An Introduction To Geometric Algebra With Some Pre
An Introduction To Geometric Algebra With Some Pre
An Introduction To Geometric Algebra With Some Pre
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Spin and pseudospin in monolayer graphene: I. Representation by geometric algebra and simple cases
A Dargys
E-mail: mail@martinerikhorn.de
Abstract. It is still a great riddle to me why Wolfgang Pauli and P.A.M. Dirac had not fully
grasped the meaning of their own mathematical constructions. They invented magnificent, fan-
tastic and very important mathematical features of modern physics, but they only delivered half
of the interpretations of their own inventions. Of course, Pauli matrices and Dirac matrices re-
present operators, which Pauli and Dirac discussed in length. But this is only part of the true
meaning behind them, as the non-commutative ideas of Grassmann, Clifford, Hamilton and
Cartan allow a second, very far reaching interpretation of Pauli and Dirac matrices.
An introduction to this alternative interpretation will be discussed. Some applications of this
view on Pauli and Dirac matrices are given, e.g. a geometric algebra picture of the plane wave
solution of the Maxwell equation, a geometric algebra picture of special relativity, a toy model
of SU(3) symmetry, and some only very preliminary thoughts about a possible geometric
meaning of quantum mechanics.
Content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distribution
of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.
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Symmetries in Science XVI IOP Publishing
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But the light, which carries the information about our three-dimensional surroundings, is absorbed by
the retina, which is a two-dimensional, curved surface. Thus the three-dimensional objects outside us
are represented as two-dimensional images in our eyes. Neurons then conduct this two-dimensional in-
formation into our brain which tries to reconstruct the three-dimensional objects around us. But as
only information about a two-dimensional distorted picture reaches our brain, the visual model our
brain constructs might be incomplete, unsuited for some situations or even faulty. And if the visual
models our brain delivers are not correct, the mathematical models we build on these visual models
might be partly wrong too.
There are strong hints that the mathematical models of sighted persons are indeed deficient. As
knot theorist Alexei Sossinsky remarks, the most complicated knots were invented by blind mathe-
maticians [2]. And “almost all blind mathematicians are (or were) geometers. (…) A blind person’s
spatial intuition is primarily the result of tactile and operational experience. It is deeper – in the literal
as well as the metaphorical sense” [2].
Children are not able to interpret two-dimensional pictures in a way we do as grown-ups. They can
not distinguish properly between illuminated convex or concave objects [3]. They have to learn it. In a
similar way we all are not able to interpret rotations correctly. We have not learned it. Usually we all
suppose objects to be in the same topological state after a rotation of 2. But this is wrong. An object
attached to its surroundings by some strings exhibits a rotation symmetry of 4 which we do not re-
cognize due to our visual deficiencies. Thus such an object has a spin of ½. Dirac’s belt trick [4], [5],
[6], [7], [8] clearly demonstrates this fact (see figure 1). We therefore have to draw two important
conclusions:
4 symmetries are nothing exclusively quantum mechanical.
4 symmetries are an essential part of our everyday world (our space) we live in.
We need an appropriate mathematical language to describe 4 symmetries.
This language is Clifford Algebra. When applied to didactical situations in physics or other sciences
Clifford Algebra Cℓ3,0 of three-dimensional Euclidean space is called Geometric Algebra [9], and Clif-
ford Algebra Cℓ1,3 of four-dimensional pseudo-Euclidean spacetime is called Spacetime Algebra [10],
[11].
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many introductory books like [14], [15], [16], [17], [18], [19], [20] from a more or less university level
perspective. But to my astonishment even university professors who should be competent with
modeling geometric situations mathematically confess that they were “finding it so difficult to under-
stand GA” (see page VII at the introduction of [17]). Therefore I decided to present the basic foun-
dations of Geometric Algebra in this paper in a way suitable for school lessons. I myself taught
Geometric Algebra and Spacetime Algebra at physics lessons of upper secondary level in school [21],
[22]. And the results of the examinations were indeed encouraging [23].
