OK, there is one more thing. Quaternions are not objects like unit vectors, even if one introduce Clifford geometric product. Quaternions are objects like bivectors (product of two vectors). Historically, Hamilton and others were trying to connect unit vectors and quaterninons, but without positive outcome. Unit vectors i, j, k are named after quternions. 

Again, unit vectors with Clifford product are base for Clifford algebra. Pauli matrices are matrix representation (one of them) of  unit vectors in 3D. Quaternions are bivectors in Clifford 3D algebra. 

Quaternions are not unit vectors-like objects, they are closer to cross product then to unit vectors.

Yes. In fact, quaternions were invented before vectors, and they have a basis (1,i,j,k), and can be represented by the 4 2x2 matrices (1, Pauli1, Pauli2, Pauli3). The product of two quaternions contains the dot product in its real part, and the cross product in the imaginary part.

Note, however, that in nature there are vectorial quantities, but their coordinates are always man-made: we choose the three directions, and we choose the units. (Othogonality, metric, etc., can be defined with light years, see e.g, the second volume of Landau and Lifshitz, on spacetime geometry). The mapping of vectors into 3 coordinates, or a Hermitean matrix (with the Pauli matrices) goes through a coordinatization, so it is at least as man-made as coordinates.

To the last part of your questions, on quantum gravity: in GR, the metric is a well-defined geometric object, and so is orthogonality (in fact, the 4D version of orthogonality, namely the causal structure, is more fundamental than the metric, it only depends on the conformal class of the metric, not the full metric). There have been attempts in quantum gravity to treat the conformal part separately, and see which sector is responsible for the difficulties. Note, however, that this has nothing to do with how you represent vectors once you have coordinates.

Dear Steve ~

Quaternions or Pauli matrices (which give a representation of the quaternion algebra) don’t eliminate the need for choosing base vectors i, j, k, they just provide a way of manipulating formulae expressed in terms of a basis i, j, k. So they are not an answer to your question, which seems to me to be “can we adequately deal with Euclidean 3-space without an orthogonal set of axes?” The answer is yes − Euclid did so long before Descartes thought of the idea of orthogonal coordinates.

Three-dimensional Euclidean geometry can be constructed axiomatically by specifying the properties satisfied by distances (given by a mapping of pairs of points to the real numbers R) and angles  (given by mappings of pairs of intersecting lines or pairs of intersecting planes to S1).

Well, instead of orthogonal vectors, one can choose only liniar independent vectors. Also one can choose spherical coordinates (one radius and two agles) or many other way to parametrise the space. Computations will be more complicated, but physical truths should not be different. They must transcend the casual representation (casual parametrisation), whatever this would be. Parametrisation is only a prerequisite, a convention necessary to start the work. The work itself (or better said, its result) must be finally independent of the chosen parametrisation.

On the other side, it is possible that working in another system would make the human researcher more sensitive to some aspects (s)he has not remarked before, in the good old orhogonal system.

Yes, Pauli matrices are just one of many matrix representations of 3D Clifford algebra, based on vectors in 3D Euclidean space. But, to obtain algebra we are introducing new product of vectors (geometric product), in fact, very intuitive and natural choice.

Literature is available on web, just search for geometric algebra (or Clifford algebra).

@Steve Faulkner

It would seem to me that what you are looking for is the mathematical concept of manifold. From my (very limited) mathematical background, it is a set with certain properties, which affords for local bases in its every points.

... every point, sorry.

Representations of Pauli spin matrices assume the standard axes of 4-space and since they represent "spin," they "mix" the different axes. I think your search will not be helpful. The purpose of unit vectors is simply pictorial. Mathematically one can do without them simply by using only coordinates and their purpose is just to make the vector nature of position in 4-space visualizable. Back in the early 19th century, 3-space partial differential equations were all written out explicitly in components. What vector operators did was to make common operations like dot product and cross product more tractable. Four space is just used as a tabula raza and carries no physics. In 4-space some of what was "dynamics" in 3-space just turns out to be more kinematics.

Euclidean 3-space with the cross product form a Lie algebra, the Lie algebra of the group SO_3 or SU_2. If we apply this isomorphism between (R^3,x) and (su_2,[ , ]), then i,j,k become the Pauli matrices.

Best regards

Jost Eschenburg

@Steve: As 'orthogonality' is a subsidiary concept of projective geometry, i.e. starting from axiomatic and projective geometry in 3-space and implementing further assumptions in order to obtain orthogonal, equidistant and/or even euclidean axes/unit 'vectors', it is also a very small and narrow 'window' only in order to treat GR. The quaternions (or Pauli matrices) in most cases are nice and handy tools to perform calculations, but often they are only needed due to some low-dimensional isomorphisms or coverings from Lie theory, some technocratical calculations and/or some accidental identifications from the Clifford Algebra construction schemes. Going deeper, you'll have to switch to non-euclidean geometry (e.g. Study's or Blaschke's work) where the quaternions (besides technical applications) have their rôle in elliptic geometry, e.g. on the sphere ij=k - understood as transformations with squares ~ -1 - the three transformations shift a point along the sides of a spherical triangle. ij shift the point to k. So the matrix rep of the quaternions and orthogonality as well are 'man-defined' only ;-)

Yes-and not only the Pauli matrices, but any representation of the rotation group can be used, for describing  membranes, in the presence of flux, that encodes the non-commutativity. This is one example of what's called non-commutative geometry.

