States as Positive Linear Functionals

A quantum state can be specified before choosing any Hilbert space: it is a positive, normalized linear functional on the algebra of observables. The Gelfand–Naimark–Segal (GNS) theorem then reconstructs a Hilbert space, a representation of the algebra, and a cyclic vector whose expectation values reproduce that functional. This order of construction is especially valuable in curved spacetime, where physically useful states can lead to unitarily inequivalent Hilbert-space representations.

Required background. Partial Differential Equations and Green Functions supplies compactly supported test functions, Green-hyperbolic equations, and the causal propagator. The bounded local bridge used here is only that the field equation can be imposed as an algebraic relation and that the propagator defines the commutator pairing; no global mode expansion is required.

Helpful background. Flat-Spacetime Quantum Field Theory provides the familiar vacuum and Fock-space examples, Causality and Global Structure defines the globally hyperbolic setting in which the field algebra is constructed, and Quantum Fields in Curved Spacetime provides the volume-level conventions and route through the subject.

States on an observable algebra

Let $\mathcal A$ be a unital complex $*$-algebra. A state is a complex-linear map

$$\omega:\mathcal A\longrightarrow\mathbb C$$

such that

$$\omega(\mathbf 1)=1, \qquad \omega(A^*A)\geq 0 \quad\text{for every }A\in\mathcal A.$$

The second condition is the algebraic form of nonnegative probabilities. It also implies $\omega(A^*)=\overline{\omega(A)}$ and the Cauchy–Schwarz inequality

$$\big\lvert\omega(A^*B)\big\rvert^2 \leq \omega(A^*A)\,\omega(B^*B).$$

To prove it, apply positivity to $(A+zB)^*(A+zB)$ for arbitrary $z\in\mathbb C$ and require the resulting quadratic polynomial to remain nonnegative. This argument is entirely algebraic. It neither chooses a basis nor assumes that $A$ and $B$ are bounded operators on a pre-existing Hilbert space.

For the Klein–Gordon field, $\mathcal A$ is generated by smeared fields $\Phi(f)$, subject to linearity, the field equation, the adjoint relation, and the canonical commutator. A state determines the distributions

$$W_n(f_1,\ldots,f_n) =\omega\!\left(\Phi(f_1)\cdots\Phi(f_n)\right).$$

One separately requires these multilinear functionals to have the distributional continuity appropriate to the chosen field algebra. Thus abstract positivity is essential, but by itself it does not provide the microlocal regularity needed for local composite observables. The algebraic definition and its relation to $n$-point functions are developed in Hollands and Wald 2015, arXiv v2, §2.1, pp. 12–14, PDF.

The null ideal and the GNS inner product

Define a positive semidefinite sesquilinear form on $\mathcal A$ by

$$\langle A,B\rangle_\omega=\omega(A^*B),$$

and let

$$\mathcal N_\omega =\{A\in\mathcal A:\omega(A^*A)=0\}.$$

Vectors in $\mathcal N_\omega$ must be removed because they have zero length. Cauchy–Schwarz shows that $A\in\mathcal N_\omega$ precisely when $\omega(B^*A)=0$ for every $B\in\mathcal A$. This characterization proves that $\mathcal N_\omega$ is a left ideal. Indeed, if $A\in\mathcal N_\omega$ and $C,B\in\mathcal A$, then

$$\omega(B^*CA)=0;$$

taking $B=CA$ gives $\omega((CA)^*CA)=0$. Therefore left multiplication descends consistently to the quotient

$$\mathcal D_\omega=\mathcal A/\mathcal N_\omega.$$

Write $[A]$ for the class of $A$. The formula

$$\langle[A],[B]\rangle=\omega(A^*B)$$

is independent of the representatives and is now positive definite. Its Hilbert-space completion is denoted $\mathcal H_\omega$. The distinction between $\mathcal D_\omega$ and $\mathcal H_\omega$ matters for polynomial field algebras: field operators are generally unbounded, so their natural common domain is the dense subspace $\mathcal D_\omega$.

Representation, cyclic vector, and reconstruction

For $C\in\mathcal A$, define

$$\pi_\omega(C)[A]=[CA].$$

The left-ideal result makes this definition unambiguous. On $\mathcal D_\omega$ it obeys

$$\pi_\omega(CD)=\pi_\omega(C)\pi_\omega(D), \qquad \langle\pi_\omega(C^*)[A],[B]\rangle =\langle[A],\pi_\omega(C)[B]\rangle.$$

Thus $\pi_\omega$ is a $*$-representation on a dense invariant domain. If $\mathcal A$ is a C*-algebra, each $\pi_\omega(C)$ extends to a bounded operator and satisfies $\lVert\pi_\omega(C)\rVert\leq\lVert C\rVert$. That bounded conclusion is not assumed for the polynomial field algebra.

Now set $\Omega_\omega=[\mathbf 1]$. Normalization gives $\lVert\Omega_\omega\rVert=1$, and

$$\langle\Omega_\omega,\pi_\omega(A)\Omega_\omega\rangle =\langle[\mathbf 1],[A]\rangle =\omega(A).$$

Moreover, $\pi_\omega(\mathcal A)\Omega_\omega=\mathcal D_\omega$, which is dense in $\mathcal H_\omega$. Hence $\Omega_\omega$ is cyclic. This proves the existence part of the GNS theorem and explains why every algebraic state admits a vector-state realization, as summarized in Hollands and Wald 2015, arXiv v2, §2.1, pp. 13–15, PDF.

