Algebraic Free Fields on Curved Spacetimes

For a free field on a curved background, the algebra is fixed before any Hilbert-space representation is chosen. On an oriented, time-oriented globally hyperbolic spacetime, the field equation and its advanced-minus-retarded propagator determine a symplectic solution space; the canonical commutation relations then turn that space into a Weyl algebra. Global hyperbolicity is doing real work: it supplies unique Green operators, causal support, and the Cauchy evolution used in the time-slice theorem.

Required background. Globally hyperbolic spacetimes and the Loc categories supplies Cauchy morphisms and causal convexity. Green-hyperbolic operators and causal propagators supplies the exact sequence for advanced and retarded Green operators. Haag–Kastler nets and locality supplies isotony and Einstein causality.

Helpful background. Hadamard states and their wavefront characterization explains the later state restriction. Locally covariant QFT as a functor supplies functorial notation. Covariant scalar fields and curvature coupling fixes the scalar operator. Spinors, tetrads, and spin connections gives the fermionic analogue. Symplectic structures and conserved inner products relates test functions to Cauchy data. Canonical quantization on curved backgrounds compares algebraic and Fock constructions. Complex structures and one-particle spaces and quasifree states and two-point functions describe representations of the algebra.

The causal-propagator phase space

Let $(M,g)$ be smooth, globally hyperbolic, without boundary, and let

$$P=\Box_g+m^2+\xi R, \qquad m^2\geq 0,$$

act on real-valued smooth functions. Here $P$ is normally hyperbolic and formally self-adjoint with respect to the metric volume form. Write $E^+$ and $E^-$ for its retarded and advanced Green operators and choose $E=E^- -E^+$. The identities $PE^\pm f=E^\pm Pf=f$ and the causal support estimates imply the exact sequence

$$C_0^\infty(M)\xrightarrow{P}C_0^\infty(M) \xrightarrow{E}C_{\mathrm{sc}}^\infty(M) \xrightarrow{P}C_{\mathrm{sc}}^\infty(M),$$

where “sc” means spacelike compact. Thus

$$\mathcal E(M)=C_0^\infty(M;\mathbb R)/P C_0^\infty(M;\mathbb R)$$

is isomorphic to the spacelike-compact solution space by $[f]\mapsto Ef$. Its symplectic form is

$$\sigma_M([f],[h])=\int_M f(Eh)\,\mathrm d\mathrm{vol}_g.$$

Formal self-adjointness makes this antisymmetric; the Green-operator exact sequence makes it independent of representatives and nondegenerate. Equivalently, on any smooth spacelike Cauchy surface $\Sigma$ with future unit normal $n$, it is the conserved Cauchy-data form $\int_\Sigma(\phi\nabla_n\psi-\psi\nabla_n\phi)\,\mathrm d\Sigma$. This equality is an independent sign check: reversing the definition of $E$ reverses both displayed conventions, but no physical relation changes if the reversal is made consistently. The construction and its causal support statement are proved in Bär, Ginoux, and Pfäffle 2007, Theorem 3.4.7 and § 3.5, pp. 110–119.

For a charged scalar one complexifies and uses a Hermitian form; for a Dirac field the conserved positive form leads to a CAR algebra. Those changes are not automatic consequences of the scalar theorem: the bundle, adjoint, field equation, and positivity statement must be supplied anew.

From the symplectic space to the Weyl algebra

The algebraic free scalar theory assigns to $M$ the unital $C^*$-algebra generated by symbols $W_M(F)$, $F\in\mathcal E(M)$, subject to

$$W_M(F)^*=W_M(-F),\qquad W_M(F)W_M(G)=e^{-i\sigma_M(F,G)/2}W_M(F+G).$$

These relations encode the field equation and CCR without selecting a vacuum. A causally convex isometric embedding $\psi:M\to N$ intertwines the propagators, so extension by zero gives a symplectic map $\mathcal E(\psi)[f]=[\psi_*f]$ and hence an injective homomorphism $\mathcal A(\psi)W_M(F)=W_N(\mathcal E(\psi)F)$. Identity and composition follow from push-forward. Causally disjoint images have zero cross-symplectic form, which yields commuting subalgebras. This is the free-field instance of the generally covariant locality principle in Brunetti, Fredenhagen, and Verch 2003, §§ 2.2–2.4, pp. 37–45.

A regular state converts the Weyl generators into self-adjoint smeared fields in its GNS representation, but the abstract algebra does not guarantee regularity, Hadamard short-distance behavior, or a preferred representation. Conversely, two unitarily inequivalent representations may represent the same algebra. This is why state selection must be kept separate from algebra construction.

The time-slice isomorphism

Suppose $O\subset M$ is open, causally convex, and contains a Cauchy surface of $M$. The inclusion induces $\mathcal E(O)\to\mathcal E(M)$. Surjectivity is the substantive point. Choose two Cauchy surfaces inside $O$ and a partition of unity adapted to their past and future. For $f\in C_0^\infty(M)$, apply the Green operators and the cutoffs to construct $h\in C_0^\infty(M)$ such that

$$f-Ph\in C_0^\infty(O).$$

Hence every class $[f]$ has a representative supported in $O$. Injectivity follows from the same exact sequence, so the induced symplectic map and therefore the Weyl-algebra map are isomorphisms. The result is algebraic Cauchy determinism, not a claim that observables at one time commute or that a state is uniquely fixed.

The worked construction on local field algebras, causality, and the time-slice property carries this argument through for the real Klein–Gordon field, including the support-moving cutoff. A quick independent check is Minkowski space: the Fourier transform of $Ef$ is supported on the mass shell, so changing $f$ by $Ph$ multiplies $\widehat h$ by the mass-shell equation and leaves the solution class unchanged.

Failure boundary: keep the quotient

If one uses $C_0^\infty(M)$ itself as the phase space, then every nonzero $Pu$ lies in the radical because $E(Pu)=0$ and therefore $\sigma(Pu,h)=0$ for all $h$. The proposed form is degenerate, and the Weyl generator labelled by $Pu$ is an unphysical central direction unless the field equation is imposed separately. Worse, choosing a complement to $P C_0^\infty(M)$ is generally noncanonical and need not be preserved by spacetime embeddings. The quotient removes exactly these directions. It does not remove genuine zero modes arising from topology or boundary conditions; those require a model-specific analysis.

Exercises

Show directly that $\sigma_M([f],[h])$ is independent of representatives.

Solution

Replace $f$ by $f+Pu$. Formal self-adjointness and $PEh=0$ give $\int_M(Pu)Eh=\int_MuP(Eh)=0$. Replacing $h$ by $h+Pv$ gives $E(h+Pv)=Eh$ because $EPv=0$. Compact support removes boundary terms. Therefore the value depends only on $[f]$ and $[h]$.

References

• Bär, Christian, Nicolas Ginoux, and Frank Pfäffle. Wave Equations on Lorentzian Manifolds and Quantization. Zürich: European Mathematical Society, 2007. DOI.
• Brunetti, Romeo, Klaus Fredenhagen, and Rainer Verch. “The Generally Covariant Locality Principle—A New Paradigm for Local Quantum Physics.” Communications in Mathematical Physics 237 (2003): 31–68. DOI; Open PDF.
• Dimock, Jonathan. “Algebras of Local Observables on a Manifold.” Communications in Mathematical Physics 77 (1980): 219–228. DOI.