Conjugation and Reflection Positivity

Hermitian conjugation in radial quantization reflects cylinder time. On the plane that reflection is inversion through the quantization sphere, followed by complex conjugation and the appropriate spin transformation. When the Euclidean theory satisfies reflection positivity, this inside–outside pairing is a positive Hilbert-space inner product; it yields $D^\dagger=D$ and $P_\mu^\dagger=K_\mu$. Without reflection positivity the algebraic radial pairing can still be defined, but its norms and squared OPE coefficients need not be nonnegative.

Required background. The State–Operator Correspondence supplies the ket prepared inside a sphere and the exterior path integral that defines a bra. Reflection Positivity within Osterwalder–Schrader Reconstruction owns the general reconstruction theorem and its hypotheses. Helpful background. Bilinear and Hermitian Forms, Adjoints, and Isometries reviews antilinear adjoints and positive forms.

Radial inversion defines the adjoint

Use a unit quantization sphere for the moment. Its inside and outside are exchanged by

$$\theta x^\mu=\frac{x^\mu}{x^2}.$$

On the cylinder, where $x=e^\tau n$, this is simply $\tau\mapsto-\tau$. For a Hermitian scalar primary of dimension $\Delta$, radial conjugation is

$$\boxed{ \mathcal O(x)^\dagger =(x^2)^{-\Delta} \mathcal O\!\left(\frac{x}{x^2}\right) }.$$

The factor is forced by the primary Weyl transformation. Applying the operation twice returns the original insertion, which is the first round-trip check:

$$\left(\mathcal O(x)^\dagger\right)^\dagger=\mathcal O(x).$$

For a primary in a complex internal representation, the operator on the right lies in the conjugate representation. For a spinning primary, every vector index is acted on by the inversion matrix

$$I_\mu{}^\nu(x) =\delta_\mu{}^\nu -2\frac{x_\mu x^\nu}{x^2},$$

and spinor indices require a lift of this orthogonal transformation to the chosen Pin or Spin structure. Fermionic products are conjugated in reverse order with their graded signs. These data cannot be recovered from the scalar prefactor alone.

The scalar bra is consequently represented by

$$\langle\mathcal O| =\langle0|\mathcal O(\infty), \qquad \mathcal O(\infty) =\lim_{|x|\to\infty}(x^2)^\Delta\mathcal O(x).$$

Rychkov 2017, §§3.1.6–3.1.7, pp. 43–45, Open PDF derives the inversion rule and uses the two-point function as its normalization check.

Reflection positivity on a sphere

Let $F$ be a finite linear combination of products of local insertions supported strictly inside the unit sphere. Define $\Theta F$ by complex conjugating coefficients, applying radial inversion and the operator reflection to each insertion, and reversing operator order. Spherical reflection positivity is the condition

$$\langle(\Theta F)F\rangle_E\geq0$$

for every such $F$, after quotienting null vectors and completing the resulting space. Through the plane–cylinder map this is ordinary reflection positivity across the cylinder slice $\tau=0$.

Several restrictions are essential:

• insertions must lie on the prescribed side of the reflection surface before the limiting construction;
• fields must transform under the full reflection, including spin, charge, and complex conjugation;
• gauge constraints and null states must be handled before interpreting the quotient as a physical Hilbert space; and
• the inequality is a property of the specified Euclidean theory and state, not a consequence of conformal covariance alone.

The Osterwalder–Schrader framework establishes how an appropriate Euclidean reflection-positive system reconstructs a positive Lorentzian Hilbert space Osterwalder and Schrader 1973, axioms E0–E4 and theorem, pp. 83–112. The CFT specialization and the conversion from planar to spherical reflection are explained in Simmons-Duffin 2017, §§7.1–7.2, pp. 31–36, Open PDF.

Norms of primaries and descendants

Normalize scalar primaries by

$$\langle\mathcal O_i(x)\mathcal O_j(0)\rangle =\frac{C_{ij}}{(x^2)^\Delta}$$

within a subspace of equal dimension and quantum numbers. Radial conjugation gives

$$\langle\mathcal O_i|\mathcal O_j\rangle=C_{ij}.$$

Reflection positivity therefore requires the Hermitian matrix $C$ to be positive semidefinite. After removing null combinations it may be made the identity by a basis change. In a nonunitary theory $C$ can be indefinite, and no basis change converts an indefinite form into a positive one.

