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Facts:

  1. The Maxwell (free) equations (4d) are invariant under the 15 dimensional conformal group.

  2. The free Schrödinger equation in 3d is invariant under the 15 dimensional group "called" Schrödinger group (15 dimensional too in 4d).

What is the relationship between those two groups (are they the same or there is any subtle difference?)?

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    $\begingroup$ Do you mean the free Schrodinger equation is invariant...? $\endgroup$ – G. Smith Sep 24 at 21:46
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    $\begingroup$ Did you read the last section of the Wikipedia article you linked? $\endgroup$ – octonion Sep 25 at 2:16
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    $\begingroup$ The conformal group is simple; the Schrödinger group is not (it is not even semi-simple). They are different groups. $\endgroup$ – AccidentalFourierTransform Sep 25 at 2:17
  • $\begingroup$ And also note that the Schrödinger group under which the 3d Schrödinger equation is invariant is only 12-dimensional, not 15-dimensional. This is not clear in your question. $\endgroup$ – M.Jo Sep 25 at 8:10
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The answer to the OP question

What is the relationship between those two groups (are they the same or there is any subtle difference?)?

is offered by Niederer, U. in his Helv.Phys.Acta article called "The connection between the Schrödinger group and the conformal group", Vol. 47, p.119 (found at http://dx.doi.org/10.5169/seals-114561 and freely download-able). From this paper you find the reference [3] which is another "bite" (same journal, but Vol. 46) at this topic by the famous Asim Barut:

"Conformal Group -> Schrödinger Group -> Dynamical Group - The Maximal Kinematical Group of the Massive Schrödinger Particle"

(the the direct link is http://doi.org/10.5169/seals-114492).

I would first read Barut's work, then Niederer's.

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