Kleinert - Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets - ed.4 - 2006 (523104), страница 46
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We choosethe gauge in which the vector potential points in the y-direction [recall (2.612)], and parametrizethe straight line between x0 and x as= x0 + s(x − x0 ),s ∈ [0, 1].(2.805)Then we findZxd A( )x0Z= B(y − y 0 )10ds [x0 + s(x − x0 )] = B(y − y 0 )(x + x0 )= B(xy − x0 y 0 ) + B(x0 y − xy 0 ).(2.806)Inserting this and (2.804) into (2.740), we recover the earlier result (2.644).Appendix 2ABaker-Campbell-Hausdorff Formula andMagnus ExpansionThe standard Baker-Campbell-Hausdorff formula, from which our formula (2.9) can be derived,readse eB̂ = eĈ ,(2A.1)whereĈ = B̂ +1Zdtg(ead A t ead B )[Â],(2A.2)0and g(z) is the functiong(z) ≡∞Xlog z(1 − z)n=,z − 1 n=0 n + 1(2A.3)and adB is the operator associated with B̂ in the so-called adjoint representation, which is definedbyadB[Â] ≡ [B̂, Â].(2A.4)One also defines the trivial adjoint operator (adB)0 [Â] = 1[Â] ≡ Â.
By expanding the exponentialsin Eq. (2A.2) and using the power series (2A.3), one finds the explicit formulaĈ = B̂ + Â +∞X(−1)nn+1n=1×Xpi ,qi ;pi +qi ≥1q1(adA)p1 (ad B)p1 !q1 !···1+1Pni=1pi(ad A)pn (adB)qn[Â].pn !qn !(2A.5)The lowest expansion terms are1 1221112 adA + adB + 6 (adA) + 2 adA adB + 2 (adB) +. . . [Â]21 1+ 3 (adA)2 + 21 adA adB + 21 adB adA + (adB)2 + .
. . [Â]3111=  + B̂ + [Â, B̂] + ([Â, [Â, B̂]] + [B̂, [B̂, Â]]) + [Â, [[Â, B̂], B̂]] . . . .21224Ĉ = B̂ + Â−(2A.6)The result can be rearranged to the closely related Zassenhaus formulaeÂ+B̂ = e eB̂ eẐ2 eẐ3 eẐ4 · · · ,(2A.7)H. Kleinert, PATH INTEGRALSAppendix 2A Baker-Campbell-Hausdorff Formula and Magnus Expansion201whereẐ2=Ẑ3=Ẑ4=...1[B̂, Â]211− [B̂, [B̂, Â]] − [Â, [B̂, Â]])3611[[[B̂, Â], B̂], B̂] + [[[B̂, Â], Â], B̂] + [[[B̂, Â], Â], Â]824(2A.8)(2A.9)(2A.10).To prove the expansion (2A.6), we derive and solve a differential equation for the operatorfunctionĈ(t) = log(eÂt eB̂ ).(2A.11)Its value at t = 1 will supply us with the desired result Ĉ in (2A.5).
Starting point is the observationthat for any operator M̂ ,(2A.12)eĈ(t) M̂ e−Ĉ(t) = ead C(t) [M̂ ],by definition of adC.Inserting (2A.11), the left-hand side can also be rewritten asÂt B̂−B̂ −Âte e M̂ e e, which in turn is equal to ead A t ead B [M̂ ], by definition (2A.4).
Hence we haveead C(t) = ead A t ead B .(2A.13)Differentiation of (2A.11) yieldsd −Ĉ(t)e= −Â.dtThe left-hand side, on the other hand, can be rewritten in general aseĈ(t)eĈ(t)d −Ĉ(t)˙e= −f (adC(t))[Ĉ(t)],dtwheref (z) ≡ez − 1.z(2A.14)(2A.15)(2A.16)This will be verified below. It implies that˙f (adC(t))[Ĉ(t)] = Â.(2A.17)We now define the function g(z) as in (2A.3) and see that it satisfiesg(ez )f (z) ≡ 1.(2A.18)˙˙Ĉ(t) = g(ead C(t) )f (adC(t))[Ĉ(t)].(2A.19)We therefore have the trivial identityUsing (2A.17) and (2A.13), this turns into the differential equation˙Ĉ(t) = g(ead C(t) )[Â] = ead A t ead B [Â],(2A.20)from which we find directly the result (2A.2).To complete the proof we must verify (2A.15).