The cyclic triple is unique up to unitary equivalence. Suppose $(\mathcal H,\pi,\Omega)$ is another cyclic representation with $\omega(A)=\langle\Omega,\pi(A)\Omega\rangle$. On the dense set of cyclic vectors define

$$U\,\pi_\omega(A)\Omega_\omega=\pi(A)\Omega.$$

Both sides have the same pairwise inner products, so $U$ is well defined and isometric. Cyclicity makes its range dense, allowing it to extend uniquely to a unitary intertwiner. The theorem therefore reconstructs the representation determined by the state, not a universal preferred representation for all states.

Mixtures, pure states, and restrictions

The state space is convex. If $\omega_1$ and $\omega_2$ are states and $0\leq\lambda\leq1$, then

$$\omega=\lambda\omega_1+(1-\lambda)\omega_2$$

is again a state. A state is pure when it is an extreme point of this convex set; otherwise it is mixed. In the C*-algebra setting, purity is equivalent to irreducibility of the GNS representation. This characterization should not be confused with choosing a density matrix in one fixed Fock space. In curved spacetime there can be inequivalent GNS representations, and an algebraic state remains meaningful even when no trace-class density operator exists in a representation chosen for another state.

If $\mathcal B\subset\mathcal A$ is a unital $*$-subalgebra, the restriction $\omega\rvert_\mathcal B$ is automatically normalized and positive. This is how a global state supplies states on local observable algebras. Restriction can turn a globally pure state into a mixed state relative to the smaller algebra; purity is therefore a property of a state together with its algebra, not of a vector label alone.

A finite-dimensional example makes the distinction concrete. On $M_2(\mathbb C)$ let $\omega(A)=\operatorname{Tr}(\rho A)$ with $\rho=\operatorname{diag}(p,1-p)$ and $0\lt p\lt1$. Its GNS Hilbert space can be identified with the four-dimensional space of matrices, equipped with the Hilbert–Schmidt inner product, through $[A]\mapsto A\rho^{1/2}$. The algebra acts by left multiplication and the cyclic vector is $\rho^{1/2}$. Right multiplication commutes with that action, so the representation is reducible, exactly as expected for a mixed state. The GNS construction has represented a mixed functional by one vector without making the functional pure.

For Bose quasifree states, these algebraic distinctions acquire concrete covariance and representation criteria. Kay and Wald 1991, pp. 49–136 develops that structure in the more restrictive setting of stationary globally hyperbolic spacetimes with a bifurcate Killing horizon; its horizon hypotheses are not needed for the abstract GNS theorem proved above.

Why positivity cannot be weakened

Normalization by itself is insufficient. On $\mathcal A=M_2(\mathbb C)$, define

$$\eta(A)=2A_{11}-A_{22}.$$

Then $\eta(\mathbf 1)=1$, but with $E_{22}=\operatorname{diag}(0,1)$ one finds

$$\eta(E_{22}^*E_{22})=-1.$$

If a Hilbert-space vector $\Psi$ represented $\eta$, this number would have to equal $\lVert\pi(E_{22})\Psi\rVert^2$, which cannot be negative. The quotient construction likewise fails because its proposed “squared norm” is negative. Positivity is therefore exactly the hypothesis that turns expectation values into Hilbert-space geometry.

There is also a clear limitation to what GNS supplies. It produces a representation and a cyclic vector, but it does not imply that the state is Hadamard, stationary, quasifree, thermal, or suitable for renormalized local products. Those are additional analytic or physical conditions. Nor does the theorem select one state: it faithfully converts whichever positive functional was supplied.

Check your understanding

Let $(\mathcal H,\pi,\Omega)$ be a cyclic representation of a unital $*$-algebra on a common invariant domain, with $\lVert\Omega\rVert=1$. Show that $\omega(A)=\langle\Omega,\pi(A)\Omega\rangle$ is a state, and identify the map from its GNS cyclic subspace to $\mathcal H$.

Solution

Linearity follows from that of $\pi$ and the inner product, while $\omega(\mathbf 1)=\langle\Omega,\Omega\rangle=1$. For every $A$, the adjoint relation on the invariant domain gives $\omega(A^*A)=\langle\pi(A)\Omega,\pi(A)\Omega\rangle\geq0$. Define $U[A]=\pi(A)\Omega$. If $[A]=0$, then $\lVert\pi(A)\Omega\rVert^2=\omega(A^*A)=0$, so $U$ is well defined. The same calculation with two classes shows that $U$ preserves inner products. Its range is dense because $\Omega$ is cyclic, and therefore its extension is unitary. It also satisfies $U\pi_\omega(B)=\pi(B)U$ on the cyclic domain.

References

• Hollands, Stefan, and Robert M. Wald. “Quantum Fields in Curved Spacetime.” Physics Reports 574 (2015): 1–35. doi:10.1016/j.physrep.2015.02.001. Open PDF.
• 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.