Unitarity, Reflection Positivity, and Causality Checks

Unitarity, Euclidean reflection positivity, and Lorentzian causal support are related but distinct. The reconstruction theorem of Osterwalder and Schrader 1973 starts from exact Euclidean correlators satisfying its axioms; it is not implied by a finite set of positive samples. Pole residues test a propagator approximation, a positive transfer matrix controls discrete evolution, and retarded support tests causality. No one finite-data diagnostic proves all four.

Required background. Perturbative and Higher-Derivative Gravity Interfaces supplies the pole problem. Reflection Positivity within Osterwalder–Schrader Reconstruction supplies the Euclidean theorem.

Helpful background. S-Matrix Unitarity, Emergence and Continuum-Limit Tests, and Validity, Unitarity, and Breakdown supply target tests.

Spectral and reflection-positivity tests

A positive-metric scalar two-point function has a Källén–Lehmann representation

$$G_E(p^2)=\int_0^\infty d\mu^2\, \frac{\rho(\mu^2)}{p^2+\mu^2},\qquad \rho\ge0.$$

A negative pole residue contradicts this form if the pole is physical. Reflection positivity requires, for every test function supported at positive Euclidean time,

$$\int dx\,dy\,f^*(\theta x)G_E(x,y)f(y)\ge0,$$

where $\theta$ reflects time. On discrete data this becomes positivity of matrices $G_E(\theta x_i,x_j)$ over increasing test sets.

Application: two inference levels

For

$$G(p^2)=\frac1{p^2}-\frac1{p^2+M^2},$$

the massive negative residue fails spectral positivity if both poles are retained as states. If it arose from a derivative expansion valid only below $M$, the appropriate conclusion is that the truncation cannot decide ultraviolet unitarity.

For a triangulated Euclidean correlator, construct reflection matrices at several volumes and lattice spacings. Positive eigenvalues within errors are necessary finite tests, not a proof for all test functions. CDT additionally admits a transfer-matrix construction under its gluing conditions; test positivity and continuum scaling of its spectrum.

Causal response

After a declared Lorentzian continuation, compute

$$G_R(x,y)=-i\Theta(t_x-t_y)\langle[O(x),O(y)]\rangle.$$

It must vanish outside the physical causal domain, modulo gauge dressing and nonlocal effective resolution. A diffusion kernel or Euclidean spectral dimension does not test this support.

Adversarial controls

Enlarge the operator basis, test multiparticle cuts and the optical theorem, vary gauge and regulator, and search for negative-norm constraint modes. A finite reflection-positive correlator can still flow to a nonunitary continuum, while a truncated violation can disappear when omitted nonlocal structure is restored. State the inference level.

Scope

Current programs possess partial checks—EFT order reduction, CDT transfer matrices, Euclidean correlators, or Lorentzian causal operators—but no single universal proof of four-dimensional quantum-gravity unitarity. The evidence summary follows on Safety, Causal, and Discrete Programs: Evidence, Obstructions, and Status.

The chapter overview contains the structure diagram and validity and failure diagram. They are embedded there once so that their shared chapter-level context is not repeated on every article.

For the chapter-wide comparison of assumptions, counterevidence, falsifiers, and claim ceilings, see the claim-domain table.

References

• Osterwalder, K., and R. Schrader. “Axioms for Euclidean Green’s Functions.” Communications in Mathematical Physics 31 (1973): 83–112. DOI.
• Stelle, K. S. “Renormalization of Higher-Derivative Quantum Gravity.” Physical Review D 16 (1977): 953–969. DOI.