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.

For two odd fermionic operators, time ordering contributes a minus sign whenever the chronological rearrangement exchanges them. For several fields the sign is the sign of a permutation restricted to the odd fields. This convention makes the free Dirac propagator a Green function of the Dirac equation with its contact term fixed by the canonical anticommutation relations.

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 a massive real scalar field in two spacetime dimensions with a polynomial potential and no derivative interactions, every connected subgraph has superficial degree of divergence $\omega =2(1-V)$, where $V$ counts interaction vertices. The only divergent loop subgraphs with external legs have one vertex and consist of self-contractions. Normal ordering removes them. With $T_{\Lambda }={\int }_{|k|<\Lambda }{d}^{2}k/[(2\pi {)}^2(k^2+m^2)]$, the ordinary-field potential representing the normal-ordered interaction is $e^{-T_{\Lambda }{\partial }_{\varphi }^2/2}V(\varphi )$. The exponential terminates for a polynomial. After these subtractions all remaining loop subgraphs have negative superficial degree of divergence, so no additional divergent field-dependent counterterms occur. Constant vacuum energy subtractions and finite on-shell renormalization scheme conditions are separate.

This correlation counts distinct detection events without the delta-function self-counting term in an un-ordered product of densities. It is nonnegative as the expectation of $A^{†}A$ with $A={\psi }_n\cdots {\psi }_1$. For a single occupied bosonic mode with sharp occupation $N$, the coincident $n$-point value contains the falling factorial $N(N-1)\cdots (N-n+1)$, rather than $N^n$.

A single-component chaotic Gaussian bosonic field has normalized coincident normally ordered correlation $n!$: every pairing contributes the same product of densities. A same-component fermionic field instead has zero coincident correlations for $n\ge 2$ by exclusion. These are state-dependent statements: a coherent bosonic field has factor one, and a single-mode number state has $N^{\underline{n}}/N^n$. Ratios to distant unnormalized correlations additionally require the compared densities to agree and distant correlations to factorize.

For a number-conserving Gaussian pairing state, six permutations give this expression. Each transposition has sign $ϵ$, positive for bosons and negative for fermions; the two three-cycles have positive sign for either statistics. Equivalently it is the permanent or determinant of the three-by-three one-body density matrix. For bosonic number states, repeated-mode factorial corrections must be included instead of assuming that Wick theorem.

For a product of bosonic number states, direct and exchange contractions correctly count distinct modes. They overcount repeated-mode terms because $⟨a^{†2}{a}^2⟩=N(N-1)$, while Wick theorem would give $2N^2$. Subtracting the excess yields the displayed exact two-point correction. At higher order the occupation decrements carry falling factorials. This distinction is essential when comparing a fixed-number condensate with independent thermal modes.

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.

For two identical even-degree $n$ scalar Feynman vertices, choose $r$ half-edges on each and pair them across the vertices. The remaining $n-r$ half-edges pair locally. This gives the displayed number of Wick contractions, with even $r$ and $(-1)!!=1$. The coefficient in the unnormalized partition function is $N_{n,r}/[2!(n!{)}^2]$. The $r=0$ contribution is disconnected and is removed from the logarithm, but remains in the partition function itself. This rule produces the quartic and sextic oscillator vacuum graph coefficients without drawing-dependent symmetry-factor guesses.

At second order in the coupling constant of a real phi cubed theory, the two-point numerator has eight fields, hence $7!!=105$ Wick contractions. Of these, six give a free external line times a vacuum theta graph, nine give a free line times a vacuum dumbbell, 36 give a two-line bubble, 36 give a connected tadpole attachment, and 18 give two separate one-point tadpoles. Dividing by $2!(3!{)}^2=72$ gives weights $1/12,1/8,1/2,1/2,1/4$. The normalized vacuum generating functional cancels the first two contributions, while the last remains in the full two-point correlation function as the product of one-point functions. Thus cancellation of vacuum bubbles does not, by itself, take the connected part.