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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.

Quantum field theory describes quantum states and interactions using fields whose smeared values act as operators. Free-field modes behave as quantum harmonic oscillators or fermionic occupation modes; their excitations supply the particle description. This allows creation operators and annihilation operators to describe processes in which particle number changes.

A relativistic quantum field theory combines this framework with special relativity, Lorentz invariance and locality conditions such as microcausality. Nonrelativistic many-body systems also have useful quantum field theories, for example a weakly interacting Bose gas described by a bosonic field. The framework is therefore broader than a list of elementary-particle models.

Correlation functions predict responses and connect through appropriate limits to scattering amplitudes. Perturbative quantum field theory expands these quantities using Feynman diagrams, while strongly interacting problems may require other methods. Composite fields and arbitrarily short distances require attention to operator definitions and renormalization, rather than treating every field product as an ordinary pointwise multiplication.

Fix a detected position and polarization of waves. Write the electric field operator as $\hat{E}(t)={\hat{E}}^{(+)}(t)+{\hat{E}}^{(-)}(t)$, where the positive-frequency part contains annihilation operators and ${\hat{E}}^{(-)}=({\hat{E}}^{(+)}{)}^{†}$. An ideal absorbing photodetector responds towith density matrix $\rho$. The diagonal gives the single-count rate up to the detector coupling. Multiple absorptions require higher, normal-ordered field products. Roy J. Glauber used these correlations to develop the quantum account of optical coherence.^[ref]

The ordering follows from a detection amplitude. If absorption takes an initial pure state $|i⟩$ to an unobserved final state $|f⟩$, its field factor is $⟨f|{\hat{E}}^{(+)}|i⟩$. Squaring its complex modulus and summing over a complete orthonormal basis of final states gives $⟨i|{\hat{E}}^{(-)}{\hat{E}}^{(+)}|i⟩$. Averaging over initial preparations replaces this expression by a trace. Thus a detector that absorbs cannot count the Fock vacuum, although the vacuum has quantum fluctuations.

For one selected normal mode, ${\hat{E}}^{(+)}(t)=\mathcal{E}u(t)a$. All Fock states with nonzero occupation, coherent states, and thermal light in this same mode havewhere both denominators are nonzero. Its magnitude is one even when the expected value of the field vanishes. Perfect first-order optical coherence therefore does not identify a coherent state. Pair detection supplies additional information.

With $p\cdot x=Et-\mathbf{p}\cdot \mathbf{x}$, placing an annihilation operator on $e^{+ipx}$ rather than $e^{-ipx}$ makes its vacuum time-ordered contraction equal to $D_F^{\ast }$, with the opposite pole prescription. For positive time separation, the zero-spatial-momentum factor is $e^{+imt}/(2m)$ rather than $e^{-imt}/(2m)$. Replacing the integration variable $p$ by $-p$ does not change the sign of $i0$ and cannot fix the physical positive-energy boundary condition.

Second quantization describes identical-particle quantum states in Fock space using creation operators and annihilation operators. Relative to an orthonormal basis of one-particle states, a one-body operator becomes ${\sum }_{\alpha \beta }{h}_{\alpha \beta }{a}_{\alpha }^{†}{a}_{\beta }$. The canonical commutation relation implements bosonic statistics, while the canonical anticommutation relations implement fermionic statistics. Expanding the quantized field in a complete basis gives a delta-function equal-time bracket; restricting the expansion to finitely many modes replaces that delta function by the corresponding projection operator kernel.

The matrix in time reversal of a Dirac field depends on the convention for the spin-flip phases and on the sign of the spinor identity used when relabeling the mode expansion of a Dirac field. If ${ϵ}_s=(-1{)}^{1/2-s}$ and ${ϵ}_s{u}^{-s\ast }(p_T)=-{\gamma }^{5}C{u}^s(p)$, the relabelled coefficient is ${ϵ}_{-s}{u}^{-s\ast }(p_T)={\gamma }^{5}C{u}^s(p)$, giving $B={\gamma }^{5}C$. A common phase changes $B$ but leaves its gamma matrix conjugation property and the induced fermion bilinear transformation unchanged. The antiunitary operator acts on both numerical coefficients and creation or annihilation operators.

The canonical commutation relation gives $[P^{\mu },a_p^{†}]=p^{\mu }{a}_p^{†}$ for a free real scalar field. Differentiate the exponential conjugation: its derivative is a scalar multiple of the same operator, yielding the displayed phase. The annihilation operator receives the opposite phase, so at equal time $e^{-i\mathbf{P}\cdot \mathbf{y}}\varphi (\mathbf{x})e^{i\mathbf{P}\cdot \mathbf{y}}=\varphi (\mathbf{x}+\mathbf{y})$. The spatial translation operator translates a state's position wavefunction in the opposite argument direction.

The normalized vacuum state annihilated by $c=ua-v{b}^{†}$ and $d=v{a}^{†}+ub$, for positive $u,v$ with $u^2+v^2=1$. Since $(a^{†}{b}^{†}{)}^2=0$, it can also be written $u\exp [(v/u)a^{†}{b}^{†}]|0⟩$. Acting with either annihilation operator makes the two terms cancel. Its original occupation numbers each have expected value $v^2$ and its pair amplitude is $⟨ab⟩=-uv$. It has even fermion parity although the original total number operator is not sharp. For the two-orbital fermionic pairing Hamiltonian, it is the unique ground state of energy $\epsilon -\sqrt{{\epsilon }^2+{\Delta }^2}$.

For a stable relativistic quantum field theory in Minkowski space, a vacuum is a lowest-energy state for the chosen time evolution; a physical vacuum is normally also invariant under spacetime translations. It need not have zero fluctuations or vanish under every field operator. A free field's Fock vacuum is annihilated by its annihilation operators, while an interacting vacuum need not equal the free-theory Fock state. Multiple degenerate vacua permit spontaneous symmetry breaking. In curved spacetime the choice of time evolution and positive-frequency modes introduces additional vacuum ambiguity in a nonstationary spacetime.

In a strictly stationary spacetime, the timelike Killing flow gives a preferred positive-frequency splitting. The corresponding vacuum is annihilated by every annihilation operator associated with a positive-frequency mode.