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Text and/or other creative content from Exchange symmetry was copied or moved into Symmetry in quantum mechanics with this edit. The former page's history now serves to provide attribution for that content in the latter page, and it must not be deleted as long as the latter page exists. |
I don't know enough about the scale invariance/renormalization group, topological conservation laws, and instantons, so will leave that for those inclined and knowledgeable. M∧Ŝ c2ħε Иτlk 07:01, 5 June 2013 (UTC)
As it stands now, the first section after the lead is "Overview of Lie group theory". I have nothing against Lie groups, but I would suggest that a better way to organize the article would be to start with an intuitive explanation of what's implied by Noether's theorem. Even experimentalists like myself get a lot of mileage out of symmetry arguments, and I have no clue what a Lie group is =p. I'll give a shot at this later, going roughly from the treatment in Sakurai. a13ean ( talk) 23:32, 5 June 2013 (UTC)
This article, as it stands, captures a lot of details about individual symmetries, and would be better described by the title Symmetries in quantum mechanics. The distinction is that with the existing title, one expects an explanation of the role of symmetry in quantum mechanics, and with the title as I've suggested one expects a list-like approach, detailing each symmetry (which is what it appears to be trying to do). Symmetry (physics) already covers the general picture pretty well, even with regard to quantum mechanics. — Quondum 13:51, 8 June 2013 (UTC)
Yes, there may well be.
The whole article needs to solidify the conventions used, as well as fixing more headache-beating notations for what means what. Some conventions are already decided, but others like which Minkowski metric needs to be chosen and used throughout.
The worst nightmare is the letter D, which could mean the Wigner matrices in the context of spin matrices (or possibly representations for the Wigner matrices themselves?), representations of group elements, or representations of generators. Perhaps something like:
would be clearer.
The next nightmare is, what are A and B? If J and K are generators, so must be A and B.
According to E. Abers, A and B start off as generators, then after some wishy-washy use of the term "representations" (with reference to what??), then finally states D(a, b)(J) are still the angular momentum operators (rotation generators) and D(a, b)(K) the boost generators. However - the D(a, b) notation refers to a representation of the boost and rotation generators, not the generators themselves. I followed E. Abers to be on the safe side.
Representation theory of the Lorentz group is vague on what A and B actually are. My consensus is that A and B are simply generators, while D(a, b)(K) and D(a, b)(J) are irreducible reps that happen to be able to be expressed in terms of the generators A and B, instead of the reps of generators D(A) and D(B), i.e. D(a, b)(K) = −iD(A) − D(B)] where the reps are just the identity transformations? Still looking for refs which make it crystal clear.
For sake of clarity, all expressions using hats will be converted to LaTeX, even if they're inline.
Will fix ERRORS soon. M∧Ŝ c2ħε Иτlk 14:01, 22 June 2013 (UTC)
A quote from C.N. Yang's Nobel lecture in 1957 [1]:
"It was however not until the development of quantum mechanics that the use of the symmetry principles began to permeate into the very language of physics. The quantum numbers that designate the states of a system are very often identical with those that represent the symmetries of the system. It indeed is scarcely possible to overemphasize the role played by the symmetry principles in quantum mechanics. To quote two examples: The general structure of the periodic table is essentially a direct consequence of the isotropy of Coulomb's law. The existence of the antiparticles -- namely the positron, the anti-proton and the anti-neutron -- were theoretically anticipated as consequences of the symmetry of physical laws with respect to Lorentz transformations..."
The article needs to do much more to explain how this close association between symmetries and states arises -- what it is in the framework of quantum mechanics specifically that makes this happen. Jheald ( talk) 18:58, 1 July 2013 (UTC)
I made a tweak of the definition in the article, effectively not defining what "reducible" means. The reason is that not irreducible isn't necessarily the same thing as reducible (as plain English suggests). It is true for semi-simple groups, but not in general. YohanN7 ( talk) 18:22, 7 November 2013 (UTC)
What is this?
and correspondingly, the boost and rotation parameters are collected into another antisymmetric four-dimensional matrix ω, with entries:
or explicitly in the standard form:
YohanN7 ( talk) 02:01, 11 November 2013 (UTC)
I appreciate the effort to keep things simple, but I believe that some care needs to be taken so that the math is right. For instance, expressions like,
The irreducible representations of D(K) and D(J), in short "irreps", can be used to build to spin representations of the Lorentz group. Defining new generators:
so A and B are simply complex conjugates of each other, it follows...:
is the reason I began working on the Lorentz rep article in the first place. This here is easy to read, and the reader may well think he understand what he is reading, but he doesn't.
I am taking the above as an example. Physics books are all, almost without exception, horrible HORRIBLE at explaining group theory adequately, and the example above is quite typical. It's something pulled out of the hat. It sounds good, but explains little.
I like the general approach in the article (as I have said before), but simplicity and volume shouldn't come at the price of moderate mathematical rigor.