The first step at these lessons has been to introduce oriented line elements (vectors) and oriented
area elements (bivectors) with the aim to immediately hit the algebraic heart of Geometric Algebra:
anti-commutativity. A simple discussion of the geometrical situation directly reveals the central idea
of Geometric Algebra (see figure 2).
x
(a) (b)
left area element
x y y y y x = – right area element
x
Figure 2. Visualization of the geometric product of two base vectors as base bivector.
If two different vectors are multiplied, we get an oriented area element. This multiplication is
called geometric product (or Clifford product if you like) and contains the complete geometric
information about the relation of the two vectors [11], [15]. Thus the product of two base vectors
results in a base bivector, which represents an oriented unit square. The orientation then of course
depends on the order of the multiplication. If we first make a step into the direction of the x-axis and
then a step into the direction of the y-axis (figure 2a), we will get a positive or counter-clockwise
orientation (which equals the direction of a traffic circle on the continent). If we first make a step into
the direction of the y-axis instead and then a step into the direction of the x-axis (figure 2b), we will
get a negative or clockwise orientation (which equals the direction of a ghost driver in a traffic circle).
This change of orientation is codified in the negative signs of equations (1).
x y = – y x y z = – z y z x = – x z (1)
Together with the condition, that unit vectors square to one
x2 = y2 = z2 = 1 (2)
these equations form the algebraic core of Geometric Algebra. This is indeed rather simple, and Parra
Serra concludes in his didactical analysis of Geometric Algebra: “The only rules to remember are that
different orthogonal generating units (vectors) anticommute and that their square is +1…”1 [13].
When I read for the first time about these rules in a paper from Cambridge, a bizarre announcement
followed: “We have now reached the point which is liable to cause the greatest intellectual shock” (see
page 1184 of [24]). And it was indeed a severe shock which influences my scientific work till today:
These rules define Pauli algebra. Therefore we are forced to conclude: Pauli matrices represent base
vectors of three-dimensional Euclidean space.
And with the help of the oriented unit volume element (or base trivector or pseudoscalar) I
I = x y z I2 = (x y z)2 = – 1 , (3)
which acts as an imaginary unit, the Pauli relations can be rediscovered in standard form:
x y = I z y z = I x z x = I y . (4)
1
…or in spacetime – 1 for timelike vectors or 0 in the case of light rays, which will be discussed later.
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Every vector r can then be written as linear combination of the Pauli matrices x, y, and z, a fact
already noticed by Cartan (see page 43 of [25]):
r = x x + y y + z z . (5)
Following the idea to describe all “geometric objects as real linear combinations of the generating
units and their geometric products” (see page 823 of [13]) the most general mathematical object
possible in Pauli algebra can be written as multivector
M = k + x x + y y + z z + Axy x y + Ayz y z + Azx z x + V x y z
(6)
M = (k + V I) + (x + Ayz I) x + (y + Azx I) y + (z + Axy I) z .
The second version of multivector (6) is called paravector [26], because the real coefficients
k, x, y, z, Axy, Ayz, Azx, V ℝ are grouped to form a complex-like vectorial structure.
r2 = y r y = – x x + y y – z z (8)
r3 = z r z = – x x – y y + z z . (9)
These are reflections: vector r3 of equation (9) equals the original vector r reflected at the z-axis. If
we multiply a vector r from the left and from the right by a base vector, the vector will be reflected at
the axis which points in the direction of the base vector. Pauli matrices represent base reflection in
three-dimensional Euclidean space.
This can be generalized for arbitrary directions. If we multiply a vector r from the left and from the
right by a unit vector n, the vector r will be reflected at the axis which points in the direction of the
unit vector n:
rref = n r n with n2 = (nx x + ny y + nz z)2 = 1 . (10)
The two interpretations of figure 3 form the geometric core of Geometric Algebra.
