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

Proposition: Finite-dimensional CCR obstruction

Carrier specification and finite realizations › Carrier correspondence and the finite-dimensional obstruction · lines 452–487 · proof included or linked

Psy02-A@1#prop:finite-CCR, Psy02-A@1#eq:compressed-CCR

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No finite-dimensional matrices $Q, P \in M_{d} (C)$ , $d \geq 1$ , satisfy

[Q, P] = i c 1_{d}

when $c \neq = 0$ .

For the oscillator truncated to $span {∣ 0 ⟩, \dots, ∣ d - 1 ⟩}$ , define

a_{d} = n = 1 \sum^{d - 1} n ∣ n - 1 ⟩ ⟨ n ∣, Q_{d} = \frac{a _{d} + a _{d} ^{†}}{2}, P_{d} = \frac{a _{d} - a _{d} ^{†}}{i 2} .

Then

[Q_{d}, P_{d}] = i (1_{d} - d ∣ d - 1 ⟩ ⟨ d - 1 ∣) .

Proof. The trace of a commutator is zero, while $Tr (i c 1_{d}) = i c d \neq = 0$ . Direct multiplication gives $[a_{d}, a_{d}^{†}] = 1_{d} - d ∣ d - 1 ⟩ ⟨ d - 1 ∣$ , which implies (eq:compressed-CCR).

Displayed formulas in this passage

Psy02-A@1#eq:compressed-CCR · line 471
$[Q_{d}, P_{d}] = i (1_{d} - d ∣ d - 1 ⟩ ⟨ d - 1 ∣) .$
unlabeled · line 456
$[Q, P] = i c 1_{d}$
unlabeled · line 463
$a_{d} = n = 1 \sum^{d - 1} n ∣ n - 1 ⟩ ⟨ n ∣, Q_{d} = \frac{a _{d} + a _{d} ^{†}}{2}, P_{d} = \frac{a _{d} - a _{d} ^{†}}{i 2} .$

Proposition: Finite-dimensional CCR obstruction

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