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Hierarchy of Self-Consistent Approximate Methods and the Parquet ...

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application I: toy model<br />

The Hamiltonian for <strong>the</strong> classical anharmonic oscillator is<br />

Its partition function is<br />

H = p2<br />

2m + 1 2 mω2 0x 2 + ux 4 (1)<br />

Z c<br />

= 1 ∫ ∞<br />

Z c,0 Z c,0<br />

−∞<br />

dp<br />

∫ ∞<br />

−∞<br />

dxe −βH =<br />

1<br />

√<br />

2πG<br />

0<br />

∫ ∞<br />

−∞<br />

dψe − ψ2<br />

2G 0 e − g 4! ψ4 (2)<br />

where G 0 = 1/(βmω 2 0), βu = g/4! <strong>and</strong> ψ ≡ x. From this, we can construct <strong>the</strong><br />

conventional perturbation or <strong>the</strong> self-consistent field formalism.<br />

<strong>Hierarchy</strong> <strong>of</strong> <strong>Self</strong>-<strong>Consistent</strong> <strong>Approximate</strong> <strong>Methods</strong> <strong>and</strong> <strong>the</strong> <strong>Parquet</strong> Formalism 35

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