Manuscript Em03-H@8 · Em03-H-v8.tex · lemma · 91e9e3bd

Lemma: CAR and number convention

Frozen microscopic model and complete preparation › Occupation representation · lines 381–405 · proof included or linked

Em03-H@8#lem:car

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These operators satisfy

{c_{j}, c_{k}^{†}} = δ_{j k} I, {c_{j}, c_{k}} = 0.

For

χ_{2 j - 1} = \frac{c _{j} + c _{j} ^{†}}{2}, χ_{2 j} = \frac{c _{j} - c _{j} ^{†}}{i 2},

one has ${χ_{a}, χ_{b}} = δ_{ab} I$ and

N_{B} = j \sum c_{j}^{†} c_{j} = \frac{n}{2} I + i j \sum χ_{2 j - 1} χ_{2 j} .

Proof. For equal labels the two orders give complementary occupation projections. For unequal labels, exchanging actions reverses the preceding-occupation sign. Expansion gives the Majorana relations. Finally, $i χ_{2 j - 1} χ_{2 j} = c_{j}^{†} c_{j} - I /2$ .

Displayed formulas in this passage

unlabeled · line 384
${c_{j}, c_{k}^{†}} = δ_{j k} I, {c_{j}, c_{k}} = 0.$
unlabeled · line 388
$χ_{2 j - 1} = \frac{c _{j} + c _{j} ^{†}}{2}, χ_{2 j} = \frac{c _{j} - c _{j} ^{†}}{i 2},$
unlabeled · line 393
$N_{B} = j \sum c_{j}^{†} c_{j} = \frac{n}{2} I + i j \sum χ_{2 j - 1} χ_{2 j} .$

Lemma: CAR and number convention

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