11 Nov

Course on vector calculus via geometric algebra

Posted by Quirino M. Sugon Jr in Uncategorized. Tagged: , , , , , .

I just started teaching two courses on geometric algebra at the Physics Department of Ateneo de Manila University.  The first is Ps 121: Vector Calculus and Complex Numbers.  The second is Ps 195: Vector Calculus in Manifolds for Physicists.

Ps 121 is for 12 sophomore physics students.  It is a good thing that vectors and complex numbers will be taught in the same course.  So I told my students that I will simultaneously teach them vector and complex analysis, because imaginary numbers are eithe bivectors or trivectors.  The first half of the course is on two-dimensions; the second half is on three-dimensions.

I began the course with \mathbf r=x\mathbf e_1+y\mathbf e_2.  We squared the vector and derived the relations \mathbf e_1^2=\mathbf e_2^2=1 and \mathbf e_1\mathbf e_2=\mathbf e_2\mathbf e_1.  We derived the properties of the unit bivector \hat i=\mathbf e_1\mathbf e_2.  We showed that \hat i is an imaginary that anticommutes with vectors and also a 90^\circ rotation operator.

We factored out \mathbf e_1 in a vector to transform it into a complex number form:  \mathbf r=\mathbf e_1(x+\hat i y)=(x-\hat i y).  This makes explicit the relationship of vectors and complex numbers.  The square of a vector then is equal to the product of a complex number and its conjugate: \mathbf r=(x-\hat y)(x+\hat i y).  For the product of two vectors, we showed that \hat a=\mathbf a\cdot\mathbf b+\mathbf a\wedge\mathbf b.  I remarked that \mathbf a\wedge\mathbf b is an oriented area in Grassmann algebra.  I promised that in the next meeting we shall discuss reflections and rotations.

Ps 195 should have been a special topics course.  But since the teacher who designed it described it as Vector Calculus on Manifolds for Physicists, I adopted the name.  This is a tutorial course for three senior students and two sit-ins.  I cannot teach the couse using the framework of differential forms and tensor analysis, because I do not understand them much.  It is difficult to teach an old dog new tricks.  But a more apt metaphor would be the parable of the finding of a treasure in a field: you sell everything you have and buy the field.  Or the parable of the pearl of great price: the man who finds it will sell everything he has and buys the pearl.  In the same way, when one knows geometric algebra, you cannot help but give up all the other algebras that you know, or at least reinterpret them in the light of the new formalism.

I told my students that the study of manifolds and differential geometry is essentially the study of maps.  There are several transformations  in map making: translation, reflection, rotation, dilation, projection, and wrapping.  What we want to know is if there are geometric properties in the original object that remains preserved by the mapping and how do other properties change.  Some of the properties are angles, lengths, areas, and volumes.  To analyze these properties, we shall use the methods of differential geometry and poisson brackets.

I followed the same introductory lecture as Ps 121.  But since these students were under me before in their Ps 41 when they were still freshmen, they already know the Pauli identity for geometric products and the generalized vector rotation expression via half-angle exponentials.  I made an introductory lecture in the complex vector algebra of the plane, so that I would not alienate my two sit-ins.  They can follow the lecture and respond to my questions.  This is a good sign.  Normally, it is difficult to teach geometric algebra to older students and faculty because they already know too many mathematics.  I really prefer to teach geometric algebra to freshmen and sophomore undergraduates because they have less mathematical baggage.

For this course I was able to reach reflections (actually only flips) before the hour ends.  But the reflection is only about the x-axis: \mathbf r'=\mathbf e_1\mathbf r\mathbf e_1.  I told them that traditional complex analysis can also do the same using complex conjugation.  But there are reflections (flips) that complex analysis cannot do, such as flip about any arbitrary axis in the xy-plane.  I shall discuss the general reflection equation next meeting.

24 Jul

Spherical basis vectors in terms of exponential rotation operators

Posted by Quirino M. Sugon Jr in Uncategorized. Tagged: , , , , .

