Matched Filter of Gaussian Signal
If you look at:

https://en.wikipedia.org/wiki/Matched_filter

the derivation does not depend on the filter being either causal or have a finite duration. The derivation is fully satisfactory for convolution for time index $k$ over $-\infty,\infty$.  It's all about what filter will maximize the Schwarz inequality.

In this case 
$$
x(t)=\exp(-\frac{(t-0)^2}{2}) = x(0-t)
$$

Functions that are symmetric are their own matched filter.

The $T$ parameter enters the discussion in terms of when does one sample the filter for the purpose of detection.  If $x(t)$ was delayed by some unknown time(nonsynchronous detection),  I would use the matched filter to estimate it's delay.