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Logical contradiction of notion "Symmetry"


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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?

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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 by ajb
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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 by ajb
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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.

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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.

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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?

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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 by Yuri Danoyan
multiple post merged
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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.

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  • 5 weeks later...
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!!

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