Manuscript Psy02-A@1 · Psy02-A.tex · section · d92ce402

Spacetime tests are not one-particle vectors

Carrier specification and finite realizations › Spacetime tests are not one-particle vectors · lines 333–364

Psy02-A@1#eq:test-map

A compact spacetime test $f$ and its associated one-particle vector $v = κ (f)$ are different objects. Support restrictions apply initially to $f$ , not automatically to a position representation of $v$ .

For an explicit flat-space example, put

ω_{m} (k) = \abs k^{2} + m^{2}, m \geq 0,

and, for real $f \in C_{c}^{\infty} (R^{4})$ , define

f (ω, k) = \int_{R^{4}} e^{i (ω t - k \cdot x)} f (t, x) d t d^{3} x,

κ_{m} (f) (k) = \frac{f ( ω _{m} ( k ) , k )}{( 2 π ) ^{3/2} 2 ω _{m} ( k )} .

The value at the single point $k = 0$ is irrelevant to the $L^{2}$ class. Coordinates in the following estimate are expressed in a fixed reference unit.

Displayed formulas in this passage

unlabeled · line 341
$ω_{m} (k) = \abs k^{2} + m^{2}, m \geq 0,$
unlabeled · line 346
$f (ω, k) = \int_{R^{4}} e^{i (ω t - k \cdot x)} f (t, x) d t d^{3} x,$
Psy02-A@1#eq:test-map · line 353
$κ_{m} (f) (k) = \frac{f ( ω _{m} ( k ) , k )}{( 2 π ) ^{3/2} 2 ω _{m} ( k )} .$

Spacetime tests are not one-particle vectors

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