Yuri Danoyan Posted November 12, 2008 Share Posted November 12, 2008 How to call symmetry between symmetry and antisymmetriy? Option 1. Symmetry (S). Option 2. Antisymmetry (AS). I think the right answer is: S-AS-S-AS-S-AS-S-AS ..... etc. I suspect that many of our questions about the World getting such oscillating answers. It look like "Liar paradox". My be it is pessimistic view? Or not, it gives dynamics to the Universe? And did him Cyclic? Link to comment Share on other sites More sharing options...
ajb Posted November 14, 2008 Share Posted November 14, 2008 I don't know what you had in mind, but if you pick everything to be [math]Z_{2}[/math] graded then the notion of symmetry and antisymmetry become the same. Link to comment Share on other sites More sharing options...
Yuri Danoyan Posted November 14, 2008 Author Share Posted November 14, 2008 then the notion of symmetry and antisymmetry become the same. That mean bosons and fermions the same? Link to comment Share on other sites More sharing options...
ajb Posted November 14, 2008 Share Posted November 14, 2008 I was not really thinking of (particle physics) supersymmetry as such, a bit more generally than that. Were you thinking of symmetries quite generally or in particle physics specifically? Link to comment Share on other sites More sharing options...
BenTheMan Posted November 14, 2008 Share Posted November 14, 2008 then the notion of symmetry and antisymmetry become the same. That mean bosons and fermions the same? AB-BA and AB+BA are the same? Link to comment Share on other sites More sharing options...
Yuri Danoyan Posted November 14, 2008 Author Share Posted November 14, 2008 AB-BA and AB+BA are the same? Fair question... Link to comment Share on other sites More sharing options...
BenTheMan Posted November 14, 2008 Share Posted November 14, 2008 That's what you're saying when you say "bosons and fermions are the same". Link to comment Share on other sites More sharing options...
Yuri Danoyan Posted November 14, 2008 Author Share Posted November 14, 2008 That follow from ajb post...It wasn't my opinion Link to comment Share on other sites More sharing options...
ajb Posted November 15, 2008 Share Posted November 15, 2008 (edited) Not what I mean, but you can treat them in a unified way. Lets look at the example of graded or super commutative algebras. Let [math]\mathcal{A} = \mathcal{A}_{0}\oplus \mathcal{A}_{1}[/math] to be a [math] Z_{2}[/math] graded commutative algebra, that is a [math]Z_{2}[/math] vector space with a product such that [math]a b = (-1)^{\widetilde{a} \widetilde{b}} ba[/math] with [math]\widetilde{a} \in Z_{2}[/math] denoting if [math]a \in \mathcal{A}_{0}[/math] or [math]\mathcal{A}_{1}[/math] etc. Then all even elements (in [math]\mathcal{A}_{0}[/math]) "commute" amongst and themselves and all odd elements (in [math]\mathcal{A}_{1}[/math]) "anticommute". Odd and even commute. In a unified way we say that the algebra is supercommutative or just commutative. If we define the commutator as [math][a,b] = ab - (-1)^{\widetilde{a} \widetilde{b}} ba[/math] we see that this is indeed zero. So in the previous language we have "symmetric" even elements and "antisymmetric" odd elements. For supervector space we have the "reverse parity functor", which we can use to define the "antialgebra" [math]\Pi \mathcal{A}[/math]. Which we define by changing the even elements to odd ones and odd ones to even ones. Basically, if [math]a \in \mathcal{A}[/math] is even/odd then [math]\Pi a \in \Pi\mathcal{A}[/math] is odd/even. We see that the antialgebra is still a (graded) commutative algebra, but now the previously "even symmetric" elements are now "odd antisymmetric". You can show that [math]\Pi[/math] is indeed a functor and really there is no fundamental difference between symmetric and antisymmetric. That was my point. Not sure what anyone else had in mind. Edited November 15, 2008 by ajb Link to comment Share on other sites More sharing options...
Yuri Danoyan Posted November 15, 2008 Author Share Posted November 15, 2008 Wavefunction for fermions-antysymmetric Wavefunction for bosons-symmetric That mean something? Link to comment Share on other sites More sharing options...
BenTheMan Posted November 16, 2008 Share Posted November 16, 2008 I think you guys are on different planets entirely Link to comment Share on other sites More sharing options...
ajb Posted November 16, 2008 Share Posted November 16, 2008 Wavefunction for fermions-antysymmetricWavefunction for bosons-symmetric That mean something? Yes. Link to comment Share on other sites More sharing options...
