Reflection Positivity and Hilbert-Space Reconstruction

Reflection positivity turns positive-time Euclidean observables into vectors, but only after zero-norm observables are identified and the resulting normed space is completed. Positive Euclidean-time translation then becomes a contraction semigroup whose self-adjoint generator is nonnegative. This is the precise origin of the reconstructed Hilbert space and Hamiltonian; it is not obtained by replacing Euclidean time with $it$ in a formula.

Required background. Osterwalder–Schrader axioms and reflection positivity supplies the reflected form. Domains, signatures, supports, and regularity supplies the support conditions used by time translations. Banach and Hilbert spaces, completion, and Riesz representation supplies quotient completion and self-adjoint semigroups.

Helpful background. The Wightman reconstruction theorem provides the Lorentzian analogue. Reflection positivity and OS reconstruction gives a shorter physical introduction.

From the reflected form to a Hilbert space

Let $\mathcal D_+$ be a dense positive-time test algebra and let

$$(F,G)_{\mathrm{OS}}=\langle \Theta F\,G\rangle_E$$

be its reflection-positive sesquilinear form. Positivity implies the Cauchy–Schwarz inequality even though the form may be degenerate. Its null space

$$\mathcal N=\{F\in\mathcal D_+:(F,F)_{\mathrm{OS}}=0\}$$

is a linear subspace, and Cauchy–Schwarz gives $(F,G)_{\mathrm{OS}}=0$ for every $G$ whenever $F\in\mathcal N$. Therefore

$$\langle[F],[G]\rangle_{\mathcal H_0} =(F,G)_{\mathrm{OS}}, \qquad \mathcal H_0=\mathcal D_+/\mathcal N,$$

is a genuine inner product. The physical Hilbert space is

$$\mathcal H=\overline{\mathcal H_0}.$$

The constant functional represents the vacuum candidate $\Omega=[1]$, normalized by $S_0=1$. This construction and the role of the null ideal are developed in Osterwalder and Schrader 1973, §4.1, pp. 90–94.

The quotient is not optional. If $F-F'\in\mathcal N$, then $F$ and $F'$ have the same scalar products with every physical vector. Any reconstructed operator must preserve this equivalence relation on its domain. Without the quotient, “zero norm but nonzero vector” remains in the space, the norm is degenerate, and operator definitions depend on the chosen representative.

Euclidean translations and the positive generator

For $a\geq0$, let $T_E(a)$ translate every field argument forward in Euclidean time:

$$(T_E(a)F)(\Phi)=F(\Phi_a), \qquad \Phi_a(f)=\Phi(f_{-a}).$$

Positive support is preserved. Euclidean covariance and reflection positivity show, on the appropriate dense domains, that $T_E(a)$ descends through $\mathcal N$, is symmetric with respect to the OS inner product, and forms a strongly continuous contraction semigroup. Hence the spectral theorem gives

$$T(a)=e^{-aH}, \qquad H=H^*,\qquad H\geq0.$$

The inequality $H\geq0$ is the reconstructed energy condition. It follows from the contraction semigroup; it is not an independent sign convention. Spatial translations preserve the time-zero plane and reconstruct as a strongly continuous unitary group $e^{-i\mathbf a\cdot\mathbf P}$. Euclidean rotations mixing time and space require more work and ultimately supply the Lorentzian boost representation after analytic continuation. The semigroup and spectral steps appear in Osterwalder and Schrader 1973, §§4.1–4.3, pp. 90–96.

Time-zero observables deserve a qualification. If smeared fields admit controlled limits as their time support approaches $\tau=0$, those limits define a time-zero algebra, and vectors obtained by applying translated time-zero observables to $\Omega$ can be dense. Distributional fields need not possess a sharp-time restriction automatically; the required trace regularity or a replacement by small positive-time smearing must be proved.

