Abstract. A paper on didactics in mathematics is presented.
The paper outlines a method, to derive angle composition in Rn
by natural algebraic modeling of some concepts of discrete planar geometry.
Extending the presented solution recipe from planes to Rn delivers
the angular composition formula together with a constructive interpretation.
This concept is well apt, to design and establish adequate teaching
material to introduce operator algebras for geometry on one hand
and to get a starting point for Clifford Algebras on the other hand.
Contents
0. Introduction
1. Dirac Notation for Vectors as a Notational Aid
2. Modeling Reflections
3. Angles in Planes
3.1. Modeling 2-Dimensional Planarity and Right Angles
3.1.1. Component Formula for Planarity Test
3.1.2. Symmetric Formula for Planarity Test
3.1.3. The Scalar Denominator as a Surface Element
3.1.4. The Operator Part as Right Angle Turn
3.2. Angles and Units for Angles
3.2.1. Modeling Angles by a Mirror Recursion
3.2.2. The Geometric Product to Model Angles
3.2.3. Some Linear Algebra Topics in a Plane
3.2.4. Planar Geometry as an Algebra in a Complex Plane
4. Decomposition of Axis Rotations by Reflections
5. Concatenation of Axis Rotations in Space
6. Extensions of the Geometric Product
6.1. Multilinear Mappings
6.2. Operator Basis and Quaternions
7. Conclusion
0. Introduction.<x|a> .This writing is known to most people, who heard once about vector calculation. Using such Dirac brackets3),4) for the inner product and also for vectors consequently, simplifies the thinking in linear algebra considerably. It is the only basic notation, we need for vector calculation: Dirac has developed the idea to allow, to break up this scalar product into its parts. Then
|a> may be considered as the vector a itself.This is the same idea as first defining a relation f(x) by a given formula and then saying, that there is a function f defined. The function f: x → <x|a> just defines the vector a by all its projections. Of course we know, that the projections to canonical unit directions |1> ... |n> are already satisfying
|a> = |1><1|a> + |2><2|a> + ... + |n><n|a> .It is just another writing of the component form for vectors. This looks like
Identity = |1><1| + |2><2| + ... + |n><n|We get the identity mapping (usually called identity matrix) without ever having introduced matrices as a separate concept. One gets a presentation of linear mappings and e.g. matrix multiplication and you do not even feel, that you have to learn something new. This is just the idea of a mnemonic aid! Do not try to see something special, not even because this notation is connected to Dirac, a famous Noble price winner. It is and remains as simple as it looks. For people having seen matrices before, we only remark, that the matrix belonging to a linear mapping A with matrix elements ai,j in an orthonormal basis, is alternatively written in the form
A = Σ|i><i|A|j><j| = Σai,j|i><j|So, in Dirac notation <i|A|j> is the Matrix element and |i><j| points to the position in the matrix, where the element belongs to. And matrix algebra can even be written down, without using some graphical typesetting system. Don't ask me, why conventional matrix notation did not already die out. Physicists and engineers, used to Einstein sum convention for tensors, may see just a version of this sum convention in Dirac notation. In matrix notation the distinction between |a> and <a| is given by writing column and row vectors.
H = |1><1| + |2><2| + ... + |k-1><k-1| - |k><k| + |k+1><k+1| + ... + |n><n|
= Identity - 2|k><k|
In one dimension this direction inversion beyond shifts is the only non-trivial isomorphy.
And one could show, that any orthogonal transformation could generally be
decomposed in reflections. So the most simple orthogonal mapping is,
up to composition, the most general too. This is the key to the considerations below. But
we will derive anything carefully.
|x> = λ1|a1> + λ2|a2>
("plane equation with parameters")
We can calculate λ1 and λ2 from |x>, |a1> and |a2>.
Multiplying the defining equation of |x> by <b1| and by <b2|
one obtains two linear equations for λ1 and λ2,
which are readily solved for λ1 and λ2:
<a1|{|b2><b1| - |b1><b2|}|x>
λ2 = - —————————————————————————————— and
<b1|a1><b2|a2> - <b1|a2><b2|a1>
<a2|{|b2><b1| - |b1><b2|}|x>
λ1 = ——————————————————————————————
<b1|a1><b2|a2> - <b1|a2><b2|a1>
We finally get rid of the two parameters, if we reinsert this
result in the equation defining |x>
-(<b1|a1><b2|a2> - <b1|a2><b2|a1>)|x> = (|a2><a1|-|a1><a2|)(|b2><b1|-|b1><b2|)|x>
I.e. at {|a1>,|a2>}-plane the following identity is valid:
-(<b1|a1><b2|a2> - <b1|a2><b2|a1>)Pa1,a2 = (|a2><a1|-|a1><a2|)(|b2><b1|-|b1><b2|)
Pa1,a2 is the Identity I on the {|a1>,|a2>}-plane.
