Eikonal Approximation
K. V. Shajesh∗ Department of Physics and Astronomy, University of Oklahoma, 440 W. Brooks St., Norman, OK - 73019, U. S. A. This is a study report on eikonal approximations which was undertaken under the guidance of Kimball Milton. This report was submitted to partially fulfill the requirements for the general examination.
Contents
I Introduction 1
II Scattering in quantum mechanics 3 A Formulation of the scattering problem ...... 3 B Identities satis ed by the scattering amplitude ...... 5 1 Dynamical reversibility theorem ...... 5 2 Optical theorem ...... 5
III Partial wave expansion 6 A Method of partial wave expansion ...... 7 B Scattering length of a potential ...... 8 C Inverted nite spherical well potential ...... 8
IV Eikonal approximation in quantum mechanics 9 A Formalism for the eikonal approximation ...... 9 B Validity of the eikonal approximation ...... 11 C Zeroth order eikonal approximation ...... 11 D Examples ...... 13 1 Inverted nite spherical well potential ...... 13 2 Yukawa potential ...... 14
V Quantum electrodynamics 15 A Schwinger’s functional di erential equations for quantum electrodynamics ...... 15 B A formal solution to Schwinger’s functional di erential equations ...... 16 C Electron-electron scattering ...... 18
VI Approximation to electron-electron scattering 19 A Quenched or ladder approximations ...... 19 µ B Eikonal approximation to G+[x, y; A ] ...... 19
I. INTRODUCTION
The etymology of the word eikon traces it back to the word eikenai which is the transliteration of the word ικναι [1] in the Greek language meaning ‘to resemble’ [2]. In the Greek language it evolved into the word eiko n which is the transliteration of the word ικoν [1] meaning ‘image’ [2]. It was borrowed into the Latin language and later into the English language as eikon. In the English language it transformed into the words icon and ikon which are the variants of the word eikon. Notice that throughout its evolution it has meant ‘image.’ The main goal of this report is to understand the eikonal approximation in quantum mechanics and quantum eld theories. Approximations play a very important role in the understanding of processes that cannot be solved
∗Email: [email protected]
1 exactly. The Born approximation in quantum mechanics is an example of an approximation that has been extensively used for studying low energy processes. In quantum eld theories involving coupling constants smaller than one we use the standard weak-coupling perturbation series which is parallel in its approach to the Born approximation in quantum mechanics. In the 1950’s and 1960’s when high energy physics was ascending towards its peak, it was realized among the high energy physicists of those times that the Born approximation is not a valid approximation for studying processes involving high energies. This period in fact was the golden age in the development of the eikonal approximation in quantum mechanics and quantum eld theories. The descendents of this era took the theory of eikonal approximation for granted in quantum mechanics and quantum eld theories. There was proli c activity in the application of eikonal approximation in high energy physics, especially in QCD. The eikonal approximation was not born in the study of quantum mechanics. It originated far back in optics. Light we know obeys Maxwell’s equations. In terms of Maxwell’s equations light is understood as a wave obeying a wave equation. But two hundred years before Maxwell wrote down his equations scientists understood re ection and refraction of light which was extensively studied under the branch of science called ray optics. Today in elementary school we learn ray optics without ever introducing Maxwell’s equations. In ray optics we assume that light travels in a straight line. This assumption works ne as long as the size of the obstacle is large compared to the wavelength of light. This is called the eikonal approximation in optics. Here we make contact with the meaning of the word eikon in the sense that eikon meaning ‘image’ is formed by light only in the straight line approximation. In processes involving di raction we encounter the limit of the validity of the eikonal approximation in optics. We quickly switch over to Maxwell’s equations to study di raction. Optics is described by Maxwell’s equations which can be written as a wave equation with the dispersion relation given by ω = kc. Similarly, quantum mechanics is described in terms of Schrodinger’s equation which is a di usion hk¯ 2 equation (in imaginary time) with a dispersion relation given by ω = 2m . With this correspondence we ask the question, can we not have a corresponding eikonal approximation in quantum mechanics? Yes we can. The eikonal approximation in quantum mechanics works for processes involving the scattering of particles with large incoming momentum and when the scattering angle is very small. In the language of di erential equations, the main advantage the eikonal approximation o ers is that the equations reduce to a di erential equation in a single variable. This reduction into a single variable is the result of the straight line approximation or the eikonal approximation which allows us to choose the straight line as a special direction. The early steps involved in the eikonal approximation in quantum mechanics are very closely related to the WKB approximation in quantum mechanics. The WKB approximation involves an expansion in terms of Planck’s constant h. The WKB approximation also reduces the equations into a di erential equation in a single variable. But the complexity involved in the WKB approximation is that this variable is described by the trajectory of the particle which in general is complicated. The advantage of the eikonal approximation is in the classical trajectory being a straight line. Thus in this manner the eikonal approximation is a very stringent semi-classical limit. A very comprehensive collection of work on scattering theory in general with a very extensive bibliography which covers scattering theory in both electromagnetism and quantum mechanics is the book by Roger G. Newton [3]. A couple of textbooks among many which I have used are [6] and [7]. An extensive list of literature on the theory of eikonal approximation in quantum mechanics and quantum eld theories developed between the years 1950 and 1970 is available in [4]. It is unanimously accepted by everyone that R. J. Glauber’s lecture notes on eikonal approximation [5] is the best available work on the subject. Glauber’s lecture notes addresses the question of the conditions for the validity of the eikonal approximations in quantum mechanics. In the section on eikonal approximations in this article starting out from his ideas we derive the conditions for the validity of the eikonal approximations more concretely. We verify our conclusions by evaluating the total scattering cross section for the simplest potential (an inverted nite spherical well) using the eikonal approximation and by comparing it with the result obtained by the partial wave expansion. After our study of eikonal approximation in quantum mechanics we concentrate on quantum electrodynamics. To study the eikonal approximation in quantum electrodynamics we need to rst write the eld theoretical equations in a suitable form. Schwinger’s formalism for quantum electrodynamics is the most suitable for this purpose. We develop Schwinger’s formalism in detail based on [15{18] in section V. In section VI we concentrate on electron-electron scattering. We introduce the quenched approximation. We outline the derivation of the electron-electron scattering in the eikonal approximation based on [19,20]. We point out the similarity of the result of electron-electron scattering obtained in the eikonal approximation to the non-relativistic scattering amplitude due to the Coulomb potential. The terminology used in this study report is the following
f(θ, φ) Scattering amplitude ≡ σ Scattering cross section scatt. ≡
2 σ Absorption cross section abs. ≡ σ σ + σ . tot. ≡ scatt. abs.
II. SCATTERING IN QUANTUM MECHANICS
A scattering process in quantum mechanics is described by the solution of the Schroedinger equation
2 h 2 ∂ + V (~r, t) ψ0(~r, t) = ih ψ0(~r, t) (1) −2m∇ − ∂t with the boundary conditions dictated by the requirement that the wave function ψ0(~r, t) must have a component that h¯2k2 involves an incident plane wave with energy E = 2m moving in the positive z direction and another component that involves a spherical outgoing wave. It should be emphasized that in scattering problems we do not require ψ0(~r, t) to go to zero at ~r . In fact the scattering amplitude which is the quantity of interest is contained in the ~r → ∞ → ∞ (asymptotic) part of ψ0(~r, t).
