Kleinert - Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets - ed.4 - 2006 (523104), страница 100
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We have to evaluate the Feynmanintegrals associated with all vacuum diagrams. These are composed of p-particle vertices carrying the coupling constant gp /p!h̄, and of lines representing the correlationfunction of the fluctuations introduced in (5.23):Dδx(τ )δx(τ 0 )h̄ 0x0G Ω (τ − τ 0 )M= a2 (τ − τ 0 , x0 ).Ex0=Ω(5.185)Since this correlation function contains no zero frequency, diagrams of the typeshown in Fig. 5.13 do not contribute.
Their characteristic property is to fall apartwhen cutting a single line. They are called one-particle reducible diagrams. A vacuum subdiagram connected with the remainder by a single line is called a tadpolediagram, alluding to its biological shape. Tadpole diagrams do not contribute tothe variational perturbation expansion since they vanish as a consequence of energyconservation: the connecting line ending in the vacuum, must have Da vanishing freEx0quency where the spectral representation of the correlation function δx(τ )δx(τ 0 )Ωhas no support, by construction.The number of Feynman diagrams to be evaluated is reduced by ignoring at firstall diagrams containing the vertices g2 .
The omitted diagrams can be recovered fromthe diagrams without g2 -vertices by calculating the latter at the initial frequency ω,and by replacing ω by a modified local trial frequency,ω → Ω̃(x0 ) ≡qΩ2 (x0 ) + g2 (x0 )/M.(5.186)After this replacement, which will be referred to as the square-root trick , all diagramsare re-expanded in powers of g [remembering that g2 (x0 ) is by (5.178) proportionalto g] up to the maximal power g 3 .5.14Applications of Variational Perturbation ExpansionThe third-order approximation W3 (x0 ) is far more accurate than W1 (x0 ). This willnow be illustrated by performing the variational perturbation expansion for the an-Figure 5.13 Structure of a one-particle reducible vacuum diagram. The dashed boxencloses a so-called tadpole diagram.
Such diagrams vanish in the present expansion sincean ending line cannot carry any energy and since the correlation function hδx(τ )δx(τ 0 )icontains no zero frequency.H. Kleinert, PATH INTEGRALS4955.14 Applications of Variational Perturbation Expansionharmonic oscillator and the double-well potential. The reason for the great increasein accuracy will become clear in Section 5.15, where we shall demonstrate that thisexpansion converges rapidly towards the exact result at all coupling strengths, incontrast to ordinary perturbation expansions which diverges even for arbitrarilysmall values of g.5.14.1Anharmonic Oscillator at T = 0Consider first the case of zero temperature, where the calculation is simplest and theapproximation should be the worst.
At T = 0, only the point x0 = 0 contributes,and δx(τ ) coincides with the path itself ≡ x(τ ). Thus we may omit the superscriptx0 in all equations, so that the interaction in (5.182) becomes writing it asAint =g4Zh̄β0dτ (r 2 x2 + x4 ).(5.187)The effect of the r 2 -term is found by replacing the frequency ω in the originalperturbation expansion for the anharmonic oscillator according to the square-roottrick (5.186), which for x0 = 0 is simplyω → Ω̃ ≡rΩ2 + ω 2 − Ω2 =qΩ2 + gr 2/2M .(5.188)After this replacement, all terms are re-expanded in powers of g.
Finally, r 2 is againreplaced by2M 2r2 →(ω − Ω2 ).(5.189)gSince the interaction is even in x, the zero-temperature expansion is automatically free of tadpole diagrams.The perturbation expansion to third order was given in Eq. (3.551). With theabove replacement it leads to the free energy 2gh̄Ω̃ g 4+ 3a +F =244 3g1+42a4h̄Ω̃8124 · 333a1h̄Ω̃2,(5.190)withh̄.(5.191)2M Ω̃The higher orders can most easily be calculated with the help of the Bender-Wurecursion relations derived in Appendix 3C.2By expanding F in powers of g up to g 3 we obtaina2 =h̄Ω+ g(3a4 − r 2 a2 )/4 − g 2(21a8 /8 + 3a6 r 2 /4 + a4 r 4 /16)/h̄Ω2+g 3 (333a12 /16 + 105a10 r 2 /16 + 3a8 r 4 /4 + a6 r 6 /32)/(h̄Ω)2 .W3Ω =(5.192)4965 Variational Perturbation TheoryTable 5.6 Second- and third-order approximations to ground state energy, in units ofh̄ω, of anharmonic oscillator at various coupling constants g in comparison with exact(0)(0)values Eex (g) and the Feynman-Kleinert approximation E1 (g) of previous section.g/40.10.20.30.40.50.60.70.80.9110501005001000(0)Eex (g)0.5591460.6024050.6379920.6687730.6961760.7210390.7439040.7651440.7850320.8037711.504972.499713.131385.319896.69422(0)E1 (g)0.5603073710.6049007480.6416298620.6733947150.7016616430.7272956680.7508578180.7727363590.7932130660.8125000001.531250002.547580403.192444045.425756056.82795331(0)E2 (g)0.5591521390.6024507130.6380887350.6689224550.6963769500.7212887890.7441994360.7654833010.7854120370.8041900951.506740002.503121333.135785305.327619696.70400326(0)E3 (g)0.5591542190.6024306210.6380357600.6688341370.6962536320.7211317760.7440103170.7652636970.7851634940.8039140531.505497502.500699633.132656565.322117096.697032860.80.75N =10.7N =20.65WN =30.60.550.50.450.4121.52.53ΩFigure 5.14 Typical Ω-dependence of approximations W1,2,3 at T = 0.
