Expanding Subscripts & Putting TeX in Board Posts

[bigplanet401]

I. What does

A_{[\alpha} B_{\beta]}

mean? How do you expand this?

II. How do you put TeX in board posts?

 

[robphy]

I. What does

A_{[\alpha} B_{\beta]}

mean? How do you expand this?

II. How do you put TeX in board posts?

[ tex ] A_{[\alpha} B_{\beta]} [ /tex ] (remove the spaces in the tags)
A_{[\alpha} B_{\beta]} (click the on the equation to see)

A_{[\alpha} B_{\beta]} =\frac{1}{2!}\left( A_{\alpha} B_{\beta} - A_{\beta} B_{\alpha} \right) the "antisymmetric part of A_{\alpha} B_{\beta}"

 

[dextercioby]

It's not necessary to use the numerical factor (we call it "weght"). See for example the em. tensor in vacuum. U can use the "no number convention" (i use it)

F_{\mu\nu}\equiv F_{\left[\mu\nu\right]}=:\partial_{[\mu}A_{\nu]}

or the one Rob exemplified, when an ugly 2 comes up

F_{\mu\nu}\equiv F_{\left[\mu\nu\right]}=:2 \partial_{[\mu}A_{\nu]}

Daniel.

 

[dextercioby]

And for symmetrizing, we use round brackets. For example, the linearized graviton field has the irreducible Lagrangian infinitesimal gauge transformations

\delta_{\epsilon}h_{\left(\mu\nu\right)}(x) =\partial_{(\mu}\epsilon_{\nu)}(x)

or with the "2", if you use an analogue convention Rob used.

Daniel.

 

[robphy]

While the combinatorial factor I used may be conventional, I believe it is the preferred convention. See, for example, http://mathworld.wolfram.com/AntisymmetricTensor.html

Note that the "symmetric part" of a matrix is \frac{1}{2}(A+A^T) and the "antisymmetric part" of a matrix is \frac{1}{2}(A-A^T). (Similarly, the "real part of a complex number z" is \frac{1}{2}(z+\bar z) and "imaginary part of a complex number z" is \frac{1}{2i}(z-\bar z).)

So, one can write the matrix equation
A= A_{SYM} + A_{ANTISYM}
and an analogous tensorial equation
\begin{align*}<br /> A_{ab} &amp;= <br /> \frac{1}{2}( A_{ab} + A_{ba} )+<br /> \frac{1}{2}( A_{ab} - A_{ba} )<br /> \\<br /> &amp;=<br /> A_{(ab)} + A_{[ab]} \\<br /> \end{align*}<br />

Note, if A is antisymmetric, then we can write
A_{ab} = A_{[ab]}.

It's not necessary to use the numerical factor (we call it "weght"). See for example the em. tensor in vacuum. U can use the "no number convention" (i use it)

F_{\mu\nu}\equiv F_{\left[\mu\nu\right]}=:\partial_{[\mu}A_{\nu]}

or the one Rob exemplified, when an ugly 2 comes up

F_{\mu\nu}\equiv F_{\left[\mu\nu\right]}=:2 \partial_{[\mu}A_{\nu]}

Daniel.

I think there is an inconsistency in your use of the brackets in the "no number convention" F_{\mu\nu}\equiv F_{\left[\mu\nu\right]}=:\partial_{[\mu}A_{\nu]}

If "bracket" means "sum the alternating permutations without dividing by the combinatorial factor", then you must write for an antisymmetric F:
F_{\mu\nu}=\frac{1}{2}F_{\left[\mu\nu\right]}=\frac{1}{2}(F_{\mu\nu}-F_{\nu\mu}) or 2F_{\mu\nu}=F_{\left[\mu\nu\right]}

 

[dextercioby]

I didn't in any place claim that

F_{[\mu\nu]}=F_{\mu\nu}-F_{\nu\mu}

So no inconsistency. Whatsoever.

Daniel.

 

[robphy]

I didn't in any place claim that

F_{[\mu\nu]}=F_{\mu\nu}-F_{\nu\mu}

So no inconsistency. Whatsoever.

Daniel.

So, maybe I am misunderstanding what your "no number convention" is.
What does \partial_{[\mu}A_{\nu]} mean in your convention?
\partial_{[\mu}A_{\nu]}\stackrel{?}{=}\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu} or
\partial_{[\mu}A_{\nu]}\stackrel{?}{=}\frac{1}{2!}\left(\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}\right)?

 

[dextercioby]

Obviously the first.

Daniel.