Can someone explain the intuition behind this moment generating function identity? If $X_i \sim N(\mu, \sigma^2) $, we know that: $\bar{X} \sim N(\mu, \sigma^2 /n)$.
But why does:
$$\exp\left({\sigma^{2}\over 2}\sum_{i=1}^{n}(t_{i}-\bar{t})^{2}\right)= M_{X_{1}-\bar{X},X_{2}-\bar{X},...,X_{n}-\bar{X}}(t_1,t_2,...,t_n)$$
Where $M$ is the moment generating function?
I have three pages of scratch work but it would be incredibly tedious to post that here, and I already know it's true...
Thanks!

This identity relies on the fact that 
$$\sum_{i=1}^nt_iX_i-\sum_{i=1}^nt_i\bar X=\sum_{j=1}^n(t_j-\bar t)X_j.$$
