This is now re-written using our component technology:
yi = xi cos θ + (1 − cos θ)nj xj ni + ǫikj nk xj sin θ ,
or
yi = Rij (θ, ~n) xj ,
with
Rij (θ, ~n) = δij cos θ + (1 − cos θ)ni nj − ǫijk nk sin θ .
Some properties:
• R is orthogonal. (Proof: direct, or use ~y 2 = ~x2 – see example sheet.)
•
1 + 2 cos θ = TrR
nk sin θ = − 12 ǫkij Rij .
So given R, then θ (angle) and axis ~n of rotation can be determined. (Note:
1 + (3 − 1) = 3 independent parameters, cf Euler angles.)
R
S
• Product of rotations ~x → ~y → ~z is given by ~z = SR~x.
• Small (infinitesimal) rotation δθ, then as cos δθ = 1 + O(δθ2 ), sin δθ = δθ +
O(δθ3 ) then
Rij = δij − ǫijk nk δθ .
A quicker (and sufficient) graphical proof is:
δθ
|~n × ~x|
|~n × ~x|δθ
~y
~n
~x
~y − ~x = ~n × ~xδθ ,
leading to the result.
• For θ 6= 0, π the only real eigenvalue of R is +1 having the only real eigenvector
~n (see example sheet).
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2.3
Miscellaneous transformations
For completeness we now consider reflections and projection operators.
2.3.1
Reflections
• Reflection of a vector in a plane (defined by ~n):
~n
~x
~y
is given by ~y = ~x − 2(~x · ~n)~n or
yi = σij xj
where σij = δij − 2ni nj .
• Inversion/Reflection of a vector in the origin is given by ~y = −~x or
yi = Pij xj
where Pij = −δij .
P is called the parity operator.
Note that for reflections, performing the operation twice yields the original vector,
ie σ 2 = I, P 2 = I.
2.3.2
Projection operators
P is a parallel projection operator onto a vector ~u if
P ~u = ~u ,
P ~v = 0 ,
where ~v is any vector orthogonal onto ~u, ie ~v · ~u = 0. Similarly Q is an orthogonal
projection to ~u if
Q~u = 0 ,
Q~v = ~v ,
(so Q = I − P ). Suitable operators are
Pij =
ui uj
,
u2
Qij = δij −
They have the properties that
P2 = P ,
Q2 = Q ,
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ui uj
.
u2
and are also unique. For example if there exists another operator T with the same
properties as P , ie T ~u = ~u, T~v = 0, then for any vector w
~ ≡ µ~u + ν~v + λ~u × ~v we
have
(P − T )w
~ = (µ~u + 0 + 0) − (µ~u + 0 + 0) = 0 ,
(as (P (~u × ~v ))i = Pij (~u × ~v )j = (ui uj /u2 ) ǫjkl uk vl = 0) and hence T = P .
2.4
Active and Passive Transformations
• rotating a vector in a fixed basis is called an active transformation, ~x → ~y with
yi = Rij xj in {~ei } basis.
• rotating the basis keeping the vector fixed is called a passive transformation,
{~ei } → {~e′i } or xi → x′i = lij xj .
Let us set Rij = lij then numerically yi = x′i , and consider (for simplicity):
1. Rotation of vector about z-axis:
~e2
~y
θ
~x
~e1
~e3
Rij (θ, ~e3 ) = δij cos θ + (1 − cos θ)δi3 δj3 − ǫijk δk3 sin θ


cos θ − sin θ 0
cos θ 0  ,
=  sin θ
0
0 1
(as ni = δi3 ). This is an active transformation through angle θ.
2. Rotation of basis about z-axis:
~e′i = lij ~ej ≡ Rij ~ej ,
or
~e′1 = cos θ~e1 − sin θ~e2
~e′2 = sin θ~e1 + cos θ~e2
~e′3 = ~e3 .
This is a passive transformation through angle −θ:
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~e2
~e′2
~x
~e1
θ
~e′1
~e3, ~e′3
Hence:
Active rotation of ~x through angle θ is the same as passive rotation of basis vectors
by an equal and opposite amount.
The general case can be built from three rotations (Euler angles).
2.4.1
Rotational and reflection symmetries and the SO(3)
group
Recalling, as discussed previously, rotation of a vector in a fixed basis is ~y = R~x
where
Rij (θ, ~n) = δij cos θ + (1 − cos θ)ni nj − ǫijk nk sin θ .
For an infinitesimal rotation δθ → 0 we thus have (upon expanding or using the
previous geometrical argument)
Rij (δθ, ~n) = δij − ǫijk nk δθ + O(δθ2)
= δij − ink (Tk )ij δθ + . . . ,
with
(Tk )ij = −iǫijk .
So

0 0
0
T1 = i  0 0 −1  ,
0 1
0


0 0 1
T2 = i  0 0 0  ,
−1 0 0


0 −1 0
0 0 .
T3 = i  1
0
0 0

The Ti are called the generators of infinitesimal rotations about basis vectors ~ei (the
−i is conventional so Ti are Hermitian, Ti† ≡ TiT ∗ = Ti ).
So infinitesimal rotation through angle δθ about ~n is
R(δθ, ~n) = 1 − iδθ ~n · T~ .
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