-(<1|a1><2|a2> - <1|a2><2|a1>)Pa1,a2 = (|a2><a1|-|a1><a2|)(|2><1|-|1><2|)
Though we already calculated with complex numbers at high school,
we did not recognize the imaginary unit |2><1|-|1><2|.
Instead we did the same solution procedure again and again.
- (<a|a> <b|b> - <a|b>2)P = ( |b><a| - |a><b| )2
where we shortened our notation a little bit by using
|a> and |b> instead of |a1> and |a2> .
Let us analyze both sides of the equation a little bit more.
<a|a> <b|b> - <a|b>2 = EG - F2
is well known as so called surface element6).
But it can be seen by a pretty elementary calculation,
that it is a complete square.
<a|a><b|b>-<a|b>2
= <a|a>(<b|b> - <a°|b>2)
= <a|a> ||b> - |a°><a°|b>|2
School kids would discover here the area formula length times height.
Mathematicians would speak of Schmidt orthogonalization procedure.
In any case, here it follows from simply playing with algebra, to
get e.g. geometric interpretations of the formula, to get some
physical imagination of pure algebra.
There is another possibility too, to represent the left hand side
as a positive quantity
<a|a><b|b>-<a|b>2
= <a|a><b|b>(1 - <a°|b°>2)
= <a|a><b|b>(1 + <a°|b°>)(1 - <a°|b°>)
= <a|a><b|b><a°+b°|a°+b°><a°-b°|a°-b°>/4
Diagonals of a parallelogram and half angles allow to calculate an area.
Only to provide a prospect, we remark, that the left hand side in general is known
as Gram's determinant G2:
G2(a1,a2;b1,b2) = <b1|a1><b2|a2> - <b1|a2><b2|a1>
By Hadamard inequality for Gn one could conclude,
that alternating tensors together with Gram's determinant deliver
vector spaces for directed areas. This is the natural switch to Grassmann algebras.
But we don't follow this way, we only remark a possible extension of this
very elementary geometric treatment.
A := |b><a| - |a><b|
turns every vector |c> in Rn by 90° as the
pure algebraic equation shows:
<c| (|b><a| - |a><b|)|c> = 0
Of course, then A also turns all vectors in {|a>,|b>}-plane by 90°.
Beyond turning, there is also a scaling in A which we already know:
<Ac| Ac> = <c| ATAc> = (<a|a> <b|b> - <a|b>2)<c|Pc>
If we scale A instead by setting
A° := A/sqrt(<a|a> <b|b> - <a|b>2), if the denominator is not 0,
then A° is a pure 90° rotation in the plane.
<c| A°c> = 0 and <A°c| A°c> = <c|c>
Just remember that P is the Identity on the plane.
Generally, in this symmetric case, P is the projection onto the plane:
P = A°TA° and P2 = P
So we know exactly, what the solution of our starting problem,
the linearity test, means geometrically: It tests, whether the projected
vector has the original direction (but an opposite directional sense) after
a turn of 180°.
|a0> := |a> 0-angle
|a1> := |a||b>/|b| = |b°><a°|a> "unit angle"
(|an+1> + |an-1>)/2 = <b°|a°> |an> (Mirror Recursion)
From geometry this recursion becomes immediately evident.
It is the bisection relation for angles.
But let us stop for a moment, to understand this recursion from a more
algebraic point of view. Then we might feel more comfortable.
Mirroring means here, changing the sign of the axis orthogonal to |an> in the {|a>,|b>}-plane.
Written down in components, this (Householder) reflection reads
<an°| an+1> = <an°| an-1>
(I-|an°><an°|)|an+1> = -(I-|an°><an°|)|an-1>
and added up we get
|an+1> = 2 |an°><an°|an-1> - |an-1>
Note that, by induction, all |an> lie in the {|a>,|b>}-plane.
<an|an+1> = <an|an-1> = ... = <a0|a1> = <a|a><a°|b°>
and by squaring we cancel out the sign change effect
<an+1|an+1> = <an-1|an-1> = ... = <a|a>
Together we observe, that the coefficient is constant
<an|an-1>/<an|an> = <b°|a°>
This derivation only moves the starting point, from a geometric construction to
more elementary geometric argumentation. There are also other realistic
problems to come to this recursion.7)
|an><b°|a°> - |an-1> = |an><an-1°|an°> - |an-1><an°|an°>
= (|an><an-1°| - |an-1><an°|)|an°>
= (|an°><an-1°| - |an-1°><an°|)|an>
All |an> lie in the {|a>,|b>}-plane
and there exists, up to a sign, only one 90° rotation. Therefore one expects, at least up to a sign
|an°><an-1°| - |an-1°><an°| = |b°><a°| - |a°><b°|
Formally this is deduced again by the recursion itself
|an+1><an| - |an><an+1| = |an><an-1| - |an-1><an|
= ...