Facts that are interesting to know are:

Quaternionic number systems exist in several versions. For example quaternions with right handed vector product exist and quaternions with left handed vector product exist. Due to the four dimensions of quaternions it is possible to define sixteen versions of quaternionic number systems that are all ordered in a different fashion.

Besides of quaternions also quaternionic functions exist and for these functions exist a very compact and quite comprehensible quaternionic differential calculus. It includes an interesting wave equation that differs essentially from the Maxwell based wave equation.

Hilbert spaces are structured storage media that are often used by quantum physicists. These Hilbert spaces can only cope with real numbers, complex numbers and quaternions. Further, separable Hilbert spaces can only store discrete data. However, each infinite dimensional separable Hilbert space owns a Gelfand triple, which is a non-separable Hilbert space and can also store continuums in the eigenspaces of some of its operators.

Some quantum physicists use tensor products of Hilbert spaces. With quaternionic Hilbert spaces that poses problems, because the tensor product of two quaternionic Hilbert spaces is a real Hilbert space.

The diversity of elementary particles can only be well understood when the existence of different versions of quaternionic number systems are accepted.

Dirac's equation for the electron and the positron can be transformed to quaternion format and gives then two equations. One treats the electron and the other treats the positron.

Quaternions are ideally suited to store dynamic geometric data.

In Hilbert spaces physical fields can be represented by eigenspaces of suitable operators. The parameter spaces of the functions that represent these fields can also be represented by eigenspaces of suitable operators. It has much sense to use quaternionic number systems as parameter spaces and quaternionic functions in order to represent fields. 

In Hilbert spaces several types of fields can coexist. It is possible to define a model wide background parameter space and a corresponding field that represents universe. Besides of that it is possible to define parameter spaces on which discrete objects reside. The ordering of these parameter spaces can be particle type dependent.

To Hans van Leunen. Dear Hans, I would like to complement your remark by reminding the audience about uses of Grassmannians. These are spaces heavily used in computer vision/image recognition and, thus, in robotics. Incidentally, they are  used in twistor theory by R.Penrose as well. Details on this topic  can be found in my book http://www.worldscientific.com/worldscibooks/10.1142/8514

Not so well known is the fact that quaternions can rotate other quaternions or complete quaternionic functions. Particular quaternions can shift the anisotropy of a quaternionic distribution to another dimension.

In fact quaternions implement much of the functionality of gluons. Gluons can switch the color charge of quarks. 

These capabilities also enable quaternions to represent tri-states. These three states can be the three color charges of a quark or the three generation flavors of fermions. Rotating tri-states might represent flavor switching neutrinos.

All this is well known.Just look at selected papers from the "Mathematical Intelligencer"

http://www.springer.com/mathematics/journal/283

OK, there is one more thing. Quaternions are not objects like unit vectors, even if one introduce Clifford geometric product. Quaternions are objects like bivectors (product of two vectors). Historically, Hamilton and others were trying to connect unit vectors and quaterninons, but without positive outcome. Unit vectors i, j, k are named after quternions. 

Again, unit vectors with Clifford product are base for Clifford algebra. Pauli matrices are matrix representation (one of them) of  unit vectors in 3D. Quaternions are bivectors in Clifford 3D algebra. 

Quaternions are not unit vectors-like objects, they are closer to cross product then to unit vectors.

@Juan

Hilbert spaces can cope with real data, with complex number based data and with quaternionic data. Quaternionic data are ideally suited for the storage of dynamic geometric data. The Hilbert space stores these data in the eigenspaces of its operators. A separable Hilbert space can only store discrete data. Every infinite dimensional separable Hilbert space owns a Gelfand triple, which is a non-separable Hilbert space that supports operators, which have continuums as their eigenspaces. These continuums can represent fields.

. See: “Division algebras and quantum theory” by John Baez. http://arxiv.org/abs/1101.5690

@Christian

The paper that you refer to takes as conclusion that Clifford algebras are the best solution for three dimensional vector systems. Most physical data are dynamic geometric data and are not three dimensional but four dimensional. These data can also be represented by Clifford algebras, but they can better be represented by quaternions. On the other hand, current physical theories use a combination of coordinate time and space, which gives a spacetime model that has a Minkowski signature. Quaternions can store proper time and space and gives then a space progression model with an Euclidean signature.

Hilbert spaces can directly cope with quaternions and can store dynamic geometric data in quaternionic eigenvalues. In this way quaternionic functions define fields that are represented as eigenspaces in non-separable Hilbert spaces.

The problem with quaternions is that due to the four dimensions of quaternions sixteen different versions of quaternionic number systems exist. Half of these quaternionic number systems are right handed. The other half features a left handed multiplication.

This can be simulated with Clifford algebras, or with a combination of spinors and matrices, but it is also possible to indicate the version of a quaternion with an extra superscript. The last method delivers compact and clear formulas.

