Manuscript I-B-VP-0@1 · I-B-VP-0-v1.tex · section · 8a9d0682

Massless infrared finiteness

Prescribed geometry, bundle, and spectral state › Massless infrared finiteness · lines 358–377

Indeed, on spacetime tests its positive form is the Gram form of

f ⟼ \int_{R} e^{i t B_{β}} (2 B_{β})^{- 1/2} f (t) d t,

which is finite by Lemma lem:IR. The standard Fock construction has nonnegative one-particle Hamiltonian $B_{β}$ Wald1994. With the conventions of I-B-G,

[Φ (X), Φ^{†} (X^{'})] = - i E_{β} (X, X^{'}), E_{β} (t) = \frac{sin ( t B _{β} )}{B _{β}} .

This is the state of I-B-VP-v2 :state, not a claim of uniqueness among all algebraic ground states CheungIBVP.

Displayed formulas in this passage

unlabeled · line 361
$f ⟼ \int_{R} e^{i t B_{β}} (2 B_{β})^{- 1/2} f (t) d t,$
unlabeled · line 369
$[Φ (X), Φ^{†} (X^{'})] = - i E_{β} (X, X^{'}), E_{β} (t) = \frac{sin ( t B _{β} )}{B _{β}} .$

Nearby in this section

lemma: Lemma: Heat kernel and compact-test covariance
definition: Definition: Distinguished charged Friedrichs ground state