Vacua, States, and Representations

An algebra specifies which quantities can be combined and adjointed; a state assigns normalized positive expectation values to that algebra; and a representation realizes the abstract algebra as operators on a Hilbert space. A vector represents a state only after a representation has been chosen. These are different mathematical objects, even when elementary notation hides the distinctions.

A vacuum is a state with additional spacetime-symmetry and spectrum properties, not a universal “empty” vector. For the massive free scalar on Minkowski spacetime, the future-mass-shell two-point function defines the standard Poincaré-invariant, positive-energy quasifree vacuum state, whose cyclic Hilbert-space realization is the usual Fock representation. Curved spacetimes, thermal states, distinct infinite-volume phases, and infinite-system limits need not select that state or any preferred Fock representation.

Helpful background. Hilbert Spaces, Completion, and the Riesz Representation Theorem supplies the representation and completion language, while Unbounded Operators: Domains, Closure, and Adjoints supplies the common-domain qualification that bounded-operator notation suppresses.

The algebraic setting for states and representations

The definitions below apply to a unital $*$-algebra of fields or observables. When bounded observables form a $C^*$-algebra, representations act by bounded operators. Polynomial field algebras contain unbounded quantities, so their representations require a common dense invariant domain $\mathcal D$ and, for all $\psi,\chi\in\mathcal D$, the adjoint relation

$$\langle\pi(A)\psi,\chi\rangle = \langle\psi,\pi(A^*)\chi\rangle.$$

The first concrete application is a massive real scalar with $m>0$ in four-dimensional Minkowski spacetime.

Algebra, state, representation, and vector

Algebra. An abstract unital $*$-algebra $\mathcal A$ records addition, multiplication, the identity $\mathbf 1$, the adjoint $A\mapsto A^*$, and the defining relations among its elements. “Abstract” means that no Hilbert space or matrix realization has yet been chosen.

State. A state is a linear functional

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

such that

$$\begin{aligned} \omega(\mathbf 1)&=1,\\ \omega(A^*A)&\geq0 \qquad(A\in\mathcal A). \end{aligned}$$

The number $\omega(A)$ is an expectation value. Positivity is the condition that turns algebraic squares into nonnegative measurement statistics; it is not the claim that $\omega(A)$ is positive for arbitrary $A$.

Representation. A representation is a $*$-homomorphism

$$\pi:\mathcal A\longrightarrow\mathcal B(\mathcal H)$$

in the bounded case. For an unbounded field algebra, read the target instead as operators preserving a declared dense domain $\mathcal D\subset\mathcal H$. The representation supplies concrete operators; it is not itself a state.

Representing vector. Once $\pi$ has been chosen, a nonzero vector $\Psi$ in the relevant domain defines

$$\omega_\Psi(A) = \frac{\langle\Psi,\pi(A)\Psi\rangle} {\langle\Psi,\Psi\rangle}.$$

Multiplying $\Psi$ by a nonzero complex scalar leaves this normalized state unchanged. Conversely, an algebraic state need not be a vector state or density-matrix state in one arbitrarily preselected representation.

The bridge is the Gelfand–Naimark–Segal statement. A state on a unital $C^*$-algebra determines a cyclic triple

$$\bigl(\mathcal H_\omega,\pi_\omega,\Omega_\omega\bigr)$$

with

$$\begin{aligned} \omega(A) &= \langle\Omega_\omega, \pi_\omega(A)\Omega_\omega\rangle,\\ \overline{\pi_\omega(\mathcal A)\Omega_\omega} &= \mathcal H_\omega. \end{aligned}$$

The triple is unique up to a unitary that intertwines the represented algebra and maps one cyclic vector to the other. The construction’s key idea is to give algebra elements the sesquilinear form $\langle A,B\rangle_\omega=\omega(A^*B)$, quotient its null vectors, complete, and let $\mathcal A$ act by left multiplication. Domain details for unbounded field algebras require the theorem-first treatment. Hollands and Wald 2015, § 2.1, arXiv v2 pp. 11–14, PDF gives the state, vector, representation, and GNS dictionary.

