Logical contradiction of notion "Symmetry"

[Yuri Danoyan]

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?

[ajb]

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.

[Yuri Danoyan]

then the notion of symmetry and antisymmetry become the same.

 

That mean bosons and fermions the same?

[ajb]

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?

[BenTheMan]

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?

[Yuri Danoyan]

AB-BA and AB+BA are the same?

 

Fair question...

[BenTheMan]

That's what you're saying when you say "bosons and fermions are the same".

[Yuri Danoyan]

That follow from ajb post...It wasn't my opinion

[ajb]

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

[Yuri Danoyan]

Wavefunction for fermions-antysymmetric

Wavefunction for bosons-symmetric

That mean something?

[BenTheMan]

I think you guys are on different planets entirely

[ajb]

Wavefunction for fermions-antysymmetric

Wavefunction for bosons-symmetric

That mean something?

 

Yes.

[Yuri Danoyan]

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.

[ajb]

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

[Yuri Danoyan]

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.

[ajb]

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.

[Yuri Danoyan]

sorry

"ongoing transformation" is wrong translations from russian

 

"continue transoformation" is correct translation from russian.

[ajb]

sorry

"ongoing transformation" is wrong translations from russian

 

"continue transoformation" is correct translation from russian.

 

Thank you Yuri, I gathered that.

[Yuri Danoyan]

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)

[ajb]

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?

[Yuri Danoyan]

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

[ajb]

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.

[Harlequinne]

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!!

[ajb]

You can use LaTex in this forum. I am sure there is a basic guide here somewhere.

[Severian]

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.