= |a1><a0| - |a0><a1| = |b><a| - |a><b|
So a part of the recursion looks like
|an><b°|a°> - |an-1> = (|b°><a°| - |a°><b°|)|an> ,
and we can reduce the original 3 term recursion to a 2 term one
|an+1> = |an><b°|a°> + (|b°><a°| - |a°><b°|) |an> for n>1
= {<b°|a°>I + (|b°><a°| - |a°><b°|)}|an>
In fact, we came to that point by using an analysis of invariants in our recursion. During our derivation we
even identified two invariants, two successful concepts. There was the index independent projection
and now we rediscovered the universal 90° turn, a 2-dimensional invariant. Further inspection
of such invariants could lead us to Grassmann spaces. But here we don't want to follow this way.
The reduction has a considerable price.
The reduced recursion is an operator recursion now.
And we remark, that the choice of the operator is not even unique.
For the moment we better restrict the operator to the {|a>,|b>} plane,
we started from. I.e. we use the projection P to the plane instead of the identity operator I above.
This is only an aid not to leave our planar angle interpretation. A priori there is
no need for that, so we just remark, that other physical interpretations like spin
live from the fact, that there are different solution operators. After this passing remark,
let us concentrate on the solution of our special mean value recursion.
The operator recursion provides the possibility, to "solve" the recursion
by continued application of this operator:
|an> = {<b°|a°>I + (|b°><a°| - |a°><b°|)}n|a>
The recursion is solved by the trace of an operator group.
In any case, our visual imagination about isotropy of a plane, makes us expect,
that there should be such a position independent operator.
Having a closed form formula does not mean, that everything is understood.
The job is now, to analyze this formula, because the algebraic properties
of a solution are still hidden in the operator product.
Formally, if one only defines the solution operator, a so called propagator,
the validity of the recursion could also be proofed by a direct calculation this way:
Denote
S(b°,a°) := <b°|a°>I + (|b°><a°| - |a°><b°|)
then squaring delivers by use of the plane identity
S(b°,a°)2 = <b°|a°>2I + 2<b°|a°>(|b°><a°| - |a°><b°|) - (1 - <b°|a°>2)P
= 2<b°|a°>S(b°,a°)P - P + <b°|a°>2(I - P)
with P the projection to {|a>, |b>}-plane .
Just for completeness, we remark, that we also know
a closed form presentation for the Projection P:
P ≡ |a°><a°| + |a'°><a'°| = -(<a|a>/<a'|a'>)(|b><a| - |a><b|)2
where |a'> := (|b><a| - |a><b|)|a> is orthogonal to |a> and
<a'|a'> = <a|a>(<a|a><b|b>-<a|b>2)
The I - P term cancels out by applying to vectors in the {|a>, |b>}-plane.
We could have worked with P instead of I, to solve our planar recursion.
Or in other words, we may use any extension of the planar propagator to
the whole space; anyhow, in the moment, we are only interested in the
planar effect. We can formally restrict our argumentation to the plane
by observing, that the form
S = SP + <b°|a°>(I - P)
is stable under repeated application:
Sn = (SP)n + <b°|a°>n(I - P)
Therefore SP fulfills the mean value equation without additional terms
(SP)2 = 2<b°|a°>SP - P
and consequently the recursion
(SP)n+1 = 2<b°|a°>(SP)n - (SP)n-1 .
Therefore Sn|a> = (SP)n|a> fulfills the |an> recursion.
All this is essentially a consequence of the plane test equation.
People working with matrices will immediately discover, that S represents an
orthogonal transformation. Physicists will see a propagator and so on.
And people working with geometric algebra will note that S is
a realization of the geometric product of a° and b°!
And in the 4 dimensional real geometric algebra it is the quaternionic product.
Analyzing the behavior of this geometric product (here with 2 factors only),
one can build up linear algebra without using any
(continuous and non-linear) trigonometric functions.
(SP)2 = (2<b°|a°>2 - 1)P + 2<b°|a°>(|b°><a°| - |a°><b°|)
Applying this result to geometrically halved angles, delivers by direct calculation
(S((a°+b°)°,a°)P)2
= (2<(a°+b°)°|a°>2 - 1)P + 2<(a°+b°)°|a°>(|(a°+b°)°><a°| - |a°><(a°+b°)°|)
= <b°|a°>P + |b°><a°| - |a°><b°|
= S(b°,a°)P
Of course, we expect this from geometric appearance. But nevertheless, one
has to confirm it algebraically.