It is well known that two succeeding reflections always result in a rotation. Or the other way round:
every rotation can be split into two (or more even numbered) reflections (see pages 1137/1138 of [5]
and pages 45/46 of [25]). Therefore rotations can be modeled by two reflections:
rrot = m rref m = m n r n m with m2 = n2 = 1 . (11)
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If we multiply a vector r from the left and from the right first by a unit vector n and then multiply
the result from the left and from the right by another unit vector m, the vector r will be rotated in the
plane spanned by the two unit vectors n and m through twice the angle between the two unit vectors n
and m (see page 111 of [9]).
This is extraordinary. Please notice the many exclamation marks Doran and Lasenby use when
discussing these points: “This is starting to look extremely simple!” (page 43 of [15]). And: “The rule
also works for any grade of multivector!” (page 44 of [15]). Modeling rotations in Geometric Algebra
is indeed a complete conceptual highlight and much easier than in other mathematical systems.
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mysterious. Unlike differential forms, which are related to areas and volumes, spinors have no such
simple explanation. They appear out of some slick algebra, but the geometrical meaning is obscure”
(see pages 113/114 of [32]).
These words are astonishing, and we should compare them with the conclusion David Hestenes
published 17 years earlier: “Thus … every spinor uniquely determines a rotation dilatation. Therein
lies the operational geometric significance of spinors. The spinors form a subalgebra of Geometric
Algebra. The algebra of spinors was discovered independent of the full Geometric Algebra by
Hamilton, who gave it the name quaternion algebra. Some readers may want to say, rather, that we
have here two isomorphic algebras, but there is no call for any such distinction. A quaternion is a
spinor. The identification of quaternions with spinors is fully justified not only because they have
equivalent algebraic properties, but more important, because they have the same geometric signi-
ficance” (see page 1022 of [33]).
An example of the ignorance with respect to such a picture of spinors can be found when solving
the vacuum Maxwell equation. Even before the re-formulation of Geometric Algebra by David
Hestenes in 1966 [34] it was known (but widely ignored) that the spinorial plane wave solution of the
Maxwell equations results in circular polarized electromagnetic waves (see page 14 of [35]). This is
shown in the following section.
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Inserting equation (15) into equation (14) results after some straightforward mathematical steps in
the following simple intermediate equation (see page 1026 of [33]):
F = z F or E + x y z B = z E + x y B . (16)
The different terms of this equation can now be split into its geometrical constituents with the help
of the inner and the outer products. The inner product of two vectors a and b or of a bivector a and a
vector b is defined as:
1 1
ab= (a b + b a) ab= (a b – b a) . (17)
2 2
It reduces the geometrical grades of the constituents: While a and b are vectors, a b is a scalar (see
left equation of (17)). While a is a bivector and b is a vector, a b is a vector (see right equation of
(17)). The outer product of two vectors a and b or of a bivector a and a vector b is defined as:
1 1
aᴧb= (a b – b a) aᴧb= (a b + b a) . (18)
2 2
It increases the geometrical grades of the constituents: While a and b are vectors, a ᴧ b is a bivector
(left equation of (18)). While a is a bivector and b is a vector, a ᴧ b is a trivector or pseudoscalar (right
equation of (18)). And please note the interesting variation of signs in equations (17) and (18). It really
depends on the geometrical grades of the factors whether a product is symmetric or antisymmetric!
The sum of inner and outer products now form the geometric product
ab = a b + a ᴧ b ab = a b + a ᴧ b . (19)
This is indeed strong stuff, and Sobczyk still today remembers the moment, when he first heard
about these products: “I remember my sense of amazement when he (David Hestenes) wrote down the
basic identity for the geometric multiplication of vectors … Why hadn't I ever heard of this striking
product, and why hadn't I ever heard of a bivector or directed plane segment, since it was the natural
generalization of a vector. Twenty-five years later I still find myself asking these same questions” (see
page 1291 of [37]).
