# Geometric Algebra

**Research Interests**

Meet the Geometric Algebra Group of Ateneo (GAGA). We are gaga over geometric algebra and we wish to rewrite all of physics in terms of geometric algebra. We know vectors very well, so well that we interpret all equations in terms of vectors. We think of imaginary numbers as oriented planes and volumes. We can unite the scalar dot and vector cross products of vectors in a single geometric vector product. We can rotate vectors, not with matrices, but with exponentials of imaginary vectors. We express rotation gtroups in crystallography in terms of exponentials of imaginary vectors. We can hyperbolically rotate events using exponentials of vectors in order to arrive at Lorentz transforms in special relativity. We write the electromagnetic field as a sum of vector electric field and imaginary vector magnetic field. We unify all the four Maxwell’s equations into a single equation. If you’ll join our group, you’ll learn more.

Time is short, but the art of geometric algebra is long. Many physics and mathematics departments around the world do not still teach geometric algebra. Many physicists and mathematicians do not still know geometric algebra. We need to convert them to the new mathematics. We need to convert them to the new way of doing physics. Why separately teach vectors, complex numbers, quaternions, matrices, tensors, spinors, Lie groups, Poisson brackets, and differential forms when geometric algebra can do what any of these algebras can do and so much more? Why insist on using several languages that cannot be seamlessly unified when there is geometric algebra which can speak several languages all at once. Geometric algebra is founded on two axioms: the square of a unit vector is unity, while the product of two perpendicular unit vectors anticommute. All geometric algebraists believe on the second axiom, but debate on the first, claiming that vectors can have a square of -1. But we believe otherwise: all vectors should square to +1. And we shall rewrite all of physics with this assumption.

**Research Supervisors**

*Fr. Daniel J. McNamara, SJ*(on leave)*Dr. Quirino Sugon Jr*.

**Students**

- Michael Andrews (1 MS Ps), Electromagnetic Lagrangian
- Javy Jalandoni (4 BS Ps), Trojan asteroids
- Uzziel Perez (4 BS Ps), Rydberg atoms and Stark effect
- Christian Laurio (4 BS Ps), Polarized light
- Francis Bayocboc (4 BS Ps), Kaluza-Klein Theory

**Alumni**

- Adler Santos, MS Physics, Stokes parameters of polarized light. (Research assistant at the SERC Subcenter, Ionosphere Research Building, Manila Observatory. Geomagnetic field interpolation using spherical harmonics.)
- Miguel Antonio Sulangi, BS Ps ’10, Geometry induced potentials of nano structures (Studying Ph.D. in Physics at John Hopkins University)
- Michelle Wynne Sze, BS Ps ’09, Oblique superposition of polarized light (Currently studying M.S. in Physics at University of the Philippines-Diliman)
- Reinabelle Reyes, BS Ps ’05, Successive Lorentz boosts. (Ph.D. in Astrophysics at Princeton University. Postdoctoral Fellow at Kavli Institute for Cosmological Physics at University of Chicago starting May 1).
- Emmanuel Sagge, BS Ps ‘o6, 3D reconstruction from stereo images
- Marvin Boni Go, BS Ps ’05,Velocity field interpolation
- Sarah Bragais, BS Ps ’05, Copernican epicycles
- Jason Angeles, BS Ps ’05,Ray tracing
- Christopher Biasbas, BS Ps ’05, Clifford algebra calculator
- Jong Silverio, BS Ps ’04, Kinematics on rotating earth
- Carlo Fernandez, BS Ps ’04, Dihedral groups
- Rael Limbitco, BS Ps ’97, Rotations and spinors
- Alfred Rodriguez, BS Ps ’96, Robertson-Walker metric

**PUBLICATIONS**

**Book Chapter**

**139**, pp. 179-224 (2006).

Quirino M. Sugon Jr. and Daniel J. McNamara, “A Hestenes spacetime algebra approach to light polarization,” in Applications of Geometric Algebra in Computer Science and Engineering, ed. by L. Dorst, C. Doran, and J. Lasenby (Birkhauser, Boston, 2002), pp. 297-315.

**Journal**

Michelle Wynne C. Sze, Quirino M. Sugon, Jr., and Daniel J. McNamara, “Oblique superposition of two elliptically polarized lightwaves using geometric algebra: is energy–momentum conserved?,” J. Opt. Soc. Am. A 27, 2468-2479 (2010). doi:10.1364/JOSAA.27.002468

Quirino M. Sugon Jr. and Daniel J. McNamara, “A geometric algebra reformulation of geometric optics,” Am. J. Phys.72(1), 92-97 (2004).

**ArXiv**

Quirino M. Sugon Jr. and Daniel J. McNamara, “Poisson commutator-anticommutator brackets for ray tracing and longitudinal imaging via geometric algebra,” arXiv:0812.2979v1 [math-ph] (16 Dec 2008). 10 pages, 9 figures.

Quirino M. Sugon Jr. and Daniel J. McNamara, “Revisiting 2×2 matrix optics: Complex vectors, Fermion combinatorics, and Lagrange invariants,” arXiv:0812.0664v1 [physics.optics] (3 Dec 2008).

Quirino M. Sugon Jr., Carlo B. Fernandez, and Daniel J. McNamara, “A geometric algebra reformulation of 2×2 matrices: the dihedral group D_4 in bra-ket notation,” arXiv:0811.3680v1 [math-ph] (24 Nov 2008).

Quirino M. Sugon Jr. and Daniel J. McNamara, “Paraxial meridional ray tracing equations from the unified reflection-refraction law via geometric algebra,” arXiv:0810.5224v1 [physics.optics] (29 Oct 2008).

Quirino M. Sugon Jr. and Daniel J. McNamara, “Taxonomy of Clifford Cl3,0 subgroups: choir and band groups,” arXiv:0809.0351v1 [math-ph] (2 Sep 2008).

Quirino M. Sugon Jr., Sarah Bragais, and Daniel J. McNamara, “Copernicus’s epicycles from Newton’s gravitational force law via linear perturbation theory in geometric algebra,” arXiv:0807.2708v1 [physics.space-ph] (17 Jul 2008).

Quirino M. Sugon Jr. and Daniel J. McNamara, “Electromagnetic energy-momentum equation without tensors: a geometric algebra approach,” arXiv:0807.1382v1 [physics.class-ph] (9 Jul 2008).