We remark, that the calculation just used the trivial relation
<(a°+b°)°|a°> = <(a°+b°)°|b°>
So we may define consistently a root of the operator S by
(S(b°,a°)P)1/2 := S((a°+b°)°,a°)P
and this root halves angles:
(S(b°,a°)P)1/2|a> = S((a°+b°)°,a°)P|a> = |a||(a°+b°)°> ,
half the way from a to b.
On the other hand we may half angles algebraically by
2<b°|a°>SP = (SP)2 + P
or applied to geometrically halved angles
2<(a°+b°)°|a°>S((a°+b°)°,a°)P = (S((a°+b°)°,a°)P)2 + P = S(b°,a°)P + P
= |a°+b°|S((a°+b°)°,a°)P
To get this formula compact and even better to memorize, one is
attempted to extend S linearly from unit vectors to all vectors to get:
S(a°+b°,a°)P ≡ |a°+b°|S((a°+b°)°,a°)P = S(b°,a°)P + P
So we are in a position to describe bisectional refinement of angle measurement
purely algebraically without reverting to geometry. By bisection
and doubling, all angles may be compared naturally.
Because SP is closed under bisection
of angles, one can refine an angular unit by halving it. So every angle
may be compared with an arbitrary angle, given by the directions a and b, by a dual
fraction representation, gained by continued bisection. From this observation
the existence and analyticity of exp, cos and sin may be derived.
And these procedures can be expressed by well defined powers of the operator S.
By binary fraction expansion of a power t, angles may be measured
by St starting with two arbitrary linear independent directions a and b.
Conventional angle measurement is based on a special choice of a and b, namely
orthogonal directions, where S is just reduced to the right angle turn part.
Doing discrete geometry here, we don't need this continuous angle measurement.
But we wanted to make this remark to show, that the given presentation
may be completed naturally in a sweeping, overall treatment; it is not a singular treatment
of only one effect.
S(c°,b°)S(b°,a°)P = S(c°,a°)P if |c> = λ|a> + μ|b>
But we better calculate this result algebraically.
S(c°,b°)S(b°,a°)P = {<c°|b°>P + (|c°><b°| - |b°><c°|)}{<b°|a°>P + (|b°><a°| - |a°><b°|)}
= <c°|b°><b°|a°>P
+ |b°> <<b°|c°>a°-<b°|a°>c°| - |<b°|c°>a°-<b°|a°>c°> <b°|
+ (|c°><b°| - |b°><c°|)(|b°><a°| - |a°><b°|)
The second term is evaluated by the fact, that <b°|c°>a°-<b°|a°>c° is orthogonal to b°
and then right angle turn normalizing according to 3.1.4. delivers
|b°> <<b°|c°>a°-<b°|a°>c°| - |<b°|c°>a°-<b°|a°>c°> <b°|
= (|<b°|c°>a°-<b°|a°>c°|2/(1-<c°|a°>2)) (|c°><a°| - |a°><c°|)
= (-<b°|{|c°><a°| - |a°><c°|}2|b°>/(1-<c°|a°>2)) (|c°><a°| - |a°><c°|)
= |c°><a°| - |a°><c°|
where we used the symmetric plane identity 3.1.2 in the last step.
The third term is evaluated readily by the general plane identity 3.1.
(|c°><b°| - |b°><c°|)(|b°><a°| - |a°><b°|)
= - (<a°|b°><b°|c°> - <a°|c°><b°|b°>)P'
(Be careful: P' generally is not the orthogonal Projection P, but P'P = P.)
Together one gets
S(c°,b°)S(b°,a°)P = <c°|a°>P + |c°><a°| - |a°><c°| = S(c°,a°)P
Obviously S(c°,b°)S(b°,a°)P = S(c°,a°)P says that SP is closed under planar angular
concatenation. This is, what we expect from an intrinsic angular unit.
From this observation, by bisectional angular refinement and analytic continuation,
the formulas of Moivre for exp, cos and sin may be
derived. The usual, opposite way to define exponential function first and abstractly,
may be a faster way to teach mathematical techniques result oriented. But if one is
honest in argumentation and prefers a didactic approach, then the algebra is
coming first and analytic properties are a consequence. And for the purposes of this paper,
except of the square root to normalize vectors, no analytic functions are needed at all.
S(b,a) := |b||a|S(b°,a°) .
