States, GNS Representations, and Folia

A state on a C*-algebra is a positive normalized functional, not a density matrix waiting to be written in a preferred Hilbert space. The Gelfand–Naimark–Segal construction produces its canonical cyclic representation. Normal states, folia, quasiequivalence, and disjointness are then comparisons relative to representations; they explain how one abstract quasilocal algebra can support vacuum, thermal, and phase-selecting realizations that are not unitarily equivalent.

Required background. Positivity, Spectrum, Covariance, and Locality Hypotheses distinguishes state positivity from spectral positivity. Quasilocal C*-Algebras and Inductive Limits supplies the algebra on which the state is defined.

Helpful background. Operator Algebras and Positive Functionals reviews the construction in finite examples. Vacua, States, and Representations gives the physical vocabulary, Bogoliubov Transformations and Unitary Implementability gives a global inequivalence test, and Crossed-Product Gravitational Algebras and Generalized-Entropy Terms is a later comparison where the algebra itself is enlarged.

The GNS theorem and its mechanism

Let $\omega$ be a state on a unital C*-algebra $\mathcal A$. Cauchy–Schwarz for positive functionals shows that

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

is a left ideal. On $\mathcal A/\mathcal N_\omega$, define

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

After completion, left multiplication gives

$$\pi_\omega(C)[A]=[CA], \qquad \lVert\pi_\omega(C)[A]\rVert_\omega \leq \lVert C\rVert\lVert[A]\rVert_\omega.$$

Thus $\pi_\omega$ is a bounded star representation on $\mathcal H_\omega$. With $\Omega_\omega=[1]$,

$$\overline{\pi_\omega(\mathcal A)\Omega_\omega}=\mathcal H_\omega, \qquad \omega(A)=\langle\Omega_\omega,\pi_\omega(A)\Omega_\omega\rangle.$$

If another cyclic triple gives the same state, the map $\pi_\omega(A)\Omega_\omega\mapsto\pi(A)\Omega$ is isometric on a dense set and extends uniquely to a unitary intertwiner. This is uniqueness for one state, not equivalence of different states. The quotient proof and the pure-state criterion trace to Segal 1947, Theorems 1.1–1.4, pp. 75–82; a QFT-oriented account appears in Fewster and Rejzner 2020, § 2.3, pp. 6–8.

The null left ideal need not equal $\ker\pi_\omega$. The latter is the largest closed two-sided ideal represented trivially. Moreover,

$$\omega\text{ is pure} \quad\Longleftrightarrow\quad \pi_\omega\text{ is irreducible}.$$

Neither faithfulness nor irreducibility follows from positivity alone.

Normal states, folia, and disjointness

For a representation $\pi:\mathcal A\to\mathcal B(\mathcal H)$, its folium is

$$\mathcal F(\pi)= \left\{ A\mapsto\operatorname{Tr}(\rho\,\pi(A)): \rho\geq0, \ \rho\text{ trace class}, \ \operatorname{Tr}\rho=1 \right\}.$$

These are precisely the states that extend normally to $\pi(\mathcal A)''$. The trace-class operator lives on the representing Hilbert space; it need not belong to the represented algebra and does not assert a spatial tensor factor.

Two representations are quasiequivalent when their normal extensions have the same state space, equivalently when the map $\pi_1(A)\mapsto\pi_2(A)$ extends to a normal star isomorphism $\pi_1(\mathcal A)''\to\pi_2(\mathcal A)''$. They are disjoint when they have no nonzero unitarily equivalent subrepresentations. Unitary equivalence implies quasiequivalence, but the converse need not select a unitary or a cyclic vector.

First application: a free thermal GNS representation

Infinite-Volume KMS States, Passivity, and Phase Multiplicity supplies the thermal interpretation and the thermodynamic-limit boundary.

