Second quantization
Second quantization extends to systems with a variable amount of particles:
- First quantization quantizes Energy modes (e.g. particle in a box)
- Second quantization interprets ervery mode as harmonic oscillator with the $n$th excitation as $n$ particle state (we can have multiple exitations of given energy)
Second quantization introduces the Fock space which is a sum of different $n$-particle hilbert spaces $$\mathcal{F} = \bigoplus^\infty_n \mathcal{H}_n.$$ If there are different particle types, they get different Hilbert spaces.
In Second quantization we have a hamiltonian with creation and annihalation operators $a_p^\dagger$ and $a_p$, which act on the Fock space. This shows the nature of second quantization: A harmonic oscillator (with creation and annihilation operators $a^{(\dagger)}$ for every $p$!) They create or destroy a particle with the momentum $p$ respectively $$a^\dagger_p\ket{0} = \frac{1}{\sqrt{2\omega_p}}\ket{\vec{p}}.$$ The hamiltonian for a free theory can be written like the one of the harmonic oscillator, but integrating over the momentum $p$ to enable the treatment of arbitrary momentum states $$H_0 = \int\frac{d^3 p}{(2\pi)^3} \omega_p \left(a^\dagger_p a_p + \frac{1}{2}V\right),$$ where $V$ is the Volume term.
The commutation relations of the creation and annihaltion operator are extended by the momentum as well $$[a_k, a^\dagger_p] = (2\pi)^3 \delta^3(\vec{p} - \vec{k})$$ One-particle states are normalized $$\braket{\vec{p}|\vec{k}} = 2\omega_p(2\pi)^3\delta^3(\vec{p}-\vec{k}), \int\frac{d^3p}{(2\pi)^3}\frac{1}{2\omega_p}\ket{\vec{p}}\bra{\vec{p}}=1$$
# Scalar field
A free scalar quantum field is defined as integral over creation and annihilation operators with different momenta
Definition: free massless scalar quantum field
$$\phi_0(\vec{x})=\int \frac{d^3p}{(2\pi)^3}\frac{1}{\sqrt{2\omega_p}}(a_pe^{ipx} + a^\dagger_pe^{-ipx}), $$ with $p^\mu=(\omega_p, \vec{p})$.
This follows from the most general solution of the equation of motion $\Box\phi = (\partial^2_t - \nabla^2)\phi = 0$, wich can be derived as the most general Lorentz invariant euqation of motion for a scalar field wich generates dynamics, or from the Lagrangian of the Scalar field theory.
$\phi_0(\vec{x})$ is a field operator. It is hermitian and creates a particle at position $\vec{x}$: $\phi_0(\vec{x})\ket{0} = \ket{\vec{x}}$. Since it is hermitian also $\bra{\vec{x}} = \bra{0}\phi_0(\vec{x})$
For massive field we would have $\omega_p=\sqrt{\vec{p}^2 + m^2}$ instead of $\omega_p=|\vec{p}|$ and the equation of motion $(\Box + m^2)\phi=0$.
Prove that $\phi_0(\vec{x})\ket{0}=\ket{x}$:
Calculate $\braket{\vec{p}|\phi_0(\vec{x})|0}$: With $\bra{\vec{p}} = \sqrt{2w_p}\ket{0}a_p$: $$\braket{\vec{p}|\phi_0(\vec{x})|0} = \int\frac{d^3k}{(2\pi)^3}\sqrt{\frac{w_p}{w_k}}\left[e^{i\vec{k}\vec{x}}\braket{0|a_pa_k|0} + e^{-i\vec{k}\vec{x}}\braket{0|a_pa^\dagger_k|0}\right] = e^{-i\vec{p}\vec{x}},$$ when using $a^\dagger_p\ket{0}=\frac{1}{\sqrt{2\omega_p}}\ket{\vec{p}}$ and $\braket{\vec{p}|\vec{k}}=2w_p(2\pi)^3\delta^3(\vec{p}-\vec{k})$. So care must be given when evaluating matrix element in the Fock space with multiple particles with different momentums as the integral can give a contribution for every momentum.
In QFT we often work in the Heisenberg picture so that the time dependence is in the operators ($\phi$ and $a$). Thus Quantum fields satisfy the Heisenberg equations of motion $$i\partial_t \phi(x) = [\phi,H]$$ The quantum mechanical limit (only first quantization not second) can be obtained by restricting to one particle states with a basis given by $\bra{x} = \bra{0}\phi(\vec{x},t)$ and $\psi(x) = \braket{x|\psi}$. This is justified in the non-relativistic limit (no creation and annihilation of particles) and the equations of motion of the quantum field then turn into the Schrödinger equation for $\psi(x)$ (calculate).
One can derive the commutation relations for equal times
$$[\phi(\vec{x}),\phi(\vec{y})]=0$$ $$[\partial_t\phi(\vec{x})\big|_{t=0},\partial_t\phi(\vec{y})\big|_{t=0}]=0$$ $$[\phi(\vec{x}),\partial_t\phi(\vec{y})\big|_{t=0}]=i\delta^3(\vec{x}-\vec{y})$$
Usually we denote $\partial_t\phi(\vec{x}) = \pi(\vec{x})$ as the conjugate momentum of $\phi$ (like the conjugate momentum in classical mechanics)