Kleinert - Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets - ed.4 - 2006 (523104), страница 47
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Am. Math. Soc. 65, 1 (1949);M. Kac, Probability and Related Topics in Physical Science, Interscience, New York, 1959, Chapter IV.A good selection of earlier textbooks on path integrals isR.P. Feynman, A.R. Hibbs, Quantum Mechanics and Path Integrals, McGraw Hill, New York 1965,L.S.
Schulman, Techniques and Applications of Path Integration, Wiley-Interscience, New York,1981,F.W. Wiegel, Introduction to Path-Integral Methods in Physics and Polymer Science, World Scientific, Singapore, 1986.G. Roepstorff, Path Integral Approach to Quantum Physics, Springer, Berlin, 1994.The path integral in phase space is reviewed byC.
Garrod, Rev. Mod. Phys. 38, 483 (1966).2062 Path Integrals — Elementary Properties and Simple SolutionsThe path integral for the most general quadratic action has been studied in various ways byD.C. Khandekar and S.V. Lawande, J. Math. Phys. 16, 384 (1975); 20, 1870 (1979);V.V. Dodonov and V.I. Manko, Nuovo Cimento 44B, 265 (1978);A.D. Janussis, G.N. Brodimas, and A. Streclas, Phys. Lett. A 74, 6 (1979);C.C.
Gerry, J. Math. Phys. 25, 1820 (1984);B.K. Cheng, J. Phys. A 17, 2475 (1984);G. Junker and A. Inomata, Phys. Lett. A 110, 195 (1985);H. Kleinert, J. Math. Phys. 27, 3003 (1986) (http://www.physik.fu-berlin.de/~kleinert/144).The caustic phenomena near the singularities of the harmonic oscillator amplitude at tb − ta =integer multiples of π/ω, in particular the phase of the fluctuation factor (2.162), have been discussed byJ.M. Souriau, in Group Theoretical Methods in Physics, IVth International Colloquium, Nijmegen,1975, ed.
by A. Janner, Springer Lecture Notes in Physics, 50;P.A. Horvathy, Int. J. Theor. Phys. 18, 245 (1979).See in particular the references therein.The amplitude for the freely falling particle is discussed inG.P. Arrighini, N.L. Durante, C. Guidotti, Am. J. Phys. 64, 1036 (1996);B.R. Holstein, Am. J. Phys. 69, 414 (1997).For the Baker-Campbell-Hausdorff formula seeJ.E. Campbell, Proc. London Math. Soc. 28, 381 (1897); 29, 14 (1898);H.F. Baker, ibid., 34, 347 (1902); 3, 24 (1905);F.
Hausdorff, Berichte Verhandl. Sächs. Akad. Wiss. Leipzig, Math. Naturw. Kl. 58, 19 (1906);W. Magnus, Comm. Pure and Applied Math 7, 649 (1954), Chapter IV;J.A. Oteo, J. Math. Phys. 32, 419 (1991);See also the internet addressE.W. Weisstein, http://mathworld.wolfram.com/baker-hausdorffseries.html.The Zassenhaus formula is derived inW.
Magnus, Comm. Pure and Appl. Mathematics, 7, 649 (1954); C. Quesne, Disentangling qExponentials, (math-ph/0310038).For Trotter’s formula see the original paper:E. Trotter, Proc. Am. Math. Soc. 10, 545 (1958).The mathematical conditions for its validity are discussed byE. Nelson, J. Math. Phys. 5, 332 (1964);T. Kato, in Topics in Functional Analysis, ed. by I. Gohberg and M. Kac, Academic Press, NewYork 1987.Faster convergent formulas:M. Suzuki, Comm. Math. Phys. 51, 183 (1976); Physica A 191, 501 (1992);H. De Raedt and B. De Raedt, Phys.
Rev. A 28, 3575 (1983);W. Janke and T. Sauer, Phys. Lett. A 165, 199 (1992).See alsoM. Suzuki, Physica A 191, 501 (1992).The path integral representation of the scattering amplitude is developed inW.B. Campbell, P. Finkler, C.E. Jones, and M.N. Misheloff, Phys. Rev. D 12, 12, 2363 (1975).See also:H.D.I. Abarbanel and C. Itzykson, Phys. Rev. Lett. 23, 53 (1969);R. Rosenfelder, see Footnote 37.The alternative path integral representation in Section 2.18 is due toM. Roncadelli, Europhys.
Lett. 16, 609 (1991); J. Phys. A 25, L997 (1992);H. Kleinert, PATH INTEGRALSNotes and ReferencesA. Defendi and M. Roncadelli, Europhys. Lett. 21, 127 (1993).207H. Kleinert, PATH INTEGRALSNovember 20, 2006 ( /home/kleinert/kleinert/books/pathis3/pthic3.tex)Aκίνητ α κινι̂ς.You stir what should not be stirred.Herodotus3External Sources, Correlations,and Perturbation TheoryImportant information on every quantum-mechanical system is carried by the correlation functions of the path x(t). They are defined as the expectation values of products of path positions at different times, x(t1 ) · · · x(tn ), to be calculated as functionalaverages. Quantities of this type are observable in simple scattering experiments.The most efficient extraction of correlation functions from a path integral proceedsby adding to the Lagrangian an external time-dependent mechanical force term disturbing the system linearly, and by studying the response to the disturbance.
