Kleinert - Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets - ed.4 - 2006 (523104), страница 55
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If two or more of the times τi are equal, the intermediate integrals areaccompanied by the corresponding power of x(τi ).Fortunately, this rather complicated-looking expression can be replaced by amuch simpler one involving functional derivatives of the thermal partition functionZ[j] in the presence of an external current j. From the definition of Z[j] in (3.233)it is easy to see that all correlation functions of the system are obtained by thefunctional formula(n)Gω2 (τ1 , . . . , τn )"#δδ= Z[j] h̄· · · h̄Z[j]δj(τ1 )δj(τn )−1.(3.299)j=0This is why Z[j] is called the generating functional of the theory.In the present case of a harmonic action, Z[j] has the simple form (3.243), (3.244),and we can write(n)Gω2 (τ1 , .
. . , τn )="h̄δδ· · · h̄δj(τ1 )δj(τn )(3.300)2503 External Sources, Correlations, and Perturbation Theory(1× exp2h̄MZ0h̄βdτZ0h̄βdτ0j(τ )Gpω2 ,e (τ00− τ )j(τ ))#,j=0where Gpω2 ,e (τ −τ 0 ) is the Euclidean Green function (3.248). Expanding the exponential into a Taylor series, the differentiations are easy to perform. Obviously, any oddnumber of derivatives vanishes. Differentiating (3.243) twice yields the two-pointfunction [recall (3.248)](2)DEGω2 (τ, τ 0 ) = x(τ )x(τ 0 ) =h̄ pG 2 (τ − τ 0 ).M ω ,e(3.301)Thus, up to the constant prefactor, the two-point function coincides with the Euclidean Green function (3.248). Inserting (3.301) into (3.300), all n-point func(2)tions are expressed in terms of the two-point function Gω2 (τ, τ 0 ):Expanding the exponential into a power series, the expansion term of order n/2 carriesthe numeric prefactors 1/(n/2)! · 1/2n/2 and consists of a product of n/2 factorsR h̄β(2)00020 dτ j(τ )Gω 2 (τ, τ )j(τ )/h̄ . The n-point function is obtained by functionally differentiating this term n times.
The result is a sum over products of n/2 factors(2)Gω2 (τ, τ 0 ) with n! permutations of the n time arguments. Most of these prod(2)ucts coincide, for symmetry reasons. First, Gω2 (τ, τ 0 ) is symmetric in its arguments. Hence 2n/2 of the permutations correspond to identical terms, their num(2)ber canceling one of the prefactors. Second, the n/2 Green functions Gω2 (τ, τ 0 )in the product are identical. Of the n! permutations, subsets of (n/2)! permutations produce identical terms, their number canceling the other prefactor.
Onlyn!/[(n/2)!2n/2 ] = (n − 1) · (n − 3) · · · 1 = (n − 1)!! terms are different. They allcarry a unit prefactor and their sum is given by the so-called Wick rule or Wickexpansion:(n)Gω2 (τ1 , . . . , τn ) =Xpairs(2)(2)Gω2 (τp(1) , τp(2) ) · · · Gω2 (τp(n−1) , τp(n) ).(3.302)Each term is characterized by a different pair configurations of the time argumentsin the Green functions. These pair configurations are found most simply by the following rule: Write down all time arguments in the n-point function τ1 τ2 τ3 τ4 . . .
τn .Indicate a pair by a common symbol, say τ̇p(i) τ̇p(i+1) , and call it a pair contraction(2)to symbolize a Green function Gω2 (τp(i) , τp(i+1) ). The desired (n − 1)!! pair configurations in the Wick expansion (3.302) are then found iteratively by forming n − 1single contractionsτ̇1 τ̇2 τ3 τ4 .
. . τn + τ̇1 τ2 τ̇3 τ4 . . . τn + τ̇1 τ2 τ3 τ̇4 . . . τn + . . . + τ̇1 τ2 τ3 τ4 . . . τ̇n ,(3.303)and by treating the remaining n − 2 uncontracted variables in each of these termslikewise, using a different contraction symbol. The procedure is continued until allvariables are contracted.H. Kleinert, PATH INTEGRALS3.10 Correlation Functions, Generating Functional, and Wick Expansion251In the literature, one sometimes another shorter formula under the name ofWick’s rule, stating that a single harmonically fluctuating variable satisfies theequality of expectations:D22eKx = eK hx i/2 .E(3.304)This follows from the observation that the generating functional (3.233) may alsobe viewed as Zω times the expectation value of the source exponentialD RZω [j] = Zω × edτ j(τ )x(τ )/h̄E.(3.305)Thus we can express the result (3.243) also asD Redτ j(τ )x(τ )/h̄E(1/2M h̄)=eRdτRdτ 0 j(τ )Gp 2 (τ,τ 0 )j(τ 0 )ω ,e.(3.306)Since (h̄/M)Gpω2 ,e (τ, τ 0 ) in the exponent is equal to the correlation function(2)DEGω2 (τ, τ 0 ) = x(τ )x(τ 0 ) by Eq.
