Quantum Mechanics for Mathematicians

If one then acts on the result with (a 1 ,R 1 ) one gets

(a 1 ,R 1 )·((a 2 ,R 2 )·v) = (a 1 ,R 1 )·(a 2 +R 2 v) =a 1 +R 1 a 2 +R 1 R 2 v

Note that this is not what one would get if one took the product group law on
R^3 ×SO(3), since then the action of (a 1 ,R 1 )(a 2 ,R 2 ) onR^3 would be

v→a 1 +a 2 +R 1 R 2 v

To get the correct group action onR^3 , one needs to takeR^3 ×SO(3) not with
the product group law, but instead with the group law

(a 1 ,R 1 )(a 2 ,R 2 ) = (a 1 +R 1 a 2 ,R 1 R 2 )

This group law differs from the standard product law by a termR 1 a 2 , which is
the result ofR 1 ∈SO(3) acting non-trivially ona 2 ∈R^3. We will denote the
setR^3 ×SO(3) with this group law by

R^3 oSO(3)

This is the group of orientation-preserving transformations ofR^3 preserving the
standard inner product.
The same construction works in arbitrary dimensions, where one has:

Definition(Euclidean group).The Euclidean groupE(d)(sometimes written
ISO(d)for “inhomogeneous” rotation group) in dimensiondis the product of
the translation and rotation groups ofRdas a set, with multiplication law

(a 1 ,R 1 )(a 2 ,R 2 ) = (a 1 +R 1 a 2 ,R 1 R 2 )

(whereaj∈Rd,Rj∈SO(d)) and can be denoted by

RdoSO(d)

E(d) can also be written as a matrix group, taking it to be the subgroup of
GL(d+ 1,R) of matrices of the form (Ris adbydorthogonal matrix,aad
dimensional column vector) (
R a
0 1

)

One gets the multiplication law forE(d) from matrix multiplication since

(

R 1 a 1

0 1

)(

R 2 a 2

0 1

)

=

(

R 1 R 2 a 1 +R 1 a 2

0 1

)

18.2 Semi-direct product groups

The Euclidean group example of the previous section can be generalized to the
following: