Manuscript XVII-A@1 · XVII-A.tex · section · 711d7ee4

Finite carrier and finite oscillator matrices

Objects fixed before the reconstruction map › Finite carrier and finite oscillator matrices · lines 227–258

XVII-A@1#eq:finite-oscillator

Fix integers $K \geq 1$ and $d \geq 2$ . Define two separately named copies

H_{bulk} = (C^{d})^{\otimes K}, H_{\partial} = (C^{d})^{\otimes K} .

On $C^{d}$ , let

a = m = 1 \sum^{d - 1} m ∣ m - 1 ⟩ ⟨ m ∣, N = m = 0 \sum^{d - 1} m ∣ m ⟩ ⟨ m ∣ .

Let $a_{n}, N_{n}$ denote these matrices on the $n$ th bulk factor. On the boundary copy use the independently named matrices $b_{n}$ with the same displayed matrix entries. The bulk Hamiltonian is

H_{bulk} = n = 1 \sum^{K} n N_{n} .

Direct evaluation on the occupation basis gives

[N_{n}, a_{n}] = - a_{n}, [a_{n}, a_{n}^{†}] = 1 - d ∣ d - 1 ⟩ ⟨ d - 1 ∣_{n} .

In particular, these are finite oscillator matrices, not an exact finite-dimensional representation of the canonical commutation relations. No continuum canonical-field theorem is inferred from them.

Displayed formulas in this passage

XVII-A@1#eq:finite-oscillator · line 236
$a = m = 1 \sum^{d - 1} m ∣ m - 1 ⟩ ⟨ m ∣, N = m = 0 \sum^{d - 1} m ∣ m ⟩ ⟨ m ∣ .$
unlabeled · line 230
$H_{bulk} = (C^{d})^{\otimes K}, H_{\partial} = (C^{d})^{\otimes K} .$
unlabeled · line 245
$H_{bulk} = n = 1 \sum^{K} n N_{n} .$
unlabeled · line 249
$[N_{n}, a_{n}] = - a_{n}, [a_{n}, a_{n}^{†}] = 1 - d ∣ d - 1 ⟩ ⟨ d - 1 ∣_{n} .$