The spherical basis vectors \mathbf e_r, \mathbf e_\theta, and \mathbf e_\phi may be expressed as

(1a)\qquad\mathbf e_r= e^{-i\mathbf e_3\phi/2} e^{-i\mathbf e_2\theta/2}\mathbf e_3 e^{i\mathbf e_2\theta/2}e^{i\mathbf e_3\phi/2},
(1b)\qquad\mathbf e_r= e^{-i\mathbf e_3\phi/2} e^{-i\mathbf e_2\theta/2}\mathbf e_1 e^{i\mathbf e_2\theta/2}e^{i\mathbf e_3\phi/2},
(1c)\qquad\mathbf e_\phi= e^{-i\mathbf e_3\phi/2} e^{-i\mathbf e_2\theta/2}\mathbf e_2 e^{i\mathbf e_2\theta/2}e^{i\mathbf e_3\phi/2},

Equations (1a) to (1c) state that \mathbf e_r, \mathbf e_\theta, \mathbf e_\phi are the unit vectors \mathbf e_3, \mathbf e_1, and \mathbf e_2 rotated counterclockwise about the the vector \mathbf e_2 by an angle \theta and then rotated counterclockwise about the vector \mathbf e_3 by an angle \phi.

Using the exponential relations

(2a)\qquad \mathbf e_1 e^{i\mathbf e_2\theta/2}=e^{-i\mathbf e_2\theta/2}\mathbf e_1,
(2b)\qquad \mathbf e_2 e^{i\mathbf e_2\theta/2}=e^{i\mathbf e_2\theta/2}\mathbf e_2,
(2c)\qquad \mathbf e_3 e^{i\mathbf e_2\theta/2}=e^{-i\mathbf e_2\theta/2}\mathbf e_3,

Eqs. (1a) to (1c) simplifies to

(3a)\qquad \mathbf e_r=e^{-i\mathbf e_3\phi/2}\mathbf e_3 e^{i\mathbf e_2\theta}e^{i\mathbf e_3\phi/2},
(3b)\qquad \mathbf e_\theta=e^{-i\mathbf e_3\phi/2}\mathbf e_1 e^{i\mathbf e_2\theta}e^{i\mathbf e_3\phi/2},
(3c)\qquad \mathbf e_\phi=e^{-i\mathbf e_3\phi/2}\mathbf e_2 e^{i\mathbf e_3\phi/2}.

Furthermore, employing the Euler identities

(4a)\qquad\mathbf e_1e^{i\mathbf e_2\theta}=\mathbf e_1\cos\theta-\mathbf e_3\sin\theta,
(4b)\qquad\mathbf e_3e^{i\mathbf e_2\theta}=\mathbf e_3\cos\theta+\mathbf e_1\sin\theta,

together with the exponential realations

(5a)\qquad\mathbf e_1 e^{i\mathbf e_3\phi/2}=e^{-i\mathbf e_3\phi/2}\mathbf e_1,
(5b)\qquad\mathbf e_2 e^{i\mathbf e_3\phi/2}=e^{-i\mathbf e_3\phi/2}\mathbf e_2,
(5c)\qquad\mathbf e_3 e^{i\mathbf e_3\phi/2}=e^{i\mathbf e_3\phi/2}\mathbf e_3,

Eqs. (3a) to (3c) becomes

(6a)\qquad\mathbf e_r=\mathbf e_3\cos\theta+\mathbf e_1 e^{i\mathbf e_3\phi}\sin\theta,
(6b)\qquad\mathbf e_\theta=\mathbf e_1 e^{i\mathbf e_3\phi}\cos\theta-\mathbf e_3\sin\theta,

(6c)\qquad\mathbf e_\phi=\mathbf e_2 e^{i\mathbf e_3\phi}

Finally, using the Euler identities

(7a)\qquad \mathbf e_1 e^{i\mathbf e_3\phi}=\mathbf e_1\cos\phi + \mathbf e_2\sin\phi,
(7b)\qquad\mathbf e_2 e^{i\mathbf e_3\phi}=\mathbf e_2\cos\phi-\mathbf e_1\sin\phi,

