Archive for the ‘Quantum Field Theory’ Category

Second Quantization

July 3, 2008

Since Melvin wrote about how I tried to introduce him to fiber bundles, I’ll try to relay what he taught me about second quantization. (I was totally lost when he was patiently explaining Feynman path integrals and diagrams :-/ ) First we start with the canonical commutation relations \text{[} x^i,p_j\text{]}=i\hbar\delta^i_j for quantum mechanics. Then we promote x^i from an “operator” to a “label” for the field \varphi(x^i,t) which we’ll assume is a scalar field. After choosing a Lagrangian which makes sense to physicists if not to me, we minimize the action to get the Klein-Gordon equation \partial^2\varphi-m^2\varphi=0. Solving the K-G equation in a Fourier-ish way gives \varphi=\int_{|k|^2=m^2}d^4 k(a(k)e^{ik\cdot x}+a^\dagger(k)e^{-ikx}). The a(k) and a^\dagger(k) are annihilation and creation operators respectively. Then we start with some Hilbert space with a vacuum state |0> and apply the creation operator to get a particle with momentum k, |k>=a^\dagger (k)|0>. So on and so forthing we can generate a Fock space maybe.

Melvin tried to expound on the difference between first and second quantization. He explained that Quantum Mechanics was when you perform canonical quantization, upgrading observables to operators (maybe) and second quantizing is Quantum Field Theory where you see particles as excitations of the quantized field.

Posted in Quantum Field Theory | 1 Comment »