the electromagnetic gauge field interpolation between the instant forma nd the front form of the...

7/24/2019 The Electromagnetic Gauge Field Interpolation Between the Instant Forma Nd the Front Form of the Hamiltonian D

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The Electromagnetic Gauge Field Interpolation between the Instant Form and the

Front Form of the Hamiltonian Dynamics

Chueng-Ryong Ji,1 Ziyue Li,1 and Alfredo Takashi Suzuki2

1Department of Physics, North Carolina State University, Raleigh, North Carolina 27695-82022Instituto de Fsica Teorica-UNESP Universidade Estadual Paulista,

Rua Dr. Bento Teobaldo Ferraz, 271 - Bloco II - 01140-070, Sao Paulo, SP, Brazil.

We present the electromagnetic gauge field interpolation between the instant form and the frontform of the relativistic Hamiltonian dynamics and extend our interpolation of the scattering ampli-tude presented in the simple scalar field theory to the case of the electromagnetic gauge field theorywith the scalar fermion fields known as the sQED theory. We find that the Coulomb gauge in theinstant form dynamics (IFD) and the light-front gauge in the front form dynamics, or the light-front dynamics (LFD), are naturally linked by the unified general physical gauge that interpolatesbetween these two forms of dynamics and derive the spin-1 polarization vector for the photon thatcan be generally applicable for any interpolation angle. Corresponding photon propagator for anarbitrary interpolation angle is found and examined in terms of the gauge field polarization and theinterpolating time ordering. Using these results, we calculate the lowest-order scattering processesfor an arbitrary interpolation angle in sQED. We provide an example of breaking the reflectionsymmetry under the longitudinal boost, Pz Pz, for the time-ordered scattering amplitude inany interpolating dynamics except the LFD and clarify the confusion in the prevailing notion of theequivalence between the infinite momentum frame (IMF) and the LFD. The particular correlationfound in our previous analysis of the scattering amplitude in the simple scalar field theory, coinedas the J-shaped correlation, between the total momentum of the system and the interpolation anglepersists in the present analysis of the sQED scattering amplitude. We discuss the singular behaviorof this correlation in conjunction with the zero-mode issue in the LFD.

I. INTRODUCTION

In 1949, Dirac [1] proposed three forms of relativis-tic dynamics: the instant form (x0 =0), the front form

(x+ =

(x0 + x3

)

2=0) and the point form (xx =a2>

0, x0 >0). While the quantization at the equal time t = x0

produces the instant form dynamics (IFD) of quantumfield theory, the quantization at equal light-front time (t + zc)2 =x+ (c is taken to be one unit in thiswork) yields the front form dynamics, known as the light-front dynamics (LFD). Although the point form dynam-ics has also been explored [2], the IFD and the LFD arestill the most popular choices.

One of the reasons why the LFD is useful may beattributed to the energy-momentum dispersion relation.For a particle of mass m that has four-momentum k =(k0, k1, k2, k3), its energy-momentum dispersion relationat equal-t (instant form) is given by

k

0=k2 + m2, (1)

where the energy k0 is conjugate to t and the three mo-mentum vector k is given by k =(k1, k2, k3). On theother hand, the corresponding energy-momentum rela-tion at equal-(light-front form) is given by

k =k2

+ m2

k+, (2)

where the light-front energy k =(k0 k3)2 is conju-

gate to, and the light-front momentak+ =(k0 + k3)2and k =(k1, k2) are orthogonal to k. In contrast tothe irrational dispersion relation Eq. (1) in the IFD, thisrational energy-momentum relation Eq. (2) in the LFDnot only makes the relation simpler, but also correlatesthe sign of k and k+. When the system is evolving tothe future direction (i.e. positive ), in order fork to bepositive,k+ also has to be positive. This feature preventscertain processes from happening in the LFD, for exam-ple, the spontaneous pair production from vacuum is for-bidden unlessk+ =0 for both particles due to the momen-tum conservation. Some dynamic processes are thereforeeliminated in the LFD and correspondingly the requiredcomputation may be simplified, although we shall discussthe zero-mode issue involving the case of k+ = 0 later.In the IFD, however, this type of sign correlation doesnot exist, and the vacuum structure appears much morecomplicated than in the case of LFD due to the quantumfluctuations. This difference in the energy-momentumdispersion relation makes the LFD quite distinct from

other forms of the relativistic Hamiltonian dynamics.

Furthermore, the Poincare algebra is drasticallychanged in the LFD compared to the IFD. In LFD, wehave the maximum number (seven) of kinematic (i.e. in-teraction independent) operators out of the ten Poincaregenerators and they leave the state at =0 unchanged.In particular, the longitudinal boost operator joins thestability group of kinematic operators in LFD. This built-in boost invariance together with the simpler vacuumproperty makes the LFD quite appealing and may save

arXiv:1412.2

726v1

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Dec2014


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substantial computational efforts to get the QCD solu-tions that reflect the full Poincare symmetries.

The light-front quantization[1,3] has been applied suc-cessfully in the context of current algebra [4] and theparton model [5] in the past. With further advances inthe Hamiltonian renormalization program [6,7], the LFDappears to be even more promising for the relativistic

treatment of hadrons. In the work of Brodsky et al. [8],it is demonstrated how to solve the problem of renormal-izing light-front Hamiltonian theories while maintainingLorentz symmetry and other symmetries. The genesis ofthe work presented in Ref. [8] may be found in Ref. [9]and additional examples including the use of LFD meth-ods to solve the bound-state problems in field theory canbe found in the review of QCD and other field theories onthe light front[10]. A possible realization of chiral sym-metry breaking in the light-front vacuum has also beendiscussed in the literature[11].

However, the transverse rotation whose direction isperpendicular to the direction of the quantization axis

z at equal becomes a dynamical problem in the LFDbecause the quantization surface is not invariant underthe transverse rotation and the transverse angular mo-mentum operator involves the interaction that changesthe particle number [12].

As an effort to understand the conversion of the dy-namical problem from boost to rotation as well as thelink between the IFD and the LFD, we interpolate thetwo forms of dynamics by introducing an interpolationangle that changes the ordinary time t to the light fronttime or vice versa. The same method of interpolatinghypersurfaces has been used by Hornbostel [13] to ana-lyze various aspects of field theories including the issueof nontrivial vacuum. The same vein of application tostudy the axial anomaly in the Schwinger model has alsobeen presented[14], and other related works [1518]canalso be found in the literature.

Our interpolation between the IFD and the LFD pro-vides the whole picture of landscape between the twoand clarifies the issue, if any, in linking them to eachother. We started out by studying the Poincare algebrafor any arbitrary interpolation angle [19], and providedthe physical meaning of the kinematic vs. dynamic oper-ators by introducing the interpolating time-ordered scat-tering amplitudes [20]. Although we want ultimately toobtain a general formulation for the QED and the QCDusing the interpolation between the IFD and the LFD,we start from the simpler theory to discuss first the bare-bone structure that will persist even in the more compli-cated theories. Since we have studied the simple scalarfield theory [20] involving just the fundamental degreesof freedom such as the momenta of particles in scatteringprocesses, we now consider involving the electromagneticgauge degree of freedom interpolated between the IFDand the LFD in the present work. We develop the elec-tromagnetic gauge field propagator interpolated betweenthe IFD and the LFD and extend our interpolation of

the scattering amplitude presented in the simple scalarfield theory to the case of the electromagnetic gauge fieldtheory but still with the scalar fermion fields known asthe sQED theory.