Then S by definition is linear in each of its arguments and one can calculate
S(c,b)S(b,a)P = λS(a,b)S(b,a)P + μS(b,b)S(b,a)P
= λ|a|2|b|2 + μ|b|2S(b,a)P
= |b|2(λS(a,a)P + μS(b,a)P)
= |b|2S(λa + μb,a)P
= |b|2S(c,a)P
So, to concatenate angles in a plane, one only has to memorize
S(a°,b°)S(b°,a°)P = P (inverse relation at the plane) and
S(a°,a°) = I
The rest is provided by linearity.
Obviously we are building up an operator algebra.
For every plane the planar rotations form a 2-dimensional algebra
with the operator basis
{ Pa,b , |b°><a°| - |a°><b°| } .
a and b are arbitrary vectors, spanning the plane.
We meanwhile know, how to transform different choices of basis into one another.
So we could start to express everything in a most convenient basis.
We do not need all these considerations for our aim, to concatenate
angles in space. But we promised, that trigonometric formulas are
contained in the formulas already evaluated and we will see, that this is true.
Let us choose x and y orthonormal in the a,b plane. Then the basis of our algebra
is especially simple:
P = |x><x| + |y><y|, R = |y><x| - |x><y|
By the expansions
|a> = |x><x|a> + |y><y|a>
|b> = |x><x|b> + |y><y|b>
we calculate the transformation of right angle turns once and for all
|b><a| - |a><b| = <b|{|y><x| - |x><y|}|a>{|y><x| - |x><y|} = <b|R|a> R
Now, by definition, S has the form
S(b°,a°) = <b°|a°> I + <b°|R|a°> R
The two coefficients are the projections of |b°> onto the orthogonal directions |a°> and R|a°>.
So they are the cos and sin components of the angle in conventional notation.
In any case, if we define
c := <b°|a°> and s := <b°|R|a°> ,
the mixed matrix elements of I and R, then S reads
S = c I + s R
The interesting part is, that the coefficients c and s are not independent.
They have to obey a form stability condition.
Beyond x and y we get another orthonormal basis |a°> and R|a°> and therefore
P = |a°><a°| - R|a°><a°|R
1 = <b°|b°> = <b°|P|b°>
= <b°|a°><a°|b°> - <b°|R|a°><a°|R|b°>
= <b°|a°>2 + <b°|R|a°>2
= c2 + s2
Here we get a natural mapping of our discrete rotation SP to points (c,s)
at the unit circle in R2. SP is built up by two
operators, projection and right angle turn, identifying the rotational plane
and a point at the 2-dimensional unit circle, identifying the conventional measure
of rotation. It can be seen easily, that any such pair (c,s) is feasible,
to define a rotation S. Just define
|a~> := |a°> and |b~> := c|a°> + sR|a°>
then one calculates
<b~|a~> = c and <b~|R|a~> = s
and from c2 + s2 = 1 one gets <b~|b~> = c2 + s2 = 1 .
Connecting rotations written in the P,R basis one obtains
S(c°,b°)S(b°,a°)P =: (c2P + s2R)(c1P + s1R) = (c1c2-s1s2)P + (s1c2+c1s2)R
This combination law for the P,R coefficients are the trigonometric
addition laws. They describe the composition law for angles in a plane.
In fact, this way one can rewrite planar geometry algebraically.
But for brevity, I will not present masses of possible details here.
Doing this we would loose the thread.
One might argue, that using complex exponentials, one gets things like combination
of planar angles by a comparably easy and even more elegant argumentation. But
our aim is an excursion in didactics of mathematics and then this is not true. Using complex
exponentials, one already used the algebraic approach to get these exponentials. And
careful analysis of the whole way, from definition of exponentials by power series until
the Moivre formulas reveals, that the logic there is turned upside down. The natural logic
and therefore didactics, always lead back to our starting point. But we are not constrained
to 2 dimensions, as we will see by connecting angles, not in a plane. And by allowing
limiting we get exponentials or Lie algebras or geometric algebra naturally, too.
(c2P + 0R)(c1P + 0R) = c1c2P + 0R
So we obtain an algebra we call complex numbers C. And, beyond linearity,
it is sufficient to know two rules, to get all calculations done in this algebra:
P2 = P , R2 = -P
Obviously, P fulfills the algebraic rules of an identity and may be identified with 1.
R is usually called the imaginary unit i. Constructed this way, C is
a realization of the rotation algebra of a really existing plane in Rn.
It is possible to embed the 2 dimensional plane in this
2 dimensional algebra too, but it is not necessary. A lot of confusion about
conformal mappings could be avoided, if everyone knew about the different
realizations of 2 dimensional vector spaces and their (different) 2 dimensional
complex transformation algebra.5) Careful distinction would never lead to the traps,
provided by concepts like axial and polar vectors. Of course, there
is no need to use different symbols to perform calculations in different spaces
along the same algebraic rules. But there should be a clear imagination if
e.g. pictorial explanations for algebraic results are discussed: not every
pure algebraic rule has a useful corresponding picture!
