Archive for August, 2011

The standard linear one-dimensional wave equation is (d^2/dt^2 – d^2/dx^2) w = 0.  If the propagation speed is c, then one puts a factor of (1/c^2) in front of the time-derivative term.  The solution to the wave equation is a superposition of a forward propagating and a backward propagating function of the form F(x+ct) + G(x-ct).

The linear wave equation on quaternions is the same with d^2/dx^2 replaced with the Laplacian.  If Dirac operator is the square root of the Laplacian, then the wave equation on quaternions can be factored as:

(1/c d/dt + Dirac) ( 1/c d/dt – Dirac ) Psi = 0

The factorization introduced 2×2 quaternion matrices in the Dirac operator because the quaternions have real four dimensions, and the Clifford algebra of R^4 is the algebra of 2×2 matrices with quaternion entries.  Thus Psi must be pairs of quaternions.

Now the Dirac operator swaps the positive and negative spinors, the two components of the pairs of spinors.   Thus if Psi = (A,B), the factorization contains the equations

1/c d/dt A + Dirac(B) = 0
1/c d/dt B – Dirac(A) = 0.

These are the Maxwell’s equations with curl replaced by the Dirac operator.  This observation is not very rigorous but is helpful for understanding the meaning of Maxwell’s equations nonetheless because they suggest the rule of thumb at least that says:  the linear wave equation on quaterions has as a square root the Maxwell’s equations on spinors.

The wave equation on quaternions is equivariant by multiplying the coordinate on the right by an SU(2) element, and therefore so is its square root system, which implies that that system (which are Maxwell’s equations with Dirac replacing curl) are PROJECTIVE lifts and their solutions are right equivariant. Therefore these equations define functions on the quaternionic projective line — functions E, B: HP^1 -> H, the electromagnetic fields.


Read Full Post »

Let us put the discussion of spinors into context.  Recall that an abundance of observed crystals of rotational symmetries of orders 5, 8, 10 and 12 in abundance tell us that the actual universe that we inhabit has not three macroscopic spatial dimensions but four macroscopic spatial dimensions.  Further, recall that observation of a uniform lower bound for the cosmic background radiation for up to 300 light years in every direction from Earth implies that the universe is not infinite and open but compact, for otherwise there cannot be a uniform lower bound on the cosmic background radiation that arises from heat diffusion.  So the actual universe we inhabit must be a compact four dimensional manifold.

On any simply connected compact four dimensional riemannian manifold, the Hodge *-operator maps 2-forms to 2-forms and squares to +1.  The two eigenspaces of * with eigenvalues +1 and -1 decompose the 2-forms into orthogonal subspaces that correspond to the Lie algebra decomposition so(4) = so(3) + so(3).  At the level of Lie groups this corresponds to the decomposition Spin(4) = SU(2) x SU(2).  Abstractly, then spinors may be introduced using the standard representation of SU(2) on two dimensional complex spaces and producing complex vector bundles on the four manifold using the representations.

The above constructions are meaningful on a general 4-manifold but the construction of the spinor spaces, the representations of SU(2), are artificial mathematically defined objects.  They leave open the questions of physical significance of spinors open.  When the riemannian four-manifold is a quaternionic projective line rather than a general four-manifold the physical meaning of spinors can be elucidated much more clearly.

The quaternionic projective line has the natural projection map HxH -> HP^1 where (q1, q2) is identified with (q1 a, q2 a) where a ranges over nonzero quaternions.  The space of spinors is precisely the pre-image H+H of the projective map which contains the standard representation of SU(2) by left multiplication on each factor.

Thus, to the question of ‘what is the physical meaning of spinors?’ we have the answer that the universe is a quaternionic projective line, and the two factors of quaternions in its pre-image decompose electricity from magnetism and are the natural space of spinors with the left-multiplication action of SU(2).  Note that the Clifford algebra in 4 real dimensions is precisely the 2×2 quaternion matrices, which is the space of endomorphisms of H x H.

Dirac’s famous act of producing the ‘square root’ of the Laplacian by introducing coefficients to d/dx1, …, d/dxn so that a linear combination of these square to the Laplacian produce the Clifford relations.  Then the spinors are traditionally introduced as mathematical objects where the Dirac operator can act.  But here we have a natural interpretation of the space of spinors as the preimage of the projective map which separates electricity from magnetism.

Read Full Post »

The human race is a single race, and biological race is a political fiction.  The Human Genome Project announced in April 2009:

According to the U.S. Department of Energy Office of Science, Office of Biological and Environmental Research, Human Genome Program,

DNA studies do not indicate that separate classifiable subspecies (races) exist within modern humans. While different genes for physical traits such as skin and hair color can be identified between individuals, no consistent patterns of genes across the human genome exist to distinguish one race from another. There also is no genetic basis for divisions of human ethnicity. People who have lived in the same geographic region for many generations may have some alleles in common, but no allele will be found in all members of one population and in no members of any other.

Source: http://www.nowpublic.com/world/human-genome-project-announces-race-does-not-exist

Nations and religions divide humanity rather than provide us with a political system that is legitimate.  Legitimacy of all acts in every scale depend on adherence to three ideals of Republic, which are Truth, Justice, and Freedom.  These are spiritual ideals.  Indeed, from observations of rotational symmetries in crystals of orders 5, 8, 10, and 12, observed since 1984 which are impossible in three spatial dimensions show that the universe has four spatial dimensions and in fact the human soul is not theory or conjecture but exists as a four dimensional electromagnetic object.  Socrates, in Plato’s Gorgias, had identified injustice as a depravity of the soul, and I agree with him completely.  Divine Love is the ideal of Justice and is more appropriately a question of our own souls rather than that of any god.

We are individuals in a single race, and while we have a variety of superficial differences, our individual differences are spiritual rather than ‘racial’.  Patriotism is a political construction.  Howard Zinn insightfully wrote:

“Around 1776, certain important people in the English colonies made a discovery that would prove enormously useful for the next two hundred years.  They found that by creating a nation, a symbol, a legal unity called the United States, they could take over land, profits, political power from the favorites of the British Empire.  In the process they could hold back a number of potential revolutions and create a consensus of political support for the rule of a new, privileged leadership.”

Source:  p. 59 http://books.google.com/books?id=Vu-hxQypyjkC&printsec=frontcover&dq=people%27s+history+of+the+united+states&hl=en&ei=Yxg8TvX5B4OJsAKfqKAu&sa=X&oi=book_result&ct=result&resnum=1&ved=0CDQQ6AEwAA#v=onepage&q&f=false

Indeed, the writers of the American Constitution hated democracy and quite possibly their “people” were just the aristocrats.  Here is a discussion on this topic by Gore Vidal:  http://therealnews.com/t2/index.php?option=com_content&task=view&id=31&Itemid=74&jumival=338

In this note of mine, I argue that America was always an empire:  https://zulfahmed.wordpress.com/2010/12/17/there-is-little-evidence-america-was-ever-a-republic/

Read Full Post »