How to present in a simple way the spinorial space-time and the privileged space direction it generates? What is the fundamental difference between space-time spinors and the conventional description of space and time? What is the relevance for the analysis of WMAP and Planck data ?

Now that cosmologists are trying to reproduce the Planck anomalies using involved inflationary patterns, it seems relevant to emphasize that the privileged space direction is an immediate and natural consequence of the spinorial space-time, and that this approach is just a description of space and time such as they are seen by fermions (electron, muon, neutrino, quarks...). The privileged space direction is then the unavoidable result of using two complex space-time coordinates instead of four real ones.

Fermions are particles with half-integer spin whose wave function changes sign by a 360 degrees rotation. Therefore, they cannot be really described with the standard rotation group SO(3) and require instead the use of its covering group SU(2) where a 360 degrees rotation acting on the fundamental (spinorial) representation is equal to -1. My suggestion, formulated already in 1996-97 [1,2] was to use directly a spinorial space-time with SU(2) instead of the standard SO(3) rotations to describe space and time in Particle Physics and in Cosmology. The SU(2) group consists of all linear and unitary transformations of the complex two-dimensional space with determinant equal to 1. A SU(2) spinor is a complex two-dimensional object transforming under SU(2) : two complex coordinates instead of the usual four real ones.

Then, as already explained in my previous articles, it turns out that SU(2) transformations of the cosmic spinorial representation of space-time will describe space translations (spinorial "rotations" around the cosmic space-time origin) and that standard space rotations will be described by suitable SU(2) transformations of these translations. See, for more details, my previous article:

A Privileged Space Direction? Spinorial Space-time, WMAP, Planck (I)

as well as my Post Scriptum after publication to my CRIS 2010 paper Cosmic rays and tests of fundamental principles:

and, posted recently to mp_arc, and HAL :

Spinorial space-time and privileged space direction (I)

What is new with spinors is that cosmic real coordinates are grouped by pairs into complex ones. It is precisely this link betwen two cosmic real coordinates which generates the privileged space direction defined by the SU(2) elements that transform them into each other. Given a cosmic spinor ξ , these transformations amount to multiplying ξ by a complex phase exp (iφ). As space translations are just the SU(2) transformations acting on the cosmic spinors, the privileged space direction is trivially generated by the phase rotations of the spinor ξ . The privileged character of this direction clearly follows from the relation :

ξ† ξ′ = | ξ′ | | ξ | exp (iφ')

where ξ† is the hermitic conjugate of ξ , exp (iφ') a complex phase and ξ′ =  exp (iφ') ξ a spinor on the privileged trajectory. No other point on the constant-time hypersphere satisfies such a relation with respect to ξ . In all other cases, the modulus of the scalar product ξ† ξ′ is smaller than the product | ξ′ | | ξ | .

I was confronted to this privileged space direction as early as February 1997 in my paper Space, Time and Superluminal Particles: .s/9702026v1.pdf

when trying to build conventional space-time coordinates from the spinorial ones. Page 3, I wrote :


Instead of four real numbers, we take space-time to be described by two complex numbers, the components of a SU(2) spinor. From a spinor ξ , it is possible to extract a SU(2) scalar, | ξ |2 = ξ† ξ (where the dagger stands for hermitic conjugate), and a vector z = ξ† σ ξ , where σ is the vector formed by the Pauli matrices. In our previous papers on the subject [4 , 5] , we proposed to interpret t = | ξ | as the time. If the spinor coordinates are complex numbers, one has: z = t2 where z is the modulus of z . It does not seem possible to interpret z as providing the space coordinates: one coordinate, corresponding to an overall phase of the spinor, is missed by t and z . Therefore, a different description of space seems necessary in this approach.

Interpreting t as the time has at first sight the drawback of positive-definiteness and breaking of time reversal, but this can be turned into an advantage if t is interpreted as an absolute, cosmic time (geometrically expanding Universe). An arrow of time is then naturally set, and space-time geometry incorporates the physical phenomenon of an expanding Universe.


(end of quote)

References [4,5] of my 1997 paper are respectively references [3] and [1] of the present article. The privileged space direction corresponds to the coordinate "missed" by z . The vector z does not correspond to a position, but to a direction : it has the same value for all points exp (iφ) ξ on this privileged trajectory. Thus, the cosmic spinor ξ naturally generates a privileged cosmic space direction z t-2 .

Together with parity violation, the spinorial space-time can thus naturally explain the directional asymmetry recently observed in Planck data.

If the Planck anomalies are finally confirmed, taking into account this simple property of the spinorial space-time may be a way to prevent standard Cosmology from becoming new epicycles.

See also my articles of this blog :

Relativity, cosmic rays, cosmology and spinorial space-time (I)

Relativity, cosmic rays, cosmology and spinorial space-time (II)*

Dark matter and dark energy, or Pre-Big Bang geometry?

and my recent two contributions to ICNFP 2013 (Crete, 28 August - 5 September) :

Pre-Big Bang, spinorial space-time, asymptotic Universe (talk)

Ultra-high energy physics and standard basic principles (poster)

[1] L. Gonzalez-Mestres, Physical and Cosmological Implications of a Possible Class of Particles Able to Travel Faster than Light, contribution to the 28th International Conference on High Energy Physics, Warsaw 1996, arXiv:hep-ph/9610474, and references therein.
[2] L. Gonzalez-Mestres, Space, Time and Superluminal ParticlesarXiv:physics/9702026
[3] L. Gonzalez-Mestres, Physics, cosmology and experimental signatures of a possible new
class of superluminal particles
, Proceedings of the International Workshop on the Identification of Dark Matter, Sheffield (England, United Kingdom), September 1996 . Paper astro-ph/9610089 .