A. Formulation of the scattering problem
For the case when the initial beam can be described by a state of de nite energy we can write
i Et ψ0(~r, t) = e− h¯ ψ(~r) (2) where ψ(~r) satis es the di erential equation
2 + k2 ψ(~r) = U(~r)ψ(~r) (3) ∇ with the boundary conditions on ψ(~r) dictated by the boundary conditions on ψ0(~r, t). We have used the notation 2 2mE 2mV (~r) k = h¯2 and U(~r) = h¯2 . The di erential equation for ψ(~r) in eqn. (3) can be rewritten as an integral equation [8] given by
3 ψ(~r) = φ(~r) + d r0 G0(~r, ~r0)U(~r0)ψ(~r0) (4) Z where φ(~r) satis es the potential free equation
2 + k2 φ(~r) = 0 (5) ∇ and the Green’s function G0(~r, ~r0) is the solution to
2 2 (3) + k G (~r, ~r0) = δ (~r ~r0). (6) ∇ 0 −
The boundary conditions on φ(~r) and G 0(~r, ~r0) are prescribed by the boundary conditions on ψ(~r). The integral 1 equation in eqn. (4) is called the Lippmann-Schwinger equation [9] . The solutions to φ(~r) and G0(~r, ~r0) are
i~k ~r i~k ~r φ(~r) = A0 e · + B0 e− · (7)
ik ~r ~r0 ik ~r ~r0 1 e | − | e− | − | G (~r, ~r0) = A + B (8) 0 −4π ~r ~r ~r ~r " | − 0| | − 0| #
1Schwinger’s view of the paper was [10]: ‘. . . I thought the importance of the paper was the variational principle . . . the so- called Lippmann-Schwinger scattering equation is to me conventional scattering theory written in operator notation. Nothing new. But that is what everybody paid attention to.’
3 with the constraint A + B = 1. Using eqn. (7) and eqn. (8) in eqn. (4) the Lippmann-Schwinger equation takes the form
ik ~r ~r0 ik ~r ~r0 i~k ~r i~k ~r 1 3 e | − | e− | − | ψ(~r) = A e · + B e− · d r0 A + B U(~r0)ψ(~r0). (9) 0 0 − 4π ~r ~r ~r ~r Z " | − 0| | − 0| # Imposing the boundary conditions that the wave function consists of a component that is a plane wave moving in the positive z direction and another that is an outgoing spherical wave we get
ik ~r ~r0 i~k ~r 1 3 e | − | ψ(~r) = A e · d r0 U(~r0)ψ(~r0). (10) 0 − 4π ~r ~r Z | − 0| As we emphasized before the information related to the scattering amplitude is contained in the asymptotic region of the wave function. In most of the problems of interest the potential V (~r) is con ned to a nite volume in space and the detectors are very far from the region containing the potential. For these cases we can safely conclude r0 r and thus approximate
2 ~r ~r0 r0 ~r ~r0 = r · + O . (11) | − | − r r " # We can thus write
~r ~r i~k ~r 1 3 1 ik(r · 0 ) ψr (~r) = A0 e · d r0 e − r U(~r0)ψ(~r0) (12) →∞ − 4π r Z ikr i~k ~r e = e · + f(θ, φ) (13) r where we have set A0 = 1 so that
1 3 i~k0 ~r0 f(θ, φ) = d r0 e− · U(~r0)ψ(~r0) (14) −4π Z ~ ~r can be interpreted as the scattering amplitude, where k0 = k r . The illustration of the variables involved is shown in g. 1.
y x ~r = (r sin θ cos φ, r sin θ sin φ, r cos θ) ~ ~k k θ φ, k θ φ, k θ b0 0 = ( sin cos sin sin cos ) ¢ ~k = (0, 0, k) ¢ ~ ~r ~r0 = (b0 cos φ0, b0 sin φ0, z0) : k0 = k ¢ φ0 r ¢ ¢ ¨θ ¢ - ~ © k, z
FIG. 1. Illustration of the various variables used in the calculation. Note that ~r, ~k0, and ~k are in spherical polar coordinates and ~r0 is in cylindrical polar coordinates.
Often it is more useful to denote f(θ, φ) as a function of ~k and ~k0 and thus write f(θ, φ) = f(~k0,~k). Observe that even though the information related to f(θ, φ) is contained in the asymptotic region of ψ(~r) the only contributions to f(θ, φ) in eqn. (14) comes from regions where the potential is not zero. To complete the formulation of the scattering problem we de ne the scattering cross section as
σ = d f(θ, φ) 2 (15) scatt. | | Z
4
1 and the total cross section as
σtot. = σscatt. + σabs. (16) where σabs. is the absorption cross section which will be explicitly de ned in eqn. (27). To summarize this section on formulation of the scattering problem, we have concluded that a prescription to evaluate the scattering amplitude is to use
1 3 i~k0 ~r0 f(θ, φ) = d r0 e− · U(~r0)ψ(~r0) (17) −4π Z where ψ(~r) is determined by solving
2 + k2 ψ(~r) = U(~r)ψ(~r) (18) ∇ under the boundary conditions described after eqn. (1).
B. Identities satisfied by the scattering amplitude
Starting from the Schroedinger equation we shall derive two identities satis ed by the scattering amplitude. The rst, the dynamical reversibility theorem was rst derived by R. J. Glauber and V. Schomaker [11] in 1953. The second, the optical theorem, is a statement of probability conservation in quantum mechanics written as a continuity equation in reference to the scattering amplitude. The optical theorem was rst derived by E. Feenberg [12] in 1932.