Now we are able to evaluate equation (16):
E + x y z B = z E + z ᴧ E + x y B + x y ᴧ B (20)
vector bivector scalar bivector vector trivector
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If the initial condition E = E0 x holds at z = 0 and t = 0, then the plane wave solution of the
Maxwell equation in empty space is
I ( t – k z)
F(t, z) = E0 x (1 + z) e = E0 (x – z x) [cos( t – k z) + I sin( t – k z)]
(26)
which is shown in figure 4 (see also figure 2 at page 510 of [36] ).
(a) (b)
The plane wave solution given in standard textbooks showing linear polarized electromagnetic
waves is thus an artificial algebraic solution which neglects the geometric structure of the space we
live in. It can only be constructed by a superposition of left and right polarized electromagnetic waves
(see also page 2027 of [33] ).
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And we have now again “reached the point which is liable to cause the greatest intellectual shock”
(page 1184 of [24] ): These rules define Dirac algebra. Therefore we are forced to conclude: Dirac
matrices represent base vectors of four-dimensional spacetime.
Every spacetime vector r can then be written as linear combination of the Dirac matrices t, x, y,
and z, a fact already noticed by Cartan (page 133 of [25]):
r = c t t + x x + y y + z z . (31)
Hestenes does not hide his astonishment about the path the history of mathematics has taken here:
“The Dirac matrices are no more and no less than matrix representations of an orthonormal frame of
spacetime vectors and thereby they characterize spacetime geometry. But how can this be? Dirac never
said any such thing!” (page 694 of [10])
Dirac invented Dirac matrices to model the strange behaviour of relativistic electrons in 1928. He
never considered them to represent spacetime base vectors. Dirac and Pauli always interpreted their
matrices as operators (or as a substantial part of an operator). This is really surprising: clearly all the
mathematics was there, and Pauli and Dirac were masters of it. But instead of following the idea of
Cartan to use the spacetime vector (31), Pauli tried to model physics with the complete spacetime
multivector (see lemma 5 at page 113 of [39])
16
X= x
A 1
A A (with A every possible product of several Dirac matrices) (32)
as the most general mathematical object possible in Dirac algebra. And of course again he used sand-
wich products like ' = S S– 1 (see equation 1 at page 110 of [39] ) or X = … (page 114 of
[39] ) in his calculations. This makes sense, as Dirac matrices possess also a double nature: They can
be interpreted as operands as well as operators which act on operands.
The sandwich products of the Dirac matrices with a spacetime vector r are:
r0 = t r t = c t t – x x – y y – z z (33)
r 1 = – x r x = – c t t + x x – y y – z z (34)
r 2 = – y r y = – c t t – x x + y y – z z (35)
r 3 = – z r z = – c t t – x x – y y + z z (36)
These are spacetime reflections. In equation (33) the vector r is reflected at the time axis, and in the
other equations (34) to (36) the vector r is reflected at the spatial axis. Dirac matrices represent base
reflections in four-dimensional spacetime.
This can be generalized for arbitrary directions. If we multiply a vector r from the left and from the
right by a unit vector n, the vector r will be reflected at the axis which points in the direction of the
unit vector n:
rref = n r n – 1 = ± n r n . (37)
2 –1 2
with n = 1 if n is a time-like unit vector (n = + n) and n = – 1 if n is a space-like unit vector
(n – 1 = – n). The two interpretations of figure 5 form the geometric core of Spacetime Algebra.
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With all that in mind, Lorentz transformations are “starting to look extremely simple” (page 43 of
[15] ), because they are nothing else than spacetime rotations or two successive spacetime reflections.
Therefore they can be modeled with the following very easy to apply formula:
rrot = m rref m – 1 = ± m n r n m with m, n spacetime unit vectors . (38)
Some Geometric Algebra examples for Lorentz transformations, special relativistic length con-
tractions or time dilations, and other special relativistic effects which can be used for secondary school
or introductory university courses can be found in [19], [21], [22], [23], [40].
