Rigorous treatment of infinitesimal reparametrizations

[arestes]

Hi!
Reading some string theory books I always find that the introductory chapters discuss the relativistic free particle (see Lüst-Theisen, or Becker-Becker-Schwarz, page 21, exercise 2.3).
Then they go on about showing that the action
$S=-m\int^{t_1}_{t_2} dx = -m \int^{t_1}_{t_2}d\tau\sqrt{-\frac{dx^\mu}{d \tau} \frac{dx^\nu}{d \tau} \eta_{\mu\nu}}$
is invariant under "infinitesimal reparametrizations"
$\tau \rightarrow \tau' = \tau + \xi(\tau)$,
for this they just Taylor expand
$x'^{\mu}(\tau')=x^{\mu}(\tau)$
around $\tau$ and drop terms of order $O(\xi(\tau)^2)$ to find the function "shift"
$\delta x^\mu (\tau)= x'^\mu (\tau)- x^\mu(\tau) = -\xi(\tau)\partial_\tau x^\mu(\tau)$
Two things I find annoying (even though that's how I learned it as a physicist):

1) Expanding $x'^{\mu}(\tau')=x^{\mu}(\tau)$ we get $x'^\mu(\tau)+ \xi(\tau)\partial_\tau x'^\mu(\tau)=x^\mu(\tau)$ therefore
$\delta x^\mu (\tau)= x'^\mu (\tau)- x^\mu(\tau) = -\xi(\tau)\partial_\tau x'^\mu(\tau)$ which is not the above result. The justification given in some lecture notes (this trick is also widely used in General Relativity) recall that
$x'^\mu(\tau)= x'^\mu(\tau'-\xi(\tau))=x'^\mu(\tau')-\xi(\tau)\partial_{\tau'} x'^\mu(\tau')=x^\mu(\tau)-\xi(\tau)\partial_{\tau'} x'^\mu(\tau')$.

However, when taking the tau derivative to this:
$\partial_\tau x'^\mu(\tau)=\partial_\tau x^\mu(\tau)-\partial_\tau\xi(\tau)\cdot\partial_{\tau'} x'^\mu(\tau') - \xi(\tau)\cdot\partial_\tau\partial_{\tau'} x'^\mu(\tau')$.

Now, multiplying by $-\xi(\tau)$ to get the shift of the function:
$\delta x^\mu (\tau)= -\xi(\tau)\partial_\tau x^\mu(\tau)+\xi(\tau)\cdot \partial_\tau\xi(\tau)\cdot\partial_{\tau'}x'^\mu(\tau')$ where I ommited the third term as it is quadratic in xi. However, the term that has the derivative of xi cannot be ommited since the derivative of an "infinitesimal" quantity doesn't necessarily have to be infinitesimal. Something being small does not imply that its derivative is small.

As you see, this all boils down to the heuristic treatment that Physics books give to the "infinitesimal" variation.

2) How can I reformulate rigorously the idea of a "shift" of the function $x^\mu$, maybe in terms of pushforwards and such (at the rigor of mathematics)? This would also clarify much of the above paragraph.Thanks for any help

 

[AndyW]

!

Hello,

Thank you for sharing your thoughts and concerns about the approach to infinitesimal reparametrizations in string theory books. I can understand your frustration with the lack of rigor in this treatment.

To address your first point, I agree that the justification given in some lecture notes for the difference between the two expressions for \delta x^\mu (\tau) is not completely satisfactory. To truly justify the second expression, we need to consider the limit as \xi(\tau) approaches 0, which would give us the correct result. However, this is often omitted in introductory discussions of string theory, as it can be quite technical and may not be necessary for understanding the main concepts.

To address your second point, I believe that a rigorous treatment of infinitesimal reparametrizations in string theory would involve the use of differential geometry and the concept of a tangent space. In this framework, the "shift" of the function x^\mu can be understood as a pushforward of the tangent vector \partial_\tau x^\mu by the infinitesimal reparametrization function \xi(\tau). This would involve defining the tangent space at each point along the worldline of the particle and then considering how the tangent vector is transformed under the reparametrization. This approach may require a deeper understanding of differential geometry, but it would provide a more rigorous and precise treatment of infinitesimal reparametrizations in string theory.

I hope this helps to clarify the issue and provides some direction for further exploration. Keep questioning and seeking to understand the foundations of string theory, as this is what drives progress and innovation in science. Best of luck in your studies!