Understand Noether's Theorem w/ Lagrangian Example

[vertices]

Hi

I was wondering if someone would be kind enough to help me understand an example in my class notes:

If we have a Lagrangian:

$$L=m(\dot{z}\dot{z^{*}})-V(\dot{z}\dot{z^{*}})$$

where z=x+iy.

Why does it follow that

$$Q=X^{i}\frac{{\partial}L}{{\partial}\dot{q}^{i}}$$

is equal to:

$$X\frac{{\partial}L}{{\partial}\dot{z}}+X^{*}\frac{{\partial}L}{{\partial}\dot{z^{*}}}$$?

I mean, mathematically that seems wrong, why are we adding the second term (the one with the complex conjugate of $$\dot{z}$$)

Also, can I check if my understanding of the superscript 'i' is correct - does it correspond to the axes, for example i=1 corresponds to the x axis, i=2 corresponds to the y axis, etc. If z = x + iy, we are no longer talking about a 3 dimensional real space, so how are the superscripts relavent?

And is there a reason why the superscript 'i' has gone in the second line?

thanks

 

[Dale]

Also, can I check if my understanding of the superscript 'i' is correct - does it correspond to the axes, for example i=1 corresponds to the x axis, i=2 corresponds to the y axis, etc. If z = x + iy, we are no longer talking about a 3 dimensional real space, so how are the superscripts relavent?

The superscripts are confusing to me. e^(0 i pi/2) is 1 which is the x-axis in the complex plane, e^(1 i pi/2) is i which is the y-axis in the complex plane, e^(2 i pi/2) is -1 which is the -x axis in the complex plane, and e^(3 i pi/2) is -i which is the -y axis in the complex plane. Are the superscripts related to that somehow?

 

[vertices]

The superscripts are confusing to me. e^(0 i pi/2) is 1 which is the x-axis in the complex plane, e^(1 i pi/2) is i which is the y-axis in the complex plane, e^(2 i pi/2) is -1 which is the -x axis in the complex plane, and e^(3 i pi/2) is -i which is the -y axis in the complex plane. Are the superscripts related to that somehow?

I am not sure about this, as the x,y, and z axes should be orthogonal to each other, in other words they are linearly independent - ie. you can't define z in the way that it has been defined (z=x+iy), as a linear combination of x and y...

 

[arunma]

Call me crazy, but don't you need to first have a transformation of the Lagrangian before you can define a Noether current and a conserved charge?

BTW, in my QFT class I never had to deal with a complex-valued coordinate. But I'm guessing that x and y in this context are both four vectors (and therefore z as well). I could be wrong...