Let $h$ be the one-particle space of the massive scalar, let $\varepsilon=(\mathbf p^2+m^2)^{1/2}$, and set

$$n_\beta=(e^{\beta\varepsilon}-1)^{-1}.$$

In compatible Weyl normalization, the quasifree KMS state has characteristic functional

$$\omega_\beta(W(f)) = \exp\!\left[-\frac14 \left\langle f,\coth\!\left(\frac{\beta\varepsilon}{2}\right)f\right\rangle \right].$$

Its GNS representation can be realized on the doubled Fock space $\Gamma_s(h\oplus\bar h)$ by

$$\pi_\beta(W(f)) = W_F\!\left((1+n_\beta)^{1/2}f\oplus n_\beta^{1/2}\bar f\right).$$

The doubled Fock vacuum $\Omega_\beta$ reproduces $\omega_\beta$. Time translations are implemented by the Liouvillean

$$L_\beta=d\Gamma(\varepsilon\oplus-\bar\varepsilon), \qquad L_\beta\Omega_\beta=0.$$

Unlike the vacuum Hamiltonian $H_0=d\Gamma(\varepsilon)\geq0$, $L_\beta$ has both signs in its spectrum: thermal equilibrium is characterized by the KMS boundary condition, not by a ground-state spectrum condition. This doubled realization is the Araki–Woods construction Araki and Woods 1963, §§ 2–4, pp. 641–655. The extension of equilibrium from Gibbs ensembles to infinite systems through the KMS condition is established in Haag, Hugenholtz, and Winnink 1967, §§ 1–3, pp. 215–225.

Failure test: a false infinite-volume trace

Writing

$$\omega_\beta(A) \stackrel{?}{=} \frac{\operatorname{Tr}_{\mathcal H_0}(e^{-\beta H_0}\pi_0(A))} {\operatorname{Tr}_{\mathcal H_0}(e^{-\beta H_0})}$$

is valid for a finite box with a trace-class Gibbs operator, but fails in infinite Minkowski volume. The partition function contains the volume divergence, $e^{-\beta H_0}$ is not trace class, and the translation-invariant thermal state is not thereby shown normal in the vacuum folium. The abstract KMS functional and its own GNS representation remain well defined.

The strongest surviving conclusion is the finite-volume Gibbs formula before the thermodynamic limit, or the algebraic KMS identity after it. A density matrix on vacuum Fock space requires a separate normality proof.

Independent checks

First compute the vacuum expectation of the doubled Weyl operator; the Fock Gaussian identity returns the factor $(1+2n_\beta)=\coth(\beta\varepsilon/2)$. Second, the opposite signs in $L_\beta$ imply the correct time action on the two summands while leaving $\Omega_\beta$ invariant. Third, test the KMS boundary relation on Weyl generators. These checks establish the stated free representation; they do not prove normality relative to the vacuum.

Exercises

Why is $\mathcal N_\omega$ a left ideal rather than merely a linear null space?

Solution

For $A\in\mathcal N_\omega$ and $C\in\mathcal A$, the C*-order inequality $C^*C\leq\lVert C\rVert^2 1$ gives

$$0\leq\omega(A^*C^*CA) \leq \lVert C\rVert^2\omega(A^*A)=0.$$

Hence $CA\in\mathcal N_\omega$. This is exactly the side required for left multiplication to descend to the quotient.

References

• Araki, Huzihiro, and E. J. Woods. “Representations of the Canonical Commutation Relations Describing a Nonrelativistic Infinite Free Bose Gas.” Journal of Mathematical Physics 4 (1963): 637–662. DOI.
• Fewster, Christopher J., and Kasia Rejzner. “Algebraic Quantum Field Theory—an Introduction.” In Progress and Visions in Quantum Theory in View of Gravity, 1–61. Cham: Birkhäuser, 2020. DOI; Open PDF.
• Haag, Rudolf, Nicolaas M. Hugenholtz, and Marinus Winnink. “On the Equilibrium States in Quantum Statistical Mechanics.” Communications in Mathematical Physics 5 (1967): 215–236. DOI.
• Segal, Irving E. “Irreducible Representations of Operator Algebras.” Bulletin of the American Mathematical Society 53 (1947): 73–88. DOI.