Microlocal QFT and Renormalized Local Fields

Wavefront sets record both where a field distribution is singular and in which cotangent directions. This chapter uses that information to decide which local operations exist and what freedom remains after renormalization.

Enter this chapter

Begin with wavefront calculus, then follow causal propagation to the Hadamard condition and composite fields. Helpful background. Singular support and wavefront sets supplies the distribution theory; Hadamard admissibility and the two-point wavefront criterion gives the physical state test; Wick polynomials and Hadamard point splitting gives the first local-observable application.

Topics

1. Microlocal Calculus for Quantum Fields — Introduces the directional singularity data used in field theory.
2. Wavefront-Set Products, Pullbacks, and Pushforwards — States the transversality criteria for products and restrictions.
3. Green-Hyperbolic Operators and Causal Propagators — Constructs causal Green operators and solution spaces.
4. Propagation of Singularities for Hyperbolic Fields — Propagates singular directions along characteristic flow.
5. Higher-Point Microlocal Spectrum Conditions — Organizes allowed singular configurations of higher correlators.
6. Hadamard States and the Wavefront-Set Characterization — Characterizes the physically admissible free-field two-point singularity.
7. Wick Polynomials under Microlocal Conditions — Defines Wick products after controlled singularity subtraction.
8. Local Covariant Wick Powers and Operator Products — Imposes covariance and scaling across backgrounds.
9. Time-Ordered Products and the Renormalized Stress Tensor — Combines causal extension with the construction of stress–energy observables.
10. Microlocal Renormalization Ambiguities and Their Classification — Classifies the finite local terms left undetermined by the axioms.

How the ideas fit together

Propagation controls how singularities move; the Hadamard condition selects their positive-frequency orientation Radzikowski 1996, Theorem 5.1. Subtracting this universal singularity permits Wick products under the microlocal hypotheses Brunetti, Fredenhagen, and Köhler 1996, § 5 (PDF). Covariance and scaling then restrict the remaining local freedom Hollands and Wald 2001, Theorems 5.1–5.2 (PDF). These analytic conditions do not supply positivity or choose a preferred state.

Hypotheses and licensed conclusions

Microlocal objects, essential hypotheses, licensed conclusions, and decisive failure tests.

Object and domain	Essential hypotheses	Licensed conclusion	Excluded converse or adversarial check
Distribution on a smooth manifold	Localized Fourier estimates in conic cotangent neighborhoods; nonzero covectors	A closed conic wavefront set with coordinate-invariant position and direction data	Equal singular supports do not distinguish a positive-frequency kernel from its transpose
Products, restrictions, and kernel compositions	No opposing covector sum for products; conormal transversality for pullback; proper support for pushforward	A canonical distributional operation with a calculable wavefront-set bound	A null worldline can have a conormal Hadamard covector; local cone compatibility does not imply proper fiber support
Green-hyperbolic solution and propagation	Globally hyperbolic spacetime; two-sided advanced and retarded inverses; real principal type in the source-free region	An exact source–solution sequence and transport of singularities along null Hamilton orbits	A timelike boundary without boundary conditions destroys uniqueness; a caustic does not terminate a wavefront
Higher-point distributions	Smooth-path graph cones with future-causal parallel labels; compatible products; for the simple derivation, quasifree Wick factorization and a Hadamard two-point function; causal- and null-path cones are stronger variants	The microlocal spectrum bound for the full quasifree hierarchy	Reversing one graph edge violates orientation; two-point data do not determine nonquasifree truncated functions
Free-field Hadamard two-point function	Smooth, oriented and time-oriented, four-dimensional boundaryless globally hyperbolic spacetime; real scalar normally hyperbolic operator; exact bisolution and fixed commutator; either local Hadamard form or the one-sided null cone; positivity only for state status	Equivalence of the Hadamard expansion and the future-directed null wavefront relation	Dynamics, the commutator, and positivity still admit a symmetric wrong-frequency admixture; the cone alone does not construct a state
Reference-state Wick powers	Quasifree Hadamard normal ordering; admissible diagonal kernels; common invariant dense domain	Operator-valued Wick distributions, an extended Wick algebra, and canonical smooth state-change isomorphisms	A positive wrong-frequency admixture retains a quadratic coincidence divergence
Local covariant Wick powers	Locally constructed Hadamard subtraction; naturality, scaling, and smooth or analytic background dependence	Natural local fields whose prescription changes are finite curvature-polynomial mixings	Reference-state normal ordering is not generally local covariant; a coordinate-dependent finite term violates naturality
Time-ordered products and stress tensor	Causal factorization; scaling-degree extension; microlocal regularity; local covariance; field identities and conservation normalization	Formal local time-ordering and a conserved renormalized stress tensor, unique up to finite local covariant terms	Flat-space subtraction misses curvature singularities; nonlocal inverse operators are not admissible counterterms

Structured table data (JSON).

Review the chapter

If two Hadamard two-point functions describe the same free field, why is their difference useful for comparing Wick prescriptions?

Solution

Their universal singular parts agree, so the difference is smooth. Its diagonal restriction and derivatives can therefore shift the finite normal-ordering terms without introducing a new ultraviolet singularity. Further covariance and scaling requirements constrain which shifts define admissible prescriptions across backgrounds.

Where to go next

Continue to curved-spacetime stress tensors and energy inequalities to apply these constructions to observable and geometric bounds.

References

• Brunetti, Romeo, Klaus Fredenhagen, and Michael Köhler. “The Microlocal Spectrum Condition and Wick Polynomials of Free Fields on Curved Spacetimes.” Communications in Mathematical Physics 180 (1996): 633–652. DOI. Open PDF.
• Hollands, Stefan, and Robert M. Wald. “Local Wick Polynomials and Time Ordered Products of Quantum Fields in Curved Spacetime.” Communications in Mathematical Physics 223 (2001): 289–326. DOI. Open PDF.
• Radzikowski, Marek J. “Micro-Local Approach to the Hadamard Condition in Quantum Field Theory on Curved Space-Time.” Communications in Mathematical Physics 179 (1996): 529–553. DOI.