Eqs. (6a) to (6c) reduces to

(8a)\qquad\mathbf e_r=\mathbf e_3\cos\theta+\mathbf e_1\sin\theta\cos\phi + \mathbf e_2\sin\theta\sin\phi,
(8b)\qquad\mathbf e_\theta=\mathbf e_1\cos\theta\cos\phi + \mathbf e_2\cos\theta\sin\phi-\mathbf e_3\sin\theta,

(8c)\qquad\mathbf e_\phi=\mathbf e_2\cos\phi-\mathbf e_1\sin\phi.

This corresponds to the standard expressions for the spherical basis vectors in rectangular coordinates (see Wikipedia).

17 Jul

Complex Geometry and Dirac Equation

Posted by Quirino M. Sugon Jr in Uncategorized. Tagged: , , , , , .

S. De Leo, WA Rodrigues, J. Vaz (Dpt Fisica,Lecce and IMECC-UNICAMP)

(Submitted on 17 May 1999)

Abstract: Complex geometry represents a fundamental ingredient in the formulation of the Dirac equation by the Clifford algebra. The choice of appropriate complex geometries is strictly related to the geometric interpretation of the complex imaginary unit $i=\sqrt{-1}$. We discuss {\em two} possibilities which appear in the multivector algebra approach: the $\sigma_{123}$ and $\sigma_{21}$ complex geometries. Our formalism permits to perform a set of rules which allows an immediate translation between the complex standard Dirac theory and its version within geometric algebra. The problem concerning a double geometric interpretation for the complex imaginary unit $i=\sqrt{-1}$ is also discussed.

Comments: 11 pages, RevTex
Subjects: High Energy Physics – Theory (hep-th)
Journal reference: Int.J.Theor.Phys. 37 (1998) 2479
Report number: IMECC RP16/98
Cite as: arXiv:hep-th/9905124v1

15 Jul

Infinite dimensional geometry and quantum field theory of strings. III. Infinite dimensional W-geometry of a second quantized free string

Posted by Quirino M. Sugon Jr in Uncategorized. Tagged: , , , , , , .

D.Juriev

(Submitted on 7 Jan 1994)

Abstract: The present paper is devoted to various objects of the infinite dimensional W-geometry of a second quantized free string. Our purpose is to include the W-symmetries into the general infinite dimensional geometrical picture related to the quantum field theory of strings, which was described in the first part of the paper (Algebras Groups Geom.11(1994)[to appear]). It is done by the change of the Lie algebra of all infinitesimal reparametrizations of a string world-sheet on the Lie quasi(pseudo)algebra of classical W-transformations (Gervais-Matsuo quasi(pseudo)algebra) as well as of the Virasoro algebra on the central extended enlarged Gervais-Matsuo quasi(pseudo)algebra. A way to obtain W-algebras from classical W-transformations (i.e. Gervais-Matsuo quasi (pseudo)algebra) is proposed. The relation of Gervais-Matsuo differential W-geometry to the Batalin-Weinstein-Karasev-Maslov approach to nonlinear Poisson brackets as well as to L.V.Sabinin program of “nonlinear geometric algebra” are mentioned.

Comments: 24 pages in AMS-TEX (style AMSPPT)
Subjects: High Energy Physics – Theory (hep-th); Algebraic Geometry (math.AG); Functional Analysis (math.FA)
Journal reference: J.Geom.Phys. 16 (1995) 275-300
Report number: LPTENS 93/53
Cite as: arXiv:hep-th/9401026v1

Submission history

From: Denis Juriev [view email]
[v1] Fri, 7 Jan 1994 11:44:58 GMT (24kb)

14 Jul

Hyperbolic rotation of an event in spacetime

Posted by Quirino M. Sugon Jr in Uncategorized.

Let \hat r^\circ be an event in spacetime,

(1)\qquad \hat r^\circ=(ct+\mathbf r)^\circ.

