Manuscript V-B@1 · V-B-v1.tex · section · 3fadc1ad

One metric oscillator and the switched identity

One metric oscillator and the switched identity · lines 1194–1222

V-B@1#eq:mixed

Let $H_{g} = Γ_{s} (C)$ , with

Q = \frac{b + b ^{†}}{2}, Π = \frac{b - b ^{†}}{i 2}, [Q, Π] = i I .

This infinite-dimensional oscillator is distinct from the finite clock introduced later.

Set

h (x) = Q e (x) + Π r (x), D_{mix} = D_{g} \otimes_{alg} D_{ϕ},

where $D_{g}$ is the finite oscillator-number span. Applying the preceding contraction calculation separately to $e, r$ gives

R_{g, f} = Q \otimes Φ (v_{e}) + Π \otimes Φ (v_{r}), v_{a} = K Q_{g a} u_{f} .

No metric–metric contraction enters this insertion pair. The oscillator metric is not a local relativistic metric field.

Displayed formulas in this passage

V-B@1#eq:mixed · line 1213
$R_{g, f} = Q \otimes Φ (v_{e}) + Π \otimes Φ (v_{r}), v_{a} = K Q_{g a} u_{f} .$
unlabeled · line 1198
$Q = \frac{b + b ^{†}}{2}, Π = \frac{b - b ^{†}}{i 2}, [Q, Π] = i I .$
unlabeled · line 1206
$h (x) = Q e (x) + Π r (x), D_{mix} = D_{g} \otimes_{alg} D_{ϕ},$