S(b,a) = <b|a> 1 + <b|R|a> i
= (bxax+byay) 1 + (byax-bxay) i
= (bx1 + byi)(ax1 - ayi)
≡ ba* and
<b|a> = " Real part of ba* "
Here we used the abbreviations ax=<x|a> etc.
Ha := I - 2|a°><a°|
Hb := I - 2|b°><b°|
We study the geometric properties of HbHa, i.e.
step by step we convert HbHa in a rotational form, so, that
most steps may also be described geometrically.
Outside the {|a>,|b>}-plane Ha as well as Hb are trivial,
and it follows that there HbHa is the identity:
HbHa = HbHaPa,b + (I - Pa,b) .
Multiplied out and ordered by symmetric and skew-symmetric parts, on gets
HbHa = I - 2(|b°><b°|-|a°><a°|)2 + 2<b°|a°>(|b°><a°|-|a°><b°|) and
HbHaPa,b = Pa,b - 2(|b°><b°|-|a°><a°|)2 + 2<b°|a°>(|b°><a°|-|a°><b°|)
The third, skew-symmetric term is, up to scaling, readily identified as a 90° rotation.
To interpret the second term, one calculates at the {|a>,|b>}-plane directly
(|b°><b°|-|a°><a°|)2|a°> = (1-<b°|a°>2)|a°>
(|b°><b°|-|a°><a°|)2|b°> = (1-<b°|a°>2)|b°>
Together we get
(|b°><b°|-|a°><a°|)2 = (1-<b°|a°>2)Pa,b and therefore
Pa,b - 2(|b°><b°|-|a°><a°|)2 = (2<b°|a°>2 - 1)Pa,b and
HbHaPa,b = (2<b°|a°>2 - 1)Pa,b + 2<b°|a°>(|b°><a°|-|a°><b°|)
We already know from 3.2.3, that the last expression is the square of a turn from |a°> to |b°>:
HbHaPa,b = S(b°,a°)2Pa,b and
HbHa ≡ (I - 2|b°><b°|)(I - 2|a°><a°|) = S(b°,a°)2Pa,b + (I - Pa,b)
is a so called axis turn.
Now the time is ready, to extend SP from the rotation plane to the whole space,
so, that it gets compatible with the composition of reflections. So we redefine S by
S(b,a) := <b|a>P + (|b><a| - |a><b|) + |b||a|(I - P)
(This S is scalable in a and b but not linear in a and b anymore!)
HbHa ≡ (I - 2|b°><b°|)(I - 2|a°><a°|) = S(b°,a°)2
We intentionally redefined S by notational abuse, to demonstrate, that there is no unique
continuation of S from unit vectors to the whole space. There are different, context dependent
continuation possibilities to the respective use.
S(b°,a°) = S((a°+b°)°,a°)2 = (I - 2|(a°+b°)°><(a°+b°)°|)(I - 2|a°><a°|)
= S(b°,(a°+b°)°)2 = (I - 2|b°><b°|)(I - 2|(a°+b°)°><(a°+b°)°|)
HcHbHbHa = HcHa
Rewritten in rotations, this looks like
S2(c°,b°)S2(b°,a°) = S2(c°,a°)
So double angles combine nicely.
This, in 3 dimensions, is all there is to angular composition.
And it is most conveniently derived from discrete geometry and vector calculation;
especially no special knowledge about trigonometric functions or about quaternions is necessary.
We could use half angles instead
S2((c°+b°)°,b°)S2(b°,(b°+a°)°) = S2((c°+b°)°,(b°+a°)°)
Then we may evaluate the squares of the operators at left hand side by doubling the angles
S(c°,b°)S(b°,a°) = S2((c°+b°)°,(b°+a°)°)
If we consider two 2-dimensional, intersecting planes in a vector space
and two axes a and b in the first plane and two axes b and c in the second plane,
the isotropy of planes ensures us, that this way on intersecting planes the most
general angles may be written down and composed.
Identity - P = |n><n| with an axis |n> in the 3-dimensional case.
This observation is basic to all the strange cross-product formulas for
3-dimensional rotation. And here a unique axis |n>, a 1-dimensional invariant
of the rotation can be identified. But the theory presented in our letter
is not restricted to a 3-dimensional space.
<b°|a°>I + (|b°><a°| - |a°><b°|) .
And in a 3-dimensional (sub-)space also concatenated rotations preserve this form.
