By Jean Zinn-Justin
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Fonction generatrice. 20) : Les valeurs moyennes (x^Xi2 ... 1 Integrate de chemin en mecanique quantique La fonction a deux points Considerons, par exemple, la fonction a deux points (xi1Xi2)x. 2 representent les contributions d'ordre A 2 . FlG. 1 - La fonction a deux points a 1'ordre A. Nous pourrions calculer de meme la fonction a quatre points, c'est-adire les valeurs moyennes des monomes de degre quatre. Nous trouverions un grand nombre de contributions. Mais les resultats se simplifient notablement si nous calculons directement les cumulants de la distribution.
On trouve les cinq valeurs moyennes suivantes En deduire les valeurs moyennes utilisera le theoreme de Wick). Quelle est la distribution gaussienne qui conduit a ces valeurs ? Solution. 2. Solent trois variables aleatoires x,y,z correlees dont on suppose la loi de probabilite gaussienne. /}, (x2y2), (x2yz). Quelle est pour a = 2 , f e = l , c = 01a distribution gaussienne qui conduit a ces valeurs ? Solution. 3. 5) par methode iterative. Le determinant d'une matrice complexe quelconque A^ n x n, d'elements A™ peut etre calcule de fagon recursive en soustrayant a toutes les lignes un multiple de la derniere ligne de fagon a annuler la derniere colonne (supposant Ann ^ 0, sinon il faut prendre une autre ligne et colonne).
1) par rapport a t et prenons la limite t" = t. Nous trouvons Quand 1'operateur H est independant du temps, 1'equation se resout formellement : U(t,t') = e-(*-*') H / R . 1) permet d'ecrire U(t",t') comme un produit de n operateurs pour des intervalles de temps £ = (t" — t')/n arbitrairement 2. 1). Quand H est independant du temps, U(t + e,i) = Q~eHln et on reconnait la formule de Trotter. Notons que les operateurs U(t + e, t) jouent un role analogue aux matrices de transfert de la mecanique statistique classique (cf.