What additional conditions define a vacuum?

Suppose the connected proper-orthochronous Poincaré group $\mathcal P^\uparrow_+$ acts point-norm continuously on a $C^*$-algebra $\mathcal A$ by automorphisms $\alpha_g$; for a topological field $*$-algebra, the corresponding continuity structure must be declared separately. A Minkowski vacuum state $\omega_0$ is invariant,

$$\omega_0\!\left(\alpha_g(A)\right) = \omega_0(A),$$

and its translation representation satisfies the spectrum condition. In the GNS representation of an invariant state, the symmetry is implemented by unitaries with

$$\begin{aligned} U_0(g)\pi_0(A)U_0(g)^{-1} &= \pi_0\!\left(\alpha_g(A)\right),\\ U_0(g)\Omega_0&=\Omega_0. \end{aligned}$$

With $U_0(a)=e^{iP\cdot a}$ and the site’s $(+---)$ metric,

$$\begin{aligned} \operatorname{sp}P&\subset\overline V_+,\\ P^\mu\Omega_0&=0,\\ \overline V_+ &= \left\{ p:\ p^0\geq0,\ p^2\geq0 \right\}. \end{aligned}$$

Thus the vacuum vector is a representative of the invariant positive-energy state in one representation. It is defined only up to an overall phase. Requiring the canonical implementers to satisfy $U_0(a)\Omega_0=\Omega_0$ fixes their otherwise available translation-dependent phase, and hence fixes the additive energy convention so that $H\Omega_0=0$.

Several properties must not be smuggled into the word “vacuum.” Uniqueness is an extra hypothesis that must be established for the chosen algebra and dynamics. The GNS vector is cyclic for the represented global algebra by construction, but cyclicity or separating properties for each local algebra are separate results. Clustering, a mass gap, and a particle interpretation likewise require their own assumptions. Hollands and Wald 2015, § 1, arXiv v2 pp. 4–8, PDF separates the algebraic theory from a preferred vacuum representation and explains why the latter need not exist in a general spacetime.

Choosing the free-scalar vacuum before Fock space

Start with the abstract canonical-commutation-relation algebra of a real Klein–Gordon field. A bounded formulation exponentiates the field to Weyl generators in a $C^*$-algebra; the polynomial notation used here is its regular-representation infinitesimal form and inherits the common-domain qualifications above. For $f,g\in C_c^\infty(\mathbb R^{1,3})$, the field generators satisfy

$$\begin{aligned} \phi(f)^*&=\phi(\overline f),\\ [\phi(f),\phi(g)] &= i\Delta(f,g)\mathbf 1. \end{aligned}$$

These relations constrain every admissible state but do not select one. For the massive Minkowski field, define

$$\begin{aligned} E_{\mathbf p} &= \sqrt{\mathbf p^2+m^2},\\ \mathrm d\Pi_{\mathbf p} &= \frac{\mathrm d^3\mathbf p} {(2\pi)^3\,2E_{\mathbf p}},\\ W_0(z) &= \int\mathrm d\Pi_{\mathbf p}\, e^{-ip\cdot z}, \qquad p^0=E_{\mathbf p}>0. \end{aligned}$$

Set the one-point function to zero and use $W_0$ as the two-point function; the quasifree, or Gaussian, rule expresses every higher $n$-point function as a sum of pairings. This defines the standard free vacuum state. Four checks show why.

Positivity. With

$$\widetilde f(p) = \int\mathrm d^4x\,e^{+ip\cdot x}f(x),$$

the quadratic expectation is

$$\omega_0\!\left(\phi(f)^*\phi(f)\right) = \int\mathrm d\Pi_{\mathbf p}\, \left|\widetilde f(p)\right|^2 \geq0.$$

For a general state, this quadratic inequality is only the first member of a hierarchy of positivity conditions. Here it is sufficient because the state is quasifree and $W_0$ also has the required antisymmetric part fixed by the CCR. Hollands and Wald 2015, § 2.1, arXiv v2 p. 12 and pp. 15–16, PDF states the general positivity condition and the quasifree criterion.

Spectrum and covariance. The measure and integrand live on the future mass shell. Proper-orthochronous Poincaré transformations preserve that shell and its measure, so $W_0$ is invariant and has positive-frequency support. In the resulting Fock representation, an $n$-particle joint spectral value is a sum $p_1+\cdots+p_n$ of future-shell momenta. Because $\overline V_+$ is a convex cone, the full joint translation spectrum—not only the two-point support—lies in $\overline V_+$.

The fixed algebraic commutator. The state-dependent symmetric part of the two-point function drops out of

$$W_0(z)-W_0(-z) = i\Delta(z),$$

leaving precisely the commutator already imposed by the algebra.

Vacuum fluctuations. Although $\omega_0(\phi(f))=0$, the positive quadratic expression above is generally nonzero. A vacuum is therefore not a configuration in which every field value or correlation vanishes.

The GNS construction for this state yields the symmetric Fock realization

$$\begin{aligned} \mathcal H_0 &= \mathcal F_s(\mathcal H_1),\\ \Omega_0 &= (1,0,0,\ldots),\\ a(h)\Omega_0&=0 \quad \text{for every }h\in\mathcal H_1. \end{aligned}$$

Here $\mathcal H_1$ is the positive-frequency one-particle space determined by $W_0$. The conceptual order is important: the CCR algebra plus the selected vacuum state determine the GNS representation; within that representation, $\Omega_0$ is the Fock vacuum and $a(h)$ is its annihilation operator. In a regulated or perturbative comparison where free operators and a physical vacuum share an ambient representation, the physical vacuum need not be annihilated by those free operators. For an exact interacting QFT, the physical representation need not be the free Fock representation at all, so $a(h)\Omega_{\mathrm{phys}}$ may not even be defined. The free construction appears explicitly in Coleman 2019, § 2.1, p. 19; § 2.4, pp. 26–30; §§ 3.3–3.4, pp. 38–45.

A state change is not necessarily a representation change

An excitation usually changes the state while staying inside the same representation. If $B\in\mathcal A$ and $0<\omega_0(B^*B)<\infty$, then

$$\omega_B(A) = \frac{\omega_0(B^*AB)} {\omega_0(B^*B)}$$

is the state represented by the normalized vector

$$\frac{\pi_0(B)\Omega_0} {\sqrt{\omega_0(B^*B)}}.$$

In the free Fock realization, $a^\dagger(h)\Omega_0$ is normalized when $h$ is normalized in $\mathcal H_1$. It lies in the same Hilbert space. The vacuum state and the excited vector state differ, but the representation has not changed.

This also shows why Fock particle-number subspaces are not automatically superselection sectors: creation and annihilation fields connect adjacent subspaces. A genuine superselection statement depends on the chosen observable algebra and a selection criterion, not merely on the eigenvalues of a convenient number operator.

Mixed states that are normal in a chosen representation can be described by density matrices there. It is false, however, that every algebraic state must be a density matrix on the vacuum Fock space. Normality is representation-relative, and infinite systems admit states outside one chosen representation’s normal-state family.

Why there is no universal Fock space

Two representations $\pi_1$ and $\pi_2$ of the same abstract algebra are unitarily equivalent only if a unitary $V:\mathcal K_1\to\mathcal K_2$ intertwines every represented element,

$$V\pi_1(A)V^{-1} = \pi_2(A) \quad \text{for all }A\in\mathcal A.$$

An abstract isomorphism between the Hilbert spaces is not enough. The GNS representations of the same state are equivalent in this precise sense, but different states can yield equivalent or inequivalent representations; difference of states alone decides neither outcome.

