Kleinert - Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets - ed.4 - 2006 (523104), страница 10
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The right-hand side shows the relative deviation from the 1/tbehavior in percent. . . . . . . . . . . . . . . . . . . . . . . . . . .140220.20 Stationary distribution of variances . . . . . . . . . . . . . . . . . .141720.21 Probability distribution of logarithm of stock price for different timescales . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . .142120.22 Universal distribution of Dow-Jones data points . . . . . . . . . . .142320.23 Slope −d log P (x t |xa ta )/dx of exponential tail of distribution . . .142420.24 Fraction f (∆t) of total probability contained in Gaussian part ofP (x t |xa ta ) . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . .142520.25 Singularities of H(p, ∆t) in complex p-plane . . . . . . . . . . . . . .142620.26 Left: Comparison of GARCH(1,1) process with S&P 500 minutedata. Right: Comparison of GARCH(1,1), Heston, and Gaussianmodels with daily market data . . . . . . . . . . . . .
. . . . . . . .142920.27 Dependence of call option price O on stock price S, strike price E,and volatility σ . . . . . . . . . . . . . . . . . . . . . . . . . . . . .144020.28 Smile deduced from options . . . . . . . . . . . . . . . . . . . . . . .144020.29 Dependence of call price O(S, t) on stock price S for truncated Lévydistribution . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .144320.30 Dependence of call price O(S, v, t) on stock price S . . . . . . . . .144620.31 Comparison of large-x expansions containing different numbers ofterms in truncated Lévy distribution . . . . . . . .
. . . . . . . . .1451List of Tables3.1 Expansion coefficients for the ground-state energy of the oscillatorwith cubic and quartic anharmonicity . . . . . . . . . . . . . . . . . . 3583.2 Expansion coefficients for the ground-state energy of the oscillatorwith cubic and quartic anharmonicity in presence of an external current3593.3 Effective potential for the oscillator with cubic and quartic anharmonicity, expanded in the coupling constant g . . .
. . . . . . . . . . 3614.1 Particle energies in purely anharmonic potential gx4 /4 for n =0, 2, 4, 6, 8, 10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4155.1 Comparison of variational energy with exact ground state energy .5.2 Example for competing leading six terms in large-B expansion . . .5.3 Perturbation coefficients up to order B 6 in weak-field expansions ofvariational parameters, and binding energy . . . . . . . . . . . . . .5.4 Approach of variational energies to Bohr-Sommerfeld approximation5.5 Energies of the nth excited states of anharmonic oscillator for variouscoupling strengths . . . .
. . . . . . . . . . . . . . . . . . . . . . . .5.6 Second- and third-order approximations to ground state energy ofanharmonic oscillator . . . . . . . . . . . . . . . . . . . . . . . . .5.7 Free energy of anharmonic oscillator for various coupling strengthsand temperatures . . . . . . . .
. . . . . . . . . . . . . . . . . . .5.8 Comparison of the variational approximations WN at T = 0 for increasing N with the exact ground state energy . . . . . . . . . . .5.9 Coefficients bn of strong-coupling expansion of ground state energy ofanharmonic oscillator . . . .
. . . . . . . . . . . . . . . . . . . . . .5.10 Equations determining coefficients bn in strong-coupling expansion .5.11 Higher approximations to excited energy with n = 8 of anharmonicoscillator at various coupling constants g . . . . . . . . . . . . . . .5.12 Numerical results for variational parameters and energy . . . . . . ..
463. 482. 484490. 491. 496. 500. 506. 510. 513. 517. 5386.1 First eight variational functions fN (c) . . . . . . . . . . . . . . . . . . 59116.1 Numbers of simple and compound knots . . . . . . . . . . . . . . . .110316.2 Tables of underpasses and directions of overpassing lines for trefoilknot and knot 41 . . . . . . .
. . . . . . . . . . . . . . . . . . . . . .110516.3 Alexander, Jones, and HOMFLY polynomials for smallest simpleknots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1106xlxli16.4 Kauffman polynomials in decomposition of trefoil knot . . .
. . . . .111016.5 Alexander polynomials A(s, t) and HOMFLY polynomials H(t, α) forsimple links of two closed curves up to 8 minimal crossings . . . . . .112117.1 Comparison between exact perturbation coefficients, semiclassicalones, and those from our variational approximation . .
. . . . . . . .122317.2 Coefficients of semiclassical expansion around classical solution . . . .122520.1 Parameters of equations with fluctuating variance obtained from fitsto Dow-Jones data . . . . . . . . . . . . . . . . . . . . . . . . . . . .1453xliiH. Kleinert, PATH INTEGRALSH.
Kleinert, PATH INTEGRALSNovember 20, 2006 ( /home/kleinert/kleinert/books/pathis3/pthic1.tex)Ay, call it holy ground,The soil where first they trod!F. D. Hemans (1793-1835), Landing of the Pilgrim Fathers1FundamentalsPath integrals deal with fluctuating line-like structures. These appear in nature in avariety of ways, for instance, as particle orbits in spacetime continua, as polymers insolutions, as vortex lines in superfluids, as defect lines in crystals and liquid crystals.Their fluctuations can be of quantum-mechanical, thermodynamic, or statistical origin.
Path integrals are an ideal tool to describe these fluctuating line-like structures,thereby leading to a unified understanding of many quite different physical phenomena. In developing the formalism we shall repeatedly invoke well-known concepts ofclassical mechanics, quantum mechanics, and statistical mechanics, to be summarized in this chapter. In Section 1.13, we emphasize some important problems ofoperator quantum mechanics in spaces with curvature and torsion. These problemswill be solved in Chapters 10 and 8 by means of path integrals.11.1Classical MechanicsThe orbits of a classical-mechanical system are described by a set of time-dependentgeneralized coordinates q1 (t), . .