If we want to rotate the event \hat r about the hyperbolic angle \beta in the direction of the unit vector \mathbf n, we write

(2)\qquad\hat r'^\circ=e^{-\mathbf n\beta/2}\,\hat r^\circ\,e^{\mathbf n\beta/2},

where

(3)\qquad e^{\pm\mathbf n\beta/2}=\cosh\frac{\beta}{2}\pm\mathbf n\sinh\frac{\beta}{2}.

Notice that Eq. (2) is similar in form to that for circular rotation of an event, except that the argument of the exponentials are vectors and not imaginary vectors.

Using the spatial inversion property of the unit time vector \mathbf e_0\equiv\,^\circ, we can show that

(4)\qquad \,^\circ e^{\mathbf n\beta/2}=e^{-\mathbf n\beta}\,^\circ.

Thus, Eq. (2) simplifies to

(5)\qquad \hat r'=e^{-\mathbf n\beta/2}\,\hat r\, e^{-\mathbf n\beta/2},

after factoring out the unit time vector \mathbf e_0\equiv\,^\circ.

Let us expand the spacetime cliffor \hat r in Eq. (5) in terms of components that commute and anticommute with $latex\mathbf n$:

(6)\qquad ct'+\mathbf r'_\parallel+\mathbf r'_\perp=e^{-\mathbf n\beta/2}(ct+\mathbf r_\parallel+\mathbf r_\perp)e^{-\mathbf n\beta/2}.

By a theorem on the exponentials of spatial cliffors, we can show that

(7a)\qquad (ct+\mathbf r_\parallel)e^{-\mathbf n\beta/2}=e^{-\mathbf n\beta/2}(ct+\mathbf r_\parallel),

(7b)\qquad \mathbf r_\perp e^{-\mathbf n\beta/2}=e^{\mathbf n\beta/2}\,\mathbf r_\perp.

Employing these relations in Eq. (6), we get

(8)\qquad ct'+\mathbf r'_\parallel+\mathbf r'_\perp=(ct+\mathbf r_\parallel)e^{-\mathbf n\beta}+\mathbf r_\perp.

Hence,

(9a)\qquad ct'+\mathbf r'_\parallel=(ct+\mathbf r_\parallel)e^{-\mathbf n\beta},

(9b)\qquad \mathbf r'_\perp=\mathbf r_\perp.

Notice that the component of the position perpendicular to the boost direction \mathbf n is unaffected by the hyperbolic rotation.

Let us rewrite Eq. (9a) by expanding the exponential:

(10)\qquad ct'+\mathbf r'_\parallel=(ct+\mathbf r_\parallel)(\cosh\beta -\mathbf n\sinh\beta).

Distributing the terms and using the Pauli identity for the products of vectors, we arrive at

(11a)\qquad ct'=ct\cosh\beta-\mathbf r_\parallel\cdot\mathbf n\sinh\beta,
(11b)\qquad \mathbf r'_\parallel=\mathbf r_\parallel\cosh\beta-\mathbf n ct\sinh\beta.

These are the expressions for the hyperbolic rotation of time t and the component of the position vector \mathbf r parallel to the boost direction \mathbf n by a hyperbolic angle \beta.

To show that the Minkowski metric is invariant under hyperbolic rotations, we simply square the rotated event \hat r'^\circ in Eq. (2):

(12)\qquad \hat r'^\circ\hat r'^\circ=e^{-\mathbf n\beta/2}\,\hat r^\circ\, e^{\mathbf n\beta/2}.

Since \hat r^\circ\hat r^\circ=c^2t^2-\mathbf r^2 is a scalar, then Eq. (12) reduces to

(13)\qquad\hat r'^\circ\hat r'^\circ=\hat r^\circ\hat r^\circ,

which is what we wish to prove.

13 Jul

Circular rotation of an event in spacetime

Posted by Quirino M. Sugon Jr in Uncategorized.

The event cliffor is defined as

(1)\qquad\hat r=(ct+\mathbf r)^\circ.

where \mathbf r and t are the position and time coordinates.

