Filetype PDF | Posted on 24 Jan 2023 | 2 years ago
Authentication
digital resources
151x Filetype PDF File size 0.53 MB Source: core.ac.uk
View metadata, citation and similar papers at core.ac.uk brought to you by CORE
provided by PhyDid - Zeitschriften (FU Berlin)
Didaktik der Physik
Frühjahrstagung – Wuppertal 2015
Sandwich Products and Reflections
Martin Erik Horn
Hochschule für Wirtschaft und Recht Berlin / Berlin School of Economics and Law
Badensche Str. 52, Fach No. 63, D – 10825 Berlin, Germany
-
BSP Business School Berlin Potsdam, Calandrellistr. 1 9, D – 12247 Berlin, Germany
mail@grassmann-algebra.de, mail@martinerikhorn.de
Abstract
As reflections are an elementary part of model construction in physics, we really should look for a
mathematical picture which allows for a very general description of reflections. The sandwich
product delivers such a picture.
Using the mathematical language of Geometric Algebra, reflections at vectors of arbitrary dimen-
sions and reflections at multivectors (i.e. at linear combinations of vectors of arbitrary dimensions)
can be described mathematically in an astonishingly coherent picture.
1. Reflections in physics and mathematics
There is a severe difference between reflections in in physics, while we are dealing with vectors and
physics and reflections in mathematics. If we reflect multivectors when discussing problems in mathe-
an object (e.g. the arrow shown in figure 1) in a matics.
plane mirror, which is a simple optical situation in
physics, the picture of the object and the original
object are always supposed to have the same dis-
tance from the plane mirror, yet lying on different
sides of the mirror. Thus we have to consider two r
kinds of information in this case: information about
the position and information about the direction of
object and picture.
In mathematics, we only consider directional infor-
mation. Identical arrows at different positions in
space represent the same vector (see figure 2). Vec- Fig.2: Some arrows which represent vector r.
tors (and multivectors in general) do not possess
positional information. They are mathematical ob- This even gives the impression that mathematics is
jects constructed purely by directions. Thus a reflec- behaving in a more natural scientific manner com-
tion of a vector at a plane will result in a picture of pared to physics, as in science one should always
the vector with arbitrary position in space (see figure change only one variable at a time when investigat-
3). An explicit value of the distance of the vector to ing problems or carrying out experiments. A mathe-
the plane does not exist. matical reflection only changes one variable (direc-
Thus we are usually dealing with position vectors tion) while a reflection in physics changes two vari-
and position multivectors when discussing problems ables (position and direction) at a time.
original arrow
(object)
shadow of the arrow (compo-
nent parallel to the r
mirror)
r
reflected arrow ref
(picture)
Fig.1: Reflection of an arrow at a plane mirror Fig.3: Reflection of vector r at a plane in
in physics. mathematics.
1
Horn
Many physicists therefore tend to discuss geometric sec. 55], [6, sec. 2.6]. In higher-dimensional spaces
situations using conformal concepts. In such situa- of dimension n (some authors like to call them hy-
tions Conformal Geometric Algebra (CGA) is espe- perspaces), n different base vectors are required.