In LFD, the light-front gauge (A+ =(A0 + A3)2 = 0)is commonly used, since the transverse polarizations ofthe gauge field can be immediately identified as the dy-namical degrees of freedom, and ghost fields can be ig-

nored in the quantum action of non-Abelian gauge the-ory[2123]. This makes it especially attractive in variousQCD applications. We find that the light-front gaugein the LFD is naturally linked to the Coulomb gaugein the IFD through the interpolation angle. The cor-responding gauge propagator that interpolates betweenthe IFD and the LFD also sheds light on the debateabout whether the gauge propagator should be the two-term form [7]or the three-term form [2426]. For exam-ple, by analyzing the lowest-order sQED Feynman am-plitude, one may typically get the corresponding threetime-ordered amplitudes, one of which corresponds tothe contribution from the instantaneous interaction of

the gauge field. This contribution from the instantaneousinteraction is, however, precisely canceled by one of theterms in the three-term gauge propagator and thus thetwo-term gauge propagator may also be used effectivelyfor the calculation of the same Feynman amplitude with-out involving the instantaneous interaction of the gaugefield. Otherwise, to maintain the equivalence to the co-variant formulation, the three-term propagator should beused including the instantaneous interaction of the gaugefield.

Our work also clarifies the singular nature of the cor-relation between the total momentum of system and theinterpolation angle and provides a deeper understandingof the treacherous zero-mode issue in the LFD. We findthat the particular correlation between the total momen-tum of the system and the interpolation angle, coinedas the J-shaped correlation in our previous analysis, per-sists even in the sQED scattering amplitude involvingthe gauge field. We discuss the universal nature of theJ-shaped correlation which appears completely indepen-dent from the nature of particles (i.e. mass, spin, etc.)involved in the scattering process.

Although the interpolation method has been intro-duced before [13,14, 19,20], it has not yet been widelyexplored and a brief description of the method is still nec-essary for the presentation of our work. In Sec.II, we thusprovide a brief review of the interpolation angle methodessential for the rest of this article. In Sec. III, we derivethe photon polarization vector for any interpolation an-gle. Using this derivation, we present the general gaugethat links the light-front gauge to the Coulomb gauge,and construct the corresponding photon propagator foran arbitrary interpolation angle. In Sec.IV,we decom-pose this interpolating gauge propagator according to thetime ordering and apply it to the lowest scattering pro-cess such as an analogue of the well-known QED processe e in sQED without involving the fermion spins.


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We also take a close look at the limiting cases ofC 0and compare the results with the exact C =0 (LFD) re-sults in this section. In Sec. V,we plot the time-orderedamplitudes in terms of the total momentum of the sys-tem and the interpolation angle to reveal both the framedependence and interpolation angle dependence of theseamplitudes. We then give a detailed discussion of the

universal J-shaped correlation curve that emerges fromthese time-ordered diagrams and how it gives rise to thezero-mode contributions at Pz = . A summary andconclusion follows in Sec.VI.

In AppendixA,we list the explicit matrix representa-tion of the boost K and rotation J generators and providethe explicit description of the steps involved in derivingthe photon polarization vectors for an arbitrary interpo-lation angle. In AppendixB,we derive the numerator ofthe photon propagator from the photon polarization vec-tors for an arbitrary interpolation angle. In AppendixC,we decompose the photon propagator on the light-frontin terms of the transverse and longitudinal components.For the completeness and the comparison with our anal-ysis of sQED scattering process (analogous to e ein QED) presented in Secs.IV and V,we present in Ap-pendices DandEour calculations for the same processe e in the scalar field theory and the process re-lated by the crossing symmetry e+e + in sQED.Together with our previous work for the scalar field the-ory discussed in[20], this work completes our study of thelowest order scattering processes related by the crossingsymmetry in the scalar field theory as well as in sQED.

II. METHOD OF INTERPOLATION ANGLE

In this section, we briefly review the interpolation anglemethod presenting just the necessary formulae for thepresent work. For more detailed introduction and reviewof this method, the readers may consult our previousworks presented in [19]and [20].

The interpolating space-time coordinates may be de-fined as a transformation from the ordinary space-timecoordinates, x =Rx

, i.e.

x+

x1

x2

x

=

cos 0 0 sin 0 1 0 00 0 1 0

sin 0 0 cos

x0

x1

x2

x3

, (3)

where 0 4 is the interpolation angle. Following[20], we use on the indices to denote the interpolatingvariables with the parameter . In the limits 0 and 4, we recover the corresponding variables in the in-stant form and the front form, respectively. For example,the interpolating coordinates x in the limit 4 be-come the light-front coordinatesx =(x0 x3)2 with-out .

In this interpolating basis, the metric becomes

g =g =

C 0 0 S0 1 0 00 0 1 0S 0 0 C

, (4)where S = sin2 and C = cos2. The covariant inter-polating space-time coordinates are then easily obtainedas

x =gx=

x+

x1

x2

x

=

cos 0 0 sin 0 1 0 00 0 1 0

sin 0 0 cos

x0

x1

x2

x3

. (5)The same transformations also apply to the momentum:

P+ =P0 cos + P3 sin , (6a)

P =P0 sin P3 cos , (6b)

P+=P0 cos P3 sin , (6c)

P=P0 sin + P3 cos . (6d)

Since the perpendicular components remain the same

(aj = aj , aj = aj , j = 1, 2), we will omit the nota-tion unless necessary from now on for the perpendicularindices j =1, 2 in a four-vector.

Using g andg, we see that the covariant and con-travariant components are related by

a+=Ca+ + Sa; a+ =Ca

+ + Sa

(7)

a=Sa+ Ca; a =Sa

+ Ca

aj = aj ,(j =1, 2).

The inner product of two four-vectors must be inter-polation angle independent as one can verify

ab =(a+b+ ab)C + (a+b + ab+)S a1b1 a2b2=ab. (8)

In particular, we have the energy-momentum dispersionrelation given by

PP =P2

+C P2

C + 2P

+PS P2

. (9)

Another useful relation

PPC = P+2

P2

P2C (10)

can also be easily verified. For the particles of massM,PP on the mass shell equals M

2 of course.


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TABLE I. Kinematic and dynamic generators for different interpolation angles

Kinematic Dynamic

=0 K1 = J2,K2 =J1, J3, P1, P2, P3 D1 = K1,D2 = K2, K3, P0

0


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+

cos 1

2 C2 1P1 + 2P2 , (16c)

P=P

cosh 3 + P

+ sinh 3 cos +

sin

C 1P1 + 2P2 , (16d)

where= C(21 + 22). Although these four-momentumcomponents (P+, P1, P2 and P) given by Eqs.(16a)-

(16d) turn out to be most convenient in carrying out ourcalculation, other choice of components such as P

+ and

P can be easily obtained using Eq. (7).