To rotate an event \hat r^\circ about the vector \mathbf n\theta by a counterclockwise angle \theta, we simply sandwich the event between two exponentials, as in the case for vector rotation in 3D:

(2)\qquad \hat r'^\circ=e^{-i\mathbf n\theta/2}\,\hat r^\circ\,e^{i\mathbf n\theta/2}.

Using the Euler identity for the exponential of an imaginary vector together with the spatial inversion property of the unit time vector \mathbf e_0\equiv\,^\circ, we can show that

(3)\qquad \,^\circ\,e^{i\mathbf n\theta/2}=e^{i\mathbf n\theta/2}\,^\circ,

so that Eq. (2) reduces to

(4)\qquad \hat r'=e^{i\mathbf n\theta/2}\,\hat r\,e^{-i\mathbf n\theta/2},

after factoring out the unit time vector \mathbf e_0\equiv\,^\circ.

Expanding the spacetime cliffor \hat r=ct+\mathbf r and distributing the exponentials, Eq. (4) becomes

(5)\qquad ct'+\mathbf r'=ct+e^{-i\mathbf n\theta/2}\,\mathbf r\,e^{i\mathbf n\theta/2}.

Separating the scalar and vector parts yields

(6a)\qquad ct'=ct,
(6b)\qquad \mathbf r'=e^{-i\mathbf n\theta/2}\,\mathbf r\,e^{i\mathbf n\theta/2}.

Notice that only the vector part is rotated, since the second equation corresponds to the expression for the vector rotation in 3D.

To check whether a circular rotation preserves the metric of an event, we take the square of the circularly rotated event \hat r'^\circ in Eq. (2)

(7)\qquad \hat r'^\circ\hat r'^\circ=e^{-i\mathbf n\theta/2}\,\hat r^\circ\hat r^\circ\,e^{i\mathbf n\theta/2},

after cancelling the inner exponentials. Since \hat r^\circ\hat r^\circ is the Minkowski metric, which is a scalar, then Eq. (7) reduces to

(8)\qquad\hat r'^\circ\hat r'^\circ=\hat r^\circ\hat r^\circ.

Therefore, the Minkowski metric is preserved under circular rotations.

13 Jul

Successive Lorentz boosts: Equivalent boost and rotation using geometric algebra

Posted by Quirino M. Sugon Jr in Uncategorized.

R. Reyes, Q. M. Sugon Jr., and D. J. McNamara

Ateneo de Manila University, Loyola Heights, Quezon City

reinabellereyes@yahoo.com, qsugon@admu.edu.ph (qsugon@yahoo.com), dmcnamara@ateneo.edu

(Note: This paper was presented at the 22nd SPP Physics Congress, Bohol Tropics Resort, Tagbilaran City, Bohol, Philippines, October 25-27, 2004. The full paper will be available soon (I am still looking for someone who has a cd containing the pdf copies of the full papers presented at the SPP National Physics Conference 2004). This paper is based on the undergraduate thesis of Reinabelle C. Reyes.  Below is only an extended abstract. )

Extended Abstract (Proceedings, p. 21)

In this paper, we derive known expressions for the equivalent Lorentz boost and rotation of two successive Lorentz boosts.  The accompanying rotation is also known as the Thomas rotation, an important but largely neglected physical consequence of special relativity [1].  We develop a Clifford algebra \mathcal G_4 [also known as the Clifford algebra \mathcal Cl_{4,0}] formalism of special relativity and use it to obtain an exact expression for the Thomas rotation.  The salient features of this formulation are: it uses only dot and cross products; and the Minkowski metric has a symmetric form.  The derivation is computationally efficient, pictorially guided, and straightforward, making it suitable for a demonstration in an undergraduate course in special relativity.