1 0 0 x 0 y 0 z
t = x = y = z = (40)
0 1 x 0 y 0 z 0
Applying the direct product of Zehfuss and Kronecker [41], [42], [43], Dirac matrices can be com-
posed of Pauli matrices. They then have the signature (+, –, –, –)2:
t = z 1 x = – (z x) x y =(z x) y z = – (z x) z (41)
The main algebraic message is: Two (2 x 2) matrices directly multiplied result in one (4 x 4)
matrix. A Dirac matrix is algebraically composed of two Pauli matrices (or Pauli matrix products). Or
colloquially: There are two Pauli matrices inside one Dirac matrix.
But there is a deep geometric relationship between Pauli algebra and Dirac algebra which contrasts
this sort of superficial algebraic picture: The spatial base vectors of Pauli algebra have to be con-
structed not only by the spatial base vectors of Dirac algebra, but also by its time-like base vector, as
the base vectors of an observer in three-dimensional space depend on the time direction of this
observer in four-dimensional spacetime. This base-vector of time is relevant for the three base vectors
of space we construct.
This can be seen, if we compare the geometric derivative of three-dimensional space (12) with the
geometric derivative of four-dimensional spacetime
□ = t + x + y + z . (42)
c t x y z
Obviously the two geometric derivatives of a multivector are not equal, if we add the time direction
according to the successful procedure of equation (14):
M □ M but: M = □ t M . (43)
c t c t
At the left equation of (43) the time derivatives have different grades. To adjust this, the geometric
derivative (also called Dirac operator) has to be post-multiplied by the time-like base vector t. This
results in the balanced equation (44).
2
Of course other systems of (4 x 4) matrices comply with Dirac algebra, too. For instance, Cartan proposed the
following matrices as spacetime base vectors (see page 133 in connection with page 82 of [25]) with signature
(+, +, +, –): A1 = z x, A2 = – z y, A3 = x 1, and A4 = c (z x) 1
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+ x + y + z M= + x t + y t + z t
M (44)
c t x y z c t x y z
The relation between Pauli and Dirac matrices can be found now by comparing the base vectors in
front of the spatial derivatives:
x = x t y = y t z = z t (45)
Geometry tells us, that there are not (colloquially spoken) two Pauli matrices inside one Dirac
matrix. There are two Dirac matrices inside every Pauli matrix instead! Of course, the Pauli matrices
of equation (45) are written as (4 x 4) matrices. Nevertheless, they totally follow the rules of Pauli
algebra3.
“Consequently, a vector in the Pauli algebra of space is a bivector in the Dirac algebra of space-
time” (page 1292 of [37] ). Whenever we move along a spatial line, we actually move along a
spacetime area element. And this fact is so important, that Sobczyk included it as special highlight in
his birthday address for David Hestenes [37].
3
In five-dimensional spacetimes or spacetimevelocities this works in a similar way. Then Pauli matrices (base
vectors of three-dimensional space) can be identified as products of three Dirac matrices (base vectors of five-
dimensional spacetimes or spacetimevelocities, see equations 24-26 of [44]) as x’ = x t v, y’ = y t v,
z’ = z t v, if together with the Dirac matrices (41) the time-like Dirac matrix v = x 1 (see equation 45
at page 497 of [45]) is used as additional base vector for the direction of the fifth coordinate.