As an example of using Eqs.(16a)-(16d), we may findthat the particle of mass M at rest gains the followingfour-momentum components under the transformationT:

P+ =(cos cosh 3 + sin sinh 3)M, (17a)P1 =1

sin

(sin cosh 3 + cos sinh 3

)M, (17b)

P2 =2sin

(sin cosh 3 + cos sinh 3)M, (17c)

P=cos (sin cosh 3 + cos sinh 3)M, (17d)

where the factor(sin cosh 3 + cos sinh 3)in the threemomentum (P1, P2, P

) is due to the first boost T3 =

ei3K3

. Our convention taking this factor to be positive,i.e.(sin cosh 3 + cos sinh 3) > 0, is consistent withthe convention taken by Jacob and Wick [27] in theirprocedure to define the helicity in the IFD. We also notethat P

=Msin , P+ =Mcos , and P1 =P2 =0 in the

particle rest frame.

Solving Eqs.(17a)-(17d) for 1, 2, 3, we further notethat

sin cosh 3 + cos sinh 3 =P

M, (18a)

cos cosh 3 + sin sinh 3 =P+

M, (18b)

and

cos =P

P , (19a)

sin =

P2C

P , (19b)

e3 =P+ + P

M(sin + cos ) , (19c)e3 =

P+ P

M(cos sin ) , (19d)j

=

PjP2C

, (j =1, 2), (19e)where P

P2

+ P2C corresponds to the magnitude of

the particles three-momentum in the IFD while it be-comes identical to P+ in the limit 4 so that = 0,i.e. cos = 1 and sin = 0, in the LFD as one can see fromEqs.(19a) and (19b), respectively. Multiplying Eqs.(19c)and (19d), we get the on-mass-shell condition consistentwith Eq. (10) for the particle of rest mass M:

P2

+ P2C =

(P+

)2

M2C. (20)

Consequently, the quantity denoted by P = P2

+ P2C

can also be written as P =(P+)2 M2C. We note the

correspondence of P to(P3)2 + P2

=P in the limit

0 (or C 1) and P to P+ in the limit 4 (orC 0).

III. THE LINK BETWEEN THE COULOMB

GAUGE AND THE LIGHT-FRONT GAUGE

We now discuss the gauge field in an arbitrary inter-

polation angle. Rather than fixing the gauge first, westart from the explicit physical polarization four-vectorsof the spin-1 particle and then identify the correspond-ing gauge that these explicit representations of the gaugefield polarization satisfy. This procedure is possible be-cause we can follow the Jacob-Wick procedure discussedin Sec. IIand apply the corresponding Ttransformation[Eq. (15)] to the rest frame spin-1 particle polarizationvectors which may be naturally given by the sphericalharmonics. Although this procedure applies to the phys-ical spin-1 particle with a non-zero mass Msuch as the-meson, we may take advantage of the Lorentz invarianceof the four-momentum squared PP =M

2 and replaceM2

by PP

to extend the obtained polarization four-vectors to the virtual gauge particle. For the real photon,of course M =0 and the longitudinal polarization vectorshould be discarded. From this procedure, we find thatthe identified gauge interpolates between the Coulombgauge in the instant form and the light-front gauge inthe front form.

A. Spin-1 Polarization Vector for Any

Interpolation Angle

We use the four-vector representation of Lorentz group

for the spin-1 particle. The polarization vector in a spe-cific frame is obtained by boosting the four-vectors tothat frame. In the rest frame, where the four-momentumis(M, 0, 0, 0), the polarization vectors are taken to be

() = 12(0, 1, i, 0), (0) =(0, 0, 0, 1), (21)

since the spherical harmonics Y11 andY0

1 naturally cor-respond to the transverse and longitudinal polarizationvectors, respectively.


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To find the polarization vectors of the spin-1 particlewith an arbitrary momentumP, we use the Jacob-Wickprocedure discussed in Sec. II and apply the T transfor-

mation [Eq.(15)] with the explicit four-vector represen-tation of the operators KandJ. The details of the calcu-lation are summarized in AppendixA.The result of thepolarization vectors written in the form of(

+, 1, 2, )

is given by

(P, +) = 12PSP,P1P iP2PP ,P2P + iP1PP , CP , (22a)

(P, ) = 12PSP,P1P + iP2PP ,P2P iP1PP , CP , (22b)

(P, 0) = P+MP

P+

M2

P+, P1, P2, P . (22c)

They satisfy the transversality and orthorgonality con-straints

(P, )P =0, (P, ) (P, ) = . (23)It is also obvious that (P, 0) is parallel to the threemomentum P, since(1, 2, ) (P1, P2, P). Noticingthat S 0, C 1, P P when 0, and S 1,C 0, P P+ when 4, one can easily checkthat these polarization vectors have correct limits of theinstant form and the light-front form at =0 and=4,respectively.

B. Interpolating Transverse Gauge

Having obtained the explicit polarization four-vectorsof the spin-1 particle with mass M, we notice that thetransverse polarizations given by Eqs. (22a) and (22b)are independent of the particle massM. Thus, they canbe used also as the transverse polarization four-vectorsof the gauge field such as the photon.

We observe that the transverse polarization vectors( = ) in Eqs.(22a) and (22b) satisfy the following con-ditions:

+

(

)= C

+

(

)+ S

(

)= 0, (24)

()P + ()PC = 0, (25)where we used () =(P, ) for convenience. So, we

can write the gauge condition for transverse photons as

A+ =0, (26)

A

P

+APC = 0, (27)

where the second condition can also be written as

A

+ AC = 0. (28)

We now demonstrate that these two conditions areclosely related to each other. First of all, the LorentzconditionA =0 is always satisfied, as already verifiedin Eq. (23). But the Lorentz condition can be rewrittenin the following way

0 = A

=+

A+C

AC +

+AS +

A+S A

=+

A+

AC +

A+S A

=+

A+ 1 S2C

A

+

A+S A

=+

A+ +S

C

(S

A

+ C

A+

)

A

C + A

=C

C+A+

+

S

CA+

A

C + A

=1

C+A+ A

C + A

=1

C+A+ (

A

+ AC) , (29)where Eq. (8)is used in the second line and the relationsin Eq. (7) are used to go between the superscript compo-nents and the subscript components. Eq. (29) indicatesthat if Eq. (26) holds, then we have Eq. (28). On theother hand, if Eq. (28) holds, then we have

+A+ =0. (30)

However, this is a differential equation, and we have thefreedom to specify the boundary condition. And we canchoose our boundary condition to make A+ = 0. Thissame trick was used in the instant form [30] where theCoulomb gauge A = 0 gives 0A

0= 0, but we can

choose our initial condition, or gauge, so that A0 = 0.Therefore, these two gauge conditions are effectivelyequivalent.

Similarly, Eq. (27) by itself does not eliminate one de-


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gree of freedom. We also need to specify a boundarycondition for this differential equation. Since we are notfocusing on writing out A explicitly, however, we willnot dwell on this subject here.

