Quantum field ($\Phi (x)$)

A quantum field is an operator-valued distribution on spacetime: smearing $\Phi (x)$ against a smooth test function gives an operator, whereas pointwise products can require renormalization. For example, a free complex scalar field contains particle annihilation operators and antiparticle creation operators. Its microcausality condition makes field commutators vanish at spacelike separation. For fermions, the corresponding locality condition uses anticommutators. This combines quantum operators with the spacetime locality of a quantum field theory.

For a quantum field with four-momentum operator $P^{\mu }$, spacetime dependence is implemented by the displayed conjugation. Taking a matrix element between momentum eigenstates gives $⟨q|\Phi (x)|p⟩=e^{i(q-p)\cdot x}⟨q|\Phi (0)|p⟩$. Differentiating gives $i{\partial }_t\Phi =[\Phi ,H]$ and $-i\nabla \Phi =[\Phi ,\mathbf{P}]$, connecting the Heisenberg equation of motion to spatial translation covariance.

For $p^2=m^2>0$, let a unitary operator $U$ implement a Lorentz boost taking $p$ to $(m,0)$. For a Lorentz scalar field, $⟨q|\varphi (x)|p⟩=⟨\Psi |\varphi (0)|m,0⟩$ with $|\Psi ⟩=e^{-i(q-p)\cdot x}U|q⟩$. The phase follows from translation covariance of a quantum field; its complex conjugate enters the bra. The transformed energy eigenvalue is $(\Lambda q{)}^0=p\cdot q/m$ by invariance of the Minkowski inner product. The reduction requires a massive initial state.