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This can be done for the other Gell-Mann matrices in a similar way. The results are in three-dimen-
sional space (see following equations on the left side) and according to transformation (45) in four-
dimensional spacetime (equations on the right side):
1 1
1(M) = (x M y + y M x) 1(M) = – t (x M y + y M x) t (50)
2 2
1 1
2(M) = (z M 1 – 1 M z) 2(M) = – t (z M t – t M z) t (51)
2 2
1 1
3(M) = (x M x – y M y) 3(M) = – t (x M x – y M y) t (52)
2 2
1 1
4(M) = (z M x + x M z) 4(M) = – t (z M x + x M z) t (53)
2 2
1 1
5(M) = (1 M y – y M 1) 5(M) = – t (t M y – y M t) t (54)
2 2
1 1
6(M) = (y M z + z M y) 6(M) = – t (y M z + z M y) t (55)
2 2
1 1
7(M) = (x M 1 – 1 M x) 7(M) = – t (x M t – t M x) t (56)
2 2
1 1
8(M) = (x M x + y M y – 2 z M z) 8(M) = – t (x M x + y M y – 2 z M z) t (57)
2 3 2 3
These results are nearly identical to results which can be found in the literature. Only the transfor-
mations for the diagonal Gell-Mann matrices 5(M) and 8(M) slightly differ from these results, see
table 1 at page 168 of [46] for the equations given above on the left side and equation 26 a-h at page 7
of [47] for the equations on the right side.
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x y z. But then we are stuck: There are two possibilities for an oriented volume element in nature. In
the preceding sections we always decided to take I = x y z as oriented volume element. But does
nature really decides for a positive sign if all base vectors are multiplied? This positive sign is nothing
else than the consequence of an arbitrary decision we as human beings make. But there is another
possibility: It might as well be that nature decides for a negatively oriented unit volume element – I.
Or in other words: Do we live in a left-handed or a right-handed coordinate system designed by
nature? Obviously we do not know. And the more interesting question is: What will happen, if even
nature does not know?
A possible answer to this question could be, that at every measurement there will be a probability
(1), that the sign is positive. And there will be a probability (–1) = 1 – (1), that the sign is nega-
tive. The consequences are clear: Whenever we measure a physical entity U which should distinguish
between the different possible signs of the oriented volume element (for example spin, magnetic
moment) we will get a statistical result which represents the mean value of U:
U = (–1) (–1) + (1) (1) (58)
Therefore the hypothesis is: It is not the spin of the electron, which is uncertain in quantum
mechanics, but the orientation or handedness of the space we live in.
As said, the algebra of quantum mechanics is understood and is completely available for all of us.
For instance you can find the algebra of equation (58) in Jordan’s book (page 73 of [50] ), where this
equation is the starting point for his discussion of quantum mechanics. Jordan develops an
interpretation of quantum mechanics, in which “all quantities (are) made from spin” (chapter 10 of
[58] ). We should try to follow his mathematical route, which really offers an “easy way to look rather
deeply into quantum mechanics” (pager VIII of [58] ). But at the same time we should also try to
change the interpretative frame of Jordan’s presentation. Quantum mechanical quantities are made
from geometry and might be a consequence of uncertainty in the handedness of the space4.
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References
[1] Rota G-C 1997 Indiscrete Thoughts (Boston: Birkhäuser) p 232–233
[2] Sossinsky A 2002 Knots. Mathematics with a Twist (Cambridge: Harvard University Press) p 12–13
[3] Stone J and Pascalis O 2010 Footprints Sticking Out of the Sand (Part I) – Children’s Percep-
tion of Naturalistic and Embossed Symbol Stimuli. Perception 39 (9) pp 1254–1260
[4] Penrose R and Rindler W 1984 Spinors and Space-Time. Vol. 1: Two-Spinor Calculus and