The above discussion should make it clear that Eq. (26)and Eq. (28) are really two sides of the same coin. How-ever, in order to be consistent with conventions used in

the instant form, where the radiation gauge conditionis specified as A0 = 0 and A = 0, here we saythat the radiation gauge condition for any interpolat-ing angle is Eq. (26) and Eq. (28). In the instant formlimit ( = 0), A

A3,C 1, and Eq. (28) becomes

A = 0 while Eq. (26) becomes A0 =0, which is the fa-miliar Coulomb gauge. In the light-front limit (=4),A A+,C 0, and the gauge conditions reduce to

A+ = 0 and +A+ = 0, which is just A+ = 0, i.e. thelight-front gauge.

C. Propagator for Transverse Photons

Choosing the transverse gauge fields as the dynamicaldegrees of freedom, we get the photon propagator in theinterpolating transverse gauge given by

0T(A(y)A(x))0 = i d4q(2)4 eiq(yx) Tq2 + i , (31)where T = ()()and() = (q, )takingthe photon momentum P =q. Here, we use the obviousfamiliar notation q2 = qq

. Although we can computeTdirectly using Eqs.(22a) and (22b) as shown in Ap-pendix B, we demonstrate here the method of vierbein

as a cross-check to our result.To construct a vierbein, we just need a temporal basis

four-vector and a longitudinal basis four-vector which wedenote as nand q, respectively, since the transverse ba-sis four-vectors are already given by (). The temporalbasis four-vector can be taken as a unit timelike four-vector given by n =

1C

n = 1C(1, 0, 0, 0) whose dual

vector is n =(C, 0, 0,SC). The 1C

in n is a nor-

malization factor to get nn=1. The longitudinal basis

four-vector q can also be rather easily found from thegauge condition given by Eq. (25) which can be writtenas

(

)q =0 with q =N

(0, q1C, q2C, q

). The nor-

malization factorN

is determined by q

q =qg

q = 1with ggiven by Eq. (4), so that we have

q =1

C(q2C + q2

)(0, q1C, q2C, q) (32)

and

q =1

C(q2C + q2

)(Sq, q1C, q2C, qC) . (33)

These four basis four-vectors are of course mutually or-thogonal: i.e. qn

= 0, ()n = 0 and ()q = 0,

where = .

Since (), n, q form a vierbein, we may start fromg = nn

=

()() qq, (34)and obtain

T =

()()= g + nn qq

= g +(q n)(qn + qn)

q2C + q2

Cqq

q2C + q2

q2nn

q2C + q2

, (35)

where we used n = 1C

n and rewrote q in terms ofqandn as

q =Cq (q n)nC(q2

C + q2

) (36)

in the last step to remove any artifact of divergence inq and n in the light-front limit. The photon propaga-tor constructed out of n and q interpolates smoothlyand correctly to the one corresponding to the light-frontgauge.

In the instant form limit, C 1, q2C +q2

q2 =(q n)2 q2, and this reduces to the well-known photon

propagator in Coulomb gauge ( A = 0)[30]:

T = + (q n)(qn + qn)(q n)2 q2

qq(q n)2 q2 q2nn(q n)2 q2 , (37)where = diag(1, 1, 1, 1), and and run for0, 1, 2, 3.

In the light-front limit, C 0, q2 q+2 =(q n)2,

and this becomes precisely the photon propagator underlight-front gauge (A+ =0), and we have

T = g +

(q n

)(qn + qn

)(q n)2

q2nn

(q n)2

, (38)

where g is given by Eq. (4) with = 4, and and run for +, 1, 2, . From this derivation, we see thatthe appropriate photon propagator for the interpolatinggauge given by Eqs. (26) and (27) (or (28)) has threeterms, consistent with [2426]. As we will see in thenext section, the last term in Eq.(38) is canceled by theinstantaneous interaction. Therefore, the two term gaugepropagator [7] can be used effectively without involvingthe instantaneous interaction.


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D. Longitudinal Photons

Having used the transverse gauge fields as the dynam-ical degrees of the freedom, we are left with the longi-tudinal degree of freedom which also deserves a physical

interpretation. This leftover longitudinal degree of free-dom is necessary to describe the virtual photon. First,to find the longitudinal polarization ( = 0) of the virtualphoton, we need to generalize Eq. (22c) replacingM2 byPP =q

2 which can be either positive (timelike) or neg-ative (spacelike). From the direct computation shown inAppendixB with this replacement, we find

L

(0)(0)=

(q n)(qn + qn)q2C + q2

+ (q+

)2qq

(q)2(q2C + q2) +q2nn

q2C + q2. (39)

Together with Tgiven by Eq. (35), it verifies the well-known completeness relation[30]

T +L = g +qq

q2 . (40)

We note that the last term in Eq. (39) without the de-pendence on q or qprovides the instantaneous contri-bution. As we show in the next section, the instanta-neous interaction appearing in the transverse gauge cor-responds to the contribution from the longitudinal polar-ization.

IV. TIME-ORDERED PHOTON EXCHANGE

As Kogut and Soper[31]regarded the theory of quan-tum electrodynamics as being defined by the usual per-turbation expansion of the S-matrix in Feynman dia-grams, we rewrite the sQED theory here by systemati-cally decomposing each covariant Feynman diagram intoa sum of interpolating x+-ordered diagrams. Since weconsider the Feynman expansion as a formal expansionas Kogut and Soper did, we also shall not be concernedin this paper with the convergence of the perturbationseries, or convergence and regularization of the integrals.

For clarity, we split this section into three subsections.In the first subsection, we decompose the covariant pho-ton propagator in an arbitrary interpolation angle as asum of x+-ordered terms. In the second part, we usethe obtained propagator to derive the x+-ordered dia-grams and their amplitudes for the lowest order photonexchange process. In the final subsection, we verify theinvariance of the corresponding total amplitude and dis-cuss about the instant form and light-front limits of the

x+-ordered photon propagators.

A. Photon Propagator Decomposition

The completeness relation given by Eq. (40) corre-sponds to the numerator of the covariant photon prop-agator in Landau gauge. We start from the covariant

photon propagator in position space

DF(x) = d4q(2)4ig + i

qq

q2

qq + i eiqx

. (41)

Due to the current conservation, however, the term in-volvingqdoesnt contribute to any physical process andour starting point is equivalent to the Feynman photonpropagator that Kogut and Soper used for their start-ing point[31]. For the same reason, the first and secondterms that involve q can be dropped in Eq. (39) and thecovariant photon propagator can be written as

DF(x) =i d4q(2)4 Tqq + i eiqx+ d

4q(2)4 nnq2C + q2

eiqx

= d2qdqdq+(2)4 exp[i(q+x+ + qx + qx)]

iTCq2

++ 2Sq

q+

Cq2

q2

+ i +

inn

q2C + q2

, (42)where we used Eq. (8) for qq in the denominator.

To get the x+-ordered contributions, we now evaluatethe q

+ integral in Eq. (42). We note here that the case

ofC =0 should be distinguished from the case ofC 0because the pole structures in the Tterm of Eq. (42)are different between the two cases, i.e. a single pole forC = 0 vs. two poles for C 0.