Yuri Danoyan Posted November 16, 2008 Author Share Posted November 16, 2008 I think you guys are on different planets entirely From different planet panorama is best. One women lives in Alaska and count Afrika as a country,but not continent. Link to comment Share on other sites More sharing options...
ajb Posted November 16, 2008 Share Posted November 16, 2008 (edited) I think you guys are on different planets entirely Maybe. Unfortunately I have always found it difficult to understand what Yuri has in mind. Yuri, you know of supersymmetries that mix fermions and bosons. Is this not what you were thinking of? Edited November 16, 2008 by ajb Link to comment Share on other sites More sharing options...
Yuri Danoyan Posted November 16, 2008 Author Share Posted November 16, 2008 My favourite quotation for definition of supersymmetryI never see before "The most important feature of supersymmetry is that it is non-trivial way combines ongoing transformation (such as translations), with a special kind of discrete transformations (such as reflection). While retaining the formal analogy between these two types of changes that are significantly different nature. It is that this analogy is «core» of supersymmetry. "L. E. Gendenshteyn, I. Krive« Supersymmetry in quantum mechanics (http://ufn.ru/ru/articles/1985/8/a/) P. 554. Link to comment Share on other sites More sharing options...
ajb Posted November 16, 2008 Share Posted November 16, 2008 Take careful note of the word analogy. In some sense supersymmetry does combine discrete with continuous, but as I said in your other thread on symmetries, it is not clear to me if you should think of supersymmetries as discrete or continuous. Link to comment Share on other sites More sharing options...
Yuri Danoyan Posted November 16, 2008 Author Share Posted November 16, 2008 sorry "ongoing transformation" is wrong translations from russian "continue transoformation" is correct translation from russian. Link to comment Share on other sites More sharing options...
ajb Posted November 16, 2008 Share Posted November 16, 2008 sorry "ongoing transformation" is wrong translations from russian "continue transoformation" is correct translation from russian. Thank you Yuri, I gathered that. Link to comment Share on other sites More sharing options...
Yuri Danoyan Posted November 16, 2008 Author Share Posted November 16, 2008 Take careful note of the word analogy. "I think, creative activity of person revealed in capability to see identity of notions,where there nobody can see."(Hideki Yukawa) Link to comment Share on other sites More sharing options...
ajb Posted November 16, 2008 Share Posted November 16, 2008 Anyway, the generally accepted term for a transformation that mixes "even/symmetric" with "odd/antisymmetric" is supersymmetry. Which in reality is nothing more that a [math]Z_{2}[/math] graded symmetry. As you know this, it cannot be what you are thinking of in your original post? or is it? Link to comment Share on other sites More sharing options...
Yuri Danoyan Posted November 16, 2008 Author Share Posted November 16, 2008 (edited) As you know this, it cannot be what you are thinking of in your original post? or is it? Just question: " Does have sense symmetry between symmetry and antisymmetry ?" Answer from Wikipedia: "Note that 'antisymmetric' is not the logical negative of 'symmetric' (whereby aRb implies bRa). (N.B.: Both are properties of relations expressed as universal statements about their members; their logical negations must be existential statements.) Thus, there are relations which are both symmetric and antisymmetric (e.g., the equality relation) and there are relations which are neither symmetric nor antisymmetric (e.g., the preys-on relation on biological species." http://en.wikipedia.org/wiki/Antisymmetric_relation Edited November 16, 2008 by Yuri Danoyan multiple post merged Link to comment Share on other sites More sharing options...
ajb Posted November 17, 2008 Share Posted November 17, 2008 I would always be very careful getting answers from Wikipedia. Just question: " Does have sense symmetry between symmetry and antisymmetry ?" I am not fully sure what you mean here. Supersymmetries are the closest thing I can think of, generically I mean [math]Z_{2}[/math] graded symmetries. We also have the reverse parity functor [math]\Pi[/math] that works on super vector spaces (and it can be extended to vector bundles). It forms a trivial group as [math]\Pi^{2} = id[/math]. That is all I can think of really. Link to comment Share on other sites More sharing options...
Harlequinne Posted December 16, 2008 Share Posted December 16, 2008 I don't know what you had in mind, but if you pick everything to be [math]Z_{2}[/math] graded then the notion of symmetry and antisymmetry become the same. HI! I was wondering if you could,you know show me how to find that script on the web someday sailor............ But seriously,I am a man and wondered if I can get the necessary stuff,because c^2 is a bit lame!! Link to comment Share on other sites More sharing options...
ajb Posted December 16, 2008 Share Posted December 16, 2008 You can use LaTex in this forum. I am sure there is a basic guide here somewhere. Link to comment Share on other sites More sharing options...
Severian Posted December 17, 2008 Share Posted December 17, 2008 I am not sure it is what Yuri was asking, but I do sympathise with ajb's point of view. To put it another way, supersymmetry is a symmetry between symmetry and antisymmetry. Link to comment Share on other sites More sharing options...
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