Wick's theorem

Wick theorem expresses a time-ordered product of free fields as its normal-ordered product plus the sum over every possible contraction. It turns the Dyson series into Feynman diagrams.

Split a free scalar field into its annihilation and creation parts. For $x^0>y^0$, changing the product into normal ordering adds $[{\varphi }^{(+)}(x),{\varphi }^{(-)}(y)]$, the positive-frequency two-point function. For the opposite ordering interchange the two points. Their combination is the Feynman propagator, equal to the vacuum expectation of the time-ordered product. Iterating this identity gives the scalar Wick theorem.

For a free real scalar field, split $\varphi ={\varphi }^{(+)}+{\varphi }^{(-)}$ into annihilation and creation parts. Moving the creation operators left gives $\varphi (x)\varphi (y)=:\varphi (x)\varphi (y):+[{\varphi }^{(+)}(x),{\varphi }^{(-)}(y)]$. The commutator is the Wightman function. Choosing either time order gives the displayed identity with the Feynman propagator. It is an identity of operator-valued distributions; coincident-point products need a suitable prescription.

Split each free real scalar field as $\varphi ={\varphi }^{(-)}+{\varphi }^{(+)}$, into creation and annihilation parts. For chronologically ordered arguments, the first field has the latest time. Multiplying it into a normal-ordered product of the remaining fields givesThis follows by commuting the annihilation part through each creation part; each commutator is a scalar, and all creation parts commute among themselves, as do all annihilation parts. The scalar commutator is the Feynman propagator for the chronological pair. Induction on the number of fields therefore lists every partial pairing exactly once: either field 1 remains unpaired or it pairs with one $j$. This proves the Wick theorem for free bosonic fields, including the empty pairing. Vacuum expectation retains only complete pairings. The identities are understood for regulated or smeared operator-valued distributions.

For a centered free real scalar field, its vacuum time-ordered product has expectation $⟨T{\varphi }_1{\varphi }_2{\varphi }_3{\varphi }_4⟩=G_{12}{G}_{34}+G_{13}{G}_{24}+G_{14}{G}_{23}$, where $G_{ij}$ is the Feynman propagator between insertions $i,j$. These are the three complete pairings in the Wick theorem. All remaining normal-ordered products have zero vacuum expectation. No fermionic interchange signs occur for a bosonic field.

A centered free field in a Gaussian vacuum has only pair contractions, so every odd correlator vanishes by Wick theorem. Equivalently, number parity reverses the field while preserving the vacuum. This statement applies to smeared or regulated fields; it does not assert that unregulated coincident products exist. Interactions or a nonzero field mean can invalidate it.

Time ordering places operators with later time arguments to the left. Interchanging fermionic operators also contributes the sign of the required permutation.

The time-ordered exponential solves $i\hslash {\partial }_{t}U=H(t)U$ with initial condition $U(t_0,t_0)=I$. Its expansion is the Dyson series. For bounded continuous Hamiltonian operators the series converges, as in differentiation of the Dyson time-ordered exponential. If the generators commute at all times, time ordering is unnecessary and an ordinary operator exponential suffices.

For $\dot{U}=A(t)U$ and $U(0)=I$, the Magnus expansion writes the logarithm of the propagator as iterated integrals of commutators. Its first terms are ${\Omega }_1={\int }_0^{T}A(t)dt$ and ${\Omega }_2=\frac{1}{2}{\int }_0^{T}dt{\int }_0^{t}ds[A(t),A(s)]$. For a Quantum Hamiltonian, $A=-iH/\hslash$, so a noncommuting time-dependent control has corrections beyond the exponential of its integrated Quantum Hamiltonian. This formula is a local convergent expansion under suitable smallness conditions, or a formal expansion; it must not be assumed globally convergent for arbitrary pulse durations.

For $H(t)={\Omega }_1(t)A+{\Omega }_2(t)B$ with fixed noncommuting Hermitian operators, $[H(t),H(s)]=[{\Omega }_1(t){\Omega }_2(s)-{\Omega }_2(t){\Omega }_1(s)][A,B]$. A constant ratio of the envelopes is sufficient to remove time ordering. Independently changing that ratio generally produces commutator corrections in the Magnus expansion, and a product of successive pulses need not equal the exponential of their summed areas.

A time-ordered product sorts operator insertions from latest time on the left to earliest time on the right. Reordering odd fermionic operators contributes the sign of their permutation. For bosonic $A,B$, $T[A(t)B(t^{\prime })]=\theta (t-t^{\prime })A(t)B(t^{\prime })+\theta (t^{\prime }-t)B(t^{\prime })A(t)$. Vacuum expectations of free time-ordered products are evaluated by the Wick theorem.

Normal ordering places creation operators to the left of annihilation operators. For free fields it removes vacuum self-contractions from a composite operator.

For the quantum harmonic oscillator $H=\hslash \omega (a^{†}a+1/2)$, its normal symbol is $H_N=\hslash \omega (\bar{\psi }\psi +1/2)$. The Weyl ordering symbol of $a^{†}a$ is $\bar{\psi }\psi -1/2$, so $H_W=\hslash \omega \bar{\psi }\psi =p^2/(2m)+m{\omega }^2{q}^2/2$. Switching from adjacent-label coherent-state time slicing to a midpoint phase-space path integral must retain this ordering correction. Both correctly normalized constructions give the thermal partition function of a quantum harmonic oscillator $Z=[2\sinh (\beta \hslash \omega /2){]}^{-1}$; retaining the normal-symbol constant after switching prescriptions would count the zero-point shift twice.

A normal-ordered product of free-field oscillator operators places all creation operators to the left of all annihilation operators, including the fermionic sign required to reorder fermions. Its vacuum expectation vanishes if any nontrivial oscillator factors remain. The Wick theorem expresses a free-field time-ordered product through these products and all Wick contractions.

A Wick contraction replaces a selected pair of free fields by their two-point function. Summing pairings computes Gaussian correlation functions and supplies the propagator lines and combinatorial factors of a Feynman diagram.