1. Dynamical reversibility theorem
In the Schroedinger equation if we write the wave function as
i Et ψ0(~r, t;~k) = e− h¯ ψ(~r;~k) (19) where 2mE = h 2k2, we can quickly derive the following identity
ψ(~r; ~k ) 2ψ(~r; k~ ) ψ(~r; k~ ) 2ψ(~r; ~k ) = (k2 k2)ψ(~r; k~ )ψ(~r; k~ ). (20) − 2 ∇ 1 − 1 ∇ − 2 1 − 2 1 2 Evaluating the above expression on a sphere of radius r for the case k~ = k~ we have → ∞ | 1 | | 2 |
2 2 dS~ ψr (~r; ~k2) ψr (~r; k~1) ψr (~r; k~1) ψr (~r; ~k2) = 0. (21) r · →∞ − ∇ →∞ − →∞ ∇ →∞ − I →∞ h i Using eqn. (13) in the above expression and evaluating the surface integrals after taking the limit r we get [5] → ∞ f(~k ,~k ) = f( ~k , ~k ) (22) 2 1 − 1 − 2 where we have used the notation we de ned for f(θ, φ) in the paragraph after eqn. (14). Thus the scattering amplitude for a process going from ~k to ~k is identical to that of a process going from ~k to ~k . This is dynamical reversibility. 1 2 − 2 − 1
2. Optical theorem
In a very similar manner as we did in the earlier section (in this case we take the complex conjugate of the wave function) we arrive at the following continuity equation
∂ρ(~r) + ~ ~j(~r) = s(~r) (23) ∂t ∇ · where
5 ρ(~r) = ψ(~r) 2 | | h ~j(~r) = ψ∗(~r) ~ ψ(~r) ψ(~r) ~ ψ∗(~r) 2im ∇ − ∇ 2 h i s(~r) = [Im V (~r)] ψ(~r) 2 . (24) h | | Integrating the continuity equation over a sphere of radius r and noting that the derivative of the ρ term does not contribute because both the states have the same energy →we∞get
~ 2 2 3 dS ψr∗ (~r) ψr (~r) ψr (~r) ψr∗ (~r) = lim d r s(~r). (25) r · →∞ ∇ →∞ − →∞ ∇ →∞ r I →∞ →∞ Z Again using eqn. (13) in the above expression and evaluating the surface integrals after taking the limit r we get [5] → ∞
2π π 2m 4π dφ sin θ dθ f(θ, φ) 2 lim d3r [Im V (~r)] ψ(~r) 2= Imf(θ = 0). (26) | | − r h 2k | | k Z0 Z0 →∞ Z Using eqn. (15) and de ning the absorption scattering cross section as
2m 3 2 σabs. = lim d r [Im V (~r)] ψ(~r) (27) − r h 2k | | →∞ Z we have the optical theorem 4π σ + σ = Imf(θ = 0). (28) scatt. abs. k
III. PARTIAL WAVE EXPANSION
In this article we will be concerned with approximation methods for evaluating the scattering amplitude. In particular we will be interested in the eikonal approximation to the scattering amplitude. After we have made an approximation we would like to be aware of how much we have deviated from the exact result. The closest we can have to an exact result in a scattering problem is the result got by the method of partial wave expansion. We shall thus nd it very useful to use the results obtained from the method of partial wave expansion as a benchmark. The method of partial wave expansion breaks down the initial wavefunction into an in nite sum over angular momentum components labeled by the quantum number l. The contribution to the scattering amplitude from each l is calculated separately and called the partial wave scattering amplitude. The complete scattering amplitude is then obtained by summing over all the partial wave scattering amplitudes. We say it is the closest we can have to an exact expression because the solution cannot in general be written in a closed form. Nevertheless for an incoming beam of a given energy characterized by the parameter k we can always de ne its angular momentum as ka where a is the scattering length de ned in section III B in terms of the partial wave scattering amplitude corresponding to l = 0. It turns out that for most of the potentials the contributions from l terms very much higher than ka converge very rapidly and thus can be neglected. Physically this is understood as, only those beams that have su cient initial energy and those that pass through the scattering length of the potential are de ected. If the method of partial wave expansion gives us the exact solution why do we need approximation methods? Firstly because for almost all potentials (even for the simple Gaussian potential) we need to depend on numerical computational methods to solve the radial di erential equation. Secondly for high energy scattering problems we need to sum over a large range of l. For potentials involving a boundary beyond which the potential is zero it sometimes becomes possible to evade the need for computational methods to a signi cant extent. For the simplest potential of this class, that of an inverted nite square well, the radial di erential equation mentioned above can be solved exactly. Thus we shall solve the inverted nite square well in detail and use it for comparison with the results obtained from other approximation methods.
6 A. Method of partial wave expansion
For a spherically symmetric potential of the form V (~r) = V (r) (29) we can write the solution to eqn. (18) as
∞ l ψ(~r) = (2l + 1)i Rl(r)Pl(cos θ) (30) Xl=0 where where Rl(r) is the solution to the radial di erential equation d2R (r) 2 dR (r) l(l + 1) l + l + k2 U(r) R (r) = 0. (31) dr2 r dr − r2 − l For potentials that die o faster than 1 we can write the solution to eqn. (31) in the r region to be r2 → ∞
lim Rl(r) = Cl cos δl lim jl(kr) sin δl lim ηl(kr) . (32) r r − r →∞ h →∞ →∞ i Substituting eqn. (32) in eqn. (30) and using the identity
ikr cos θ ∞ l e = (2l + 1)i jl(kr)Pl(cos θ) (33) Xl=0 in eqn. (14) we have
ikr e ∞ l f(θ, φ) = (2l + 1)i Pl(cos θ) (Cl cos δl 1) lim jl(kr) Cl sin δl lim ηl(kr) . (34) r r r − →∞ − →∞ Xl=0 h i Using 1 π lim jl(kr) = cos kr (l + 1) r →∞ kr − 2 1 h(l+1) ikr +(il+1) ikr = i− e + i e− (35) 2kr 1 h π i lim ηl(kr) = sin kr (l + 1) r →∞ kr − 2 1 h(l+2) ikr l i ikr = i− e i e− (36) 2kr − ikr ikr h i and equating the coe cients of e and e− in eqn. (34) we get
∞ 1 l (l+1) +iδ f(θ) = (2l + 1)i P (cos θ)i− C e l 1 (37) 2k l l − l=0 X 1 ∞ l +(l+1) iδ 0 = (2l + 1)i P (cos θ)i C e− l 1 . (38) 2k l l − l=0 X iδl The second equation above further reduces to Cl = e which when used in eqn. (37) to eliminate Cl gives
1 ∞ f(θ) = (2l + 1)P (cos θ) e2iδl 1 . (39) 2ik l − l=0 X This is the standard expression for the scattering amplitude given as a sum of partial waves. δl’s are de ned as the phase shifts in Bessel functions (which are trigonometric functions in the r limit) as introduced in eqn. (32). → ∞ Observe that the phase shifts δl’s in eqn. (39) are in principle determined by solving the radial di erential equation in the presence of U(r) which can be solved analytically only for special cases. In practice the phase shifts are evaluated by solving the radial equation numerically. For problems involving high energies a further complication arises because we need to sum over a su ciently high number of partial waves. For the case of a inverted square well potential it is possible to solve the radial equation exactly. In the subsequent section we shall write down an expression for δl for the inverted square well potential. We shall nd it useful to compare the results later when we are doing eikonal approximations.
7 B. Scattering length of a potential
The scattering cross section is obtained by squaring eqn. (39) and integrating over all directions. Using the orthogonality of Legendre polynomials it turns out to be
4π ∞ σ = (2l + 1) sin2 δ . (40) scatt. k2 l 0 X The contribution to the scattering cross section from the s-wave (l = 0) is 4π σ = sin2 δ . (41) { scatt.}l=0 k2 0 The scattering length ‘a’ is de ned as the radius of the sphere that will give a contribution in a classical scattering problem equal to the s-wave contribution above. Correspondingly we write σ = 4πa2. (42) { scatt.}l=0 Thus we get 1 a = sin δ . (43) k 0
Observe that sin δ0 will require the solution to the di erential equation in eqn. (31) for l = 0 which normally can only be solved numerically.
C. Inverted finite spherical well potential
For a spherically symmetric potential of the form V (r) for r < a V (~r) = (44) 0 for r > a we can write
Rl<(r) for r < a Rl(r) = (45) Cl [cos δljl(kr) sin δlηl(kr)] for r > a − where Rl<(r) is the solution to eqn. (31) for r < a. Requiring the wave function and its derivative to be continuous and taking the ratio of the equations got by requiring the continuity we get
dR (r) l< d d a dr ka j (ka) tan δ η (ka) r=a dka l − l dka l = . (46) Rl<(a ) [jl(ka) tan δlηl(ka)] − The above expression can be solved for tan δl if Rl<(r) is known at r = a. For an inverted square well potential de ned by V for r < a V (~r) = (47) 0 for r > a we further have
Rl<(r) = Bljl(αr) (48) where α = k 1 V with V < 1. For this case we can write the explicit expression for the phase shifts to be − E E q N 2(ka, αa) sin2 δ = (49) l N 2(ka, αa) + D2(ka, αa) where
N(ka, αa) = αa jl 1(αa)jl(ka) ka jl(αa)jl 1(ka) − − − D(ka, αa) = αa jl 1(αa)ηl(ka) ka jl(αa)ηl 1(ka). (50) − − − V In g. 2 we plot σtot. versus ka E for the inverted spherical well potential for ka = 500.