Relativistic Fields (Cambridge: Cambridge University Press) p 43
[5] Misner C W, Thorne K S and Wheeler J A 1973 Gravitation (New York: Freeman) p 1149
[6] Feynman R and Weinberg S 1987 Elementary Particles and the Laws of Physics. The 1986
Dirac Memorial Lectures (Cambridge: Cambridge University Press) pp 29, 30, 58
[7] Snygg J 1997 Clifford Algebra. A Computational Tool for Physicists (New York: Oxford
University Press) p 12
[8] Staley M 2010 Understanding Quaternions and the Dirac Belt Trick. European Journal of
Physics 31 (3) pp 467–478
[9] Hestenes D 2003 Oersted Medal Lecture 2002: Reforming the Mathematical Language of Phy-
sics. American Journal of Phyics 71 (2) pp 104–121
[10] Hestenes D 2003 Spacetime Physics with Geometric Algebra. American Journal of Physics 71
(7) pp 691–714
[11] Jung F 2006 Geometrische Algebra und die Rolle des Clifford-Produkts in der Klassischen und
Quantenmechanik. Diploma thesis, written at the Institute of Physics, University of Mainz
[12] Lounesto P 1998 Book review of [7]. Foundations of Physics 28 (6) p 1021
[13] Parra Serra J M 2009 Clifford Algebra and the Didactics of Mathematics. Advances in Applied
Clifford Algebras 19 (3/4) pp 819–834
[14] Hestenes D 2002 New Foundations for Classical Mechanics. 2nd ed. (New York: Kluwer
Academic Publishers)
[15] Doran C and Lasenby A 2003 Geometric Algebra for Physicists (Cambridge: Cambridge Uni-
versity Press)
[16] Chisolm E 2012 Geomertric Algebra. arXiv:1205.5935v1 [math-ph] May 27th, 2012
[17] Vince J 2008 Geometric Algebra for Computer Graphics (London: Springer) p VII
[18] Hildenbrand D 2013 Foundations of Geometric Algebra Computing (Berlin: Springer)
[19] Horn M E 2012 Pauli-Algebra und S3-Permutationsalgebra – Eine algebraische und geometri-
sche Einführung. Published at www.bookboon.com/de (London: Ventus Publishing ApS)
[20] González Calvet R 2001 Treatise of Plane Geometry Through Geometric Algebra. Published at
http://campus.uab.es/~PC00018 (Barcelona: University of Barcelona)
[21] Horn M E 2009 Die Spezielle Relativitätstheorie im Kontext der Raumzeit-Algebra & Arbeits-
bögen zur Geometrischen Algebra und zur Raumzeit-Algebra. Published in Nordmeier V
and Grötzebauch H (Eds) Didaktik der Physik. CD-ROM of the Spring Meeting of the Ger-
man Physical Society in Bochum, paper 30.40 (Berlin: LOB – Lehmanns Media)
[22] Horn M E 2011 Grassmann, Pauli, Dirac: Special Relativity in the Schoolroom. Published in
Petsche H-J, Lewis A C, Liesen J and Russ S (Eds) From Past to Future – Graßmann’s
Work in Context, pp 435–450 (Basel: Birkhäuser)
[23] Horn M E 2010 Die Raumzeit-Algebra im Abitur. PhyDid B – Didaktik der Physik, paper 28.4
[24] Gull S, Lasenby A and Doran C 1993 Imaginary Numbers are not Real – The Geometric Alge-
bra of Spacetime. Foundations of Physics 23 (9) pp 1175–1201
[25] Cartan É 1981 The Theory of Spinors. Unabridged republication of the complete English
translation first published in 1966 (New York: Dover Publications)
[26] Baylis W E 2002 Electrodynamics. A Modern Geometric Approach (Boston: Birkhäuser) p 22
[27] Pauli W 1927 Zur Quantenmechanik des magnetischen Elektrons. Zeitschrift für Physik 43
(9/10) pp 601–623
[28] Dyson F J 1972 Missed Oppertunities. Printed version of the Josiah Willard Gibbs Lecture of
Jan 17th, 1972. Bulletin of the American Mathematical Society 78 (5) pp 635–652
14
Symmetries in Science XVI IOP Publishing
Journal of Physics: Conference Series 538 (2014) 012010 doi:10.1088/1742-6596/538/1/012010
[29] Peirce C S 1877 Note on Grassmann’s Calculus of Extension. Proceedings of the American
Academy of Arts and Sciences 13 pp 115–116
[30] Clifford W K 1878 Applications of Grassmann's Extensive Algebra. American Journal of
Mathematics 1 (4) pp 350–358
[31] Dirac P A M 1945 Applications of Quaternions to Lorentz Transformations. Proceedings of the
Royal Irish Academy, Vol L, Sect A, pp 261–270
[32] Atiyah M F 1998 The Dirac Equation and Geometry. Published in Goddard P (Ed) Paul Dirac.