My advice would be to take math (and even mathematical terminology, provide a translation table) as a baseline. A reader should be able to leave the article with ,for instance, at least some understanding of the interconnections between su(2), SU(2), so(3), SO(3), so(3;1), SO(3;1)+, etc, and SL(2;C) for that matter since spin is involved, even if it makes the article a little "harder" to read (and write). YohanN7 ( talk) 02:49, 11 November 2013 (UTC)
What would it look like? Let's present it below. M∧Ŝ c2ħε Иτlk 16:59, 18 November 2013 (UTC)
For internal usage we should define what we mean by a representation. "Representation" is probably one of the most (mis)used word (both i math and physics). For a framework, we should explicitly assume a Hilbert space H (the elements being quantum states), and the set of linear operators on it, End(H).
In our case, a representation then could be a subset of End(H) endowed with the appropriate structures. Here we need to be precise. Some would mean that a representation is a subset of H itself. A formalist might argue that the representation is the map from the abstract group or algebra to End(H). Neither is wrong, but we should be precise by what we mean.
Generators are of course elements of a Lie algebra, but here it becomes murky. Should we let generators be elements of representations of Lie algebras? Logically, yes, since the reps are as much Lie algebras as the abstract Lie algebra itself. Also, when plural is used, it is often meant a complete set of basis vectors for the Lie algebra.
There's much more to define, but lets cook this table slowly. YohanN7 ( talk) 17:27, 18 November 2013 (UTC)
And now I got a brilliant idea: Make a picture with the Hilbert space, the operators, and the groups/algebras. If you come up with just something here to start with, we can find a way to make it very illustrative. YohanN7 ( talk) 17:32, 18 November 2013 (UTC)
So, being in a speculative mode, I fleshed out a bit. I can (and will in due time) back up all with refs, except the last statement:
I made this up, but think that it is true. Does someone know of a ref where this is discussed? If not, it has to be scrapped. At any rate, it needs reformulation. In essence, "difficult to balance..." -> extremely small cross section. (This is why we don't see it.) YohanN7 ( talk) 17:45, 11 November 2013 (UTC)
With a general title like this, there should also be discussion about symmetries typically encountered in condensed matter systems and chemistry. General discussion about symmetry breaking is also needed. Most likely, discussion about such topics already exist somewhere in WP, and could be just summarized here. Jähmefyysikko ( talk) 08:26, 27 February 2024 (UTC)
This
level-5 vital article is rated C-class on Wikipedia's
content assessment scale. It is of interest to the following WikiProjects: | |||||||||||
|
Text and/or other creative content from Exchange symmetry was copied or moved into Symmetry in quantum mechanics with this edit. The former page's history now serves to provide attribution for that content in the latter page, and it must not be deleted as long as the latter page exists. |
I don't know enough about the scale invariance/renormalization group, topological conservation laws, and instantons, so will leave that for those inclined and knowledgeable. M∧Ŝ c2ħε Иτlk 07:01, 5 June 2013 (UTC)
As it stands now, the first section after the lead is "Overview of Lie group theory". I have nothing against Lie groups, but I would suggest that a better way to organize the article would be to start with an intuitive explanation of what's implied by Noether's theorem. Even experimentalists like myself get a lot of mileage out of symmetry arguments, and I have no clue what a Lie group is =p. I'll give a shot at this later, going roughly from the treatment in Sakurai. a13ean ( talk) 23:32, 5 June 2013 (UTC)
This article, as it stands, captures a lot of details about individual symmetries, and would be better described by the title Symmetries in quantum mechanics. The distinction is that with the existing title, one expects an explanation of the role of symmetry in quantum mechanics, and with the title as I've suggested one expects a list-like approach, detailing each symmetry (which is what it appears to be trying to do). Symmetry (physics) already covers the general picture pretty well, even with regard to quantum mechanics. — Quondum 13:51, 8 June 2013 (UTC)
Yes, there may well be.
The whole article needs to solidify the conventions used, as well as fixing more headache-beating notations for what means what. Some conventions are already decided, but others like which Minkowski metric needs to be chosen and used throughout.
The worst nightmare is the letter D, which could mean the Wigner matrices in the context of spin matrices (or possibly representations for the Wigner matrices themselves?), representations of group elements, or representations of generators. Perhaps something like:
would be clearer.
The next nightmare is, what are A and B? If J and K are generators, so must be A and B.
According to E. Abers, A and B start off as generators, then after some wishy-washy use of the term "representations" (with reference to what??), then finally states D(a, b)(J) are still the angular momentum operators (rotation generators) and D(a, b)(K) the boost generators. However - the D(a, b) notation refers to a representation of the boost and rotation generators, not the generators themselves. I followed E. Abers to be on the safe side.
