Quasifree States and n-Point Functions

For a free field, a centered quasifree state is determined completely by its two-point function, but not every bidistribution qualifies. The field equation, reality, canonical relation, continuity, and positivity must all hold. Wick pairing then reconstructs every higher correlation function. This economy concerns the state on an algebra; it does not by itself choose a preferred state or guarantee the Hadamard short-distance condition.

Required background. Flat-Spacetime Quantum Field Theory supplies vacuum correlations and Wick pairing, while Variational and Symplectic Geometry defines the reduced classical pairing that fixes the antisymmetric part of the covariance.

Helpful background. Partial Differential Equations and Green Functions provides the test-function quotient and distributional field equation, and Causality and Global Structure establishes the causal support used in the commutator condition.

For immediate use, imposing the wave equation in each argument makes a bidistribution descend from test functions to solution classes, while the causal propagator fixes its antisymmetric part. Its symmetric part remains state-dependent.

Conditions on the scalar two-point function

Work first with the real scalar CCR algebra over $(\mathcal V,\sigma)$, where $\sigma([f],[g])=E(f,g)$ and $E=G_{\mathrm{ret}}-G_{\mathrm{adv}}$. For real $v,w\in\mathcal V$, define

$$\Lambda(v,w)=\omega\bigl(\Phi(v)\Phi(w)\bigr).$$

The necessary hypotheses on a candidate two-point form are:

• field equation and continuity: when lifted to test functions, $\Lambda(Pf,g)=\Lambda(f,Pg)=0$, and $\Lambda$ is a bidistribution;
• reality: $\overline{\Lambda(v,w)}=\Lambda(w,v)$;
• canonical antisymmetric part: $\Lambda(v,w)-\Lambda(w,v)=i\sigma(v,w)$;
• positive type: for finite families, $\sum_{j,k}\overline{c_j}c_k\Lambda(v_j,v_k)\geq0$.

The last condition is exactly $\omega(A^*A)\geq0$ for $A=\sum_jc_j\Phi(v_j)$. Normalizing the state adds $\omega(\mathbf1)=1$. Under these conditions, the Wick construction below defines a centered quasifree state on the polynomial CCR algebra. The structural result for Bose quasifree states and their one-particle realizations is developed in Kay and Wald 1991, §2, pp. 57–79.

Write the symmetric real covariance as

$$\mu(v,w)=\frac12\bigl(\Lambda(v,w)+\Lambda(w,v)\bigr).$$

Then

$$\Lambda(v,w)=\mu(v,w)+\frac i2\sigma(v,w).$$

Positive type implies the uncertainty inequality

$$\lvert\sigma(v,w)\rvert^2 \leq4\mu(v,v)\mu(w,w).$$

Indeed, positivity makes $\Lambda$ a positive semidefinite Hermitian form, so Cauchy–Schwarz bounds its imaginary part. Conversely, a real symmetric $\mu$ satisfying the corresponding positive-type condition makes $\mu+i\sigma/2$ a valid two-point form. The commutator fixes only the imaginary part; the freedom in $\mu$ is the freedom to choose the state.

Wick reconstruction of four and six points

A centered bosonic quasifree state has vanishing odd $n$-point functions. For an even number of fields,

$$\omega_{2n}(v_1,\ldots,v_{2n}) =\sum_{\mathcal P}\prod_{(i,j)\in\mathcal P}\Lambda(v_i,v_j),$$

where the sum runs over pairings and each pair retains its original order $i<j$. At four points this gives

$$\omega_4(1,2,3,4) =\Lambda_{12}\Lambda_{34} +\Lambda_{13}\Lambda_{24} +\Lambda_{14}\Lambda_{23}.$$

At six points it is clearer to expose the recursion than to hide terms in an ellipsis:

$$\omega_6(1,2,3,4,5,6) =\sum_{j=2}^{6}\Lambda_{1j}\, \omega_4(2,\ldots,\widehat{j},\ldots,6).$$

There are $5\times3=15$ products, the required number $(6-1)!!$. Commuting two adjacent arguments changes the Wick expression by $i\sigma$ times the lower-point function, so the reconstruction respects the CCR rather than treating the fields as commuting random variables.

The proof architecture is algebraic. Define the moments by the pairing formula, use the canonical antisymmetric part to show they descend through the CCR ideal, and use the one-particle completion of the positive form $\Lambda$ to realize them as vacuum expectations on Fock space. Positivity on all polynomial elements then follows from that representation. This construction establishes existence; it does not imply any particular ultraviolet wavefront set.

In particular, for a quadratic test element $B=\sum_{j,k}c_{jk}\Phi(v_j)\Phi(v_k)$, Wick’s four-point formula gives $\omega(B^*B)=\lVert\pi(B)\Omega\rVert^2\geq0$. The vacuum and two-particle components of $\pi(B)\Omega$ provide an independent Gram-norm check of the three pairings rather than a new assumption.

Purity, mixtures, and coherent displacement

If a compatible complex structure $J$ exists with

$$\mu(v,w)=\frac12\sigma(v,Jw),$$

the resulting quasifree state is pure under the standard nondegeneracy and continuity assumptions. Its GNS representation is the Fock representation built from $J$. Mixed quasifree states have a larger covariance and can be represented by a doubled one-particle construction; a thermal oscillator is the elementary example. Purity is an operator-level extremality condition, not the claim that the scalar uncertainty bound happens to saturate for one selected pair.