The expression is not simply equal to˙˙Ĉ(t) ˙−eĈ(t)M e−Ĉ(t) since Ĉ(t) does not. in general, commute with Ĉ(t). To account for thisconsider the operatord(2A.21)Ô(s, t) ≡ eĈ(t)s e−Ĉ(t)s .dt2022 Path Integrals — Elementary Properties and Simple SolutionsDifferentiating this with respect to s gives∂s Ô(s, t) ===eĈ(t)s Ĉ(t)d −Ĉ(t)s d Ĉ(t)e−Ĉ(t)se− eĈ(t)sdtdt˙−eĈ(t)s Ĉ(t)e−Ĉ(t)s˙−ead C(t)s [Ĉ(t)].(2A.22)HenceÔ(s, t) − Ô(0, t) =ZÔ(1, t) = eĈ(t)ds0 ∂s0 Ô(s0 , t)0= −from which we obtains∞Xsn+1n ˙(adC(t)) [Ĉ(t)],(n+1)!n=0d −Ĉ(t)˙e= −f (adC(t))[Ĉ(t)],dt(2A.23)(2A.24)which is what we wanted to prove.Note that the final form of the series for Ĉ in (2A.6) can be rearranged in many different ways,using the Jacobi identity for the commutators.
It is a nontrivial task to find a form involving thesmallest number of terms.38The same mathematical technique can be used to derive useful modification of the NeumannLiouville expansion or Dyson series (1.240) and (1.252). This is the so-called Magnus expansion 39 ,in which one writes one writes Û (tb , ta ) = eÊ , and expands the exponent Ê as 2 Z tbZZ t2ihi tb1 −iÊ = −dt1 Ĥ(t1 ) +dt2dt1 Ĥ(t2 ), Ĥ(t1 )h̄ ta2 h̄tata 3 (Z t2Z t3Z tbhhii1 −idt1 Ĥ(t3 ), Ĥ(t2 ), Ĥ(t1 )dt2dt3+4 h̄tatata)Z tbZ tbZ tbiihh1++ ...
,(2A.25)dt3dt2dt1 Ĥ(t3 ), Ĥ(t2 ) , Ĥ(t1 )3 tatatawhich converges faster than the Neumann-Liouville expansion.Appendix 2BDirect Calculation of Time-SlicedOscillator AmplitudeAfter time-slicing the amplitude (2.138), it becomes a multiple integral over short-time amplitudes[using the action (2.185)]1i M (xn − xn−1 )21(xn |xn−1 0) = p− ω 2 (x2n + x2n−1 ) .(2B.26)exph̄ 222πh̄i/MWe shall write this as(xn |xn−1 0) = N1 exp38i a1 (x2n + x2n−1 ) − 2b1 xn xn−1 ,h̄(2B.27)For a recent discussion see J.A. Oteo, J. Math. Phys. 32 , 419 (1991).See A. Iserles, A.
Marthinsen, and S.P. Nørsett, On the implementation of the method ofMagnus series for linear differential equations, BIT 39, 281 (1999) (http://www.damtp.cam.ac.uk/user/ai/Publications).39H. Kleinert, PATH INTEGRALSAppendix 2B Direct Calculation of Time-Sliced Oscillator Amplitude203with ω 2 M1−2,221N1 = p.2πh̄i/Ma1 =b1 =M,2(2B.28)When performing the intermediate integrations in a product of N such amplitudes, the result musthave the same general formi aN (x2N + x20 ) − 2bN xN x0 .(2B.29)(xN |xN −1 0) = NN exph̄Multiplying this by a further short-time amplitude and integrating over the intermediate positiongives the recursion relationsriπh̄,(2B.30)NN +1 = N1 NNaN + a1a2N − b2N + a1 aNa2 − b21 + a1 aNaN +1 == 1,(2B.31)a1 + aNa1 + aNb1 bNbN +1 =.(2B.32)a1 + aNFrom (2B.31) we finda2N = b2N + a21 − b21 ,(2B.33)and the only nontrivial recursion relation to be solved is that for bN .
With (2B.32) it becomesbN +1 =a1 +or1bN +11=b1b1 bNp,b2N − (b21 − a21 )a1+bNsb 2 − a21− 1 2 1bN!(2B.34).(2B.35)We now introduce the auxiliary frequency ω̃ of Eq. (2.156). Thena1 =Mcos ω̃,2(2B.36)and the recursion for bN +1 reads1bN +1cos ω̃2=+bNMs1−M 2 sin2 ω̃.42 b2N(2B.37)By introducing the reduced quantitiesβN ≡2bN ,M(2B.38)withβ1 = 1,(2B.39)the recursion becomes1βN +1cos ω̃=+βNs1−sin2 ω̃.2βN(2B.40)2042 Path Integrals — Elementary Properties and Simple SolutionsFor N = 1, 2, this determinesp1sin 2ω̃= cos ω̃ + 1 − sin2 ω̃ =,β2sin ω̃ssin 3ω̃1sin2 2ω̃sin 2ω̃== cos ω̃+ 1 − sin2 ω̃.β3sin ω̃sin ω̃sin2 ω̃(2B.41)We therefore expect the general result1βN +1=sin ω̃(N + 1).sin ω̃(2B.42)It is easy to verify that this solves the recursion relation (2B.40). From (2B.38) we thus obtainbN +1 =sin ω̃M.2 sin ω̃(N + 1)(2B.43)Inserting this into (2B.30) and (2B.33) yieldsaN +1NN +1Mcos ω̃(N + 1)sin ω̃,2sin ω̃(N + 1)ssin ω̃,= N1sin ω̃(N + 1)=(2B.44)(2B.45)such that (2B.29) becomes the time-sliced amplitude (2.192).Appendix 2CDerivation of Mehler FormulaHer we briefly sketch the derivation of Mehler’s formula.40 It is based on the observation that theleft-hand side of Eq.
