Consider the free Schrödinger equation in {d} spatial dimensions, which I will normalise as

\displaystyle i u_t + \frac{1}{2} \Delta_{{\bf R}^d} u = 0 \ \ \ \ \ (1)

where {u: {\bf R} \times {\bf R}^d \rightarrow {\bf C}} is the unknown field and {\Delta_{{\bf R}^{d+1}} = \sum_{j=1}^d \frac{\partial^2}{\partial x_j^2}} is the spatial Laplacian. To avoid irrelevant technical issues I will restrict attention to smooth (classical) solutions to this equation, and will work locally in spacetime avoiding issues of decay at infinity (or at other singularities); I will also avoid issues involving branch cuts of functions such as {t^{d/2}} (if one wishes, one can restrict {d} to be even in order to safely ignore all branch cut issues). The space of solutions to (1) enjoys a number of symmetries. A particularly non-obvious symmetry is the pseudoconformal symmetry: if {u} solves (1), then the pseudoconformal solution {pc(u): {\bf R} \times {\bf R}^d \rightarrow {\bf C}} defined by

\displaystyle pc(u)(t,x) := \frac{1}{(it)^{d/2}} \overline{u(\frac{1}{t}, \frac{x}{t})} e^{i|x|^2/2t} \ \ \ \ \ (2)

for {t \neq 0} can be seen after some computation to also solve (1). (If {u} has suitable decay at spatial infinity and one chooses a suitable branch cut for {(it)^{d/2}}, one can extend {pc(u)} continuously to the {t=0} spatial slice, whereupon it becomes essentially the spatial Fourier transform of {u(0,\cdot)}, but we will not need this fact for the current discussion.)

An analogous symmetry exists for the free wave equation in {d+1} spatial dimensions, which I will write as

\displaystyle u_{tt} - \Delta_{{\bf R}^{d+1}} u = 0 \ \ \ \ \ (3)

where {u: {\bf R} \times {\bf R}^{d+1} \rightarrow {\bf C}} is the unknown field. In analogy to pseudoconformal symmetry, we have conformal symmetry: if {u: {\bf R} \times {\bf R}^{d+1} \rightarrow {\bf C}} solves (3), then the function {conf(u): {\bf R} \times {\bf R}^{d+1} \rightarrow {\bf C}}, defined in the interior {\{ (t,x): |x| < |t| \}} of the light cone by the formula

\displaystyle conf(u)(t,x) := (t^2-|x|^2)^{-d/2} u( \frac{t}{t^2-|x|^2}, \frac{x}{t^2-|x|^2} ), \ \ \ \ \ (4)

also solves (3).

There are also some direct links between the Schrödinger equation in {d} dimensions and the wave equation in {d+1} dimensions. This can be easily seen on the spacetime Fourier side: solutions to (1) have spacetime Fourier transform (formally) supported on a {d}-dimensional hyperboloid, while solutions to (3) have spacetime Fourier transform formally supported on a {d+1}-dimensional cone. To link the two, one then observes that the {d}-dimensional hyperboloid can be viewed as a conic section (i.e. hyperplane slice) of the {d+1}-dimensional cone. In physical space, this link is manifested as follows: if {u: {\bf R} \times {\bf R}^d \rightarrow {\bf C}} solves (1), then the function {\iota_{1}(u): {\bf R} \times {\bf R}^{d+1} \rightarrow {\bf C}} defined by

\displaystyle \iota_{1}(u)(t,x_1,\ldots,x_{d+1}) := e^{-i(t+x_{d+1})} u( \frac{t-x_{d+1}}{2}, x_1,\ldots,x_d)

solves (3). More generally, for any non-zero scaling parameter {\lambda}, the function {\iota_{\lambda}(u): {\bf R} \times {\bf R}^{d+1} \rightarrow {\bf C}} defined by

\displaystyle \iota_{\lambda}(u)(t,x_1,\ldots,x_{d+1}) :=

\displaystyle \lambda^{d/2} e^{-i\lambda(t+x_{d+1})} u( \lambda \frac{t-x_{d+1}}{2}, \lambda x_1,\ldots,\lambda x_d) \ \ \ \ \ (5)

solves (3).