For C 0, the Tterm of Eq. (42) has two poles at

A i =Sq

+

q2

+ Cq2 C i, (43a)

B + i =Sq

q2

+ Cq2 C + i, (43b)

where >0. In calculating the contour integration, we

close the contour in the lower (upper) half plane forx+ > 0 (x+ < 0). This produces a term proportional

to the step function (x+) and the other term propor-tional to (x+). We then make changes of the vari-ables q q and q q which lead to B A forthe (x+) term and simplify the result expressing q

+

in terms of A. The last term in Eq. (42) immediately

gives a delta function of x+ after the q+

integration andprovides the instantaneous contribution. We note thatthe instantaneous contribution stems from the longitudi-


9/22

9

nal polarization of the virtual photon. Putting all termstogether, we obtain the following result for the case ofC 0:

DF(x) = d2q(

2

)3

dq

T

2

q2

+ Cq2

(x+)eiqx

+ (x+)eiqx

+ i(x+) d2q(2)3 dq nnq2

C + q2

ei(qx+qx

), (44)

whereq+

in the exponent of the first two terms should betaken as A given by Eq. (43a).

Now lets look at the light-front case. Because C =0,theres only one pole at q2

2q

i2q

=q2

2q+ i2q+.

It depends on the sign of q+ whether this pole is in theupper half plane or the lower half plane. As the integra-tion overq needs to be done inq+ >0 andq+


10/22

10

For the T integration, we use D given by Eq. (47) aswell as the following relations:

dT+

(T+

)eiP+T

+

=i

P+

, (57)

dT

+

(T+)eiP+T+ = iP+ , (58)

dT+eiP+T+

=2(P+). (59)

After the x and T integration, we get

iM =(ie)2(p4

+p2)(p3 +p1)

(2

)44

(p4 +p3 p2 p1

), (60)

where

= d2qdq(q)2

q2

+ Cq2

iTp4+ p2+ A

(p4 p2 q)2(p4 p2 q)

iT

p4+ p2+ + A(p4 p2 + q)2(p4 p2 +q)

+ d2qdq innq2C + q2

(p4 p2 q

)2

(p4 p2 q

). (61)

The three terms in corresponds to three different

time orderings y+ > x+, y+ < x+ and y+ = x+ respec-tively 1. The associated delta functions provide the mo-mentum conservation at each vertex as well as the conser-vation of total energy and momentum between the initialand final particles. In Fig.1, the three time ordereddiagrams are depicted with the momentum conservationat each vertex.

The corresponding photon propagators for T+ >0 (or

y+

>x+

) and T+


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11

p1 p2

p3

p4

x

y z

p1

p2

p3

p4

p1 p1 p3p3

p4 p4p2 p2

x+

(a) (b) (c)

initial inter med iate initial inter mediate

x

y

x

y

x

y

q q q

FIG. 1. The scattering of two particles in their center of mass frame (left most figure) and the three corresponding x+-ordereddiagrams.

with the familiar time-ordered perturbation theory in theIFD, although we now see it as a generalization to anyinterpolation angle. We note here that the propagatingphoton as the dynamical degree of freedom has only thetransverse polarization. This is consistent with the in-terpretation that the intermediate particles are now on-mass-shell or physical.

Besides the two propagating terms, there is a thirdterm in Eq. (61) that represents the instantaneous con-tribution. For the sake of consistency in our notation, wedenote this part of as

(c)

=inn

q2C + q2

, (69)

where q

and q are given by Eqs. (66) and (67), respec-tively. As noted earlier, the instantaneous contributionstems from the longitudinal polarization of the virtual

photon.We thus have three time-ordered diagrams as shown

in FIG.1, the first two of which represent time-orderedexchanges of propagating photon and the third of whichrepresents the instantaneous interaction. The invariantamplitude is then the sum of all three time-ordered am-plitudes:

iM = j=a,b,c

iM(j)

=(ie)2 j=a,b,c

(p4

+p2)(j)(p3 +p1), (70)

where the total energy-momentum conservation factor(2)44(p4 + p3 p2 p1) is implied.C. LFD vs. the Limit to LFD

Since the time-ordered gauge propagators have the in-terpolating step function given by Eq. (46), we take aclose look at the limiting cases ofC 0 and compare theresults with the exact C = 0 (LFD) results in this subsec-

tion. For convenience and simplicity of our discussion,we drop the(i) superscript in Eq. (64) and have

Q+(a) =Q+(b) =(p1 p3)2 + (p1 p3)2C Q+.

(71)

Because the terms proportional toqor q

in

Tcan bediscarded due to the current conservation in the physical

process, we may also replace T by

T g q2nn

q2C + q2

. (72)

Then, Eq. (62), Eq. (63) and Eq. (69) can be rewrittenas

(a)

=1

2Q+i T(p1 p3)p1+ p3+ Q

(a)+

, (73a)

(b

) =

1

2Q+i T

(p3 p1

)p1+ p3+ + Q(b)+ , (73b)

(c)

=inn(Q+)2 , (73c)

where Q(a)+

andQ(b)+

are given by Eq. (65).

For C 0, assigning the four-momentum transfer qas

qp1 p3 =q

(a)

= q(b)

, (74a)

q p1 p3 =q(a) = q

(b)

, (74b)

q+p1+ p3+ = (p2+ p4+), (74c)we have

Q(a)+

=

Sq

+

q2

+ Cq2

C =

Sq

+ Q+

C , (75)

Q(b)+

=

Sq

+

q2

+ Cq2

C =

Sq

+ Q+

C . (76)

Since both (q) and (q

) are unity for C 0, (a)


12/22

12

and (b)

can be rewritten as

(a)

=1

2Q+i TC

Cq+

+ Sq

Q+=

1

2Q+i TC

q+ Q+, (77a)

(b) =

1

2Q+i TC

Cq+

+ Sq

+ Q+=

1

2Q+i TC

q+ + Q+. (77b)

Summing all contributions, we use Eqs. (10), (71) and(72) to verify

(a)

+ (b)

+ (c)

=i TC(q+)2 (Q+)2 + innq2C + q2

=i T

q2 +

inn

q2C + q2

=ig

q2 . (78)

It is clear from this derivation that the contribution fromthe instantaneous interaction is cancelled by the corre-sponding nn term in the transverse photon propaga-tor and the sum of all the contributions is totally Lorentzinvariant.