S(b°, a°)|x> = Pa,bT'(b°, a°, x) + (I-Pa,b)|x> ,
with the abbreviation
T'(b°, a°, x) := <b°|a°>|x> + (|b°><a°| - |a°><b°|)|x>
Obviously the rotation intelligence is hidden in the mapping T'.
T' has a nice building law. Every term in T' looks like a product of the same 3 vectors,
just differently arranged.
No one can forbid, to evaluate T' everywhere in space. What we mean is, that T'
may be linearly continued to a
trilinear mapping T': (b,a,x) → T'(b, a, x) from Rn×Rn×Rn to Rn
Now three vectors are mapped (multi-)linearly to a vector.
Beyond linearity, T' algebraically represents the geometric idea, we started from; the idea,
to construct discrete rotations from continued reflections:
(I) T'(b, a, x) is a trilinear mapping in Rn
(II) T'(b, a, a) = |a|2|b> ("S(b,a) maps a in b")
(III) T'(b°, a°, b) = 2<b°|a°>|b> - |b||a°> ("S(b,a) mirrors a on b",
"b gets the angle bisector")
= -(I-2|b°><b°|)|b||a°>
Two reflections: an inversion of sign for a and
a reflection at the hyperplane
orthogonal to b.
Originally (using unit vectors only) our considerations were based on (II) and (III).
But, after linear continuation, (III) is already a consequence of the two
more simple relations (I), (II) and
(II)' T'(b, b, a) = |b|2|a>
This can be seen by the calculation
T'(b°, a°, b) = T'(b°,a°, b) + T'(b°, b, a°) - T'(b°, b, a°)
= T'(b, a°+b°, a°+b°) - T'(b, a°, a°) - T'(b, b°, b°) - T'(b°, b, a°)
= (<a°+b°|a°+b°> - <a°|a°> - <b°|b°>)|b> - |b||a°>
= 2<b°|a°>|b> - |b||a°> .
In words, because the symmetric part of our original operator is given by the scalar
product of its defining vectors, the trilinear mapping T' can be used to describe
reflections, regardless what the skew-symmetric part of the operator looks like.
So we found a trilineare mapping T' obeying the fundamental relations
T'(b, a, a) = T'(a, a, b) = |a|2|b>
and such a mapping necessarily encodes a reflection:
-T'(b°, a°, b°) = (I-2|b°><b°|)|a°> .
To be precise, any trilinear mapping T from Rn×Rn×Rn to Rn
with the property
T(b, a, a) = T(a, a, b) = -|a|2|b>
encodes reflections:
T(b°, a°, b°) = (I-2|b°><b°|)|a°> .
Such a mapping exists; we could choose -T' for T.
T(b°,T(a°, x, a°),b°) = (I-2|b°><b°|)(I-2|a°><a°|)|x> .
We could look for the most convenient T, to perform calculations in various contexts.
In order to formulate rules for multilinear maps more suggestive,
one is inclined, to write such a mapping T
by (product) composition instead of a function:
Instead of T(b,a,x) we write bax.
-T' seems to be the most simple such T, but it is not necessarily the
most convenient one, to calculate with. With T' the product notation is
not much simpler, especially because only 3-factor products are
defined neatly and one has to set a lot of brackets, because the
product is not associative. So it is worthwhile, to look out for
other possibilities.
aa = -|a|2 and therefore also
baa = aab = -|a|2|b>
was done about 100 years ago by Clifford. So, today, we know in fact
all these "mirror-algebras" and we can represent them as operator algebras.
It is impressing , that in proper designed operators,
building laws of formulas can be written down and treated in a stand alone fashion.
This is one of the secrets of success, contained in the context free examination
of algebraic structures. But these successes do not mean, that abstract operator
algebras have to be applied to real world models. Formalistic treatments
are, also from the mathematical point of view, only allowed, if corresponding
objects really exist. One should notice, that we did not try to generally proof the
existence of some universal Clifford algebra8); we did not even define the term. Instead,
we used and manipulated well defined objects, namely vectors and multilinear functions
of them. And we are very well allowed to analyze and discuss calculation rules
for such objects. One should only avoid the slippery ground of manipulations
on formal objects, where neither a realization nor a existence proof is available.
e'0 := |1><1|+|2><2|+|3><3|
e'1 := |1><2|-|2><1|
e'2 := |1><3|-|3><1|
e'3 := |3><2|-|2><3|
<b|a>I + (|b><a| - |a><b|) = <b|a>e'0 + <3|b×a>e'1 - <2|b×a>e'2 - <1|b×a>e'3
Although, products of the e' operators do not reproduce themselves, we know
from our concatenation result, that they are extendable to rotations and these can
be expanded in the e' again. Therefore the four e', in some sense, form an operator basis for
the algebra, formed by three dimensional rotations. In other words, the 3-dimensional rotations
are characterized by four parameters. In order to simplify
calculations using such a basis, one is inclined either to write down
once forever rules for the calculation of coefficients, or to reengineer
the operator product or the operator basis.
