I was studying Tao's book on Nonlinear Dispersive Equations and came upon an exercise (2.28) where I am asked to link the pseudoconformal transformation for the Schrodinger equation: $$ (i \partial_t v + \frac{1}{2}\Delta v)(t,x) = \frac{1}{t^2}\frac{1}{(it)^{d/2}}\overline{(i\partial_t u+\frac{1}{2}\Delta u)\Bigl(\frac{1}{t},\frac{x}{t}\Bigr)}e^{i|x|^2/2t}, $$ where $$ v(t,x) = \frac{1}{(it)^{d/2}}\overline{u\Bigl( \frac{1}{t},\frac{x}{t} \Bigr)}e^{i|x|^2/2t} $$ with the conformal invariance of the wave equation (in $d$ spatial dimensions): $$ \square \tilde{u}(t,x) = (t^2-|x|^2)^{-\frac{d-1}{2}-1}\square u\Bigl( \frac{t}{t^2-|x|^2},\frac{x}{t^2-|x|^2} \Bigr), $$ where $$ \tilde{u}(t,x) := (t^2-|x|^2)^{-\frac{d-1}{2}}u\Bigl( \frac{t}{t^2-|x|^2},\frac{x}{t^2-|x|^2} \Bigr). $$
To do this, we are told to use Exercise 2.11 (which has a very straightforward solution), which describes an embedding of the Schrodinger equation in $d$ spatial dimensions into the wave equation in $d+1$ spatial dimensions. More precisely, if $u$ is a smooth function on $\mathbb{R}\times \mathbb{R}^d$, $u$ solves the Schrodinger equation $$ i \partial_t u + \frac{1}{2}\Delta u $$ if and only if $v$ solves the $1+(d+1)$-dimensional wave equation: $$ \square v = -\partial_t^2 v + \Delta v = 0, $$ where $v$ is defined by $$ v(t,x,x_{d+1}) = e^{-i(t+x_{d+1})}u\Bigl( \frac{t-x_{d+1}}{2},x \Bigr). $$
I have tried the combining the conformal invariance of the wave equation with this embedding, but I don't seem to be getting anywhere. I also haven't been able to find any other reference where this connection is explained.
Any help would be greatly appreciated.
EDIT: Tao actually posted an explanation of precisely this problem on his blog just now. Here is the link: http://terrytao.wordpress.com/2013/02/14/the-pseudoconformal-and-conformal-transformations/