8 σtot. πa2
3.5 partial wave (ka = 500) 3
2.5
2
1.5
1
0.5 V ka 20 40 60 80 100 E
V FIG. 2. σtot. versus ka E for ka = 500 using partial wave method.
IV. EIKONAL APPROXIMATION IN QUANTUM MECHANICS
In this section we shall convince ourselves that the eikonal approximation is valid for processes involving small angle scattering and very large incoming momentum. More rigorously the conditions are V 1 and 1 ka 1 . E V/E (V/E)2 In this stringent parameter zone the expression for scattering amplitude takes the form
k 2 i~k0 ~b0 iχ(~b0) f(θ, φ) = d b0 e− · e 1 (51) 2πi − Z h i where
+ 1 2m ∞ χ(~b0) = dz0 V (~b0, z0). (52) −2k h 2 Z−∞ The scattering cross section takes the form
+ 2 ∞ 2 k ∞ σscatt. = 8πa dt t sin dz0 V (~b0, z0) . (53) 0 E Z Z−∞ We shall derive the above formulas in this section.
A. Formalism for the eikonal approximation
In eqns. (17) and (18) we found the prescription for calculating the scattering amplitude to be
1 3 i~k0 ~r0 f(θ, φ) = d r0 e− · U(~r0)ψ(~r0) (54) −4π Z where ψ(~r) is determined by solving
2 + k2 ψ(~r) = U(~r)ψ(~r). (55) ∇ To solve for ψ(~r) let us write
i~k ~r ψ(~r) = e · φ(~r). (56)
With the above substitution and with the choice ~k along the z direction eqn. (55) takes the form
9 ∂ 2ik U(~b, z) φ(~b, z) = 2φ(~b, z) (57) ∂z − −∇ where we have used the notation ~r (~b, z). We can write the formal solution to eqn. (57) as ≡ 2 ∞ 2 φ(~b, z) = η(~b, z) d b0 dz0 G (~b, z, b~ , z0) 0 φ(~b0, z0) (58) − e 0 ∇ Z Z−∞ where η(~b, z) satis es ∂ 2ik U(~b, z) η(~b, z) = 0 (59) ∂z − ~ ~ and Ge(b, z, b0, z0) satis es
∂ (2) 2ik U(~b, z) G (~b, z, b~ , z0) = δ (~b ~b0) δ(z z0) (60) ∂z − e 0 − − ~ ~ ~ with boundary conditions on η(b, z) and Ge(b, z, b0, z0) prescribed by the boundary conditions on ψ(~r). The solutions to eqns. (59) and (60) are z 1 du U(~b,u) ~ 2ik η(b, z) = e −∞ (61) R where we imposed the boundary condition η (~b) = η(~b, z = ) = 1, and 0 −∞ z 1 1 du U(~b,u) ~ ~ (2) ~ ~ 2ik G (b, z, b , z0) = δ (b b0) θ(z z0) e z0 (62) e 0 2ik − − R 1 (2) 1 = δ (~b ~b0) θ(z z0) η(~b, z) η− (~b, z0). (63) 2ik − − Using eqns. (61) and (63) in eqn. (58) we have z 2 ~ ~ 1 1 ~ 2 ∂ ~ φ(b, z) = η(b, z) 1 dz0 η− (b, z0) b + 2 φ(b, z0) (64) − 2ik ∇ ∂z0 Z−∞ which after iteration takes the form z ∂ z ∂ z0 ∂ φ(~b, z) = η(~b, z) 1 + dz0K(~b, z0, ~ b, ) + dz0K(~b, z0, ~ b, ) dz00K(~b, z00, ~ b, ) + . . . (65) ∇ ∂z0 ∇ ∂z0 ∇ ∂z00 " Z−∞ Z−∞ Z−∞ # where the expression for K(~b, z, ~ , ∂ ) acting on an arbitrary function g(z) is given by ∇b ∂z 2 ∂ 1 1 2 ∂ K(~b, z, ~ , ) g(z) = − η− (~b, z) + η(~b, z) g(z). (66) ∇b ∂z 2ik ∇b ∂z2 Using the above series expansion for φ(~b, z) in eqn. (56) and plugging it in eqn. (54) we can write the scattering amplitude as f(θ, φ) = f (0)(θ, φ) + f (1)(θ, φ) + f (2)(θ, φ) + . . . (67) where + (0) 1 2 ∞ i(~k ~k0) ~r0 f (θ, φ) = d b0 dz0 e − · U(~b0, z0) η(~b0, z0) −4π Z Z−∞ + z0 (1) 1 2 ∞ i(~k ~k0) ~r0 f (θ, φ) = d b0 dz0 e − · U(~b0, z0) η(~b0, z0) dz00K(~b0, z00) −4π Z Z−∞ Z−∞ + z0 z00 (2) 1 2 ∞ i(~k ~k0) ~r0 f (θ, φ) = d b0 dz0 e − · U(~b0, z0) η(~b0, z0) dz00K(~b0, z00) dz000K(~b0, z000). (68) −4π Z Z−∞ Z−∞ Z−∞ ~ ~ ∂ ~ ~ ~ We have used K(b, z, b, ∂z ) K(b, z) for compactness. The power in the exponent i(k k0) ~r0 in the above expressions can be explicitly∇ written≡ as − ·
2 θ i(~k ~k0) ~r0 = ikb0 sin θ cos(φ φ0) + ikz02 sin . (69) − · − − 2 The coordinates used are pictorially described in g. 1.