The couplingconstant has the value g = 0.4. The second-order approximation W2Ω has no extremum.Here the minimal Ω-dependence lies at the turning point, and the condition ∂ 2 W2Ω /∂Ω2renders the best approximation to the energy (short dashes).After the replacement (5.189), we minimize W3 in Ω, and obtain the third-order(0)approximation E3 (g) to the ground state energy. Its accuracy at various coupling2The Mathematica program is available on the internet under http://www.physik.fu-berlin/.de~kleinert/294/programs.H.
Kleinert, PATH INTEGRALS4975.14 Applications of Variational Perturbation Expansionstrengths is seen in Table 5.6 where it is compared with the exact values obtainedfrom numerical solutions of the Schrödinger equation. The improvement with respectto the earlier approximation W1 is roughly a factor 50. The maximal error is nowsmaller than 0.05%.We shall see in the last subsection that up to rather high orders N, the minimumhappens to be unique.Observe that when truncating the expansion (5.183) after the second order andworking with the approximation W2Ω (x0 ), there exists no minimum in Ω, as canbe seen in Fig.
5.14. The reason for this is the alternating sign of the cumulantsin (5.183). This gives an alternating sign to the highest power a and thus to thehighest power of 1/Ω in the g n -terms of Eq. (5.192), causing the trial energy of orderN to diverge for Ω → 0 like (−1)N −1 g N × (1/Ω)3N −1 . Since the trial energy goes forlarge Ω to positive infinity, only the odd approximations are guaranteed to possessa minimum.The second-order approximation W2 can nevertheless be used to find an improvedenergy value. As shown by Fig. 5.14, the frequency of least dependence Ω2 is welldefined.
It is the frequency where the Ω-dependence of W2 has its minimal absolutevalue. Thus we optimize Ω with the condition∂ 2 W2Ω= 0.∂Ω2(5.193)(0)This leads to the energy values E2 (g) listed in Table 5.6. They are more accurate(0)than the values E1 (g) by an order of magnitude.5.14.2Anharmonic Oscillator for T > 0Consider now the anharmonic oscillator at a finite temperature, where the expansion(5.183) consists of the sum of one-particle irreducible vacuum diagrams1W3Ω̃ =β(1−2Ω̃+3Ω̃Ω̃1−2!1+3!Ω̃+72+2592+624+1728+3456!Ω̃Ω̃Ω̃Ω̃Ω̃(5.194)Ω̃+1728Ω̃!)+648.648The vertices represent the couplings g3 (x0 )/3!h̄, g4 (x0 )/4!h̄, whereas the lines standfor the correlation function G0xΩ̃0 (τ, τ 0 ).
The numbers under each diagram are theirmultiplicities acting as factors.4985 Variational Perturbation TheoryOnly the five integrals associated with the diagrams;;;;need to be evaluated explicitly; all others arise by the expansion of Ω̃ or by factorization. The explicit form of three of these integrals can be found in the first, forth,and sixth of Eqs. (3.547).
The results of the integrations are listed in Appendix 5A.Only the second and fourth diagrams are new since they involve vertices withthree legs. They can be found in Eqs. (3D.7) and (3D.8).In quantum field theory one usually calculates Feynman integrals in momentumspace. At finite temperatures, this requires the evaluation of multiple sums overMatsubara frequencies.
The present quantum-mechanical example corresponds toa D = 1 -dimensional quantum field theory. Here it is more convenient to evaluatethe integrals in τ -space. The diagrams;,for example, are found by performing the integralsh̄βa2 ≡1h̄βω2a123≡Z0Z0h̄βdτ G2 (τ, τ ),h̄βZ h̄βZ h̄β00G2 (τ1 , τ2 )G2 (τ2 , τ3 )G2 (τ3 , τ1 ) dτ1 dτ2 dτ3 .The factor h̄β on the left-hand side is due to an overall τ -integral and reflectsthe temporal translation symmetry of the system; the factors 1/ω arise from theremaining τ -integrations whose range is limited by the correlation time 1/ω.In general, the β- and x0 -dependent parameters a2LV have the dimension of alength to the nth power and the associated diagrams consist of m vertices and n/2lines (defining a21 ≡ a2 ).We now use the rule (5.188) to replace ω by Ω̃ and expand everything in powersof g2 up to the third order. The expansion can be performed diagrammatically ineach Feynman diagram.
Letting a dot on a line indicate the coupling g2 /2h̄, theone-loop diagram is expanded as follows:Ω̃=−2+−13.(5.195)The other diagrams are expanded likewise:Ω̃=−+16+16,H. Kleinert, PATH INTEGRALS4995.14 Applications of Variational Perturbation ExpansionΩ̃12=12−16,=12−16,=12Ω̃12Ω̃12−16−16.In this way, we obtain from (5.194) the complete graphical expansion for W3Ω (x0 )including all vertices associated with the coupling g2 (x0 ):βW3Ω (x0 )1=−21 1−2! 21+3!1+81+21+2"1+81+21+341+ 341+243+ 16+1+41+61+618++1+21434++1834#.(5.196)In the latter diagrams, the vertices represent directly the couplings gn /h̄. The denominators n! of the previous vertices gn /n!h̄ have been combined with the multiplicities of the diagrams yielding the indicated prefactors.
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