Representation theory of the Lorentz group is vague on what A and B actually are. My consensus is that A and B are simply generators, while D(a, b)(K) and D(a, b)(J) are irreducible reps that happen to be able to be expressed in terms of the generators A and B, instead of the reps of generators D(A) and D(B), i.e. D(a, b)(K) = −iD(A) − D(B)] where the reps are just the identity transformations? Still looking for refs which make it crystal clear.
For sake of clarity, all expressions using hats will be converted to LaTeX, even if they're inline.
Will fix ERRORS soon. M∧Ŝ c2ħε Иτlk 14:01, 22 June 2013 (UTC)
A quote from C.N. Yang's Nobel lecture in 1957 [1]:
"It was however not until the development of quantum mechanics that the use of the symmetry principles began to permeate into the very language of physics. The quantum numbers that designate the states of a system are very often identical with those that represent the symmetries of the system. It indeed is scarcely possible to overemphasize the role played by the symmetry principles in quantum mechanics. To quote two examples: The general structure of the periodic table is essentially a direct consequence of the isotropy of Coulomb's law. The existence of the antiparticles -- namely the positron, the anti-proton and the anti-neutron -- were theoretically anticipated as consequences of the symmetry of physical laws with respect to Lorentz transformations..."
The article needs to do much more to explain how this close association between symmetries and states arises -- what it is in the framework of quantum mechanics specifically that makes this happen. Jheald ( talk) 18:58, 1 July 2013 (UTC)
I made a tweak of the definition in the article, effectively not defining what "reducible" means. The reason is that not irreducible isn't necessarily the same thing as reducible (as plain English suggests). It is true for semi-simple groups, but not in general. YohanN7 ( talk) 18:22, 7 November 2013 (UTC)
What is this?
and correspondingly, the boost and rotation parameters are collected into another antisymmetric four-dimensional matrix ω, with entries:
or explicitly in the standard form:
YohanN7 ( talk) 02:01, 11 November 2013 (UTC)
I appreciate the effort to keep things simple, but I believe that some care needs to be taken so that the math is right. For instance, expressions like,
The irreducible representations of D(K) and D(J), in short "irreps", can be used to build to spin representations of the Lorentz group. Defining new generators:
so A and B are simply complex conjugates of each other, it follows...:
is the reason I began working on the Lorentz rep article in the first place. This here is easy to read, and the reader may well think he understand what he is reading, but he doesn't.
I am taking the above as an example. Physics books are all, almost without exception, horrible HORRIBLE at explaining group theory adequately, and the example above is quite typical. It's something pulled out of the hat. It sounds good, but explains little.
I like the general approach in the article (as I have said before), but simplicity and volume shouldn't come at the price of moderate mathematical rigor.
My advice would be to take math (and even mathematical terminology, provide a translation table) as a baseline. A reader should be able to leave the article with ,for instance, at least some understanding of the interconnections between su(2), SU(2), so(3), SO(3), so(3;1), SO(3;1)+, etc, and SL(2;C) for that matter since spin is involved, even if it makes the article a little "harder" to read (and write). YohanN7 ( talk) 02:49, 11 November 2013 (UTC)
What would it look like? Let's present it below. M∧Ŝ c2ħε Иτlk 16:59, 18 November 2013 (UTC)
For internal usage we should define what we mean by a representation. "Representation" is probably one of the most (mis)used word (both i math and physics). For a framework, we should explicitly assume a Hilbert space H (the elements being quantum states), and the set of linear operators on it, End(H).
In our case, a representation then could be a subset of End(H) endowed with the appropriate structures. Here we need to be precise. Some would mean that a representation is a subset of H itself. A formalist might argue that the representation is the map from the abstract group or algebra to End(H). Neither is wrong, but we should be precise by what we mean.
Generators are of course elements of a Lie algebra, but here it becomes murky. Should we let generators be elements of representations of Lie algebras? Logically, yes, since the reps are as much Lie algebras as the abstract Lie algebra itself. Also, when plural is used, it is often meant a complete set of basis vectors for the Lie algebra.
There's much more to define, but lets cook this table slowly. YohanN7 ( talk) 17:27, 18 November 2013 (UTC)
And now I got a brilliant idea: Make a picture with the Hilbert space, the operators, and the groups/algebras. If you come up with just something here to start with, we can find a way to make it very illustrative. YohanN7 ( talk) 17:32, 18 November 2013 (UTC)
So, being in a speculative mode, I fleshed out a bit. I can (and will in due time) back up all with refs, except the last statement:
I made this up, but think that it is true. Does someone know of a ref where this is discussed? If not, it has to be scrapped. At any rate, it needs reformulation. In essence, "difficult to balance..." -> extremely small cross section. (This is why we don't see it.) YohanN7 ( talk) 17:45, 11 November 2013 (UTC)
With a general title like this, there should also be discussion about symmetries typically encountered in condensed matter systems and chemistry. General discussion about symmetry breaking is also needed. Most likely, discussion about such topics already exist somewhere in WP, and could be just summarized here. Jähmefyysikko ( talk) 08:26, 27 February 2024 (UTC)