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Gamma Matrices

Gamma Matrices. Rewrite:. Feymann slash notation:. Lorentz Transform (Overview). Repeated index convention. Differential of proper time Frame invariant. Proper Lorentz Transforms:. Improper Lorentz Transforms:. Useful formulae:. CM transform:. Lorentz Transforms. Differential Change:.

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Gamma Matrices

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  1. Gamma Matrices Rewrite: Feymann slash notation:

  2. Lorentz Transform (Overview) Repeated index convention Differential of proper time Frame invariant Proper Lorentz Transforms: Improper Lorentz Transforms: Useful formulae: CM transform:

  3. Lorentz Transforms Differential Change: with Let us assume that we are transforming around axis n. where (for example) Finite change as sequence of differential changes

  4. Covariance Lorentz Unitrary transform to original γ,s Free to define as necessary Covariant if: Equation to find S.

  5. Differential S(a) Parameterize S in terms of ω: Properties: Now must find σ tensor of this.

  6. σ Tensor σ must be antisymmetric; therefore try: Q.E.D. Differential S: Note: Still 4x4 matrix Note:

  7. Finite Transforms Differential ω: Finite ω: For example.

  8. Boosted Column Vector Plus exponential factor:

  9. Properties of S(a) Lorentz Boosts Rotations Since

  10. Probability Current must transform as a four vector. Proof:

  11. Parity Transformation Parity Transformation: x→-x, t→t Note: Where phase has no consequences

  12. Other Bilinear Combinations Lorentz scaler: Lorentz vector Lorentz tensor: Lorentz pseudoscaler: Lorentz axial vector: All possible combinations (16 possible – 10 from symmetric matrices, 6 from antisymmetric)

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