For C = 0, (q) and (q

) are neither unity nor

same to each other. Thus, we need to look into the de-tails of each contribution in Eqs. (73a), (73b) and (73c)very carefully to understand the Lorentz invariance of thetotal amplitude. TakingC =0 in Eqs. (73a), (73b) and(73c), we get :

(a) =1

2q+

i

(g

q2nn

(q+

)2

)

(q+

)p1 p3 q22q+, (79a)

(b) = 1

2q+i(g q2nn(q+)2)(q+)

p1

p3

+q2

2q+, (79b)

(c) =inn

q+2 , (79c)

where q+ =p+1 p+

3 and of course the conservation of to-tal LF energy in the initial and final states is imposedsuch as p1 p

3 =p

4 p

2 . In this case, the contributionfrom the LF time-ordered process (a) or (b) in Fig. 1depends on the values of external momenta p+1 and p

+

3 ,i.e. whether p+1 >p

+

3 or p+

1 p

+

3 , then only (a) contributes.Similarly, if p+1 < p

+

3 , then only (b) contributes. This isdramatically different from the C 0 case where both (a)and (b) contributes regardless ofp1 >p3 or p1 p

+

3 , then the sum of allcontributions is given by

(a) + (c) =

1

2q+

i(g q2nn(

q+

)2)

p1

p3

q2

2q

+

+

inn

q+2

=

i(g q2nn(q+)2)2q+q q2

+

inn

q+2

=ig

q2 , (80)

where p1 p

3 = q since the four-momentum transfer q

is assigned as p1 p3. As in the C 0 case, the con-tribution from the instantaneous interaction is cancelledby the corresponding nn term in the transverse pho-ton propagator and the sum of all the contributions istotally Lorentz invariant. Similarly, if p+1 < p

+

3 , then

(a) is replaced by (b) withq+ = q+ and the sum

(b) +

(c) =

ig

q2 is totally Lorentz invariant.

Now, lets consider the limit C 0 (i.e., 4 orS 1) from the C 0 case given by Eqs. (77a) and(77b),

where we take q+ q+ =q

, q q =q

+, Q+ Q+ =q+.To obtain the correct limits, we need to make a carefulexpansion. For q+ =p+1 p

+

3 >0, we get

Cq+

+ Sq

Q+ =Cq+

+ Sq

q2

+ Cq2

Cq + q+

q+2 + Cq2

Cq Cq2

2q++O(C2), (81)

Cq+

+ Sq

+ Q+ 2q+ + Cq+ + Cq2

2q++O(C2), (82)

T g 2q

q2

2q+

q+nn. (83)

Thus, in the limitC 0 forp+1 >p+

3 , Eq. (77a), Eq. (77b)as well as Eq. (73c) become

(a) =i T

2q+1

q

q2

2q+

= ig

2q+1

q

q2

2q+

inn

(q+

)2

, (84a)

(b) =limC0

i T

2q+C

2q+ + Cq+ + Cq2

q+

0, (84b)

(c) =inn(q+)2. (84c)

This shows the agreement between the C =0 result andthe C 0 result for the kinematic situationp+1 >p

+

3 . Note

here that (b) =0 was given in LFD (C = 0) via the factor


13/22

13

(q+)in Eq. (79b) while (b) 0 as C 0 in Eq. (84b)is obtained without the factor (q+). Similarly, theagreement between the C = 0 result and the C 0 resultis also found for the kinematic situation p+1 p

+

3 ,p+1


14/22

14

the particles that are going back to back at angle inCMF are depicted in FIG. 1. Correspondingly, the four-momenta of the initial and final particles in CMF aregiven by

p1 =(1, 0, 0, p), (89a)p2 =

(2, 0, 0, p

), (89b)

p3 =(1, p sin , 0, p cos ), (89c)p4 =(2, p sin , 0, p cos ). (89d)

To discuss the longitudinal boost (T3) effect on time-ordered scattering amplitudes, lets now boost the wholesystem to the total momentum Pz. From the Lorentztransformation for a composite free particle system, weknow that the total energy in the new frame is E =

(Pz)2

+

M

2

. The Lorentz transformation for each pi(i = 1, 2, 3, 4) can then be described in terms of=EMand =PzM:

p0

i =p0

i + pzi (90a)

pz

i =pzi + p

0

i (90b)

p

i =p

i (90c)

The boosted four momentum transfer can therefore bewritten as

FIG. 2. (color online) Time-ordered amplitudesM(a),M(b),

M(c) for m1 =1,m2 =2, p =3, =3, and their sumM

(tot).


15/22

15

qx=p

x1 p

x3 = p sin , (91a)

qy=p

y

1 p

y

3 =0, (91b)

qz=p

z1 p

z3 =

E

M

p

(1 cos

), (91c)

q0=p

0

1 p0

3 =Pz

Mp(1 cos ), (91d)

where M and E are functions of m1, m2, p and Pz.

We can use these four-momentum components given by

Eq. (91) and apply Eq. (6) to form q

and q for anyinterpolation angle. Eq. (90) can also be used togetherwith Eq. (6) and Eq. (87) to express all the componentsof the boosted four-momenta. Computingp24 p

13 and

plugging it with other factors such as p+

24 p+

13 and q

into Eq. (71) and subsequently Eq. (86), we get the in-

terpolating time-ordered amplitudes M(j)

in a boosted

frame. In CMF, all of these amplitudes are then given byfunctions of each particles initial momentum p(p), in-dividual particles rest massm1 and m2, scattering angle, the total momentum Pz and the interpolation angle. For given values of m1, m2, p and , the x

+-orderedamplitudes are dependent on the frame (Pz) and the in-terpolation angle (or C).

To exhibit this feature quantitatively, we choose m1 =1, m2 =2, p = 3, in the same energy unit (i.e., m2 and pscaled by m1), and =3. For simplicity, we also takethe charge e to be 1 unit in our calculation. In FIG.2,we plot M(a), M(b), M(c) as well as the sum of all threeamplitudes M(tot) as functions of both the interpolationangle and the total momentum Pz to reflect not onlythe interpolation angle dependence but also the frame de-pendence. The total amplitude M(tot) shown in FIG.2is both frame independent and interpolation angle inde-pendent as it should be.

The detailed structures of these three time-ordered di-agrams are very interesting. Just like in the3 toy modeltheory studied in[20], thePz >0 region is smooth for allthree amplitudes, while a J-shaped correlation curve ex-ists in the Pz < 0 region, which we plotted as the redsolid line in FIG.2 (color online). This is the curve thatstarts out in the center of mass frame (Pz = 0) in the = 0 limit, but maintains the same value of amplitudethroughout the whole range of interpolation angle. The

values maintained by these curves can be found for arbi-trarym1, m2, p and , and given by

M(a)

=M(b)

= cot2

2

2 , (92a)

M(c)

= 12

p2 sin2 2

. (92b)

Again, i =

q2 + m2i ,(i = 1, 2) and the charge e inEq. (86) is taken as 1 unit.

FIG. 3. (color online) Time-ordered amplitudesM(a),M(b)

and M(c) as a function of the total momentum Pz at theinstant form limit ( = 0). The red dots on thePz = 0 axisdenotes the starting position of the J-curves on each surface.

In FIG.3, we plot the profiles ofM(a), M(b)and M(c)

in IFD (C =

1 or =

0) for the given values ofm1

, m2

, pand and depict the starting point of the J-curve cor-responding to the values ofM(a), M(b) and M(c) atPz = 0 given by Eqs. (92). The maxima for M(a) andM

(b) stay in the Pz >0 and Pz


16/22

16

all the plus components of the particle momenta vanishin the limit Pz , we now should look closely atthe kinematic situation p+1 =p

+

3 =q+=0 which we have

postponed its discussion in Sec. IV C.

Since the details of each contribution depend on thevalues ofC and Pz, the results are in general dependenton the order of taking the limits to C 0 andPz .