10 B. Validity of the eikonal approximation
Starting from the expression for f (0)(θ, φ) in eqn. (68) and switching to dimensionless variables z at, ~b aw~ → → and V (~b, z) V v(~b, z) where ‘a’ is the scattering length, we can write → a V V V 2 f(θ, φ) = (ka)2 g(0)(ka) + g(1)(ka) + ka g(1)(ka) + . . . (70) −4π E E A E B " # (n) (0) (1) (1) where we have suppressed the θ and φ dependence in g ’s. The explicit expressions for the g , gA and gB are + 2 θ (0) 2 ∞ ikaw0 sin θ cos(φ φ0)+ikat0 2 sin g = d w0 dt0 e− − 2 v(w~ 0, t0) η(w~ 0, t0) Z Z−∞ + (0) 2 ∞ g d w0 dt0 v(w~ 0, t0) (71) | | ≤ | | Z Z−∞ + 2 θ (1) 1 2 ∞ ikaw0 sin θ cos(φ φ0)+ikat02 sin 2 g = d w0 dt0 e− − 2 v(w~ 0, t0) η(w~ 0, t0) A −4 Z Z−∞ + t0 t00 2 θ 1 2 ∞ ikaw0 sin θ cos(φ φ0)+ikat02 sin 2 d w0 dt0 e− − 2 v(w~ 0, t0) η(w~ 0, t0) dt00 dt000 v(w~ 0, t000) −4 ∇w0 Z Z−∞ Z−∞ Z−∞ + + t0 t00 (1) 1 2 ∞ 2 1 2 ∞ 2 g d w0 dt0 v(w~ 0, t0) + d w0 dt0 v(w~ 0, t0) dt00 dt000 v(w~ 0, t000) | A | ≤ 4 | | 4 | | ∇w0 Z Z−∞ Z Z−∞ Z−∞ Z−∞ + t0 2 θ (1) i 2 ∞ ikaw0 sin θ cos(φ φ0)+ikat02 sin 2 g = d w0 dt0 e− − 2 v(w~ 0, t0) η(w~ 0, t0) dt00v(w~ 0, t00) B −8 Z Z−∞ Z−∞ 2 + t0 t00 2 θ i 2 ∞ ikaw0 sin θ cos(φ φ0)+ikat02 sin 2 d w0 dt0 e− − 2 v(w~ 0, t0) η(w~ 0, t0) dt00 dt000 v(w~ 0, t000) −8 ∇w0 Z Z−∞ Z−∞ "Z−∞ # + t0 (1) 1 2 ∞ 2 g d w0 dt0 v(w~ 0, t0) dt00 v(w~ 0, t00) | B | ≤ 8 | | | | Z Z−∞ Z−∞ 2 + t0 t00 1 2 ∞ 2 + d w0 dt0 v(w~ 0, t0) dt00 dt000 w v(w~ 0, t000) . (72) 8 | | ∇ 0 Z Z−∞ Z−∞ Z−∞
It is easy to verify from the above expressions that for potentials that fall away faster than 1 the contributions to t√w g(n)’s are nite. For these potentials it can thus be concluded that major contribution to the scattering amplitude V V 2 comes from the zeroth order for E 1 and ka E 1. We shall later observe that the expression for the scattering amplitude obtained from the zeroth order eikonal approximation satis es the optical theorem only inthe limit ka V . This requirement can be achieved only if we E → ∞ additionally require ka 1 . V/E Thus to summarize, the conditions under which the eikonal approximation holds are V 1 1 1 and ka . (73) E V/E (V/E)2
C. Zeroth order eikonal approximation
Thus for potentials that die out su ciently fast and for conditions V 1 and 1 ka 1 the zeroth E V/E (V/E)2 order contribution to f(θ, φ) is the most signi cant. From eqns. (67), (68) and (61) we can write
+ (0) 1 2 ∞ i(~k ~k0) ~r0 f (θ, φ) = d b0 dz0 e − · U(~b0, z0) η(~b0, z0) −4π Z Z−∞ + z0 2 θ 1 ~ 1 2 ∞ ikb0 sin θ cos(φ φ0)+ikz0 2 sin 2ik du U(b0,u) = d b0 dz0 e− − 2 U(~b0, z0) e . (74) −4π −∞ Z Z−∞ R
11 For large incoming momentum and for small scattering angles most of the contribution to the scattering cross section V comes from angles less than or of order E . Thus in the zeroth order eikonal approximation we can approximate
2 θ ikb0 sin θ cos(φ φ0) + ikz02 sin ikb0θ cos(φ φ0). (75) − − 2 ≈ − − Using eqn. (75) in eqn. (74) we have
2π + z 1 ∞ ∞ 1 0 du U(b~ ,u) (0) ikb0θ cos(φ φ0) ~ 2ik 0 f (θ, φ) = b0db0 dφ0 e− − dz0U(b0, z0) e −∞ . (76) −4π 0 0 Z Z Z−∞ R We notice that the approximation involved in eqn. (75) allows us to carry out the z integral by substituting z0 ~ du U(b0, u) for a new integration variable. After evaluating the z integral we get −∞ R 2π (0) k ∞ ikb0 θ cos(φ φ0) iχ(~b0 ) f (θ, φ) = b0db0 dφ0 e− − e 1 (77) 2πi 0 0 − Z Z h i where + k 1 ∞ χ(~b0) = dz0 V (~b0, z0). (78) − 2 E Z−∞ For the case when the potential is independent of φ variable such that the scattering is symmetric about the z-axis we can further carry out the φ0 integral and thus get
(0) k ∞ iχ(b0 ) f (θ) = b0db0 J0(kb0θ) e 1 (79) i 0 − Z h i where we have used the identity
2π 1 it cos φ J (t) = dφ e− (80) 0 2π Z0 and J0(t) = J0( t). For making the above expressions more illustrative we scale the integral variables with respect to the scattering−length of the potential de ned in section III B by introducing the dimensionless integral variables u and t de ned as z = au and b = at, where a is the scattering length of the potential. We also use V to signify the maximum value of the function V (~r). In term of these variables scattering amplitude in eqn. (79) takes the form
1 ∞ ika V ξ(t) f(θ) = a ka t dt J0(t kaθ) e E 1 (81) i 0 − Z h i where + 1 1 ∞ ξ(t) = du V (at, au). (82) −2 V Z−∞ Using eqn. (15) we have the expression for the scattering cross section to be
π σscatt. ∞ ∞ V V 2 +ika E ξ(t) ika E ξ(t0) 2 = 2(ka) t dt t0 dt0 e 1 e− 1 sin θ dθ J0(t kaθ)J0(t0 kaθ). (83) πa 0 0 − − 0 Z Z h i h i Z V V We said earlier that E is a good estimate for the upper limit of the scattering angle θ. Thus in the limit E 1 we V can write sin θ θ and limit our range of integration in θ from 0 to E . Further introducing the integration variable x = kaθ we hav≈e
ka V σscatt. ∞ ∞ V V E +ika E ξ(t) ika E ξ(t0) 2 = 2 t dt t0 dt0 e 1 e− 1 x dx J0(tx)J0(t0x). (84) πa 0 0 − − 0 Z Z h i h i Z Once we have the above expressions for the scattering amplitude in eqn. (81) and scattering cross section in eqn. (84) we would like to check if the optical theorem is satis ed in the eikonal approximation. For potentials that are not complex the optical theorem can be stated as
12 4π σ = Im f(0). (85) scatt. k Using eqn. (81) in eqn. (85) we have
σ ∞ V scatt. = 8 t dt sin2 ka ξ(t) . (86) πa2 E Z0 V V 2 Comparing eqn. (84) and eqn. (86) we conclude that under the conditions E 1 and ka E 1 alone the eikonal approximation does not satisfy the optical theorem. The physical content of the optical theorem is the statement of probability conservation in quantum mechanics. Can we save the eikonal approximation from violating the optical theorem? Yes. We recognize that in eqn. (84) if we take the limit ka V and use the identity E → ∞
∞ δ(t t0) x dx J (tx)J (t0x) = − (87) 0 0 t Z0 we get exactly the expression required for the optical theorem to be satis ed. Thus we observe that even though V the expression for the scattering amplitude in the eikonal approximation is derived under the conditions E 1 and V 2 V ka E 1 alone, we still need to put an additional condition ka E 1 for it to satisfy the optical theorem. Thus we have the eikonal approximation valid under the conditions V 1 1 1 and ka . (88) E V/E (V/E)2
For a given ka, the second inequality above puts an upper and lower bound on V given by 1 V 1 . This can E ka E √ka be rewritten in the form V 1 ka √ka. (89) E
D. Examples
1. Inverted finite spherical well potential
An inverted nite spherical well potential is de ned as
V for r < a V (~r) = . (90) 0 for r > a Noting that b2 + z2 = r2 and thus integrating z from 0 to √r2 b2 we have for this potential − ξ(t) = 1 t2 − − 1 p1 ika V √1 t2 f(θ) = a ka t dt J0(t kaθ) e− E − 1 i 0 − 1 Z h i σscatt. 2 V 2 2 = 8 t dt sin ka 1 t . (91) πa 0 E − Z p V In gs. 3 and 4 we plot σscatt. versus ka E and compare it with the plots got using the partial wave method. Using V eqn. (89) for ka = 50, we have the region of validity to be 1 ka E 7, and for ka = 750, we have the region of V validity to be 1 ka E 27. These estimates for the region of validity are in agreement with the region where the curves obtained from the eikonal approximation t the curve obtained from partial wave method in gs. 3 and 4.