cially helpful, as positional information of three- These base vectors can be seen as generalized Pauli
dimensional space or four-dimensional spacetime is matrices , , , … , as they obey Pauli alge-
encoded as directional information of a five- or six- bra. 1 2 3 n
dimensional conformal spacetime [1], [2], [3, chap. In four-dimensional spacetime four base vectors are
10], [4] in CGA. In this way the two different varia- required to describe all possible vectors as linear
bles are condensed into one. combinations of base vectors. These base vectors
Therefore only mathematical reflections will be can be represented by Dirac matrices , , , [5,
t x y z
considered in the following. sec. 156], [6, chap. 5]. In higher-dimensional spaces
2. Geometric representations of scalars of dimension n (some authors like to call them hy-
per-spacetimes), n different base vectors are re-
In the preceding section the difference between quired. These base vectors can be seen as general-
vectors and position vectors was discussed. A simi- ized Dirac matrices , , , … , as they obey
lar relation exists between scalars and position sca- Dirac algebra. 1 2 3 n
lars. Thus we live in mathematical worlds with the fol-
A scalar is a dimensionless vector without direction, lowing mathematical ingredients. In three-dimensio-
called a point in geometry. nal space there are:
We are usually dealing with points at fixed positions Scalars (0-vectors) k, ℓ
in physics. Often scientists represent a scalar by a dimensionless points,
very special point of a coordinate system: the origin. Vectors (1-vectors) r, n
In this paper scalars are represented by all points of oriented one-dimensional line elements,
space. These mathematical scalars do not possess a Bivectors (2-vectors) A, N
position (see figure 4). Every point of space or oriented two-dimensional area elements,
spacetime will represent a given scalar. Trivectors (3-vectors) V, T
Different vectors might not only have a different oriented three-dimensional volume elements.
direction, but a different 1-dimensional volume,
7
2 7 k = 7
7 ref
7 2 k = 7
2 7 ref
k = 7
7 ref
k = 7
ref
7 7 2
7 2 k = 7
2
2 k = 7
7 ref
7 7 k = 7
2 ref
2 2 k = 7 k = 7
2 ref ref
Fig.4: Geometric representation of the scalar 2 (blue Fig.5: Reflection of scalar k = 7 at a plane in ma-
points) and the scalar 7 (black points). thematics which results in k = 7.
ref
called length, too. In a similar way different scalars In four-dimensional spacetime there are:
possess different 0-dimensional volumes, called Scalars (0-vectors) k, ℓ, vectors (1-vectors) r, n,
value of the scalars. bivectors (2-vectors) A, N, trivectors (3-vec-
Thus scalars cannot be distinguished by their posi- tors) V, T,
tions (which are not defined), but only by their val- Quadvectors (4-vectors) Q, Q
ues. oriented four-dimensional hyper volume ele-
And as a scalar is a dimensionless object or point ments.
without direction, its picture will be the same dimen- And in higher-dimensional spaces or spacetimes
sionless object without direction. Thus scalars (or there are:
points) of value k will always be reflected into the Scalars (0-vectors) k, ℓ, vectors (1-vectors) r, n,
identical scalar of same value k = kref (see figure 5). bivectors (2-vectors) A, N, trivectors (3-vec-
3. Basic entities of Geometric Algebra tors) V, T, quadvectors (4-vectors) Q, Q,
In three-dimensional space three base vectors are Pentavectors (5-vectors) P, P
required to describe all possible vectors as linear oriented five-dimensional hyper volume ele-
combinations of the base vectors. These base vectors ments,
can be represented by Pauli matrices , , [5, Hexavectors (6-vectors) H, H
x y z
2
Sandwich Products and Reflections
oriented six-dimensional hyper volume ele- The axis at which a vector is reflected at is written in
ments, the same way as the vector which is reflected. This
Septavectors (7-vectors) S, S makes it easy to find parallel and orthogonal com-
oriented seven-dimensional hyper volume ponents of these vectors (see section 7).
elements, Of course we live in three-dimensional space. There-
… etc … fore only equations {1} to {4}, {9} to {12}, {17} to
k-dimensional hypervectors (k-vectors) {20}, and {25} to {28} are directly accessible to
oriented k-dimensional hyper volume ele- experiments in physics. We will never be able to
ments. hold a five-dimensional hypercube in front of a
These higher-dimensional entities are required to three-dimensional mirror and look at the picture of
describe Special Relativity as a four-dimensional this hypercube. But we are able to generalize obser-
geometry, Cosmological Special Relativity [7] or vations we made and mathematical ideas we formed
Conformal Geometric Algebra as five-dimensional in our very narrow, constricted three-dimensional
geometries, Conformal Spacetime Algebra as a six- world we live in.
dimensional geometry and Conformal Cosmological These generalizations are inventions, and as long as
Algebra as a seven-dimensional geometric world these higher-dimensional mathematical inventions
(see figure 6). contain all known relations of three-dimensional
space, they can be considered as reasonable or ra-
tional.
