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Special Unitary Groups and the Standard Model - Part 1

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Standard Model Lagrangian

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The Nature of the Weak Interaction

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Last modified: January 26, 2018


Conjugate Pair Theorem
----------------------

If P is a polynomial in one variable with real coefficients, 
and a + bi is a root of P with a and b real numbers, then 
its complex conjugate a - bi  is also a root of P.

Example: 

P(x) = x³ - 9x² + 16x - 14 = 0

With x = 1 + i

∴ (x - (1 + i))(x - (1 - i))(x ± ?) = x³ - 9x² + 16x - 14 

∴ (x - 1 - i)(x - 1 + i)g(x) = x³ - 9x² + 16x - 14 

∴ (x² - x + ix - x + 1 - i - ix + i + 1)g(x) = x³ - 9x² + 16x - 14 

∴ (x² - 2x + 2)g(x) = x³ - 9x² + 16x - 14 

∴ g(x) = (x³ - 9x² + 16x - 14)/(x² - 2x + 2) 

∴ g(x) = (x - 7) after long division.

The factors are:

 (x - (1 + i))(x - (1 - i))(x - 7)