13 σtot. πa2
3.5 - - - partial wave (ka = 50) 3 —– eikonal
2.5
2
1.5
1
0.5 V ka 20 40 60 80 100 E
FIG. 3. Comparison of eikonal and partial wave method (ka = 50) for the inverted finite spherical well.
σtot. πa2
3.5 - - - partial wave (ka = 750) 3 —– eikonal
2.5
2
1.5
1
0.5 V ka 20 40 60 80 100 E
FIG. 4. Comparison of eikonal and partial wave method (ka = 750) for the inverted finite spherical well.
2. Yukawa potential
Consider the Yukawa potential
e r/a V (r) = V − (92) − r/a
e2 2 2 2 where V = a and r = b + z . For this potential we get
+ √t2+u2 1 ∞ e− ∞ t cosh θ ξ(t) = du = dθ e− = K0(t) (93) 2 2 2 √t + u 0 Z−∞ Z where K0(t) is modi ed Bessel function of order zero. Using this in eqn. (81) we have
∞ V 1 ika K0(t) f(θ) = a ka t dt J0(t kaθ) e E 1 . (94) i 0 − Z h i 14 The contribution to the above expression from J0(t kaθ) dies o very fast. Thus to a good approximation we can say that we get non-zero contribution to the integral for t kaθ 1. Noting that in the eikonal approximation θ < V and E 1 ka V 1 we conclude that non zero contributions to the integral comes from t 1. In the limit t∼ 1 the E V/E modi ed Bessel function takes the form 2 K (t) ln γ (95) 0 ≈ t − where γ = 0.577 . . . is the Euler’s constant. Further we observe that one of the terms in eqn. (94) contributes only at θ = 0 and thus is a delta function. Overall after substituting x = kaθ we get
2 ika V (ln 2+ln(kaθ) γ) 1 ∞ 1 ika V f(θ) = ika δ(θ) ie E − dxJ (x)x − E . (96) − kθ2 0 Z0 In terms of the gamma functions we have
n ∞ (1 + + iα) dxJ (x)x1+2iα = 22iα+1 2 . (97) n ( n iα) Z0 2 − Thus we can write
2 1 i ka V 1 2 − E f(θ) θ=0= (98) | | 6 θ 2 ka V 2 2k 2 E 1 + i V ka E 1 V e2 e2 where 2 ka E = hv¯ using V = a and hk = p = mv. Observe that the above result is identical to the scattering amplitude due to a Coulomb potential for small scattering angles using the approximation sin θ θ. It is worth observing the interesting fact that the above result is independent of ‘a’. ≈
V. QUANTUM ELECTRODYNAMICS
A. Schwinger’s functional differential equations for quantum electrodynamics
We de ne the vacuum to vacuum persistence amplitude for electrodynamics in the presence of external sources (x) J (x), η(x), η (x) to be J ≡ { µ }
Z[ ] = 0+ 0 J . (99) J h | −i Using Schwinger’s quantum variational principle we can write
δZ[ ] = i 0+ δW [ ; ] 0 J (100) J h | A J | −i where W [ ; ] is the action for quantum electrodynamics given in terms of the elds (x) A (x), F A(xJ), ψ(x), ψ (x) interacting with external sources is (x) A ≡ { µ µν } J
W [ ; ] = S[ ] + d4x J µ(x)A (x) + η (x)ψ(x) + ψ (x)η(x) (101) A J A µ Z where 1 1 S[ ] = d4x F µν (x)(∂ A (x) ∂ A (x)) + F µν (x)F (x) + ψ (x)[iγµ∂ m + eγµA (x)]ψ(x) . (102) A −2 µ ν − ν µ 4 µν µ − µ Z µ Gauge invariance requires us to constrain the external source Jµ(x) to satisfy the condition ∂ Jµ(x) = 0. Using Schwinger’s quantum variational principle to vary Z[ ] with respect to the elds and the external sources we get J A J
15 δW [ , ] δW [ , ] µ δW [ , ] δZ[ ] = i δη (x) 0+ A J 0 J + i δη(x) 0+ A J 0 J + i δJ (x) 0+ A J 0 J J h | δη (x) | −i h | δη(x) | −i h | δJ µ(x) | −i δW [ , ] δW [ , ] + i δψ (x) 0+ A J 0 J + i δψ(x) 0+ A J 0 J h | δψ (x) | −i h | δψ(x) | −i
µ δW [ , ] µν δW [ , ] + i δA (x) 0+ A J 0 J + i δF (x) 0+ A J 0 J h | δAµ(x) | −i h | δF µν (x) | −i µ = i δη (x) 0+ ψ(x) 0 J + i δη(x) 0+ ψ (x) 0 J + i δJ (x) 0+ Aµ(x) 0 J h | | −i h | | −i h | | −i µ µ + i δψ (x) (iγ ∂µ m) 0+ ψ(x) 0 J + e 0+ γ Aµ(x)ψ(x) 0 J + η(x) 0+ 0 J − h | | −i h | | −i h | −i µ µ + i 0+ ψ(x) 0 J ( iγ ←−∂ µ m) + e 0+ ψ(x)γ Aµ(x) 0 J + η (x) 0+ 0 J δψ (x) h | | −i − − h | | −i h | −i h µ µ i + i δA (x) ∂ 0+ Fµν (x) 0 J + e 0+ ψ(x)γµψ(x) 0 J + Jµ(x) 0+ 0 J − h | | −i h | | −i h | −i µν 1 + i δF (x) 0+ Fµν (x) 0 J ∂µ 0+ Aν (x) 0 J ∂ν 0+ Aµ(x) 0 J . (103) 2 h | | −i − { h | | −i − h | | −i } In the above equation we have treated the variations in the elds to be c-number variations. Schwinger’s quantum action principle states that Z[ ] is stationary with respect to variations in the dynamical parameters ( elds (x) in the present case). Thus we canJwrite A
δZ[ ] ( i) J = 0+ Aµ(x) 0 J − δJ µ(x) h | | −i δZ[ ] ( i) J = 0+ ψ(x) 0 J − δη (x) h | | −i δZ[ ] ( i) J = 0+ ψ (x) 0 J − δη(x) h | | −i µ µ 0 = (iγ ∂µ m) 0+ ψ(x) 0 J + e 0+ γ Aµ(x)ψ(x) 0 J + η(x) 0+ 0 J − h | | −i h | | −i h | −i µ µ 0 = 0+ ψ (x) 0 J ( iγ ←∂−µ m) + e 0+ ψ (x)γ Aµ(x) 0 J + η (x) 0+ 0 J h | | −i − − h | | −i h | −i µ 0 = ∂ 0+ Fµν (x) 0 J + e 0+ ψ(x)γµψ(x) 0 J + Jµ(x) 0+ 0 J − h | | −i h | | −i h | −i 0 = 0+ Fµν (x) 0 J ∂µ 0+ Aν (x) 0 J ∂ν 0+ Aµ(x) 0 J . (104) h | | −i − { h | | −i − h | | −i } Eliminating the elds in the above equations we get
δ δZ[ ] iγµ∂ m + eγµ( i) ( i) J = η(x) Z[ ] (105) − µ − − δJ µ(x) − δη (x) J δZ[ ] µ µ ←−δ ( i) J iγ ←∂−µ m + eγ ( i) µ = η (x) Z[ ] (106) − − δη(x) "− − − δJ (x) # J δZ[ ] δ δZ[ ] gµν ∂2 ∂µ∂ν ( i) J = J µ(x) Z[ ] + e ( i) γµ ( i) J . (107) − − − δJ µ(x) J − δη(x) − δη (x) These are the functional di erential equations for Z[ ]. This was rst written down in this form by Julian Schwinger [13,14]. Since Z[ ] is the generating functional for theJ Green functions it has the information regarding all possible processes in quanJtum electrodynamics. Conversely a solution to the above functional di erential equations solves quantum electrodynamics completely.
