Spin 0 field
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Real and complex scalar fields. Klein-Gordon equation. Plane-wave (normal mode) solutions. Generation and anihilation operators. Hamiltonian. Commutation relations.
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Real and complex scalar fields.
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The equations of motion for a real scalar field ϕ {\displaystyle \phi } can be obtained from the following lagrangian densities
L = 1 2 ∂ μ ϕ ∂ μ ϕ − 1 2 M 2 ϕ 2 = − 1 2 ϕ ( ∂ μ ∂ μ + M 2 ) ϕ {\displaystyle {\begin{matrix}{\mathcal {L}}&=&{\frac {1}{2}}\partial _{\mu }\phi \partial ^{\mu }\phi -{\frac {1}{2}}M^{2}\phi ^{2}\\&=&-{\frac {1}{2}}\phi \left(\partial _{\mu }\partial ^{\mu }+M^{2}\right)\phi \end{matrix}}}
and the result is
( ◻ + M 2 ) ϕ ( x ) = 0 {\displaystyle \left(\Box +M^{2}\right)\phi (x)=0} .
The complex scalar field ϕ {\displaystyle \phi } can be considered as a sum of two scalar fields: ϕ 1 {\displaystyle \phi _{1}} and ϕ 2 {\displaystyle \phi _{2}} , ϕ = ( ϕ 1 + i ϕ 2 ) / 2 {\displaystyle \phi =\left(\phi _{1}+i\phi _{2}\right)/{\sqrt {2}}}
The Langrangian density of a complex scalar field is
L = ( ∂ μ ϕ ) + ∂ μ ϕ − M 2 ϕ + ϕ {\displaystyle {\mathcal {L}}=(\partial _{\mu }\phi )^{+}\partial ^{\mu }\phi -M^{2}\phi ^{+}\phi }
Klein-Gordon equation
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Klein-Gordon equation is precisely the equation of motion for the spin-0 particle as derived above:
( ◻ + M 2 ) ϕ ( x ) = 0 {\displaystyle \left(\Box +M^{2}\right)\phi (x)=0}
Spin 1/2 field
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Dirac equation
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The Dirac equation is given by:
( i γ μ ∂ μ − m ) ψ ( x ) = 0 {\displaystyle \left(i\gamma ^{\mu }\partial _{\mu }-m\right)\psi \left(x\right)=0}
where ψ {\displaystyle \psi } is a four-dimensional Dirac spinor. The γ {\displaystyle \gamma } matrices obey the following anticommutation relation (known as the Dirac algebra):
{ γ μ , γ ν } ≡ γ μ γ ν + γ ν γ μ = 2 g μ ν × 1 n × n {\displaystyle \left\{\gamma ^{\mu },\gamma ^{\nu }\right\}\equiv \gamma ^{\mu }\gamma ^{\nu }+\gamma ^{\nu }\gamma ^{\mu }=2g^{\mu \nu }\times 1_{n\times n}}
Notice that the Dirac algebra does not define a priori how many dimensions the matrices should be. For a four-dimensional Minkowski space, however, it turns out that the matrices have to be at least 4 × 4 {\displaystyle 4\times 4} .
Plane-wave (normal mode) solutions. Generation and annihilation operators. Hamiltonian. Anticommutation relations.
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Spin 1 field
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Massive spin 1 field. Additional (Lorentz) condition to eliminate spin-0.
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Massless spin 1 field. Gauge invariance. Quantization within Coulomb (radiation) gauge.
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Spin-statistics theorem. Discrete symmetries (C,P,T). CPT theorem.
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