Such a system of higher-dimensional, reasonable
(Standard) Geometric Algebra 3 dimensions
relations is given in the following section 5. They
Spacetime Algebra 4 dimensions help us to understand the geometry of four-
dimensional spacetime, which is a very reasonable
Cosmological Spacetimevelocity Algebra and rational invention, too. We are (at least at pre-
& Conformal Geometric Algebra 5 dimensions
sent) not able to hold a 3d spacetime cube with two
Conformal Spacetime Algebra 6 dimensions spacelike and one timelike edges in front of a two-
dimensional spacetime mirror with one spacelike
Conformal Cosmological Algebra 7 dimensions
and one timelike dimension and look at the picture
Higher-dimensional Conformal Geo- of this 3d spacetime cube.
metric Algebras 8 and more dimensions
But physicists are able to do some experiments in 4d
spacetime. And they will find the mathematical
Fig.6: Dimensionalities of different Geometric Algebras.
structure of the equations given in section 5 helpful
to describe the outcome of these experiments in a
4. Central message of this paper very coherent, clear and well-structured mathemati-
cal way.
Every reflection can be modeled mathematically as a In a similar way I hope that the structure of these
threefold multiplication forming a sandwich product. equations will one day help to describe and under-
Using Clifford Algebra, matrix multiplication is not stand conformal geometric algebras in a didactically
required to find a reflected object. well-structured way.
And it makes sense to reverse this sentence: Every Today the presentation of reflections sometimes
sandwich product can be considered as a reflection – lacks the coherence and clearness seen in equations
at least in a formal way, e.g. a rotation equals a re- {1} to {64}. For instance standard textbooks of
flection at an oriented parallelogram. Thus sand- Geometric Algebra often use a dual description for
wich products describe reflections (see figure 7). reflecting a vector. They describe reflections of a
This is the central message of this paper. vector at a plane or at a hyper-plane by using the
This description of reflections has a tremendous dual n of the plane or hyper-plane. Then equations
didactical advantage over other mathematical con- like a = – n a n [3, eq. 2.99, p. 40] may lead to a
structions: Both operators and operands are modeled ref
mathematical picture, which is confusing as it is
within the same mathematical language. They are written as a reflection at a vector, followed by the
always considered as mathematical objects of Geo- reflection at a point. In contrast to that, equations
metric Algebra – linear combinations of vectors or {1} to {64} are given without any reference to the
linear combinations of geometric products of vec- dual of the operators.
tors.
multiplied multiplied –1
reflected operand = operator operand operator
by by
Fig.7: Sandwich products describe reflections: An operand is multiplied from left and right by an operator.
