum Theory of Fields (3 vols.) Cambridge University Press

and complementary in covering all the subjects swept under the rug by Weinberg (including so disturbing findings like Haag's theorem and the like) and at equal level of brilliance as Weinberg:

A. Duncan, The Conceptual Framework of Quantum Field Theory, Oxford University Press

#### Jimmy87

No! There's a subtle difference between "Relativistic Quantum Mechanics", which is outdated and, in my opinion, should not be taught in the standard physics courses anymore. It's more or less defined by the content of volume 1 of Bjorken&Drell's classic.

Then there is "Relativistic Quantum (Field) Theory", which is the modern exposition of the subject and basis of the Standard Model of elementary particles, which is such a successful theory that it withstands all tries of the particle experimentalists to find "physics beyond the Standard Model". There are some 2-$3\sigma$ signals of deviations always present, but so far all struggles of the experimentalists to make the hint or evidence a discovery, i.e., reaching $5\sigma$ confidence level, lead to another triumph of the SM. So it's really fair to say that the SM is among the most successful theories ever discovered, as incomplete as it is (since it doesn't incorporate gravity) and a bit unsatisfactory for some physicists (hierarchy/fine-tuning problem etc). That's by the way is treated in Bjorken's and Drell's other classic "Relativistic Quantum Field Theory", which is also quite old but in many parts still among the best introductions. Particularly you find a very good treatment of the LSZ reduction formalism, which always keeps me puzzling as often as I try to explain it to somebody. On the downside is that the part on renormalization theory is outdated, because it was written before the final answer by BPHZ. Another one is that it was written before the importance non-Abelian gauge theories and the Higgs mechanism has been fully recognized.

For a modern introduction to the subject my favorite is

M. D. Schwartz, Quantum field theory and the Standard Model, Cambridge University Press, Cambridge, New York, 2014.

The ultimate books are

S. Weinberg, Quantum Theory of Fields (3 vols.) Cambridge University Press

and complementary in covering all the subjects swept under the rug by Weinberg (including so disturbing findings like Haag's theorem and the like) and at equal level of brilliance as Weinberg:

A. Duncan, The Conceptual Framework of Quantum Field Theory, Oxford University Press
Great thanks. So do both QED and QFT use special relativity?

Mentor

#### bhobba

Mentor
Not there. The wiki shows several different measurements, converted to measurements of alpha so they can be compared, but they are not a comparison between theory and experiment. There is, as far as I know, no ab initio calculation of alpha.
Of course you are correct - its measured and not an actual prediction. However the high precision the fine structure constant has been measured to is one of the reasons the calculations that use it have such a good agreement with experiment.

Thanks
Bill

#### vanhees71

Science Advisor
Gold Member
Great thanks. So do both QED and QFT use special relativity?
Yes, QED is the paradigmatic example of a relativistic quantum field theory. Today it's part of the Standard Model, unified with the weak interaction to the socalled quantum flavor dynamics (Glashow-Salam-Weinberg model). The other building block of the Standard Model is Quantum Chromodynamics, which describes the strong interaction.

#### Jimmy87

Yes, QED is the paradigmatic example of a relativistic quantum field theory. Today it's part of the Standard Model, unified with the weak interaction to the socalled quantum flavor dynamics (Glashow-Salam-Weinberg model). The other building block of the Standard Model is Quantum Chromodynamics, which describes the strong interaction.
Does the astounding precision of QED & QFT also mean that special relativity has just as good precision if it is a key part of QFT?

#### bhobba

Mentor
Does the astounding precision of QED & QFT also mean that special relativity has just as good precision if it is a key part of QFT?
Yes.

But the thing with SR is something would have to be very screwy with our understanding of the world if it was wrong:
http://www2.physics.umd.edu/~yakovenk/teaching/Lorentz.pdf

Its nearly, but not quite the same with GR as well - it has a couple of other assumptions that are a bit more open to doubt than things in SR; like if you are in intergalactic space away from any influences like gravity and you conduct an experiment at one place and exactly the same experiment at another you would get the same result. Science as we know it, which is based on repeatable experiments, would be vastly more difficult. GR has an assumption called the principle of general invariance that goes a bit beyond SR - SR deals with inertial frames which by definition have the property all points, direction and times are the same. The principle of invariance says even the coordinate systems you use should not make a difference - its pretty intuitive, but is an assumption rather than a defining characteristic like what an inertial frame is

Thanks
Bill

#### Jimmy87

Yes.

But the thing with SR is something would have to be very screwy with our understanding of the world if it was wrong:
http://www2.physics.umd.edu/~yakovenk/teaching/Lorentz.pdf

Its nearly, but not quite the same with GR as well - it has a couple of other assumptions that are a bit more open to doubt than things in SR; like if you are in intergalactic space away from any influences like gravity and you conduct an experiment at one place and exactly the same experiment at another you would get the same result. Science as we know it, which is based on repeatable experiments, would be vastly more difficult.

Thanks
Bill
Thanks, sorry for such a basic question but roughly how much SR is there in QFT? (e.g. 50%) Or is that difficult to answer. I haven't done that much on QM and nothing on relativistic QM so sorry for such a crude question.

#### bhobba

Mentor
Thanks, sorry for such a basic question but roughly how much SR is there in QFT? (e.g. 50%) Or is that difficult to answer. I haven't done that much on QM and nothing on relativistic QM so sorry for such a crude question.
Its basically SR, QM and the concept of a fields. You cant write down proper field equations without SR. That is due to some deep theorems about something called spin. I cant explain it to you but once your math is better the following book will give the detail:
http://jakobschwichtenberg.com/physics-from-symmetry/

Thanks
Bill

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