Last modified on 10 January 2010, at 18:02

Quantum Field Theory/Quantization of free fields

Spin 0 fieldEdit

Real and complex scalar fields. Klein-Gordon equation. Plane-wave (normal mode) solutions. Generation and anihilation operators. Hamiltonian. Commutation relations.Edit

Real and complex scalar fields.Edit

The equations of motion for a real scalar field  \phi can be obtained from the following lagrangian densities

 \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 \left( \Box+M^2 \right)\phi(x)=0 .

The complex scalar field  \phi can be considered as a sum of two scalar fields:  \phi_1 and  \phi_2 ,  \phi=\left(\phi_1+i\phi_2\right)/ \sqrt{2}

The Langrangian density of a complex scalar field is

  \mathcal{L} =  (\partial_{\mu} \phi)^+ \partial^{\mu}\phi  - M^2 \phi^+ \phi

Klein-Gordon equationEdit

Klein-Gordon equation is precisely the equation of motion for the spin-0 particle as derived above: \left( \Box+M^2 \right)\phi(x)=0

Spin 1/2 fieldEdit

Dirac equationEdit

The Dirac equation is given by:

\left(i\gamma^\mu\partial_\mu - m\right)\psi\left(x\right) = 0

where \psi is a four-dimensional Dirac spinor. The \gamma matrices obey the following anticommutation relation (known as the Dirac algebra):

\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\times 4.

Plane-wave (normal mode) solutions. Generation and anihilation operators. Hamiltonian. Anticommutation relations.Edit

Spin 1 fieldEdit

Massive spin 1 field. Additional (Lorentz) condition to eliminate spin-0.Edit

Massless spin 1 field. Gauge invariance. Quantization within Coulomb (radiation) gauge.Edit

Spin-statistics theorem. Discrete symmetries (C,P,T). CPT theorem.Edit