For unbounded field representations, one must additionally require $V\mathcal D_1=\mathcal D_2$ and impose the intertwining equation on those common invariant domains.

For finitely many regulated canonical degrees of freedom, regular irreducible representations obey a uniqueness theorem under its stated hypotheses. A continuum field has infinitely many degrees of freedom, and the corresponding representation question can have a different answer. Thermal states, different long-range behaviors, and different admissible positive-frequency choices can lead to representations not unitarily equivalent to the standard vacuum Fock representation. A general curved or time-dependent spacetime may not provide any symmetry that selects a preferred positive-frequency split in the first place. Hollands and Wald 2015, § 1, arXiv v2 pp. 4–8; § 2.1, pp. 14–19, PDF gives the curved-spacetime, Gaussian, and thermal qualifications.

One can still begin from the same abstract CCR algebra and compare two states through their correlation functions. Equality of restricted correlators, unitary equivalence, local quasiequivalence, and membership in the same folium are different questions. No one of them should be inferred from another without the appropriate theorem.

Boundaries and handoffs

• Continue to One-Particle States: Mass, Spin, and Relativistic Normalization for the Poincaré labels carried by excitations, then to Multiparticle States, Statistics, and Fock Organization for their sector structure.
• Construct the model explicitly in Quantizing the Real Scalar Field and develop the chosen representation in Fock Space, Vacuum, and Particle Number.
• Use States, GNS Representations, and Folia for proofs, domain hypotheses, normal states, folia, and sectors.
• Keep unitary equivalence and local quasiequivalence with Equivalence, Uniqueness, and Comparison Notions.
• See Vacuum Ambiguity and Observer Dependence when no global symmetry selects a preferred state or particle splitting.

Common pitfalls

“A state is a vector.” A vector defines a state only after a representation is chosen. The same abstract state generates a preferred cyclic realization through GNS, but the objects remain distinct.

“Vacuum means no particles.” This is true only relative to the annihilation operators of a selected free or asymptotic Fock representation. The invariant-state and spectrum conditions used here do not presuppose a free particle-number basis.

“The vacuum is automatically unique.” Uniqueness is an additional property that must be proved for the chosen algebra, state conditions, and dynamics.

“Every state is a density matrix on vacuum Fock space.” Density matrices describe states normal to that representation. Infinite systems can have states whose GNS representations are inequivalent.

“All isomorphic Hilbert spaces give equivalent physics.” Representation equivalence also requires a unitary that intertwines the action of every algebra element.

“Particle-number sectors are superselection sectors.” Free field operators move between Fock number subspaces. Superselection depends on what the physical observable algebra can connect.

Check your understanding

1. Classify the symbols in $\omega(A)=\langle\Omega,\pi(A)\Omega\rangle$. A satisfactory answer identifies $\mathcal A$ as the algebra, $A$ as an algebra element, $\omega$ as a state functional, $\pi$ as a representation, and $\Omega$ as a vector in that representation.
2. Why does $a(h)\Omega_0=0$ not define a universal vacuum? The operator $a(h)$ exists only after a positive-frequency choice and Fock representation have been made; a general vacuum is characterized instead by symmetry and spectrum conditions.
3. Does $a^\dagger(h)\Omega_0$ define a new representation? No. It is a different vector state inside the same free Fock representation.
4. What extra evidence is needed to call two representations equivalent? One needs an intertwining criterion for the represented algebra; matching a finite list of correlators or merely identifying the Hilbert spaces is insufficient.

References

• Coleman, Sidney. Lectures of Sidney Coleman on Quantum Field Theory. Edited by Bryan Gin-ge Chen, David Derbes, David Griffiths, Brian Hill, Richard Sohn, and Yuan-Sen Ting. Singapore: World Scientific, 2019. DOI.
• Hollands, Stefan, and Robert M. Wald. “Quantum Fields in Curved Spacetime.” Physics Reports 574 (2015): 1–35. DOI. Open PDF.