Manuscript I-B-VP@3 · I-B-VP-v3.tex · proposition · 93c04eb4

Proposition: Friedrichs state and symmetries

Friedrichs heat kernel and the massless state · lines 629–679 · proof included or linked

I-B-VP@3#prop:state

The Friedrichs prescription is a continuous positive covariance on compact smooth spacetime tests and defines a spectral ground state. With

E_{β} (t) = \frac{sin ( t B _{F} )}{B _{F}},

the real and charged commutators are respectively

[ϕ (X), ϕ (X^{'})] = - i E_{0} (X, X^{'}), [Φ (X), Φ^{†} (X^{'})] = - i E_{β} (X, X^{'}) .

The state is invariant under $t, z$ translations, rotations, and proper time-oriented $(t, z)$ boosts. Its covariance has scaling degree two:

W_{F} (λ t, λ r, φ, λ z; λ t^{'}, λ r^{'}, φ^{'}, λ z^{'}) = λ^{- 2} W_{F} (X, X^{'}) .

Proof. Lemma lem:infrared bounds the one-particle vector on fixed compact supports. The Gram form proves positivity and continuity; the spectral Fock construction gives a nonnegative Hamiltonian Wald1994. The difference of time orientations is

\frac{e ^{- i t B} - e ^{i t B}}{2 B} = - i \frac{sin ( t B )}{B} .

Differentiating in the second time variable at equal times gives the canonical $+ i id$ sign.

Translations and rotations commute with the operator. For boosts use

W_{F} = \frac{1}{2 π α} n \sum e^{i q_{n} (φ - φ^{'})} \int_{0}^{\infty} k d k J_{ν_{n}} (k r) J_{ν_{n}} (k r^{'}) \times \int_{R} \frac{d p}{2 π} \frac{e ^{i p Δ z - i k^{2} + p^{2} Δ t}}{2 k ^{2} + p ^{2}} .

The inner mass-shell measure is the boost-invariant $1 + 1$ -dimensional positive-frequency measure of mass $k$ . The infrared estimate justifies compact testing. Rescaling $k, p$ proves the dilation identity.

Displayed formulas in this passage

unlabeled · line 667
$W_{F} = \frac{1}{2 π α} n \sum e^{i q_{n} (φ - φ^{'})} \int_{0}^{\infty} k d k J_{ν_{n}} (k r) J_{ν_{n}} (k r^{'}) \times \int_{R} \frac{d p}{2 π} \frac{e ^{i p Δ z - i k^{2} + p^{2} Δ t}}{2 k ^{2} + p ^{2}} .$
unlabeled · line 634
$E_{β} (t) = \frac{sin ( t B _{F} )}{B _{F}},$
unlabeled · line 638
$[ϕ (X), ϕ (X^{'})] = - i E_{0} (X, X^{'}), [Φ (X), Φ^{†} (X^{'})] = - i E_{β} (X, X^{'}) .$
unlabeled · line 645
$W_{F} (λ t, λ r, φ, λ z; λ t^{'}, λ r^{'}, φ^{'}, λ z^{'}) = λ^{- 2} W_{F} (X, X^{'}) .$
unlabeled · line 658
$\frac{e ^{- i t B} - e ^{i t B}}{2 B} = - i \frac{sin ( t B )}{B} .$

Proposition: Friedrichs state and symmetries

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