- #1

Jason Bennett

- 49

- 3

- Homework Statement
- see below

- Relevant Equations
- see below

1) Likely an Einstein summation confusion.

Consider Lorentz transformation's defined in the following matter:

Please see image [2] below.

I aim to consider the product [tex]L^0{}_0(\Lambda_1\Lambda_2).[/tex] Consider the following notation [tex]L^\mu{}_\nu(\Lambda_i) = L_i{}^\mu{}_\nu.[/tex] How then, does [tex]L^0{}_0(\Lambda_1\Lambda_2) = L_1{}^0{}_\mu L_2{}^\mu{}_0?[/tex]

2) To right the J and K generators of the Lorentz group in a compact way, one can write [tex](M^{lm})^j{}_k=i (g^

{lj}g^m{}_k - g^{mj}g^l{}_k)[/tex] where on the left hand side, it is helpful to think of l and m as indices/labels, and the j and k as rows/columns for the whole matrix.

(Apparently) One can write this in an operator representation as [tex]M^{\mu\nu} = i(x^\mu \partial^\nu-x^\nu\partial^\mu).[/tex]

a) where does this come from?

b) why and how is it used? What is it operating on, [tex]x^\mu?[/tex]

c) how does [tex]\partial^\nu x^\sigma= \frac{\partial}{\partial x_\nu} x^\sigma[/tex] equal [tex]g^{\nu\sigma}?[/tex]

[2]: https://i.stack.imgur.com/uPsLc.png

Consider Lorentz transformation's defined in the following matter:

Please see image [2] below.

I aim to consider the product [tex]L^0{}_0(\Lambda_1\Lambda_2).[/tex] Consider the following notation [tex]L^\mu{}_\nu(\Lambda_i) = L_i{}^\mu{}_\nu.[/tex] How then, does [tex]L^0{}_0(\Lambda_1\Lambda_2) = L_1{}^0{}_\mu L_2{}^\mu{}_0?[/tex]

2) To right the J and K generators of the Lorentz group in a compact way, one can write [tex](M^{lm})^j{}_k=i (g^

{lj}g^m{}_k - g^{mj}g^l{}_k)[/tex] where on the left hand side, it is helpful to think of l and m as indices/labels, and the j and k as rows/columns for the whole matrix.

(Apparently) One can write this in an operator representation as [tex]M^{\mu\nu} = i(x^\mu \partial^\nu-x^\nu\partial^\mu).[/tex]

a) where does this come from?

b) why and how is it used? What is it operating on, [tex]x^\mu?[/tex]

c) how does [tex]\partial^\nu x^\sigma= \frac{\partial}{\partial x_\nu} x^\sigma[/tex] equal [tex]g^{\nu\sigma}?[/tex]

[2]: https://i.stack.imgur.com/uPsLc.png