B. A formal solution to Schwinger’s functional differential equations
Using Schwinger’s quantum variation principle in eqn. (100) for the variation of Z[ ] with respect to e we have J ∂Z[ ] ∂W [ , ] ( i) J = 0+ A J 0 J − ∂e h | ∂e | −i 4 µ = d x 0+ ψ (x)γ Aµ(x)ψ(x) 0 J . (108) h | | −i Z
16 Using eqn. (104) to eliminate the elds we get the rst order di erential equation
∂Z[ ] δ δ δ J = i d4x ( i) γµ( i) ( i) Z[ ] (109) ∂e − δη(x) − δJ µ(x) − δη (x) J Z Which can be integrated to yield the result
ie d4x ( i) δ γµ( i) δ ( i) δ Z[ ] = e − δη(x) − δJµ(x) − δη¯(x) Z [ ] (110) J e=0 J R Where Ze=0[ ] is the generating function for the case when the photon eld is not coupled to the fermion eld. It is fairly straighJt forward to show [19] that the solution to eqns. (105), (106) and (107) for the e = 0 case is
i 4 4 µν 4 4 d x d x0Jµ(x)D (x x0)Jν (x0) i d x d x0η¯(x)S (x x0)η(x0) Z [ ] = e 2 + − e + − (111) e=0 J µν R R R R where the Green functions D (x x0) and S (x x0) satisfy the di erential equations + − + − 2 µν µν (4) (g ∂ ∂ ∂ )D (x x0) = g δ (x x0) (112) − µν − µ ν + − − µ (4) (γ ∂ m)S (x x0) = δ (x x0) (113) − µ − + − − with the boundary conditions on the Green functions prescribed by the requirement Z 2 1. Using eqn. (111) in eqn. (110) we have | | ≤
4 4 δ (4) µ δ δ 4 4 i 4 4 µν e d x d x0 δη(x) δ (x x0)γ δJµ(x) i d x d x0η¯(x)S+(x x0)η(x0) d x d x0Jµ(x)D (x x0)Jν (x0) Z[ ] = e− − δη¯(x0) e − e 2 + − . (114) J R R R R R R For xi, yi being Grassmann variables we have the identity
∂ ∂ ij T 1 amn xib yj ~x Bˆ (1+ˆ Aˆ Bˆ)− ~y Tr ln(1+ˆ Aˆ Bˆ) e ∂ym ∂xn e = e · · · · e · (115)
i where we have used the symbolic notation, ~a (vector) for a and A^ (matrix) for Aij and Tr is the trace over the i, j indices. Generalizing the above result for the case of functionals dependent on Grassmann variables we have
4 4 δ δ 4 4 4 4 1 d x d x0 δη(x) M(x x0) d x d x0η¯(x)N(x x0)η(x0) d x d x0η¯(x)[N(δ+MN)− (x,x0)]η(x) Tr ln(δ+MN) e − δη¯(x0) e − = e e (116) R R R R R R where we have used the corresponding symbolic notation to suppress the integrals, Tr is the trace over both the spinor and coordinate index, and δ is the Dirac delta function. Using the above identity, the expression in eqn. (114) simpli es to
4 4 δ δ i 4 4 µν i d x d x0η¯(x)G+[x,x0;( i) µ ]η(x) L[( i) µ ] d x d x0Jµ(x)D (x x0)Jν (x0) Z[ ] = e − δJ e − δJ e 2 + − (117) J R R R R where
µ L[A] = Tr ln G [x, x0; A ] + Tr ln S (x x0) (118) − + + − µ with Tr denoting the trace over both spinor and coordinate index and the Green function G+[x, x0; A ] satis es the di erential equation
µ µ µ (4) [ (iγ ∂ m) + eγ A (x)] G [x, x0; A ] = δ (x x0) (119) − µ − µ + − with the boundary condition prescribed by the initial condition
µ lim G+[x, x0; A ] = S+(x x0). (120) e 0 → − To get prepared for the next step we write down the following identity involving the Gaussian function and an arbitrary function F (x) in terms of the real variable x
∂ 1 xbx 1 xbx 1 ∂ b ∂ F ( ) e 2 = e 2 e 2 ∂bx ∂bx F (bx). (121) ∂x Generalizing the above identity for functionals we have
17 i 4 4 µν i 4 4 µν δ d x d x0Jµ(x)D (x x0)Jν (x0) d x d x0Jµ(x)D (x x0)Jν (x0) F [( i) ] e 2 + − = e 2 + − − δJ µ R R R i R 4 4 δ µν δ d x d x0 µ D (x x0) e 2 δA (x) + − δAν (x) F [A] (122) × A = DJ n R R o µ 4 µν R where A = DJ stands for A (x) = d x0D (x x0)J (x0). Using the above identity in eqn. (117) we have + − ν
i 4 4 µν R d x d x0Jµ(x)D R(x x0)Jν (x0) Z[ ] = e 2 + − J i 4 4 δ µν δ 4 4 µ d x d x0 µ D (x x0) ν i d x d x0η¯(x)G+[x,x0;A ]η(x) L[A] Re 2 R δA (x) + − δA (x) e e . (123) × A = DJ n R R R R o We observe that our problem of determining Z[ ] for quantum electrodynamics reduces to the evRaluation of the Green µ µ J µ function G+[x, x0; A ] for an arbitrary A (x). Any kind of approximate solution to G+[x, x0; A ] suggests a possible non-perturbative approximate solution to Z[ ]. J
C. Electron-electron scattering
We can expand Z[ ] around Z[0] and thus write J δ δ Z[ ] = Z[0] + d4x η (x) Z[ ] + d4x Z[ ] η(x) J δη (x) J =0 δη(x) J =0 Z J Z J 4 µ δ + d x J (x) µ Z[ ] + . . . + T + . . . (124) δJ (x) J =0 Z J where T is the quantity of interest in the electron-electron scattering process and is given by the expression
( i)4 T = − d4x d4x d4y d4y η (x )η (x )η (y )η (y ) G (x , y , x , y ) (125) 2! 2! 1 2 1 2 A 2 B 1 C 2 D 1 ABCD 1 1 2 2 Z Z Z Z where we have explicitly introduced the Dirac indices for clarity and
4 δ δ δ δ GABCD(x1, y1, x2, y2) = ( i) Z[ ] − δη A(x2) δη B(x1) δηA(y2) δηA(y1) J =0 J M[ δ ] µ µ L[A] = e δA G+[x1, y1; A ] G+[x2, y2; A ] e (126) A = DJ,J=0 n h io where we have introduced the short form for the operator R
δ i 4 4 δ µν δ M = d x d x0 D (x x0) . (127) δA 2 δAµ(x) + − δAν (x) Z Z T will be interpreted as the probability amplitude for the electron-electron scattering process. Every process in quantum electrodynamics can be generated out of the expansion in eqn. (124). It is due to this reason that Z[ ] is also called the generating function. J For the electron-electron process we shall be interested in the case when the sources of the process η (x1), η (x2), η(y1) and η(y2) are interaction free states with well de ned initial or nal momentums. Thus we choose
ip1x1 µ η (x1) = e− u (p1)[γ p1µ + m]
ip10 y1 µ η(y1) = e [γ p10 µ + m]u(p10 )
ip2x2 µ η (x2) = e− u (p2)[γ p2µ + m]
ip20 y2 µ η(y2) = e [γ p20 µ + m]u(p20 ). (128)
Accordingly T will be denoted as T (p1, p10 , p2, p20 ). In the next section we shall concentrate on approximate solutions to T (p1, p10 , p2, p20 ).