In this formulation, we define the product of vectors as \mathbf a\mathbf b=\mathbf a\cdot\mathbf b+i(\mathbf a\times\mathbf b) and the spacetime element \hat r^\circ=(ct+\mathbf r)^\circ [where \,^\circ\equiv\mathbf e_0\equiv\mathbf e_4] to represent the event (\mathbf r, t).  Spatial rotations about \mathbf n_\theta\theta and Lorentz boosts along \mathbf n_\beta\beta then take the following compact forms

(1a)\qquad\hat r'^\circ = e^{-i\mathbf n_\theta\theta/2}\,\hat r^\circ \,e^{i\mathbf n_\theta\theta/2},

(1b)\qquad\hat r'^\circ = e^{-\mathbf n_\beta\beta/2}\,\hat r^\circ \,e^{\mathbf n_\beta\beta/2},

where the exponentials can be expanded as

(2a)\qquad e^{\pm i\mathbf n_\theta\theta/2}=\cosh\frac{\theta}{2}\pm i\mathbf n_\theta\sinh\frac{\theta}{2}.

(2b)\qquad e^{\pm\mathbf n_\beta\beta/2}=\cosh\frac{\beta}{2}\pm\mathbf n_\beta\sinh\frac{\beta}{2}.
Here, the velocity parameter \mathbf n_\beta\beta is related to the relative velocity \vec v by \mathbf n_\beta\beta=\mathbf n_v\tanh \frac{v}{c}.

Now, we can express the product of two successive boosts, say \mathbf n_\beta\beta_1 followed by \mathbf n_\beta\beta_2, as the product of an equivalent boost along \mathbf L and rotation about \mathbf n_\theta\theta, in a boost-rotation identity

(3)\qquad e^{\mathbf n_\beta\beta/2}e^{\mathbf n_\beta\beta/2}=e^{\mathbf L/2}e^{i\mathbf n_\beta/2}.

By expanding this using Eqs. (2a) and (2b) and simplifying, we obtain four working equations from the scalar, vector, imaginary vector, and imaginary scalar parts of the equation, respectively:

(4a)\qquad\cosh\frac{\beta_1}{2}\cosh\frac{\beta_2}{2}+\cos\phi\sinh\frac{\beta_1}{2}\sinh\frac{\beta_2}{2}=\cosh\frac{L}{2}\cos\frac{\theta}{2},

(4b)\ \mathbf n_{\beta 1}\sinh\frac{\beta_1}{2}\cosh\frac{\beta_2}{2}+\mathbf n_{\beta 2}\cosh\frac{\beta_1}{2}\sinh\frac{\beta_2}{2}=\mathbf n_L\sinh\frac{L}{2}\cos\frac{\theta}{2}-\mathbf n_L\times\mathbf n_\theta\sin\frac{L}{2}\sin\frac{\theta}{2},

(4c)\qquad i(\mathbf n_{\beta 1}\times\mathbf n_{\beta 2})\sin\phi\sinh\frac{\beta_1}{2}\sinh\frac{\beta_2}{2}=i\mathbf n_\theta\cosh\frac{L}{2}\sin\frac{\theta}{2},

(4d)\qquad 0=i(\mathbf n_L\cdot\mathbf n_\theta)\sinh\frac{L}{2}\sin\frac{\theta}{2},

where we let \phi be the angle between \mathbf n_{\beta 1} and \mathbf n_{\beta 2}.

We find that we can easily extract a geometric picture from these equations. Since \mathbf n_L\cdot\mathbf n_\theta=0 from Eq. (4d), and \mathbf n_\theta=\mathbf n_{\beta 1}\times\mathbf n_{\beta 2} from Eq. (4c), we conclude that the axis of rotation \mathbf n_\theta is normal to the plane containing all three boosts.

Finally, we obtain the Thomas rotation angle from Eqs. (4a) and (4c), and the equivalent boost from manipulation of Eq. (4b),

(5a)\qquad \theta =2\tan^{-1}[(\sin\phi\tanh\frac{\beta_1}{2}\tanh\frac{\beta_2}{2})/(1+\cos\phi\tanh\frac{\beta_1}{2}\tanh\frac{\beta_2}{2})],

(5b)\qquad L=\cosh^{-1}(\cosh\beta_1\cosh\beta_2+\sinh\beta_1\sinh\beta_2\cos\phi).