First QFT application: the free scalar one-particle space

For the massive Gaussian covariance, a positive-time one-field vector $\Phi(f)$ has the norm below. This is the explicit free-field realization of the quotient described conceptually in reflection positivity and OS reconstruction.

$$\lVert[\Phi(f)]\rVert^2 =\int\frac{d^{d-1}\mathbf p}{(2\pi)^{d-1}} \frac{1}{2\omega_{\mathbf p}} \left\lvert \int_0^\infty d\tau\, e^{-\omega_{\mathbf p}\tau}\widetilde f(\tau,\mathbf p) \right\rvert^2,$$

where $\omega_{\mathbf p}=\sqrt{\mathbf p^2+m^2}$. Thus the equivalence class depends only on the on-shell Laplace transform

$$Kf(\mathbf p)=\int_0^\infty e^{-\omega_{\mathbf p}\tau}\widetilde f(\tau,\mathbf p)\,d\tau.$$

After quotient and completion, the one-particle space is naturally

$$L^2\!\left(\mathbb R^{d-1}, \frac{d^{d-1}\mathbf p}{(2\pi)^{d-1}2\omega_{\mathbf p}}\right).$$

Forward Euclidean translation sends $Kf$ to $e^{-a\omega_{\mathbf p}}Kf$. Therefore

$$(H\psi)(\mathbf p)=\omega_{\mathbf p}\psi(\mathbf p), \qquad H=\sqrt{\mathbf P^2+m^2}$$

on its standard multiplication-operator domain. The full Gaussian Hilbert space is the symmetric Fock space over this one-particle space, and $H$ is its second quantization. This calculation checks the sign, spectrum, and relativistic dispersion relation independently of an abstract reconstruction theorem.

Where the construction can stop

The Hilbert-space step proves less than the full OS theorem. From reflection positivity one obtains a positive inner product and time semigroup. Local Lorentzian fields, Poincaré covariance, and Wightman distributions require the rest of the hierarchy, its Euclidean covariance and symmetry, suitable uniform growth, and controlled analytic continuation.

The adversarial failure is immediate in the free example. Choose a nonzero $f$ whose on-shell Laplace transform $Kf$ vanishes almost everywhere. Then $\Phi(f)$ is nonzero as a test-algebra symbol but lies in $\mathcal N$. Retaining it as a physical vector produces a degenerate norm; declaring two operator actions on it independently can make the same equivalence class acquire two answers. Quotienting by $\mathcal N$ removes precisely this ambiguity.

Independent checks

• Representative independence: verify that each proposed operator maps $\mathcal N$ into $\mathcal N$ on its declared domain.
• Semigroup direction: positive Euclidean time gives $e^{-aH}$, not $e^{+aH}$; the former is contractive when $H\geq0$.
• Vacuum: $T(a)\Omega=\Omega$, hence $H\Omega=0$ when $\Omega$ lies in the generator domain.
• Free spectrum: $\omega_{\mathbf p}\geq m$ on the one-particle subspace, while the Fock vacuum remains at zero energy.

Exercise

Prove that $\mathcal N$ is orthogonal to all of $\mathcal D_+$ and hence that the quotient inner product is well defined.

Solution

For a positive semidefinite sesquilinear form, positivity of $(F+\lambda G,F+\lambda G)_{\mathrm{OS}}$ for every $\lambda\in\mathbb C$ implies Cauchy–Schwarz:

$$\lvert(F,G)_{\mathrm{OS}}\rvert^2 \leq(F,F)_{\mathrm{OS}}(G,G)_{\mathrm{OS}}.$$

If $F\in\mathcal N$, the right-hand side vanishes, so $(F,G)_{\mathrm{OS}}=0$ for all $G$. Replacing either representative by a null vector therefore leaves the quotient inner product unchanged.

References

• Osterwalder, Konrad, and Robert Schrader. “Axioms for Euclidean Green’s Functions.” Communications in Mathematical Physics 31 (1973): 83–112. doi:10.1007/BF01645738. Open PDF.
• Osterwalder, Konrad, and Robert Schrader. “Axioms for Euclidean Green’s Functions II.” Communications in Mathematical Physics 42 (1975): 281–305. doi:10.1007/BF01608978. Open PDF.