(3.301), we may also writeD Redτ j(τ )x(τ )/h̄ER=edτRdτ 0 j(τ )hx(τ )x(τ 0 )ij(τ 0 )/2h̄2.(3.307)Considering now a discrete time axis sliced at t = tn , and inserting the special sourcecurrent j(τn ) = Kδn,0 , for instance, we find directly (3.304).The Wick theorem in this form has an important physical application. Theintensity of the sharp diffraction peaks observed in Bragg scattering of X-rays oncrystal planes is reduced by thermal fluctuations of the atoms in the periodic lattice.The reduction factor is usually written as e−2W and called the Debye-Waller factor .In the Gaussian approximation it is given bye−W ≡ he−∇·u(x) i = e−Σk h |k·u(k)]2 i/2,(3.308)where u(x) is the atomic displacement field.If the fluctuations take place around hx(τ )i =6 0, then (3.304) goes obviously overintoheP x i = eP hx(τ )i+P3.10.12 hx−hx(τ )ii2 /2.(3.309)Real-Time Correlation FunctionsThe translation of these results to real times is simple. Consider, for example, theharmonic fluctuations δx(t) with Dirichlet boundary conditions, which vanish at tband ta .
Their correlation functions can be found by using the amplitude (3.23) as agenerating functional, if we replace x(t) → δx(t) and xb = xa → 0. Differentiatingtwice with respect to the external currents j(t) we obtain(2)DEGω2 (t, t0 ) = x(t)x(t0 ) = ih̄G 2 (t − t0 ),M ω(3.310)2523 External Sources, Correlations, and Perturbation Theorywith the Green function Gω2 (t − t0 ) of Eq. (3.36), which vanishes if t = tb or t = ta .The correlation function of ẋ(t) isDEẋ(t)ẋ(t0 ) = ih̄ cos ω(tb − t> ) cos ω(t< − ta ),Mω sin ω(tb − ta )(3.311)h̄cot ω(tb − ta ).M(3.312)and has the valuehẋ(tb )ẋ(tb )i = iAs an application, we use this result to calculate once more the time evolutionamplitude (xb tb |xa ta ) in a way closely related to the operator method in Section 2.23.We observe that the time derivative of this amplitude has the path integral representation [compare (2.739)]ZiDih̄∂tb (xb tb |xa ta ) = − D x L(xb , ẋb )eR tbtadt L(x,ẋ)/h̄= −hL(xb , ẋb )i (xb tb |xa ta ),(3.313)and calculate the expectation value hL(xb , ẋb )i as a sum of the classical LagrangianL(xcl (tb ), ẋcl (tb )) and the expectation value of the fluctuating part of the LagrangianhLfl (xb , ẋb )i ≡ h [L(xb , ẋb ) − L(xcl (tb ), ẋcl (tb ))] i.
If the Lagrangian has the standardform L = M ẋ2 /2 − V (x), then only the kinetic term contributes to hLfl (xb , ẋb )i, sothatMhδ ẋ2b i.(3.314)hLfl (xb , ẋb )i =2There is no contribution from hV (xb ) − V (xcl (tb ))i, due to the Dirichlet boundaryconditions.The temporal integral over − [L(xcl (tb ), ẋcl (tb )) − hLfl (xb , ẋb )i] agrees with theoperator result (2.766), and we obtain the time evolution amplitude from the formulaiA(xb ,xa ;tb −ta )/h̄(xb tb |xa ta ) = C(xb , xa )eiexph̄ZtbtaMdtb0 hδ ẋ2b0 i ,2(3.315)where A(xb , xa ; tb − ta ) is the classical action A[xcl ] expressed as a function of theendpoints [recall (4.80)].
The constant of integration C(xb , xa ) is fixed as in (2.752)by solving the differential equation−ih̄∇b (xb tb |xa ta ) = hpb i(xb tb |xa ta ) = pcl (tb )(xb tb |xa ta ),(3.316)and a similar equation for xa [compare (2.753)]. Since the prefactor pcl (tb ) on theright-hand side is obtained from the derivative of the exponential eiA(xb ,xa ;tb −ta )/h̄in (3.315), due to the general relation (4.81), the constant of integration C(xb , xa )is actually independent of xb and xa . Thus we obtain from (3.315) once more theknown result (3.315).As an example, take the harmonic oscillator.
