where is the unknown field and 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 (if one wishes, one can restrict 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 solves (1), then the pseudoconformal solution defined by
for can be seen after some computation to also solve (1). (If has suitable decay at spatial infinity and one chooses a suitable branch cut for , one can extend continuously to the spatial slice, whereupon it becomes essentially the spatial Fourier transform of , but we will not need this fact for the current discussion.)
where is the unknown field. In analogy to pseudoconformal symmetry, we have conformal symmetry: if solves (3), then the function , defined in the interior of the light cone by the formula
also solves (3).
There are also some direct links between the Schrödinger equation in dimensions and the wave equation in dimensions. This can be easily seen on the spacetime Fourier side: solutions to (1) have spacetime Fourier transform (formally) supported on a -dimensional hyperboloid, while solutions to (3) have spacetime Fourier transform formally supported on a -dimensional cone. To link the two, one then observes that the -dimensional hyperboloid can be viewed as a conic section (i.e. hyperplane slice) of the -dimensional cone. In physical space, this link is manifested as follows: if solves (1), then the function defined by
solves (3). More generally, for any non-zero scaling parameter , the function defined by
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 (or more generally ) to explicitly connect together the pseudoconformal transformation and the conformal transformation . It turns out that this connection is a little bit unusual, with the “obvious” guess (namely, that the embeddings intertwine and ) 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
for some energy level and some function . (The function then obeys the time-independent Schrödinger equation , although we will not use this equation here.) The connection between and is then as follows:
for any non-zero .
Thus, the conformal transformation is not intertwined with the pseudoconformal transformation via the embeddings , but instead the conformal transformation relates different embeddings of the pseudoconformal transform of a solution to (1) to each other, where the precise embeddings involved depend on the energy level of the original solution .
and thus by (5)
where we abbreviate and also introduce the null coordinates , . Applying (4) and noting that , we conclude that
is equal to
We can cancel out the factors of to get
Next, we can combine
to simplify the previous expression as
which we can rearrange a little as
Comparing this with (7), we see that this expression is equal to as desired.