Galilean

Galilean spacetime is the nonrelativistic limit of Minkowski spacetime. It is the vector space $\mathbb {R} \times \mathbb {R} ^{d}$ equipped with the following structures:

• Absolute duration: given two events $E=(t,x)$ and $E'=(t',x')$ in the spacetime, one can establish the time difference $|t'-t|$ between the two. In particular, one can determine when two events are simultaneous.
• Instantaneous distance: given two simultaneous events $E=(t,x)$ and $E'=(t,x')$ in the spacetime, one can determine the distance $|x'-x|$ between the two.
• Inertial motion: given a trajectory $t\mapsto x(t)$, one can tell if this trajectory is inertial (i.e. $\partial _{tt}x(t)=0$) or not.

Galilean spacetime has a number of symmetries. In addition to the "obvious" symmetries of space and time translation, time reversal, and spatial rotation and reflection, one also has the independent dilation symmetries of space and time, as well as the (classical) Galilean symmetry

$(t,x)\mapsto (t,x-vt)$

for any fixed velocity $v\in \mathbb {R} ^{d}$.

The pseudoconformal transform

$(t,x)\mapsto ({\frac {1}{t}},x-vt)$

is a pseudoconformal transformation of Galilean spacetime; it affects infinitesimal duration and distance by a scalar factor, and preserves inertial motion.

The classical Galilean symmetry induces a corresponding quantum Galilean symmetry for ($U(1)$-invariant) Schrodinger equations, namely

$u(t,x)\mapsto e^{iv\cdot x/2}e^{i|v|^{2}t/4}u(t,x-vt).$