The terms linear in δx(t) = x(t) −xcl (t) vanish since they are they are odd in δx(t) while the exponent in (3.313) isH. Kleinert, PATH INTEGRALS2533.11 Correlation Functions of Charged Particle in Magnetic Field . . .even. Inserting on the right-hand side of (3.314) the correlation function (3.311),we obtain in D dimensionshLfl (xb , ẋb )i =h̄ωMhδ ẋ2b i = i D cot ω(tb − ta ),22(3.317)which is precisely the second term in Eq. (2.765), with the appropriate oppositesign.3.11Correlation Functions of Charged Particlein Magnetic Field and Harmonic PotentialIt is straightforward to find the correlation functions of a charged particle in amagnetic and an extra harmonic potential discussed in Section 2.19.
They areobtained by inverting the functional matrix (2.669):(2)Gω2 ,B (τ, τ 0 ) =h̄ −1Dω2 ,B (τ, τ 0 ).M(3.318)By an ordinary matrix inversion of (2.672), we obtain the Fourier expansion(2)GB (τ, τ 0 ) =∞01 XG̃ω2 ,B (ωm )e−iωm (τ −τ ) ,h̄β m=−∞(3.319)with(2)G̃ω2 ,B (ωm )h̄1=2222M (ωm + ω+ )(ωm+ ω−)2ωm+ ω 2 − ωB22ωB ωm2−2ωB ωmωm+ ω 2 − ωB2 ,!. (3.320)2222Since ω++ ω−= 2(ω 2 + ωB2 ) and ω+− ω−= 4ωωB , the diagonal elements can bewritten as122(ωm+22ω+)(ωm1=2("+2ω−)h2222(ωm+ ω+) + (ωm+ ω−) − 4ωB2#"i111ω1+ B22 + 2222 − 22ω ωm + ω+ ωm + ω−ωm + ω+ ωm + ω−#). (3.321)Recalling the Fourier expansion (3.245), we obtain directly the diagonal periodiccorrelation function(2)Gω2 ,B,xx=cosh ω+ (|τ −τ 0 |−h̄β/2) cosh ω− (|τ −τ 0 | − h̄β/2)h̄+,4Mωsinh(ω+ h̄β/2)sinh(ω− h̄β/2)"#(3.322)(2)which is equal to Gω2 ,B,yy . The off-diagonal correlation functions have the Fouriercomponents"#112ωB ωmωm2222 =22 − 22 .2ω ωm + ω+ ωm + ω−(ωm + ω+ )(ωm + ω− )(3.323)2543 External Sources, Correlations, and Perturbation TheorySince ωm are the Fourier components of the derivative i∂τ , we can write(2)(2)Gω2 ,B,xy (τ, τ 0 ) = −Gω2 ,B,yx (τ, τ 0 ) =h̄i∂2M τ1 cosh ω+ (|τ − τ 0 | − h̄β/2)2ω+sinh(ω+ h̄β/2)#1 cosh ω− (|τ − τ 0 | − h̄β/2)−.(3.324)2ω−sinh(ω−h̄β/2)"Performing the derivatives yields(2)(2)Gω2 ,B,xy (τ, τ 0 ) = Gω2 ,B,yx (τ, τ 0 ) =h̄(τ −τ 0 )2Mi1 sinh ω+ (|τ −τ 0 | − h̄β/2)2ω+sinh(ω+h̄β/2)#1 sinh ω− (|τ −τ 0 | − h̄β/2)−,(3.325)2ω−sinh(ω− h̄β/2)"where (τ −τ 0 ) is the step function (1.312).For a charged particle in a magnetic field without an extra harmonic oscillator wehave to take the limit ω → ωB in these equations.
Due to translational invariance ofthe limiting system, this exists only after removing the zero-mode in the Matsubarasum. This is done most simply in the final expressions by subtracting the hightemperature limits at τ = τ 0 . In the diagonal correlation functions (3.322) thisyields1(2) 0(2) 0(2)Gω2 ,B,xx(τ, τ 0 ) = Gω2 ,B,yy(τ, τ 0 ) = Gω2 ,B,xx −,(3.326)βMω+ ω−where the prime indicates the subtraction. Now one can easily go to the limitω → ωB with the result(2) 0Gω2 ,B,xx0(τ, τ ) =(2) 0Gω2 ,B,yyh̄cosh 2ω(|τ − τ 0 | − h̄β/2)1(τ, τ ) =.
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