18 VI. APPROXIMATION TO ELECTRON-ELECTRON SCATTERING
A. Quenched or ladder approximations
To start with we shall assume that eL[A] in eqn. (126) does not contribute. Further we assume that the operator M in eqn. (127) act in such a way that they connect the two fermion propagators via photon lines. In the diagrammatic language this amounts to omitting all those graphs that involve fermion loops and keeping only those graphs that have photons exchanged between the fermion lines. This is called the quenched or ladder approximation. In this approximation eqn. (126) reads
δ M12[ ] µ µ GABCD(x1, y1, x2, y2) = e δA [G+[x1, y1; A ] G+[x2, y2; A ]] (129) A= DJ, J=0 n o where we introduced M12 in place of M to signify that it connects the two fermion lines.R Using the identities
1 M12 λM12 e = 1 + dλ e M12 (130) Z0 and δ G [x , y ; Aµ] = G [x , z; Aµ] eγ G [z, y ; Aµ] (131) δAµ(z) + 1 1 + 1 µ + 1 we thus get
1 λM 4 4 µν T (p , p0 , p , p0 ) = i dλ e 12 d z d z D (z z ) 1 1 2 2 − 1 2 + 1 − 2 Z0 Z Z 4 ip x µ µ 4 ip0 y µ µ d x e− 1 1 u (p )[γ p + m]G [x , z ; A ] eγ d y e 1 1 G [z , y ; A ][γ p0 + m]u(p0 ) × 1 1 1µ + 1 1 1 µ 1 + 1 1 1 1µ 1 Z Z 4 ip x µ µ 4 ip0 y µ µ d x e− 2 2 u (p )[γ p + m]G [x , z ; A ] eγ d y e 2 2 G [z , y ; A ][γ p0 + m]u(p0 ). (132) × 2 2 2µ + 2 2 2 ν 2 + 2 2 2 2µ 2 Z Z
µ B. Eikonal approximation to G+[x, y; A ]
µ µ Next step is to evaluate G+[x, y; A ] in the eikonal approximation. We de ned G+[x, y; A ] in eqn. (119) to satisfy the di erential equation given by
[ (iγµ∂ m) + eγµA (x)] G [x, y; Aµ] = δ(4)(x y). (133) − µ − µ + − In the eikonal approximation we have [19{22]
s 1 µ p µ ∞ ims p ie dτ p Aµ(x τ ) G [x, y; A ] = i ds e− δ(x y s ) e 0 m − m . (134) + − − m Z0 R µ Using the above eikonal approximation to G+[x, y; A ] in eqn. (132) it becomes possible to evaluate T (p1, p10 , p2, p20 ). Here we shall simply state the result without the detailed steps in the derivation. We de ne
4 (4) T (p , p0 , p , p0 ) = (2π) δ (p + p0 p p0 )M(p , p0 , p , p0 ) (135) 1 1 2 2 1 1 − 2 − 2 1 1 2 2 and in the eikonal approximation we have the result [19]
2 e2 (p + p )2 (1 i e ) 1 2 − 4π M(p1, p10 , p2, p20 ) = 2 2 e2 . (136) | | 2m (p1 p0 ) (1 + i ) − 1 4π We notice the close similarity of the above expression to the non-relativistic scattering amplitude due to the Coulomb potential. This is not surprising because eikonal approximation is a stringent semi-classical limit.
19 ACKNOWLEDGMENTS
Most of the work presented here evolved during my regular meetings with Kimball Milton and Ines Cavero. The meetings played a very signi cant role in my learning. I would like to thank Kimball Milton for agreeing to be my PhD thesis advisor. I have learnt from him the approach to attend to every problem by starting from scratch. Most of all, I would like to thank him for being the devil’s advocate2 during all the stimulating discussions we have had. Almost always the discussions led to such remote regions of thought which would have been left completely unexplored otherwise. I am also grateful to him for scrutinizing this report for errors and defects. I have tried my best to incorporate his suggestions. I am grateful to Bahman Roostaei for allowing me to use the condensed matter groups computer (IBM RS/6000/7028-6E4 which uses powerpc multiprocessors and has a 4GB memory). The plots reproduced here would have been impossible to draw without such a giant memory. Thanks are also due to Pravin Chaubey for helping me during the search on the etymology of the word eikon. I would like to thank David Branch, Leonid Dickey, Ronald Kantowski, and Chung Kao for agreeing to be in my PhD advisory committee. I would also like to thank Shankar for all the nice discussions and April Hendley for being a patient listener.
[1] Translated by Margarita Kamilieri who is originally from Greece. She is majoring in social work at the University of Oklahoma. [2] Merriam-Webster’s Collegiate dictionary, 10th edition. [3] Roger G. Newton, Scattering Theory of Waves and Particles, 2nd ed., Springer-Verlag New York, Inc., (1982). [4] Page 610, ref. [3]. [5] R. J. Glauber, Lectures in Theoretical Physics, ed. by W. E. Brittin and L. G. Dunham, Interscience Publishers, Inc., New York, Volume I, page 315, (1959). [6] Charles J. Joachain, Quantum Collision Theory I, II, North Holland Publishing Company, (1975). [7] Leonard I. Schiff, Quantum Mechanics, 3rd ed., McGraw-Hill Book Company, New York, (1968). [8] F. Smithies, Integral equations, Cambridge Tracts in Mathematics and Mathematical Physics, ed. by P. Hall and F. Smithies, Cambridge University Press, No. 49, (1958). [9] B. A. Lippmann and Julian Schwinger, Phys. Rev. 79 469 (1950). [10] Jagdish Mehra and Kimball A. Milton, Climbing the Mountain: The Scientific Biography of Julian Schwinger, Oxford University Press, Inc., New York, (2000), page 171. [11] R. J. Glauber and V. Schomaker, Phys. Rev. 89 667 (1953). [12] E. Feenberg, Phys. Rev. 40 40 (1932). [13] Julian Schwinger, On the Green’s Functions of Quantized Fields. I, II, Proc. Natl. Acad. Sci. U.S.A. 37 452, 455 (1951). [14] Julian Schwinger, The Theory of Quantized Fields. I-V, Phys. Rev. 82 914 (1951), 91 713, 721 (1953), 92 1283 (1953), 93 615 (1954). [15] Julian Schwinger, Particles, Sources and Fields, Addison-Wesley, Reading, MA, (1970). [16] Kimball A. Milton, Lecture Notes, University of Oklahoma, (2002). [17] H. M. Fried, Green’s Functions and Ordered exponentials, Cambridge University Press, (2002). [18] H. M. Fried, Functional Methods and Models in Quantum Field Theory, The MIT Press, Cambridge, MA, (1972). [19] Leonard Gamberg and Kimball A. Milton, Phys. Rev. D 61 075013 (2000). [20] F. Guerin and H. M. Fried, Phys. Rev. D 33 3039 (1986). [21] Henry D. I. Abarbanel, Lectures in Theoretical Physics XIV A, University of Colorado, page 3 (1971). [22] Robert L. Sugar, Lectures in Theoretical Physics XIV A, University of Colorado, page 47 (1971).
2Webster’s dictionary meaning: a person who champions the less accepted cause for the sake of argument.
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