(2.290), let us call it F (x, x0 ), is the Fourier transform of the functionF̃ (k, k 0 ) = π e−(k2+k02 +akk0 )/2,(2C.46)as can easily be verified by performing the two Gaussian integrals in the Fourier representationZ ∞Z ∞dk dk 0 ikx+ik0 xF (x; x0 ) =eF̃ (k, k 0 ).(2C.47)2π2π−∞ −∞We now consider the right-hand side of (2.290) and form the Fourier transform by recognizing the2exponential ek /2−ikx as the generating function of the Hermite polynomials41ek2/2−ikx=∞X(−ik/2)nHn (x).n!n=0(2C.48)This leads toF̃ (k, k 0 ) =×Z∞Z∞−∞ −∞Z ∞Z ∞−∞0dx dx0 F (x, x0 )e−ikx−ik x = e−(k−∞dx dx0 F (x, x0 )2+k02 )/20∞ X∞X(−ik/2)n (−ik 0 /2)nHn (x)Hn0 (x).n!n0 !n=0 0n =0Inserting here the expansion on right-hand side of (2.290) and using the orthogonality relation ofHermite polynomials (2.299), we obtain once more (2C.47).40See P.M.
Morse and H. Feshbach, Methods of Theoretical Physics, McGraw-Hill, New York,Vol. I, p. 781 (1953).41I.S. Gradshteyn and I.M. Ryzhik, op. cit., Formula 8.957.1.H. Kleinert, PATH INTEGRALSNotes and References205Notes and ReferencesThe basic observation underlying path integrals for time evolution amplitudes goes back to thehistoric articleP.A.M. Dirac, Physikalische Zeitschrift der Sowjetunion 3, 64 (1933).He observed that the short-time propagator is the exponential of i/h̄ times the classical action.See alsoP.A.M. Dirac, The Principles of Quantum Mechanics, Oxford University Press, Oxford, 1947;E.T.
Whittaker, Proc. Roy. Soc. Edinb. 61, 1 (1940).Path integrals in configuration space were invented by R. P. Feynman in his 1942 Princeton thesis.The theory was published in 1948 inR.P. Feynman, Rev. Mod. Phys. 20, 367 (1948).The mathematics of path integration had previously been developed byN. Wiener, J. Math. Phys. 2, 131 (1923); Proc. London Math. Soc. 22, 454 (1924); Acta Math. 55,117 (1930);N.
Wiener, Generalized Harmonic Analysis and Tauberian Theorems, MIT Press, Cambridge,Mass., 1964,after some earlier attempts byP.J. Daniell, Ann. Math. 19, 279; 20, 1 (1918); 20, 281 (1919); 21, 203 (1920);discussed inM. Kac, Bull. Am. Math. Soc. 72, Part II, 52 (1966).Note that even the name path integral appears in Wiener’s 1923 paper.Further important papers areI.M. Gelfand and A.M. Yaglom, J.
Math. Phys. 1, 48 (1960);S.G. Brush, Rev. Mod. Phys. 33, 79 (1961);E. Nelson, J. Math. Phys. 5, 332 (1964);A.M. Arthurs, ed., Functional Integration and Its Applications, Clarendon Press, Oxford, 1975,C. DeWitt-Morette, A. Maheshwari, and B.L. Nelson, Phys. Rep. 50, 255 (1979);D.C. Khandekar and S.V. Lawande, Phys. Rep.
137, 115 (1986).The general harmonic path integral is derived inM.J. Goovaerts, Physica 77, 379 (1974); C.C. Grosjean and M.J. Goovaerts, J. Comput. Appl.Math. 21, 311 (1988); G. Junker and A. Inomata, Phys. Lett. A 110, 195 (1985).The Feynman path integral was applied to thermodynamics byM. Kac, Trans.
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