0  1  n  n + 1 a^{+} |1 a|1 a^{+} |\sqrt{[n + 1]_{q}} a|\sqrt{[n + 1]_{q}} a_{+} a a|0

Figure 1: Classical Fock space

0  1  n  n + 1 a^{+} |1 a|1 a^{+} |1 a|n + 1 a_{+} a a|0

Figure 2: Bargman-Fock representation (flat). L’état “n” est z^{n} et a^{+} → z ; a → D = \frac{d}{dz}

W_{k}^{(i)} = {w ∈ {a, a^{+}}^{∗}|π_{e}(w) = k and |w| = i}  (1)

with k = m − n and π_{e}(w) = |w|_{a^{+}} − |w|_{a}.

0  1  n  n + 1 a^{+} |1 a|1 a^{+} |n + 1 a|1 a_{+} a a|0

Figure 3: Représentation de Bargman-Fock avec états normalisés. L’tat “n” is \frac{z^{n}}{n!} and a^{+} → z ; a → D = \frac{d}{dz}

0  1  n  n + 1 a^{+} |1 a|\sqrt{n + 1} a^{+} |\sqrt{n + 1} a|1 a_{+} a a|0

Figure 4: Représentation de Bargman-Fock avec états normalisés. L’tat “n” is \frac{z^{n}}{n!} and a^{+} → z ; a → D = \frac{d}{dz}

0  1  n  n + 1 a^{+} |1 a|1 a^{+} |1 a|[n + 1]_{q} a_{+} a a|0

Figure 5: Bargman-Fock representation. State “n” is z^{n} and a^{+} → z ; a → D_{q} with D_{q}(f) = \frac{f(qz)−f(z)}{z(q−1)}

0  1  n  n + 1 a^{+} |β_{1} a|α_{1} a^{+} |β_{n+1} a|α_{n+1} a_{+} a a|0

Figure 6: General setting: in order that the Fock space be bounded below, one must have α_{0} = 0.

0  1  n  m a^{+} |β_{1} a|α_{1} w |ω_{w} a_{+} a a|0

Figure 7: The transfer packet problem

[figure: a^{†}; a^{†}; a; a^{†}; a; a^{†}; a^{†}; a^{†}; a^{†}; a]

Figure 8: A word which transfers from n to n + 4.

e_{n}.W_{m−n}^{(i)} = ω_{n→m}^{(i)} e_{m} ; T_{n→n+k} := ∑_{i≥0} t^{i}ω_{n→n+k}^{(i)}  (2)

(D_{+} + D_{−}^{(n)})^{∗}  (3)

a^{+}(D_{+})^{∗}  (4)

D_{+} = a^{+}(D_{+})^{∗}a ; D_{−}^{(n)} = a(D_{−}^{(n−1)})^{∗}a^{+} ; D_{−}^{(0)} = ∅  (5)

T_{n→n}[t] = \frac{1}{1 − _{\frac{t^{2}α_{n+1}β_{n+1}}{1− \frac{t^{2}α_{n+2}β_{n+2}}{1−···}}} − _{\frac{t^{2}α_{n}β_{n}}{1− \frac{t^{2}α_{n−1}β_{n−1}}{1−···}}}}  (6)

T_{n→n}^{+}[t] = \frac{1}{1 − _{\frac{t^{2}α_{n+1}β_{n+1}}{1− \frac{t^{2}α_{n+2}β_{n+2}}{1−···}}}}  (7)