Adopt the Euclidean algebra convention

$$[D,P_\mu]=P_\mu, \qquad P_\mu^\dagger=K_\mu, \qquad D^\dagger=D, \qquad M_{\mu\nu}^\dagger=-M_{\mu\nu}.$$

For a scalar primary $|\Delta\rangle$ with $K_\mu|\Delta\rangle=0$, the level-one Gram matrix is

$$\begin{aligned} \langle\Delta|K_\mu P_\nu|\Delta\rangle &=\langle\Delta|[K_\mu,P_\nu]|\Delta\rangle\\ &=2\Delta\,\delta_{\mu\nu} \langle\Delta|\Delta\rangle. \end{aligned}$$

Hence a nontrivial scalar primary in a positive theory must have $\Delta>0$; $\Delta=0$ is the identity multiplet after the usual uniqueness assumptions on the vacuum. At level two, the scalar descendant obeys

$$\|P^2|\Delta\rangle\|^2 =8d\,\Delta \left(\Delta-\frac{d-2}{2}\right) \langle\Delta|\Delta\rangle.$$

Its positivity gives the scalar unitarity bound

$$\Delta\geq\frac{d-2}{2}$$

for a nonidentity scalar in $d>2$. At saturation $P^2|\Delta\rangle$ is null, the representation shortens, and the corresponding operator obeys a free wave equation in correlation functions away from contact terms. The full spin-dependent bounds and their exceptions belong to Unitarity Bounds and Null States; Descendant States and Gram Matrices derives the level-by-level method.

Positivity of OPE data

Choose Hermitian operators and an orthonormal basis of exchanged primaries in a reflection-positive sector. Then the coefficient multiplying a conformal block in an identical-scalar four-point function is

$$\lambda_{\phi\phi\mathcal O} \lambda_{\phi\phi\mathcal O}^* =|\lambda_{\phi\phi\mathcal O}|^2\geq0.$$

This familiar nonnegativity has conditions: the external operators are related by the radial adjoint, the intermediate states have positive norm, and phases and degeneracies are treated consistently. For mixed correlators, the exchanged coefficients form positive-semidefinite matrices rather than independent nonnegative numbers. For charged, non-Hermitian, logarithmic, or nonunitary operators, the correct pairing can be complex, indefinite, or nondiagonal; writing an unexplained square is then wrong.

The shared geometry figure connects the reflection surface to its two plane images. Inspect in particular that the bra lies outside the sphere, that inversion reverses radial time, and that positivity is shown as an additional condition rather than as a geometric identity.

Radial conjugation and positivity. Inversion plus complex conjugation defines the adjoint; reflection positivity makes the resulting inside–outside pairing nonnegative. The figure is schematic and uses a scalar label—spin and internal indices require their own reflection matrices.

Failure tests

Proposed inference	Required check	What failure means
$C_\mathcal O>0$ from conformal covariance	Evaluate the reflected two-point pairing	Covariance fixes the form, not the sign
$P_\mu^\dagger=K_\mu$	Include inversion, Weyl weight, and generator convention	A missing factor or alternate $i$ convention changes the displayed adjoint relation
$\lambda^2\geq0$	Verify Hermiticity, reflection positivity, and basis phases	The coefficient may be complex or part of a matrix pairing
Null descendant implies an operator equation	Quotient the null state and control contact terms	A zero determinant alone can signal reducibility without the claimed local equation in every distributional context
Euclidean positivity implies every Lorentzian ordering is positive	Carry out the specified analytic continuation	Reflection positivity constrains a particular reflected configuration, not arbitrary Wightman orderings

The supported conclusion is exact but conditional: spherical reflection positivity turns radial conjugation into a positive Hilbert-space adjoint, making $D$ self-adjoint and descendant Gram matrices positive semidefinite. If the condition is absent, conformal representation theory survives but bootstrap positivity must be withdrawn or replaced.

References

• Osterwalder, Konrad, and Robert Schrader. “Axioms for Euclidean Green’s Functions.” Communications in Mathematical Physics 31 (1973): 83–112. doi:10.1007/BF01645738.
• Rychkov, Slava. EPFL Lectures on Conformal Field Theory in $D\geq3$ Dimensions. SpringerBriefs in Physics. Cham: Springer, 2017. doi:10.1007/978-3-319-43626-5. Open PDF.
• Simmons-Duffin, David. “The Conformal Bootstrap.” In New Frontiers in Fields and Strings, 1–74. Singapore: World Scientific, 2017. doi:10.1142/9789813149441_0001. Open PDF.