A coherent displacement introduces a real linear functional $\ell(v)=\omega(\Phi(v))$. Its higher truncated correlations above second order still vanish, but its odd ordinary moments need not. Terminology varies: some authors call these displaced states quasifree, while “centered quasifree” means $\ell=0$. Declaring the convention prevents Wick formulas with silently missing one-point terms.

The algebra, state, and representation remain distinct. The same CCR algebra admits many quasifree states; each state has a GNS representation; unitary equivalence of two such representations is an additional global question. Quasifree also does not mean physically admissible on a curved spacetime. The Hadamard condition, treated later, restricts the singular part of $\Lambda$ and is not implied by positivity and Wick factorization alone.

Fermionic quasifree covariances

For the gauge-invariant CAR algebra on a one-particle Hilbert space $\mathcal K$, a bounded operator $C$ defines the two-point function by

$$\omega_C(a^*(f)a(g))=\langle g,Cf\rangle.$$

The necessary and sufficient covariance condition is

$$0\leq C\leq\mathbf1.$$

The ordering of $f,g$ matters: the chapter uses $a(f)$ antilinear in $f$, $a^*(f)$ linear in $f$, and a Hilbert product antilinear in its first entry. The CAR therefore fixes the complementary ordering as

$$\omega_C(a(g)a^*(f))=\langle g,(\mathbf1-C)f\rangle.$$

For a normalized eigenvector $Cf=nf$, the two expectations are $n$ and $1-n$, explaining both bounds on $C$. Higher even correlations are signed Wick sums, equivalently determinants in the gauge-invariant case; odd correlations vanish. Such a state is pure exactly when $C$ is a projection. More general fermionic quasifree states may also have anomalous pairings and are described efficiently by the self-dual CAR covariance. Araki’s classification supplies the precise operator framework and equivalence criteria Araki 1970, §§2–4, pp. 389–418.

The bound above is fermionic and should not be imported into the scalar covariance problem. For bosons, positivity is constrained by the symplectic uncertainty inequality; for fermions, exclusion appears as the upper bound on $C$ and the signs in Wick expansion.

Thermal covariance and a failed positivity test

For a dimensionless oscillator with Hamiltonian $H=\omega(a^*a+1/2)$, the thermal state at inverse temperature $\beta\gt0$ has mean occupation $n_\beta=(e^{\beta\omega}-1)^{-1}$. Writing $Q=(a+a^*)/\sqrt2$ and $P=(a-a^*)/(i\sqrt2)$ gives

$$\mu(Q,Q)=\mu(P,P)=n_\beta+\frac12, \qquad \mu(Q,P)=0.$$

As $\beta\to\infty$, this becomes the pure vacuum covariance $\mu=\tfrac12\mathbf1$. At finite temperature it is larger and the state is mixed. This concrete example explains why Wick factorization describes both vacuum and thermal states: their pairing rule agrees, while the two-point data differ.

Consider a dimensionless oscillator with $\sigma(q,p)=1$ and propose

$$\mu(q,q)=\mu(p,p)=\frac14, \qquad \mu(q,p)=0.$$

In the ordered basis $(q,p)$, the two-point matrix is

$$\Lambda= \begin{pmatrix} 1/4&i/2\\ -i/2&1/4 \end{pmatrix}.$$

Its eigenvalues are $3/4$ and $-1/4$. Equivalently, the uncertainty inequality would require $1\leq4(1/4)(1/4)=1/4$. The candidate has the correct antisymmetric part but is not positive, so it cannot define a state. This counterexample shows why imposing only the field equation and commutator is not sufficient.

Exercise: reconstruct and test a four-point function

For a centered quasifree oscillator state let $Q=\Phi(q)$, $P=\Phi(p)$, and suppose $\Lambda(q,q)=A$, $\Lambda(p,p)=B$, and $\Lambda(q,p)=C+i/2$. Compute $\omega(Q^2P^2)$ and state the positivity restriction on $A$, $B$, and $C$.

Solution

The three Wick pairings give $\omega(Q^2P^2)=AB+2(C+i/2)^2$ with the displayed operator order. Reality also gives the reversed covariance $\Lambda(p,q)=C-i/2$; it should not replace either ordered contraction in this expression. Positivity of the two-point matrix requires nonnegative diagonal entries and nonnegative determinant, namely $A\geq0$, $B\geq0$, and $AB-C^2\geq1/4$. The final inequality is the oscillator uncertainty relation. It is independent of the four-point calculation and therefore provides a useful check on any proposed numerical values.

References

• Araki, Huzihiro. “On Quasifree States of CAR and Bogoliubov Automorphisms.” Publications of the Research Institute for Mathematical Sciences 6, no. 3 (1970): 385–442. DOI 10.2977/PRIMS/1195193913.
• Kay, Bernard S., and Robert M. Wald. “Theorems on the Uniqueness and Thermal Properties of Stationary, Nonsingular, Quasifree States on Spacetimes with a Bifurcate Killing Horizon.” Physics Reports 207, no. 2 (1991): 49–136. DOI 10.1016/0370-1573(91)90015-E.