Deriving the relativistic point particle action from Polchinski

[alfredblase]

I am an MPhys graduate currently reading Joseph Polchinski’s, String Theory, Vol. 1. Unsurprisingly I’m stuck on the first real bit of maths… :p
I quote from page 10, heh:
“The simplest Poincaré invariant action that does not depend on the parametrization would be proportional to the proper time along the world line,
S_pp = -m∫dτ(- δX¬μ/δτ δX_μ/δτ )^1/2 “
Where X¬μ is a covariant tensor of time dependent equations describing the position of the particle in all space time dimensions and X_μ is the contravariant tensor. (I don't know how to write subscripts or superscripts in this btw).
Now I understand basic tensor rules, and know that a Poincare invariant action is given by: dS = L dt. My questions are: why does the negative mass come into the equation? ; why does the negative in front of the derivative come into it?; why is the term in brackets square rooted?; and where does the pp subscript come from? Perhaps someone could point me to an article or web page that could help, or if you know the answers and would be so kind, perhaps explain this equation to me.
Any help will be much appreciated. Thanks.

 

[Physics Monkey]

I replied to your other thread in Classical Physics.

 

[alfredblase]

Thankyou very much for your speedy reply. I'm happy with the negative in the brackets now and the dimensionally necessary m. I'm looking into the other points you mentioned as I write, but I strongly suspect you have given me much needed pointers with the word "metric", the kinetic energy equation and the word "Lagrangian". :) Hopefully my physics brain will warm up once the cobwebs accumulated over long disuse have been brushed away, heh. Thanks again.

P.S. should this thread be in the homework or in the classical physics section?

 

[Physics Monkey]

Let's continue the discussion here in the homework section since this is technically a textbook question. I posted a few edits to my other post that you might want to look at, but I will post here from now on.

 

[alfredblase]

Ok this is my attempt at the question after digging up my old General Relativity notes: (p¬μ is the covariant momentum vector)
S_pp=∫L dτ
Dimensionally the energy of the particle, L = (p¬μ p_μ * X¬μ;τ X_μ;τ)^1/2
which leads to: L=(-p¬μ p_μ c^2)^1/2
as all objects travel at the speed of light in space time.
Since we work in hbar = c = 1 units then
L=-m(-X¬μ;τ X_μ;τ)^1/2
and so we have the answer!
Not sure whether I was fine finding the Lagrangian in the first instance through dimensinal analysis though... I would appreciate if someone would tell me whether that was ok or not. But I guess it must be no? if it came up with the "right" answer, I dunno...Thanks :).