e0 := |1><1|+|2><2| + |3><3|+|4><4|
e1 := (|1><2|-|2><1|) + (|4><3|-|3><4|)
e2 := (|1><3|-|3><1|) + (|2><4|-|4><2|)
e3 := e1e2 = (|3><2|-|2><3|) + (|1><4|-|4><1|)
(We will remark, that we have chosen the Pauli matrix form for consistency.)
e12 = e22 = e32 = -e0
This algebra is a minimum dimension algebra
to encode rotations. It is a realization of the quaternions, as one sees from the following suite:
e1e3 = e1e1e2 = -e2
e3e2 = e1e2e2 = -e1
e3e1 = -e3e3e2 = e2 = -e1e3
e2e3 = -e1e3e3 = e1 = -e3e2
e2e1 = e2e2e3 = -e3 = -e1e2
And this algebra even is a Clifford algebra. But we had to pay a price for the easy
multiplication law: the operator needs a 4-dimensional vector space as a numbering
reservoir for calculation. So one has, if possible with a reasonable cost, to find the rules,
how to shrink the calculation results properly by e0-|4><4| to
the 3-dimensional space. For the product of two quaternions one calculates with the above rules:
qa := a1e1 + a2e2 + a3e3
qb := b1e1 + b2e2 + b3e3
qbqa = -<b|a>e0 + <1|b×a>e1 + <2|b×a>e2 + <3|b×a>e3
And for three such quaternions qa, qa and qx
one then obtains from the two elementary vector equations
qbqaqx = - <b|a>qx - <b×a|x>e0
+ <1|(b×a)×x>e1 + <2|(b×a)×x>e2 + <3|(b×a)×x>e3
= - <b|a>qx - det(b,a,x)e0
- <1|(|b><a|-|a><b|)|x>e1 - <2|(|b><a|-|a><b|)|x>e2 - <3|(|b><a|-|a><b|)|x>e3
= - det(b,a,x)e0
- <1|T'(b,a,x)>e1 - <2|T'(b,a,x)>e2 - <3|T'(b,a,x)>e3
where, as above T'(b,a,x) := <b|a>|x> + (|b><a| - |a><b|)|x>
So the quaternion product delivers at least the same components as the spin S(b,a). If x
lies in the {a,b}-plane, then the components of the triple quaternionic product are exactly
the components of the rotated vector x. The e0 term is identified as an oriented volume.
For notational convenience, the vectors x are embedded
into the quaternions by identifying x and qx. But with this embedding of R3 one
has to be similar careful with the transformation of "axial" and "polar" vectors as in the
case of embedding R2 in C. But using this embedding,
one gets the simple relation
bax = - det(b,a,x)e0 - T'(b,a,x)
Note that both parts are trilinear. This formula reflects the geometric content
in the quaternionic product. Moreover, for x=b, the volume factor is zero
and b°ab° encodes just a reflection. Now we know, that reflections and then also
axial rotations may be formulated in the framework of quaternions.1) J. E. Gilbert & M. A. M. Murray,
Clifford algebras and Dirac Operators in harmonic analysis,
Cambridge Univ. Press 1991
Even though P. Lounesto constructed
a counter example to theorem (5.16) of Gilbert's and Murray's book,
it is doubtless a brilliant, self-contained overview
on the state of art in Clifford algebras.
2) D. Hestenes
Vectors, Spinors, and Complex Numbers in Classical and Quantum Physics
American Journal of Physics, Vol. 39/9, September 1971
D. Hestenes and G. Sobczyk,
Clifford Algebra to Geometric Calculus
Kluwer; reprinted with corrections in 1992
3) Steve Pollock (University of Colorado at Boulder, U.S.A.)
Lecture Notes on Quantum Mechanics,1997
Dirac notation, some linear algebra
4) K.Allinger and M.Ratner:
Influence Functionals: General Methodology for Subsystem Calculations.
Phys. Rev. A39 (1989) p.864 - 880
5) Ramon González Calvet
Treatise of plane geometry through the geometric algebra, 2007
6) Eric Weisstein's World of Mathematics
Surface Area
7) R.P.Feynman, Statistical Mechanics
Addison-Weseley, 13th Printing (1990), Chap.7.3
8) S. Lang, Algebra, Addison-Wesley 1974, Chap.XIV, §8
9) Laura Downs, University of California at Berkeley:
Using Quaternions to Represent Rotation