The Man and His Work. Chap. 4, pp 108–124 (Cambridge: Cambridge University Press)
[33] Hestenes D 1971 Vectors, Spinors, and Complex Numbers in Classical and Quantum Physics.
American Journal of Physics 39 (9) pp 1013–1027
[34] Hestenes D 1966 Space-Time Algebra. (New York: Gordon & Breach)
[35] Moses H E 1958 The Solution of Maxwell’s Equations in Terms of a Spinor Notation. Part I:
The Initial Value Problem in Terms of Field Strengths and the Inverse Problem. New York
University, Institute of Mathematical Sciences, Research Report No. EM-114
[36] Vold T 1993 An Introduction to Geometric Calculus and its Application to Electrodynamics.
American Journal of Physics 61 (6) pp 505–513
[37] Sobczyk G 1993 David Hestenes: The Early Years. Foundations of Physics 23 (10) pp 1291–
1293
[38] Minkowski H 1909 Space and Time. English version published in Lorentz H A, Einstein A,
Minkowski H and Weyl H 1923 The Principle of Relativity. Chap 5, pp 73–91 (New York:
Dover Publications)
[39] Pauli W 1936 Contributions Mathématique à la Théorie des Matrices de Dirac. Annales de
l’Institut Henri Poincaré 6 (2) pp 109–136
[40] Horn M E 2010 Die Spezielle Relativitätstheorie in der Mathematikerausbildung & Pauli-
Algebra und Dirac-Algebra. OH-Folien zur Einführung in die Spezielle Relativitätstheorie
im Rahmen eines Kurses zur Physik für Mathematiker. PhyDid B – Didaktik der Physik,
paper 19.35 & attachment 19.35, URL: http://www.phydid.de [30 Nov 2013]
[41] Zehfuss J G 1858 Über eine gewisse Determinante. Zeitschrift für Mathematik und Physik 3
pp 298–301
[42] Steeb W-H 1991 Kronecker Product of Matrices and Applications (Mannheim, Wien, Zürich:
Bibliographisches Institut/Wissenschaftsverlag)
[43] Henderson H V, Pukelsheim F and Searle S R 1983 On the History of the Kronecker Product.
Linear and Multilinear Algebra 14 (2) pp 113–120
[44] Horn M E 2011 Die fünfdimensionale Raumzeit-Algebra am Beispiel der Kosmologischen
Relativität. PhyDid B – Didaktik der Physik, paper 17.1
[45] Horn M E 2012 Translating Cosmological Special Relativity into Geometric Algebra. Published
in Sivasundaram S (Ed) Proceedings of the 9th International Conference on Mathematical
Problems in Engineering, Aerospace and Sciences in Vienna (ICNPAA). American Institute
of Physics, AIP Conference Proceedings, 1493 pp 492–498
[46] Hestenes D 1982 Space-Time Structure of Weak and Electromagnetic Interactions. Foundations
of Physics 12 (2) pp 153–168
[47] Vroegindewey P G 1989 Gauge Theories and Space-Time Algebra. Reports on Applied and
Numerical Analysis (RANA 89-06). Department of Mathematics and Computing Science,
Eindhoven University of Technology, May 1989.
[48] Hestenes D 1992 Mathematical Viruses. Published in Micali A, Boudet R and Helmstetter J
(Eds) Clifford Algebras and Their Applications in Mathematical Physics, pp 3–16 (Dord-
recht: Kluwer Academic Publishers)
[49] Feynman R P 2008 QED – The Strange Theory of Light and Matter. First Princeton paperback
printing (Princeton, New Jersey: Princeton University Press)
[50] Jordan T 2005 Quantum Mechanics in Simple Matrix Form (New York: Dover Publications)
[51] Seneca Letter to Lucilius, letter 104, section 26
15