Hilbert spaces appear to be capable to employ different versions of the quaternionic number systems within the same Hilbert space.

The diversity of the quaternionic number system plays a role in the behavior of elementary particles and the version of the number system determines part of the properties of the particle such as electric charge and color charge.

See: "On the Origin of Electric Charge" ; http://vixra.org/abs/1507.0185

you can equally well do vector analysis using exterior algebra and p forms

In roger penrose, the road to reality has a very nice intro and does maxwell using

it. It is due to Elie Cartan

Quaternions again, and again ... Ok, let as form Clifford basis, take scalar part (grade 0) and bivector part (grade 2) and we have even multivectors basis for quaternions (just check multiplication table) . With my Mathematica code for Clifford algebra I can do any calculation on quaternions. Any! Bivectors have natural geometric interpretation and they are consequence of vector multiplication. Quaternions are  ingenious step forward from 19 century, they are very suitable for rotations in 3D, but the same formalism doesn't apply to higher dimensions, there is no 5D quaternions. Clifford algebra deals easily with rotations in any dimension, using the same formalism as in 3D. 

For complex quaternions just use even part of 3D Clifford algebra over complex field (but Clifford 3D algebra possess complex and hyper-complex structure over real field). 

"Four dimensional dynamics" (van Leunen) follows naturally and easily  from 3D Clifford algebra (Baylis, Sobczyk, Chappell, ...).  There is no need for time dimension. 

Question here, as I understand it, was about Pauli matrices and vectors in 3D, and, again, quaternions are not vectors in 3D, they are squared to -1. 

Geometric (Clifford) algebra gives amassing possibility to express great extent of mathematical physics in one unique language and even simplify it, with many surprising consequences. Rotations are extremely simple, spinors are there, geometry is intuitive and powerful,  one can forget about matrices, tensors ... Fundamental theorem of calculus in geometric algebra contains all well known theorems from vector calculus in one formula and is richer then all them together, etc. 

All You need is to start with is simple and intuitive:

1) there is a vector product and it is not commutative

2) vectors are squared to reals.

This is legacy from 19-th cent., from Hamilton, Grassman and Clifford. Clifford died young, Gibbs published his famous book about vectors and mathematica started to grow as wild tree without proper care: many branches with too many languages. Geometric algebra contains all that branches and much more, but within just one powerful language. It is very easy to implement  it on computer. Forget about nonintuitive matrices, forget about coordinates, they are easy to include after your calculations. 

So, dear physicists, especially young students, we all should to think seriously about our choice: to follow  complicated mathematics with so many languages, or just rethink about vectors multiplication and just one powerful language. Hestenes and his followers  are done so many nice work since 1960s.

Here is may challenge to readers: give me any "branch" from mathematical physics and I will give You an answer within geometric algebra, or at  least suggest You were to find answer, as a rule, simpler one and with unexpected consequences.  

For example, what is origin of a spin in quantum mechanic? Well, look at rotations in 3D geometric algebra. Spin formulas appear as geometrical consequence of properties of rotations. Famous physicist Pauli introduced his matrices to describe spin in quantum mechanics. But there is no need for matrices, just learn how to properly multiply vectors and use them: ordinary well known unit vectors are suited here much better than Pauli's matrices. This is proven mathematical fact!

Multiply all base unit vectors from 3D and find interesting object: commutes with all elements of algebra, squares to -1, so, it is nice imaginary unit, but this time with geometrical interpretation. Now enjoy in quantum mechanic formalism, every appearance of "imaginary unit" will give You something interesting and   geometrically clear.

My prophecy for future is use of Clifford product of vectors (and Gibbs's book will be known  as sideway in history of physics).

Christian,

I agree, but my "problem" is that I firmly believe that time is not a 4-th dimension. 

And that is not just my arbitrary choice, but something that came to me after years of rethinking about time. 

Algebra of Dirac matrices (STA) is nice, useful, but there is no need for it: Cl3 is enough (Baylis and Sobczyk). 

My "prophecy" is intended to possibly motivate someone, not to be taken seriously. 

But, since first excitements doing geometry as young boy, there is almost nothing that really is "breath taken away" for me in mathematics. So, forgive me, please, my "prophecy".

Thanks for Your comment! 

Miroslav, to my opinion, your only argument in favour of your Hestenes and 'Geometric Algebra' evangelism  above is the way to use it on a computer. Sounds like a mechanism for people who do not know about the respective background and just want to calculate some formalism...?! Of course you can do a lot of formalism in N dimensions, however, nature decided to prefer certain dimensions. And time in conjunction with velocities (derived from actions) have to be related to Klein's Erlanger program and some resulting consequences. That's a very basic and still valid issue in (Lie) group theory. Clifford algebras (or Hankel's hypercomplex numbers or Raschewskis multi-vectors or ... - to be continued...) provide you representations as a lot of basic ingredients are already put in 'by hand', e.g. 'the metric'. So whether you count time as 4th dimension has relevance only when being stuck to point concepts such as manifolds and differentials... and the respective number systems (or 'Clifford algebras') are strictly related to the respective geometries which has at least consequences for the related physics...