Why Hadamard states?
Abstract
In QFT on curved spacetime, the Hadamard condition is often presented as a necessary condition on 'physically reasonable' states of the quantum field, and plays a central role in many theoretical and foundational applications - ranging from proofs of the renormalizability of Wick polynomials to derivations of the Hawking temperature. Yet despite this, the philosophical and foundational underpinnings of the Hadamard condition remain murky. I critically discuss existing motivations for the Hadamard condition in the literature, before arguing in favour of an alternative motivation for the Hadamard condition, according to which it is best understood as a necessary and sufficient condition for the existence of a well-defined operator product of a certain form on a sufficiently large space of observables of the quantum field, satisfying a variety of further conditions. This clarifies the role and status of the Hadamard condition, including its relationship to the equivalence principle, to well-definedness of physical quantities such as Wick polynomials and the expectation value of the stress-energy operator, and the sense in which Hadamard states are 'vacuum-like'.
Related
- Hadamard states for linearized gravity on asymptotically flat spacetimes
- The Hadamard parametrix on globally hyperbolic spacetimes with Robin boundary conditions: Fundamental solutions and Hadamard states
- Hadamard states for spacetimes with timelike boundaries
- A Hadamard Cube of Order 22
- Isoperimetric and total curvature inequalities in Cartan-Hadamard manifolds with nullity
BibTeX
@misc{march2026hadamard,
title = {{Why Hadamard states?}},
author = {Eleanor March},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2606.22767},
eprint = {2606.22767},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.22767},
}
RIS
TY - UNPB
TI - Why Hadamard states?
AU - Eleanor March
PY - 2026
DA - 2026/10/19
PB - arXiv
DO - 10.48550/arxiv.2606.22767
UR - https://arxiv.org/abs/2606.22767
AB - In QFT on curved spacetime, the Hadamard condition is often presented as a necessary condition on 'physically reasonable' states of the quantum field, and plays a central role in many theoretical and foundational applications - ranging from proofs of the renormalizability of Wick polynomials to derivations of the Hawking temperature. Yet despite this, the philosophical and foundational underpinnings of the Hadamard condition remain murky. I critically discuss existing motivations for the Hadamard condition in the literature, before arguing in favour of an alternative motivation for the Hadamard condition, according to which it is best understood as a necessary and sufficient condition for the existence of a well-defined operator product of a certain form on a sufficiently large space of observables of the quantum field, satisfying a variety of further conditions. This clarifies the role and status of the Hadamard condition, including its relationship to the equivalence principle, to well-definedness of physical quantities such as Wick polynomials and the expectation value of the stress-energy operator, and the sense in which Hadamard states are 'vacuum-like'.
ER -