Spinors and Pythagorean triples

[AntonioLao]

Reference: Elie Cartan, “The Theory of Spinors,” Chap. III, Dover,
1981

There is an underlying similarity between the conceptual definition of spinors and pairs of positive integers for finding a Pythagorean triple.
If m < n where n=2, 3, 4, 5, … and m=1, 2, 3, 4, … then there exist sets of Pythagorean triples given by

[math]n^2-m^2[/math]

[math]2mn[/math]

[math]n^2+m^2[/math]

For a given null vector (x1, x2, x3) whose components satisfy

[math]x^{2}_1+x^{2}_2+x^{2}_3=0[/math]

These components can be expressed as functions of a pair of spinors [math]\xi_0[/math], [math]\xi_1[/math] given by

[math]x_1=\xi^2_0 - \xi^2_1[/math]

[math]x_2= -2\xi_0\xi_1[/math]

[math]x_3=i ( \xi^2_0 + \xi^2_1 )[/math]

These similarities imply that orthogonality exists in real space as well as in abstract imaginary and complex spaces as long as the dimensionality remains at the 2nd order. In lower or higher dimensionality, the equal sign must be replaced using signs of inequalities.

[Guille]

Why would higher or lower dimensions mean inequalities?

[AntonioLao]

Why would higher or lower dimensions mean inequalities?

Please see http://www.toequest.com/forum/showthread.php?t=320

[AntonioLao]

Why would higher or lower dimensions mean inequalities?

as a follow-up to your question, quantum mechanics is based on the following three basic uncertainties:

[math]\Delta x \cdot \Delta p_x \geq \hbar [/math]

[math]\Delta \phi \cdot \Delta J_z \geq \hbar [/math]

[math]\Delta t \cdot \Delta E \geq \hbar [/math]

these imply that the dimension of quantum domain should be exactly zero.

Einstein's field equations implying higher dimension greater than 2 should be an inequality given by

[math]G_{\mu \nu} + g_{\mu \nu} \Lambda \leq 8 \pi T_{\mu \nu} [/math]

For superstring theory of 11th dimension the following is true

[math]G_{\mu\nu}+\Lambda \g_{\mu\nu} \leq 8\pi T_{\mu\nu}[/math]

and for 26th dimensional theory

[math]\alpha^{26} + \beta^{26} &lt;&lt;&lt;&lt;&lt;&lt;&lt; \gamma^{26} [/math]

where alpha, beta, and gamma are parameters of the theories.

[Guille]

as a follow-up to your question, quantum mechanics is based on the following three basic uncertainties:

[math]\Delta x \cdot \Delta p_x \geq \hbar [/math]

[math]\Delta \phi \cdot \Delta J_z \geq \hbar [/math]

[math]\Delta t \cdot \Delta E \geq \hbar [/math]

these imply that the dimension of quantum domain should be exactly zero..

So, is it accepted a zero?

Einstein's field equations implying higher dimension greater than 2 should be an inequality given by

[math]G_{\mu \nu} + g_{\mu \nu} \Lambda \leq 8 \pi T_{\mu \nu} [/math]

For superstring theory of 11th dimension the following is true

[math]G_{\mu\nu}+\Lambda \g_{\mu\nu} \leq 8\pi T_{\mu\nu}[/math]

and for 26th dimensional theory

[math]\alpha^{26} + \beta^{26} &lt;&lt;&lt;&lt;&lt;&lt;&lt; \gamma^{26} [/math]

where alpha, beta, and gamma are parameters of the theories.

Do you know what these paraemters are?

and how you can calculate them?

[AntonioLao]

So, is it accepted a zero?

theoretically but not experimentally. The limit of experiments is set at Planck's constant, h.

Even the big bang theory is limited at the singularity (zero dimension) to which theories can never be able to reach.

Do you know what these paraemters are?

my educated guess is that these parameters are helicity, convection current, mass, charge and energy but not in any order of priority or importance.

[AntonioLao]

and how you can calculate them?

i can only calculate mass ratios such as the ratio of proton mass to electron mass. My calculation is within .22% of the experimental value.