CCR and CAR Algebras

The classical equations determine more of the quantum theory than a mode expansion suggests. For a free field on a globally hyperbolic spacetime, the causal propagator supplies the invariant bilinear form in the canonical relations. Quantization first produces an abstract algebra; a state and its Hilbert-space representation are additional choices. This order keeps locality visible and does not presuppose a vacuum.

Required background. Flat-Spacetime Quantum Field Theory supplies smeared fields and canonical brackets, while Variational and Symplectic Geometry defines the covariant symplectic form and the reduction by degenerate directions used below.

Helpful background. Partial Differential Equations and Green Functions provides the advanced and retarded solution concept, and Causality and Global Structure establishes the causal support and domain-of-dependence language used to prove locality.

The scalar solution space from the causal propagator

Let $M$ be oriented, time-oriented, globally hyperbolic, and without timelike boundary. For a real scalar field,

$$P=\Box-m^2-\xi R$$

is formally self-adjoint on real compactly supported test functions. We fix

$$E=G_{\mathrm{ret}}-G_{\mathrm{adv}}, \qquad \operatorname{supp}(G_{\mathrm{ret}}f)\subset J^+(\operatorname{supp}f), \qquad \operatorname{supp}(G_{\mathrm{adv}}f)\subset J^-(\operatorname{supp}f).$$

Here is the short Green-function bridge needed for the construction. Each $G_{\mathrm{ret/adv}}$ solves $PGf=f=GPf$ on test functions, with the displayed future or past support. Their difference therefore solves the homogeneous equation. Formal self-adjointness gives $E^*=-E$, and exactness on a globally hyperbolic spacetime gives

$$\ker E=P C_0^\infty(M,\mathbb R).$$

Consequently

$$\mathcal V= C_0^\infty(M,\mathbb R)/P C_0^\infty(M,\mathbb R), \qquad \sigma([f],[g])=E(f,g):=\int_M f(Eg)\,d\mathrm{vol}_g$$

is a real symplectic space. Changing $f$ by $Ph$ changes neither $Ef$ nor $E(f,g)$. Skew-adjointness makes $E(f,g)=-E(g,f)$, and exactness removes its degeneracy. Equivalently, $E$ identifies test-function classes with spacelike-compact solutions and $E(f,g)$ equals the conserved Cauchy-surface symplectic form. This quotient and its functorial quantization are developed in Bär, Ginoux, and Pfäffle 2007, §§4.2–4.7, pp. 116–153.

If a timelike boundary is present, or global hyperbolicity fails, these statements are not automatic: boundary conditions and a choice of Green operators enter before the algebra can be defined. Quotienting by $PC_0^\infty$ alone is then not sufficient.

CCR as polynomial and Weyl relations

The unital polynomial field algebra is generated by symbols $\Phi(f)$ subject to complex linearity after complexification and

$$\Phi(Pf)=0, \qquad \Phi(f)^*=\Phi(\overline f), \qquad [\Phi(f),\Phi(g)]=iE(f,g)\mathbf 1.$$

For real $f$, $\Phi(f)$ is formally Hermitian. The last equation is the covariant canonical commutation relation (CCR). It answers the central question directly: the same causal propagator that evolves a compact source fixes the field commutator. If the supports of $f$ and $g$ are causally disjoint, the support properties of $E$ give $E(f,g)=0$, hence the smeared fields commute. This is microcausality, not a claim that timelike-separated commutators must be nonzero.

Because polynomial fields are unbounded in Hilbert-space representations, it is often cleaner to use the Weyl algebra generated by unitaries $W(v)$, $v\in\mathcal V$:

$$W(0)=\mathbf 1, \qquad W(v)^*=W(-v), \qquad W(v)W(w)=e^{-i\sigma(v,w)/2}W(v+w).$$

The minus sign in the phase is forced by our sign for $E$. Indeed, in any regular representation set $W(tv)=e^{it\Phi(v)}$. Since the commutator is central, the Baker–Campbell–Hausdorff formula gives

$$e^{i\Phi(v)}e^{i\Phi(w)} =e^{i\Phi(v+w)+[i\Phi(v),i\Phi(w)]/2} =e^{-i\sigma(v,w)/2}e^{i\Phi(v+w)}.$$

Differentiating the Weyl relation at the identity recovers $[\Phi(v),\Phi(w)]=i\sigma(v,w)\mathbf1$. An abstract Weyl algebra need not assume continuity in $v$; a regular representation is one in which $t\mapsto\pi(W(tv))$ is strongly continuous, so Stone’s theorem supplies the field generators.

CAR from a positive Hermitian space

Fermions do not arise by replacing a commutator with an anticommutator on the same symplectic space. A Dirac-type equation and its causal propagator first produce a complex solution space $\mathcal K$ with a positive Hermitian form $q$. The sign in the conserved Dirac current is fixed so that $q(u,u)\geq0$ on a future-oriented Cauchy surface, and any null directions are removed. Positivity is essential: an indefinite form does not define the CAR $C^*$-algebra.