In particular, ifC = 0 is taken first, then the results mustbe independent of the Pz values due to the longitudinalboost invariance in LFD as discussed before and the re-sult at Pz = is identical to that for any other Pz

value. However, if we take Pz first and considerC 0, then we should first examine the C 0 results inthe Pz limit with a great care. We begin with thediscussion on the C =1 (i.e. IFD) results and vary theinterpolation angle from =0 to 4.

As shown in FIGs. 2and3, each of the time-orderedamplitudesM(a), M(b)and M(c)at C = 1(=0), are notsymmetric under the reflection of Pz Pz. Althoughthe IFD results coincide with the LFD results asPz ,

they do not agree in the limit Pz as shown in

FIG. 2. Thus, the prevailing notion of the equivalencebetween IFD and LFD in the infinite momentum frame(IMF) should be taken with a great caution since it worksfor the limit of Pz but not in the limit Pz .We have already discussed the treachery in taking thelimit Pz even for the amplitudes with the reflec-tion symmetry underPz Pz in IFD such as the scalarannihilation process analogous to QED e+e + inour previous work [20]. The present analysis of the sQEDscattering process analogous to QED e eprovides aclear example of breaking the reflection symmetry underPz Pz in IFD and fortify the clarification of the con-fusion in the folklore of the equivalence between the IMFand the LFD. As we vary the interpolation angle from=0 and approach to=4, this broken reflection sym-metry under Pz Pz persists while the LFD resultsare symmetric under Pz Pz due to the longitudinalboost invariance discussed previously. The J-curve cor-relation between Pz and C given by Eq. (93) provides asimultaneous limit of Pz 1C12 as C 0 andthe corresponding results ofM(a), M(b) and M(c) areuniquely given by Eqs. (92a) and (92b). As mentionedearlier, the numerical values along the J-curve plotted asthe red solid line in FIG.2(color online) are identical allthe way to the Pz limit and thus the clear distinc-tion is manifest between the results in the C 0 limitwith the J-curve correlation and the LFD results withthe exact C = 0. Since the J-curve exists only for Pz


17/22

17

provided a clear example of breaking the reflection sym-metry underPz Pz in IFD and clarified the confusionin the folklore of the equivalence between the IMF andthe LFD. As we vary the interpolation angle from =0(C = 1) and approach to = 4 (C = 0), this brokenreflection symmetry under Pz Pz persists while theLFD results are symmetric under Pz Pz due to thelongitudinal boost invariance. Moreover, the universalcorrelation between Pz and C given by Eq. (93) whichwas coined as the J-curve in our previous work [20] is

intact in each of thesex+-ordered diagrams as plotted bythe red solid line in FIG. 2. The J-curve starts out fromthe center of mass frame in the instant form and goes toPz = as it approaches to the light-front limit. Thiscorrelation is independent of the specific kinematics ofthe scattering process and manifests the difference in theresults between the light-front limit and the exact light-front (LFD). Since the J-curve exists only for Pz


18/22

18

(P, 0) =

cos (sin cosh 3 + cos sinh 3) sin cos (cos cosh 3 + sin sinh 3)C

1 sin (sin cosh 3 + cos sinh 3)

2 sin

(sin cosh 3 + cos sinh 3

)

cos cos (cos cosh 3 + sin sinh 3) sin (sin cosh 3 + cos sinh 3)C

. (A3)

Using the relations listed in Eqs. (18) and (19), and multiplying the transverse polarization vectors (P, ) with thephase factor P

1P

2

P to simplify, we arrive at the following polarization vectors for the four-momentum P:

(P, +) = 12Psin P,P1P iP2PP ,P2P + iP1PP , cos P , (A4)

(P, ) = 12Psin P,P1P + iP2PP ,P2P iP1PP , cos P , (A5)

(P, 0

)=

1

MPP2 cos P

P+ sin

C , P1P+, P2P+,

P+P

cos P2 sin

C . (A6)

The polarization vectors listed above are written in the form (P, ) =(0, 1, 2, 3). We then change the basis to(+

, 1, 2, ) using the relations listed in Eq. (7). Finally, we obtain the polarization vectors given by Eq. (22):(P, +) = 1

2PSP,P1P iP2PP ,P2P + iP1PP , CP , (A7a)

(P, ) = 12PSP,P1P + iP2PP ,P2P iP1PP , CP , (A7b)

(P, 0) = P+MP

P+

M2

P+, P1, P2, P , (A7c)

where the relation P2 =(P+

)2

M2C is used to derive the first component of (P, 0).

Appendix B: T and L derived directly from the

interpolating polarization vectors

In this Appendix, we derive the numerator of the pho-ton propagator directly from the polarization vectorslisted in Eq. (22) and Eq. (A7).

We evaluate the relevant matrix elements. With thefour-momentum P =q for virtual photon, these are, for

= +1:

+(q, = +)

+(q, = +) = S2q2

2Q2 , (B1)

+(q, = +)1(q, = +) = S (qq1 iq2Q)

2Q2 , (B2)

+(q, = +)2(q, = +) = S (qq2 + iq1Q)

2Q2 , (B3)

+(q, = +)

(q, = +) = CSq2

2Q2 , (B4)

(q, = +)

+(q, = +) = CSq2

2Q2 , (B5)

(q, = +)1(q, = +) = C (qq1 iq2Q)

2Q2 , (B6)

(q, = +)2(q, = +) = C (qq2 + iq1Q)

2Q2 , (B7)

(q, = +

)

(q, = +

)=C2

q

2

2Q2 , (B8)

1(q, = +)+(q, = +) = S (qq1 + iq2Q)2Q2 , (B9)1(q, = +)1(q, = +) = q2q21 + q22Q2

2q2Q2 , (B10)1(q, = +)2(q, = +) = Cq1q2 + iqQ2Q2 , (B11)


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19

1(q, = +)(q, = +) = C(qq1 + iq2Q)2Q2 , (B12)2

(q, = +

)+

(q, = +

)=S (q

q2 iq1Q)2Q2

, (B13)

2(q, = +)1(q, = +) = Cq1q2 iqQ2Q2 , (B14)2(q, = +)2(q, = +) = q2q22 + q21Q2

2q2Q2 , (B15)2(q, = +)(q, = +) = C(qq2 iq1Q)2Q2 , (B16)

and for = 1:

+(q, = )

+(q, = ) = S2q2

2Q2 , (B17)

+

(q, =

)1

(q, =

)=S

(q

q1 + iq2Q

)2Q2

, (B18)

+(q, = )2(q, = ) = S (qq2 iq1Q)

2Q2 , (B19)

+(q, = )

(q, = ) = CSq2

2Q2 , (B20)

(q, = )

+(q, = ) = CSq2

2Q2 , (B21)

(q, = )1(q, = ) = C (qq1 + iq2Q)

2Q2 , (B22)

(

q, =

)2

(q, =

)=

C

(q

q2 iq1Q

)2Q2 , (B23)

(q, = )

(q, = ) = C2q2

2Q2 , (B24)

1(q, = )

+(q, = ) = S (qq1 iq2Q)

2Q2 , (B25)

1(q, = )1(q, = ) = q2q21 + q22Q2

2q2Q2 , (B26)1(q, = )2(q, = ) = Cq1q2 iqQ2Q2 , (B27)

1(q, = )(q, = ) =C

(q

q1 iq2Q

)2Q2 , (B28)2(q, = )+(q, = ) = S (qq2 + iq1Q)2Q2 , (B29)2(q, = )1(q, = ) = Cq1q2 + iqQ2Q2 , (B30)2(q, = )2(q, = ) = q2q22 + q21Q22q2Q2 , (B31)

2(q, = )

(q, = ) = C (qq2 + iq1Q)

2Q2 , (B32)

where Q

q2

+q2C.

