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Chapter III Dirac Field Lecture 4 Books Recommended: Lectures on Quantum Field Theory by Ashok Das Advanced Quantum Mechanics by Schwabl
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Pauli Exclusion Principle
Quantized Klein Gordon Field theory is Used for the Spin 0 boson particles. To construct the field theory for Fermions we need to incorporate the Pauli exclusion principle. As per Pauli principle: at the most be one fermions in a given state.
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Consider an oscillator having annihilation
And creation operator Corresponding number operator -----(1) Above oscillator will obey Fermi Dirac Statistics if annihilation and creation operators obey anti-commutation relation.
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We have anti-commutation relations
-----(2) We write, ---(3)
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From (3), we can write -----(4) Eigenvalues of number operator ----(5) which is Pauli exclusion principle. With anti-commutation relations, wave function will be antisymmetric and therefore, describe fermions.
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Quantization of Dirac Field
Dirac Eq ----(6) Adjoint Eq ---(7) Where adjoint spinor ---(8)
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Dirac field operators belong to the Spin ½
representation of Lorentz group and hence, are fermions and should be described by Anti-commutation relations. The Lorentz invariant Lagragian density for Dirac field ------(9)
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Using (9) and Euler Lagrange Eq., we can find
Eqs (6) and (7) ----(10) ----(11)
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Lagrangian given by (9) is not hermitian. First
Term of (9) is not hermitian 2nd term of (9) is hermitian (12)
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We can write a Hermitian and Lorentz invariant
Lagrangian ---- (13) Lagrangian (9) and (13) are differ by total divergence only: -----(14)
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Dynamical Eqns derived using (9) and (13)
will be same and we will use Lagrangian given by (9). Momenta conjugate to and will be ----(15) (16)
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Equal time anti-commutation relation will be
---(17) WE can also write using (15): ----(18) ----(19)
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Hamiltonian density -----(20) Total Hamiltonian ---(21)
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Using Heisenberg Eq, we can derive Dirac equat-
ion of motion. (1)
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In deriving above, in first step on last slide we
used Where,
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(2)
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