fermions and the dirac equation in 1928 dirac proposed the following form for the electron wave...
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![Page 1: Fermions and the Dirac Equation In 1928 Dirac proposed the following form for the electron wave equation: The four µ matrices form a Lorentz 4-vector,](https://reader035.vdocuments.us/reader035/viewer/2022062300/56649d2d5503460f94a03a69/html5/thumbnails/1.jpg)
Fermions and the Dirac Equation
In 1928 Dirac proposed the following form for the electron wave equation:
The four µ matrices form a Lorentz 4-vector, with components, µ. That is, they transform like a 4-vector under Lorentz transformations between moving
frames. Each µ is a 4x4 matrix.
4x4 matrix 4x4 unit matrix
4-row column matrix
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Dirac wanted an equation with only first order derivatives:
But the above must satisfy the Klein=Gordon equation:
![Page 3: Fermions and the Dirac Equation In 1928 Dirac proposed the following form for the electron wave equation: The four µ matrices form a Lorentz 4-vector,](https://reader035.vdocuments.us/reader035/viewer/2022062300/56649d2d5503460f94a03a69/html5/thumbnails/3.jpg)
This must =
Klein-Gordon equation
10 conditions must be satisfies
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Pauli spinors:
=
1 =
2 =
3 =
They must be 4x4 matrices!
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The 0 and 1,2,3 matrices anti-commute
The 1,2,3, matrices anti-commute with each other
The square of the 1,2,3, matrices equal minus the unit matrix
The square of the 0 matrixequals the unit matrix
4x4 unit matrix
All of the above can be summarized in the following expression:
Here gµ is not a matrix, it is a component
of the inverse metric tensor.
4x4 matrices
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With the above properties for the matrices one can show that if satisfies the Dirac equation, it also satisfies the Klein Gordon equation. It takes some work.
4 component column matrix
4x4 unit matrix
Klein-Gordon equation:
Dirac derived the properties of the matrices by requiring that the solution tothe Dirac equation also be a solution to the Klein-Gordon equation. In the process
it became clear that the matrices had dimension 4x4 and that the was a column matrix with 4 rows.
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The Dirac equation in full matrix form
0 1 2
3
spin dependence
space-timedependence
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p0
After taking partial derivatives … note 2x2 blocks.
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Writing the above as a 2x2 (each block of which is also 2x2)
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Incorporate the p and E into the matrices …
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The exponential (non-zero) can be cancelled out.
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Finally, we have the relationships between the upper and lower spinor components
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p along z and spin = +1/2
p along z and spin = -1/2
Spinors for the particle with p along z direction
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p along z and spin = +1/2
p along z and spin = -1/2
Spinors for the anti-particle with p along z direction
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Field operator for the spin ½ fermion
Spinor for antiparticlewith momentum p and spin s
Creates antiparticle with momentum p and spin s
Note: pµ pµ = m2 c2
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Creates particle with momentum p and spin s
r, s = 1/2
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Lagrangian Density for Spin 1/2 Fermions
1. This Lagrangian density is used for all the quarks and leptons – only the masses will be different!
2. The Euler Lagrange equations, when applied to this Lagrangian density, give the Dirac Equation!
Comments:
3. Note that L is a Lorentz scalar.
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Dirac equation
The Euler – Lagrange equations applied to L:
[ A B ]†= B † A †
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Lagrangian Density for Spin 1/2 Quarks and Leptons
Now we are ready to talk about the gauge invariance that leads tothe Standard Model and all its interactions. Remember a “gaugeinvariance” is the invariance of the above Lagrangian under
transformations like e i . The physics is in the -- which can be a matrix operator and depend on x,y, z and t.