3
Horn
5. Overview: Reflection formulas
Reflection at a 5d hyperspace, hyperspacetime or
Reflection at a point (represented by scalar ℓ): spacetimevelocity (represented by pentavector P):
–1 –1
Scalars: k = ℓ k ℓ {1} Scalars: k = P k P {41}
ref ref
–1 –1
Vectors: r = – ℓ r ℓ {2} Vectors: r = P r P {42}
ref ref
–1 –1
Bivectors: A = ℓ A ℓ {3} Bivectors: A = P A P {43}
ref ref
–1 –1
Trivectors: V = – ℓ V ℓ {4} Trivectors: V = P V P {44}
ref ref
–1 –1
Quadvectors: Q = ℓ Q ℓ {5} Quadvectors: Q = P Q P {45}
ref ref
–1 –1
Pentavectors: P = – ℓ P ℓ {6} Pentavectors: P = P P P {46}
ref ref
–1 –1
Hexavectors: H = ℓ H ℓ {7} Hexavectors: Href = P H P {47}
ref
–1 –1
Septavectors: S = – ℓ S ℓ {8} Septavectors: S = P S P {48}
ref ref
Reflection at an axis (represented by vector n): Reflection at a 6d hyperspace or hyperspacetime
–1
(represented by hexavector H):
Scalars: k = n k n {9}
ref –1
–1
Scalars: k = H k H {49}
Vectors: r = n r n {10} ref
ref –1
–1
Vectors: rref = – H r H {50}
Bivectors: Aref = n A n {11} –1
–1
Bivectors: A = H A H {51}
Trivectors: Vref = n V n {12} ref
–1
–1
Trivectors: V = – H V H {52}
Quadvectors: Qref = n Q n {13} ref
–1
–1
Quadvectors: Q = H Q H {53}
Pentavectors: P = n P n {14} ref
ref –1
–1
Pentavectors: Pref = – H P H {54}
Hexavectors: H = n H n {15}
ref –1
–1
Hexavectors: Href = H H H {55}
Septavectors: S = n S n {16}
ref –1
Septavectors: S = – H S H {56}
Reflection at a plane (represented by bivector N): ref
–1
Scalars: k = N k N {17} Reflection at a 7d hyperspace or hyperspacetime
ref (represented by septavector S):
–1
Vectors: r = – N r N {18}
ref –1
–1 Scalars: k = S k S {57}
Bivectors: Aref = N A N {19} ref
–1
–1 Vectors: rref = S r S {58}
Trivectors: Vref = – N V N {20} –1
–1 Bivectors: A = S A S {59}
Quadvectors: Qref = N Q N {21} ref
–1
–1 Trivectors: V = S V S {60}
Pentavectors: P = – N P N {22} ref
ref –1
–1 Quadvectors: Q = S Q S {61}
Hexavectors: H = N H N {23} ref
ref –1
–1 Pentavectors: Pref = S P S {62}
Septavectors: S = – N S N {24}
ref –1
Hexavectors: Href = S H S {63}
–1
Reflection at a 3d space or reduced spacetime Septavectors: S = S S S {64}
(represented by trivector T): ref
–1
Similar equations with the same sandwich product
Scalars: k = T k T {25}
ref
–1 structure can be found for reflections at higher-
Vectors: r = T r T {26}
ref dimensional hyperspaces and hyper-spacetimes.
–1
Bivectors: Aref = T A T {27}
–1
6. Reflection of points (scalars) and at points
Trivectors: Vref = T V T {28}
–1
As points or scalars do not have any direction there
Quadvectors: Qref = T Q T {29}
–1 will be no change of the non-existing direction, if
Pentavectors: P = T P T {30}
ref points are reflected at an arbitrary geometrical ob-
–1
Hexavectors: H = T H T {31}
ref ject. And the value of the point or scalar does not
–1
Septavectors: S = T S T {32}
ref change either.
Reflection at a 4d hyperspace or spacetime Therefore equations {1}, {9}, {17}, {25}, {33},
(represented by quadvector Q): {41}, {49}, and {57} simply state that
–1 k = k {65}
ref
Scalars: k = Q k Q {33}
ref
–1 because scalars commute with every other mathe-
Vectors: r = – Q r Q {34}
ref matical object. Therefore the operator and its inverse
–1
Bivectors: Aref = Q A Q {35} cancel if the operand (see figure 7) is a scalar.
–1
Trivectors: Vref = – Q V Q {36} A point and its reflected picture are always identical.
–1
Quadvectors: Qref = Q Q Q {37} So it is not misleading to regard points as being
–1
Pentavectors: P = – Q P Q {38}
ref somehow “parallel” to every other geometrical ob-
–1
Hexavectors: H = Q H Q {39} ject. It is always possible to place them completely
ref
–1
inside other mathematical objects.
Septavectors: S = – Q S Q {40}
ref
4
no reviews yet
Please Login to review.
