Defiunition of kroneker delta as a tensor

[tamiry]

hi,

the delta symbol as a tensor (in the minkovski space, in case one has to be specific), what is it exactly?
is it
$\delta^a_b = \frac{\partial{x^a}}{\partial{x^b}}$

is it
$\delta^a_b = g^{ac} g_{cb}$or is there some other definition?thanks

 

[jambaugh]

The Kroneker delta is the the indexed components of the Identity operator which looks the same in all bases.

$$\mathbf{1}\mathbf{x} = \mathbf{x}: x^\mu = \delta^\mu_\nu x^\nu$$

 

[Chestermiller]

hi,

the delta symbol as a tensor (in the minkovski space, in case one has to be specific), what is it exactly?



is it
$\delta^a_b = \frac{\partial{x^a}}{\partial{x^b}}$

is it
$\delta^a_b = g^{ac} g_{cb}$


or is there some other definition?


thanks

It is both. The kroneker delta represents the mixed (covariant/contravariant) components of the metric tensor, and applies to both flat as well as curved spacetime.

 

[DimReg]

I believe the technical definition is:

$\delta^a_b = 0 (a != b), \delta^a_a = 1$

In other words, it's just the identity matrix. The two "definitions" you gave don't define δ, rather you can show those two expressions equal δ as I defined it.