Asimilar linear term is used extensively in quantum field theory, for instance in quantum electrodynamics where it is no longer a mechanical force, but a source of fields,i.e., a charge or a current density. For this reason we shall call this term genericallysource or current term.In this chapter, the procedure is developed for the harmonic action, where alinear source term does not destroy the solvability of the path integral. The resultingamplitude is a simple functional of the current. Its functional derivatives will supplyall correlation functions of the system, and for this reason it is called the generatingfunctional of the theory.
It serves to derive the celebrated Wick rule for calculatingthe correlation functions of an arbitrary number of x(t). This forms the basis forperturbation expansions of anharmonic theories.3.1External SourcesConsider a harmonic oscillator with an actionAω =ZtbtadtM 2(ẋ − ω 2 x2 ).2(3.1)Let it be disturbed by an external source or current j(t) coupled linearly to theparticle coordinate x(t). The source action isAj =Ztbtadt x(t)j(t).208(3.2)2093.1 External SourcesThe total actionA = Aω + Aj(3.3)is still harmonic in x and ẋ, which makes it is easy to solve the path integral inthe presence of a source term. In particular, the source term does not destroythe factorization property (2.146) of the time evolution amplitude into a classicalamplitude eiAj,cl /h̄ and a fluctuation factor Fω,j (tb , ta ),(xb tb |xa ta )jω = e(i/h̄)Aj,cl Fω,j (tb , ta ).(3.4)Here Aj,cl is the action for the classical orbit xj,cl (t) which minimizes the total actionA in the presence of the source term and which obeys the equation of motionẍj,cl (t) + ω 2 xj,cl (t) = j(t).(3.5)In the sequel, we shall first work with the classical orbit xcl (t) extremizing the actionwithout the source term:xcl (t) =xb sin ω(t − ta ) + xa sin ω(tb − t).sin ω(tb − ta )(3.6)All paths will be written as a sum of the classical orbit xcl (t) and a fluctuation δx(t):x(t) = xcl (t) + δx(t).(3.7)Then the action separates into a classical and a fluctuating part, each of whichcontains a source-free and a source term:A = Aω + Aj ≡ Acl + Afl= (Aω,cl + Aj,cl ) + (Aω,fl + Aj,fl).(3.8)The time evolution amplitude can be expressed as(xb tb |xa ta )jω(i/h̄)Acl= eZiADx exph̄ fl(i/h̄)(Aω,cl +Aj,cl )= eZiDx exp (Aω,fl + Aj,fl) .h̄(3.9)The classical action Aω,cl is known from Eq.
(2.152):Aω,cl =hiMω(x2b + x2a ) cos ω(tb − ta ) − 2xb xa .2 sin ω(tb − ta )(3.10)The classical source term is known from (3.6):Aj,cl ==Ztbtadt xcl (t)j(t)1sin ω(tb − ta )Z(3.11)tbtadt[xa sin ω(tb − t) + xb sin ω(t − ta )]j(t).2103 External Sources, Correlations, and Perturbation TheoryConsider now the fluctuating part of the action, Afl = Aω,fl + Aj,fl. Since xcl (t)extremizes the action without the source, Afl contains a term linear in δx(t). Aftera partial integration [making use of the vanishing of δx(t) at the ends] it can bewritten asZ tbM Z tb000dt δx(t)j(t),dtdt δx(t)Dω2 (t, t )δx(t ) +Afl =2 tata(3.12)where Dω2 (t, t0 ) is the differential operatorDω2 (t, t0 ) = (−∂t2 − ω 2 )δ(t − t0 ) = δ(t − t0 )(−∂t20 − ω 2 ),t, t0 ∈ (ta , tb ).(3.13)It may be considered as a functional matrix in the space of the t-dependent functionsvanishing at ta , tb .
The equality of the two expressions is seen as follows. By partialintegrations one hasZtbtadt f (t)∂t2 g(t) =Ztbtadt ∂t2 f (t)g(t),(3.14)for any f (t) and g(t) vanishing at the boundaries (or forperiodic functions in theR tbinterval). The left-hand side can directly be rewritten as ta dtdt0 f (t)δ(t−t0 )∂t20 g(t0 ),Rthe right-hand side as ttab dtdt0 ∂t2 f (t)δ(t − t0 )g(t0), and after further partial integraRtions, as dtdt0 f (t)∂t2 δ(t − t0 )g(t).The inverse Dω−12 (t, t0 ) of the functional matrix (3.13) is formally defined by therelationZtbtadt0 Dω2 (t00 , t0 )Dω−12 (t0 , t) = δ(t00 − t),t00 , t ∈ (ta , tb ),(3.15)which shows that it is the standard classical Green function of the harmonic oscillatorof frequency ω:Gω2 (t, t0 ) ≡ Dω−12 (t, t0 ) = (−∂t2 − ω 2 )−1 δ(t − t0 ),t, t0 ∈ (ta , tb ).(3.16)This definition is not unique since it leavesroom for an additional arbitrary solutionRH(t, t0 ) of the homogeneous equation ttab dt0 Dω2 (t00 , t0 )H(t0 , t) = 0.
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