In the ordinary formulation, annihilation and creation symbols obey

$$\{a(u),a(v)\}=0, \qquad \{a(u),a^*(v)\}=q(u,v)\mathbf1,$$

with $q$ antilinear in its first entry, $a(u)$ antilinear in $u$, and $a^*(u)$ linear in $u$. For example, replacing $u$ by $iu$ multiplies both sides of the second relation by $-i$; this is a useful check before writing fermionic two-point functions. The self-dual formulation packages particles and antiparticles into a Hilbert space with an antiunitary involution $\Gamma$. Its linear generators satisfy

$$B(u)^*=B(\Gamma u), \qquad \{B(u)^*,B(v)\}=q(u,v)\mathbf1.$$

The Dirac equation is again imposed by passing to the appropriate test-section quotient. Causal support yields graded locality: odd fields anticommute at spacelike separation, while even observables commute. The causal propagator determines the Hermitian form only together with the field’s adjoint and current conventions; unlike the scalar case, writing a bare real antisymmetric pairing would lose the positivity needed by CAR.

Algebra, state, and representation are different data

The CCR or CAR relations specify an algebra $\mathcal A$. A state is a normalized positive linear functional $\omega:\mathcal A\to\mathbb C$. The GNS construction then produces a representation $\pi_\omega$ on a Hilbert space $\mathcal H_\omega$ and a cyclic vector $\Omega_\omega$ such that

$$\omega(A)=\langle\Omega_\omega,\pi_\omega(A)\Omega_\omega\rangle.$$

None of $\mathcal H_\omega$, $\Omega_\omega$, or a particle number operator belongs to the abstract algebra by itself. In finitely many degrees of freedom, regular irreducible CCR representations are unitarily equivalent under the Stone–von Neumann hypotheses. A field has infinitely many degrees of freedom, so those hypotheses no longer provide one universal Hilbert space. This is why defining the algebra before choosing a state is more than formal tidiness. Wald’s separation of algebraic relations, states, and Hilbert-space realizations is especially useful here Wald 1994, Chapter 3, pp. 23–52.

For CAR, the generators are bounded in the $C^*$-norm; for bosonic polynomial CCR fields, they are generally unbounded. The Weyl generators remain bounded. These analytic differences do not erase the shared logical order: classical form, abstract algebra, state, then representation.

Sign, star, and one-mode checks

Take one classical oscillator with $v=(q,p)$ and

$$\sigma(v,w)=qp'-pq'.$$

Represent $\Phi(v)=qQ+pP$ with $[Q,P]=i\mathbf1$. Directly,

$$[\Phi(v),\Phi(w)] =q p'[Q,P]+p q'[P,Q] =i(qp'-pq')\mathbf1 =i\sigma(v,w)\mathbf1.$$

Thus the commutator and Weyl phase have the advertised signs. Star compatibility follows because $E$ is real: taking the adjoint of the CCR reverses the commutator and complex-conjugates $i$, producing the same relation with $f$ and $g$ exchanged. For one fermionic mode, $\{a,a^*\}=1$ implies $\lVert a^*\Omega\rVert^2=\langle\Omega,aa^*\Omega\rangle=1$ when $a\Omega=0$, an immediate positivity and normalization check. Neither calculation selects the representation of the full field.

Exercise: recover the Weyl cocycle

Assume two self-adjoint operators obey $[Q,P]=i\mathbf1$. Let $W(q,p)=e^{i(qQ+pP)}$. Determine the phase in $W(q,p)W(q',p')$ and explain what would change if the causal propagator were defined with the opposite sign.

Solution

Set $A=i(qQ+pP)$ and $B=i(q'Q+p'P)$. Their commutator is central and equals $[A,B]=-i(qp'-pq')\mathbf1$. The Baker–Campbell–Hausdorff series therefore terminates after $[A,B]/2$, giving $W(q,p)W(q',p')=e^{-i(qp'-pq')/2}W(q+q',p+p')$. The antisymmetric combination is precisely the oscillator symplectic form, so this calculation independently checks both its ordering and its normalization. Replacing $E$ by $-E$ would reverse both the field commutator and this phase. Reversing only one of them would make differentiation of the Weyl relation disagree with the canonical commutator, so the resulting definitions would be inconsistent.

References

• Bär, Christian, Nicolas Ginoux, and Frank Pfäffle. Wave Equations on Lorentzian Manifolds and Quantization. ESI Lectures in Mathematics and Physics. EMS Press, 2007. DOI 10.4171/037.
• Wald, Robert M. Quantum Field Theory in Curved Spacetime and Black Hole Thermodynamics. Chicago Lectures in Physics. University of Chicago Press, 1994. Publisher record.