Galilean

Galilean spacetime is the nonrelativistic limit of Minkowski spacetime. It is the vector space Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \R \times \R^d} equipped with the following structures:

• Absolute duration: given two events Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle E = (t,x)} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle E' = (t',x')} in the spacetime, one can establish the time difference Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle |t'-t|} between the two. In particular, one can determine when two events are simultaneous.
• Instantaneous distance: given two simultaneous events Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle E = (t,x)} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle E' = (t,x')} in the spacetime, one can determine the distance Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle |x'-x|} between the two.
• Inertial motion: given a trajectory Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle t \mapsto x(t)} , one can tell if this trajectory is inertial (i.e. Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \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

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle (t,x) \mapsto (t, x - vt) }

for any fixed velocity Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle v \in \R^d} .

The pseudoconformal transform

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle (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 (Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle U(1)} -invariant) Schrodinger equations, namely

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle u(t,x) \mapsto e^{ i v \cdot x / 2} e^{i |v|^2 t / 4} u(t, x - vt).}