Thus, we have obtained standard results for equivalent boost and rotation using a computationally efficient and pictorially-guided derivation, with simple and straightforward steps which any physics student familiar with vector algebra and basic trigonometry can easily follow and carry out herself.

Reference:

[1] J. P. Costella, B. H. J. McKellar, and A. A. Rawlinson, “Thomas Rotation,” Am. J. Phys. 69, 837-846 (2001).

13 Jul

An introduction to geometric algebra with an application in rigid body mechanics

Posted by Quirino M. Sugon Jr in Uncategorized.

American Journal of Physics — June 1993 — Volume 61, Issue 6, pp. 491-504
Issue Date: June 1993

Terje G. Vold
Department of Physics and Astronomy, Swarthmore College, Swarthmore, Pennsylvania 19081
This paper presents a tutorial of geometric algebra, a very useful but generally unappreciated extension of vector algebra. The emphasis is on physical interpretation of the algebra and motives for developing this extension, and not on mathematical rigor. The description of rotations is developed and compared with descriptions using vector and matrix algebra. The use of geometric algebra in physics is illustrated by solving an elementary problem in classical mechanics, the motion of a freely spinning axially symmetric rigid body.

©1993 American Association of Physics Teachers

History: Received 8 September 1992; accepted 16 October 1992
Permalink: http://dx.doi.org/10.1119/1.17201

13 Jul

An introduction to geometric calculus and its application to electrodynamics

Posted by Quirino M. Sugon Jr in Uncategorized.

American Journal of Physics — June 1993 — Volume 61, Issue 6, pp. 505-513
Issue Date: June 1993

Terje G. Vold
Department of Physics and Astronomy, Swarthmore College, Swarthmore, Pennsylvania 19081
A tutorial of geometric calculus is presented as a continuation of the development of geometric algebra in a previous paper. The geometric derivative is defined in a natural way that maintains the close correspondence between geometric algebra and the algebra of real numbers. The use of geometric calculus in physics is illustrated by expressing some basic results of electrodynamics.

©1993 American Association of Physics Teachers

History: Received 8 September 1992; accepted 16 October 1992
Permalink: http://dx.doi.org/10.1119/1.17202

13 Jul

Light polarization: A geometric-algebra approach

Posted by Quirino M. Sugon Jr in Uncategorized.

American Journal of Physics — June 1993 — Volume 61, Issue 6, pp. 534-545
Issue Date: June 1993

W. E. Baylis, J. Bonenfant, J. Derbyshire, and J. Huschilt
Department of Physics, University of Windsor, Windsor, Ontario, N9B 3P4 Canada
The geometric algebra of three-dimensional space (the “Pauli algebra”) is known to provide an efficient geometric description of electromagnetic phenomena. Here, it is applied to the three-dimensional Stokes subspace to describe the polarization of an approximately monochromatic collimated beam of electromagnetic radiation. The coherency density rho is a real element of the algebra whose components are the four Stokes parameters: a scalar representing the total photon flux density plus a three-dimensional vector whose direction and length in the Poincaré sphere give the type and degree of polarization. The detection of the radiation and the incoherent and coherent modification of the polarization by various optical elements are calculated by algebraic multiplication which has faithful representations in 2×2 matrices. One matrix representation of rho is the coherency matrix with which Jones and Mueller matrices are related whereas another representation is the spin density matrix. However, the calculations are simplest to perform and interpret in the algebraic form independent of any particular matrix representation. It is shown that any possible change in the Stokes parameters can be treated algebraically by a combination of attenuation, depolarization, polarization, and rotation transformations of rho. The geometric algebra thus unifies Stokes parameters, the Poincaré sphere, Jones and Mueller matrices, and the coherency and density matrices in a single, simple formalism.

©1993 American Association of Physics Teachers

History: Received 29 July 1992; accepted 10 October 1992
Permalink: http://dx.doi.org/10.1119/1.17205
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