. , qN (t). A LagrangianL(qi , q̇i , t)(1.1)depending on q1 , . . . , qN and the associated velocities q̇1 , . . . , q̇N governs the dynamics of the system. The dots denote the time derivative d/dt. The Lagrangian is atmost a quadratic function of q̇i . The time integralA[qi ] =Ztbtadt L(qi (t), q̇i (t), t)(1.2)of the Lagrangian along an arbitrary path qi (t) is called the action of this path. Thepath being actually chosen by the system as a function of time is called the classicalpath or the classical orbit qicl (t). It has the property of extremizing the action incomparison with all neighboring pathsqi (t) = qicl (t) + δqi (t)1Readers familiar with the foundations may start directly with Section 1.13.1(1.3)21 Fundamentalshaving the same endpoints q(tb ), q(ta ). To express this property formally, oneintroduces the variation of the action as the linear term in the Taylor expansion ofA[qi ] in powers of δqi (t):δA[qi ] ≡ {A[qi + δqi ] − A[qi ]}lin .(1.4)The extremal principle for the classical path is thenδA[qi ]qi (t)=qicl (t)=0(1.5)for all variations of the path around the classical path, δqi (t) ≡ qi (t) − qicl (t), whichvanish at the endpoints, i.e., which satisfyδqi (ta ) = δqi (tb ) = 0.(1.6)Since the action is a time integral of a Lagrangian, the extremality property canbe phrased in terms of differential equations.
Let us calculate the variation of A[qi ]explicitly:δA[qi ] = {A[qi + δqi ] − A[qi ]}lin=Ztb=Ztb=Ztbtatatadt {L (qi (t) + δqi (t), q̇i (t) + δ q̇i (t), t) − L (qi (t), q̇i (t), t)}lin)dt(∂L∂Lδqi (t) +δ q̇ (t)∂qi∂ q̇i idt(tbd ∂L∂L∂L−δqi (t) +δqi (t) .∂qi dt ∂ q̇i∂ q̇ita)(1.7)The last expression arises from a partial integration of the δ q̇i term. Here, as in theentire text, repeated indices are understood to be summed (Einstein’s summationconvention).
The endpoint terms (surface or boundary terms) with the time t equalto ta and tb may be dropped, due to (1.6). Thus we find for the classical orbit qicl (t)the Euler-Lagrange equations:d ∂L∂L=.(1.8)dt ∂ q̇i∂qiThere is an alternative formulation of classical dynamics which is based on aLegendre-transformed function of the Lagrangian called the HamiltonianH≡∂Lq̇ − L(qi , q̇i , t).∂ q̇i i(1.9)Its value at any time is equal to the energy of the system.
According to the generaltheory of Legendre transformations [1], the natural variables which H depends onare no longer qi and q̇i , but qi and the generalized momenta pi , the latter beingdefined by the N equations∂L(qi , q̇i , t).(1.10)pi ≡∂ q̇iH. Kleinert, PATH INTEGRALS31.1 Classical MechanicsIn order to express the Hamiltonian H (pi , qi , t) in terms of its proper variables pi , qi ,the equations (1.10) have to be solved for q̇i ,q̇i = vi (pi , qi , t).(1.11)This is possible provided the Hessian metricHij (qi , q̇i , t) ≡∂2L(qi , q̇i , t)∂ q̇i ∂ q̇j(1.12)is nonsingular. The result is inserted into (1.9), leading to the Hamiltonian as afunction of pi and qi :H (pi , qi , t) = pi vi (pi , qi , t) − L (qi , vi (pi , qi , t) , t) .(1.13)In terms of this Hamiltonian, the action is the following functional of pi (t) and qi (t):A[pi, qi ] =Ztbtahidt pi (t)q̇i (t) − H(pi (t), qi (t), t) .(1.14)This is the so-called canonical form of the action.
The classical orbits are now specclified by pcli (t), qi (t). They extremize the action in comparison with all neighboringorbits in which the coordinates qi (t) are varied at fixed endpoints [see (1.3), (1.6)]whereas the momenta pi (t) are varied without restriction:qi (t) = qicl (t) + δqi (t),δqi (ta ) = δqi (tb ) = 0,(1.15)pi (t) = pcli (t) + δpi (t).In general, the variation isδA[pi , qi ] =Ztb=Ztbtata"∂H∂Hdt δpi (t)q̇i (t) + pi (t)δ q̇i (t) −δpi −δq∂pi∂qi idt("#"#∂H∂Hq̇i (t) −δpi − ṗi (t) +δqi∂pi∂qi)#(1.16)tb+ pi (t)δqi (t) .tbclSince this variation has to vanish for the classical orbits, we find that pcli (t), qi (t)must be solutions of the Hamilton equations of motion∂H,∂qi∂H=.∂piṗi = −q̇i(1.17)These agree with the Euler-Lagrange equations (1.8) via (1.9) and (1.10), as caneasily be verified.
The 2N-dimensional space of all pi and qi is called the phasespace.41 FundamentalsAs a particle moves along a classical trajectory, the action changes as a functionof the end positions (1.16) byδA[pi, qi ] = pi (tb )δqi (tb ) − pi (ta )δqi (ta ).(1.18)An arbitrary function O(pi (t), qi (t), t) changes along an arbitrary path as follows:∂O∂Od∂Oṗi +q̇i +O (pi (t), qi (t), t) =.dt∂pi∂qi∂t(1.19)If the path coincides with a classical orbit, we may insert (1.17) and find∂H ∂O ∂O ∂H ∂OdO−+=dt∂pi ∂qi∂pi ∂qi∂t∂O≡ {H, O} +.∂t(1.20)Here we have introduced the symbol {. . .
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