As an “extra challenge” posed in an exercise in one of my books (Exercise 2.28, to be precise), I asked the reader to use the embeddings {\iota_1} (or more generally {\iota_\lambda}) to explicitly connect together the pseudoconformal transformation {pc} and the conformal transformation {conf}. It turns out that this connection is a little bit unusual, with the “obvious” guess (namely, that the embeddings {\iota_\lambda} intertwine {pc} and {conf}) being incorrect, and as such this particular task was perhaps too difficult even for a challenge question. I’ve been asked a couple times to provide the connection more explicitly, so I will do so below the fold.

To state the connection, it turns out that one must first use separation of variables and restrict to a subclass of solutions to (1), namely those of the form

\displaystyle u(t,x) = e^{-iEt} \phi(x) \ \ \ \ \ (6)

for some energy level {E \in {\bf R}} and some function {\phi: {\bf R}^d \rightarrow {\bf C}}. (The function {\phi} then obeys the time-independent Schrödinger equation {-\frac{1}{2} \Delta_{{\bf R}^d} u = E u}, although we will not use this equation here.) The connection between {pc} and {conf} is then as follows:

Proposition 1 Let {u} be a solution to (1) of the form (6) for some energy {E}. Then we have

\displaystyle conf( \iota_{\lambda}( pc(u) ) ) = \iota_{-2E/\lambda}( pc( u ) )

for any non-zero {\lambda}.

Thus, the conformal transformation is not intertwined with the pseudoconformal transformation via the embeddings {\iota}, but instead the conformal transformation relates different embeddings of the pseudoconformal transform of a solution {u} to (1) to each other, where the precise embeddings involved depend on the energy level {E} of the original solution {u}.

We can verify this proposition by a straightforward calculation. From (6) and (2) we have

\displaystyle pc(u)(t,x) = \frac{1}{(it)^{d/2}} \overline{\phi}(\frac{x}{t}) e^{iE/t} e^{i|x|^2/2t}

and thus by (5)

\displaystyle \iota_{\lambda}(pc(u))(t,x,x_{d+1}) = \frac{e^{-i\lambda a}}{(ib/2)^{d/2}} \overline{\phi}(\frac{2x}{b}) e^{2iE / \lambda b} e^{i\lambda |x|^2/b} \ \ \ \ \ (7)

where we abbreviate {x = (x_1,\ldots,x_d)} and also introduce the null coordinates {a := t+x_{d+1}}, {b := t-x_{d+1}}. Applying (4) and noting that {t^2-|(x,x_{d+1})|^2 = ab - |x|^2}, we conclude that

\displaystyle conf(\iota_{\lambda}(pc(u)))(t,x,x_{d+1})

is equal to

\displaystyle (ab-|x|^2)^{-d/2} \frac{e^{-i\lambda a/(ab-|x|^2)}}{(ib/2(ab-|x|^2))^{d/2}} \overline{\phi}(\frac{2x}{b}) e^{2iE (ab-|x|^2)/ \lambda b} e^{i\lambda |x|^2/b(ab-|x|^2)}.

We can cancel out the factors of {(ab-|x|^2)^{-d/2}} to get

\displaystyle \frac{e^{-i\lambda a/(ab-|x|^2)}}{(ib/2)^{d/2}} \overline{\phi}(\frac{2x}{b}) e^{2i\lambda E (ab-|x|^2)/ \lambda b} e^{i\lambda |x|^2/b(ab-|x|^2)}.

Next, we can combine

\displaystyle e^{-i\lambda a/(ab-|x|^2)} e^{i\lambda |x|^2/b(ab-|x|^2)} = e^{-i\lambda /b}

to simplify the previous expression as

\displaystyle \frac{e^{-i\lambda /b}}{(ib/2)^{d/2}} \overline{\phi}(\frac{2x}{b}) e^{2iE (ab-|x|^2)/ \lambda b}

which we can rearrange a little as

\displaystyle \frac{e^{2iE a / \lambda}}{(ib/2)^{d/2}} \overline{\phi}(\frac{2x}{b}) e^{-i\lambda /b} e^{-2i E |x|^2/\lambda b}.

Comparing this with (7), we see that this expression is equal to {\iota_{-2E/\lambda}(pc(u))(t,x,x_{d+1})} as desired.