By defining T ==+, (q, )(q, ), we obtainthe following matrix form:

T =1

Q2S2

q

2 Sq

q1 Sqq2 CSq

2

Sq

q1 q2

+ Cq22 Cq1q2 Cqq1

Sq

q2 Cq1q2 q2

+ Cq21 Cqq2

CSq2 Cqq1 Cqq2 C2q2 .

(B33)

This can be written in general as

T = g + X(qn + nq) + Y nn + Zqq, (B34)where the coefficients X, Y, Zcan be fixed by compar-ing this equation with the explicit matrix form given byEq. (B33). We find

X=q+

Q2 = (q n)Q2 , (B35a)

Y = q2

Q2, (B35b)

Z= C

Q2. (B35c)

Thus, our result agrees with Eq.(35) derived in the maintext.

Similar manipulation can be done for the longitudinalpart, L =

(q, 0

)

(q, 0

), yielding

L =(q+)2(q)2Q2

L2

Lq1 Lq2 LqLq1 q

21 q1q2 qq1

Lq2 q1q2 q2

2 q

q2

Lq

q

q1 qq2 q2

, (B36)where we introduced for convenience,

L = q+

(q)2q+

.

Notice that because we are dealing with virtual photons,we have replaced M2 by PP = q

2 in the longitudinalpolarization vector given by Eq. (22c). Written in a gen-

eral form, this gives Eq. (39).

Appendix C: Photon propagator decomposition on

the light front

BecauseC = 0, theres only one pole at q22q+ i2q+

for the first term in Eq. (42). Whenq+ >0, the pole is inthe lower half plane, and when q+


20/22

20

separate the q+ integral into two regions:

d2q(2)4 0 dq+ei(q+xqx)I

+ 0

dq+ei(q+xq

x)I , (C1)

where

I=

dqeiqx+ T

2qq+ q2

+ i. (C2)

To evaluateI, we close the contour in the plane where thearc contribution is zero. This means we close the contourin the upper (lower) half plane when x+ < 0 (x+ > 0).For the factor I in the first term of Eq. (C1), the onlynone-zero contribution comes from the pole thats in thelower half plane, which is picked up when we close thecontour from below (x+ > 0). Similarly, for the factorI in the second term of Eq. (C1), the only none-zerocontribution comes from the pole thats in the upper half

plane, which is picked up when we close the contour fromabove (x+


21/22

21

M(b)

= 1

2Q+1

q+ + Q+, (D1b)

where Q+ is defined in Eq. (71).

We use the same kinematics used for the sQED case,i.e. q = 0. Because the kinematics is the same,

the boosted q

satisfies the same equations as listed inEq. (91). Using Eq. (6), we can also get the boosted qandq

, which can then be used to calculate the scatter-

ing amplitudes in the boosted frame.

The amplitudes are given by the functions ofm1, m2,p and . For comparison, we use the same parametervalues that we used for plotting FIG. 2 in sQED, i.e.m1 =1, m2 =2, p = 3, and = 3. M(a) and M(b) withthe J-curve depicted by the solid red line are plotted inFIG. 4. Comparing it to FIG. 2, we see that the pho-ton propagator modifies the profiles of the time-orderedamplitudes. Nevertheless, the J-curve remains the sameand is still given by Eq. (93).

Appendix E: Time-ordered annihilation amplitudes

in sQED

For the sQED annihilation process analogous toe+e + in QED, we can still use FIG. 1 for the kinemat-ics. Three time-ordered diagrams corresponding to thelowest order amplitudes are shown in FIG.5. Here, theamplitude of Eq. (70) changes to

iM = j=a,b,c

iM(j)

=(ie)2 j=a,b,c

(p3 p4 )(j)(p1 p2). (E1)The time-ordered amplitudes are given by

M(a)

= (ie)2 [p34 p12 + p+34p+12q2(Q+)2]C2Q+(q+ Q+) , (E2a)

M(b)

=(ie)2 [p34 p12 + p+34p+12q2Q+)2]C2Q+(q+ + Q+) , (E2b)

M(c)

=

(ie

)2p

+

34p+12

(Q+

)2

, (E2c)

where

p34 =p3 p4, (E3a)

p12 =p1 p2, (E3b)

and q =p1 +p2. We use the same kinematics (CMF) asbefore so that pis are given by

p1 =(, 0, 0, p), (E4a)p2 =(, 0, 0, p), (E4b)

p1p1

p2 p4

p3p1

p2 p3

p4

q

q

(a) (b) (c)

p1

p2 p3

p4

q

FIG. 5. Time-ordered annihilation diagrams and their ampli-tudes for sQED

p3 =(, pfsin , 0, pfcos ), (E4c)p4 =(, pfsin , 0, pfcos ). (E4d)

Since we are considering the annihilation process, theinitial particles 1 and 2 should have the same mass, andthe final particles 3 and 4 should have the same mass.So, we set m1 =m2 =m and m3 =m4 =mf. Due to theenergy conservation, all four particles should have thesame energy in CMF, and p and pfare therefore related

by pf =

q2 + m2 m2f. The boosted p

is still follow

Eq. (90). Thus, q in the annihilation process is rathersimple:

qx=p

x1 +p

x2 =0, (E5a)

qy=p

y

1 +p

y

2 =0, (E5b)

qz=p

z1 +p

z2 =P

z, (E5c)

q0=p

0

1 +p

0

2 =E. (E5d)

Applying Eq. (90) to Eqs. (E3a) and (E3b) with pi


22/22

22

given by Eq. (E4), we find that

p34 p

12 = 4p

m2 m2f

+p2 cos , (E6a)

q2p

+

34p+

12(Q+)2 =4pm2 m2f +p2 cos . (E6b)So the numerators in Eqs. (E2a) and (E2b) are 0, regard-less of the frame or the interpolation angle . Therefore,M

(a) = M

(b) = 0, and the only non-zero contribution

comes from the instantaneous interaction. One can ver-

ify that indeed the covariant total amplitude is the sameas M(c):

M(c)

=M = p34 p12

q2 =

p

m2 m2f

+p2 cos

m2 +p2 , (E7)

where the coupling constant e is taken as 1. We cansee that this result is also independent of the frame andinterpolation angle, just as expected.

In FIG. 5, we again use the same parameter valuesas before: m1 = 1, m2 = 2, p = 3, and = 3. Thecorresponding time-ordered amplitudes shown here areall flat planes.

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the electromagnetic gauge field interpolation between the instant forma nd the front form of the...

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