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Journal of Modern Physics, 2010, 1, 385-392 doi:10.4236/jmp.2010.16055 Published Online December 2010 (http://www.SciRP.org/journal/jmp) Copyright © 2010 SciRes. JMP Light-Front Hamiltonian, Path Integral and BRST Formulations of the Chern-Simons Theory under Appropriate Gauge-Fixing Usha Kulshreshtha1, Daya Shankar Kulshreshtha2, James P. Vary3 1Department of Physics, Kirori Mal College, University of Delhi, Delhi, India 2Department of Physics and Astrophysics, University of Delhi, Delhi, India 3Department of Physics and Astronomy, Iowa State University, Iowa, USA E-mail: ushakulsh@gmail.com, dskulsh@gmail.com, jvary@iastate.edu Received August 24, 2010; revised September 17, 2010; accepted October 20, 2010 Abstract The Chern-Simons theory in two-space one-time dimensions is quantized on the light-front under appropriate gauge-fixing conditions using the Hamiltonian, path integral and BRST formulations. Keywords: Hamiltonian Quantization, P ath Integral Quantization, BRST Quantization, Ch ern-Simons Theories, Light-Cone Quantization, Light-Front Quantization, Constrained Dynamics, Quantum Electrodynamics Models in Lower Dimensions, Light-Cone Quantization 1. Introduction Studies of the models of quantum electrodynamics in two-space one-time dimensions involving the Chern- Simons (CS) theories [1-10] are of wide interest and form a rather broad field of investigations in various contexts. Effective theories with excitations, with frac- tional statistics are supposed to be described by gauge theories with Chern-Simons term. The statistics (Bose- Fermi) transmutation has some important experimental consequences in the physics of high -c supercon- ductivity [3]. W ilczek studied [5] the possib ility of exotic statistics appearing in two-space one-time dimensions where the objects obeying this unusual statistics are called anyons [3,5]. The above studies are of very wide interests [1-10] and they provide rather natural motiva- tions for our present studies. T Very recently, we have studied [8-11] the CS theory [8] and the CS-Higgs (CSH) theory in the symmetry phase of the Higgs potential [9] as well as the CSH theory in the so-called broken (or frozen) phase of the Higgs po- tential [10] using the usual instant-form (IF) of dynamics (on the hyperplanes: 0== x t constant ), under appro- priate gauge-fixing conditions. In the present work we quantize the pure CS theory on the light-front (LF) using the Hamiltonian, path integral and BRST formulations [8-16] under appropriate gauge- fixing, using the LF dynamics (on the hyperplanes defined by the LC time: ( 01 == 2 xx x constant ) [17,18]. It may be important to mention here that because the LF coordinates are not related to the conventional IF coordinates by a finite Lorentz transformation, the des- criptions of the same physical result may be different in the IF and LF dynamics and the LF quantization (LFQ) often has some advantages over the conventional IF quantization (IFQ) and a study of both the IFQ and the LFQ of a theory determines the canonical structure and constrained dynamics of a theory rather completely [8-18]. Different aspects of this theory have been studied by several authors in various contexts [1-10]. For further details of the motivations for a study of the different aspects of the Chern-Simons theories by various authors including a comparative description of different studies, we refer to our earlier work of Reference [8-11]. In the next section, we study its LF Hamiltonian and path integral formulations and its BRST formulation is studied in Section 3. The summary and discussion is finally given in Section 4. U. KULSHRESHTHA ET AL. 386 2. Hamiltonian and Path Integral Formulations In this section we quantize the pure CS theory on th e LF, using the Hamiltonian, path integral formulations under appropriate gauge-fixing. The pure Chern-Simons theory in two-space one-time dimensions is defined by the action [1-10]: 3 11 12 =, =, = 22 SAdx AA (1) 012 012 :1,1,1; ,0,1,2; 1gdiag (2) Here is the Chern-Simons parameter. The LF [5] action of the theory reads: 2 222 222 = := 2 Sdxdxdx AAAAAAAAAAAA (3) In the following, we would consider the Hamiltonian formulation of the theory described by the above action. The canonical momenta obtained from the above equa- tion are: 2 22 :==0, :==, := 22 A E A AA A (4) Here , and are the momenta canonically conjugate respectively to 2 :=E A , A and 2 A . The above equations however, imply that the theory po- ssesses three primary constraints: 12 23 =0; =0; =0 22 AEA (5) The symbol here denotes a weak equality (WE) in the sense of Dirac [12,13], and it implies that these above constraints hold as strong equalities only on the reduced hypersurface of the constraints and not in the rest of the phase space of the classical theory (and similarly one can consider it as a weak operator equality (WOE) for the correspondi n g quantum theor y ). The canonical Hamiltonian density corresponding to is: 2 22 22 := =2 cAAEA AAAAAAAA (6) After including the primary constraints 1 , 2 and 3 in the canonical Hamiltonian density c with the help of the Lagrange multiplier field H u s , and the total Hamiltonian density could be written as: v T H 2 22 22 =22 2 TsAuEA AAAAAAAA v (7) The Hamilton’s equations of motion of the theory that preserve the constraints of the theory in the course of time could be obtained from the total Hamiltonian (and are omitted here for the sake of bravity): 2 = TT H dxdx (8) The preservation of 1 , 2 and 3 for all times does not give rise to any further constraints. The theory is thus seen to possess only three constraints i (with i = 1, 2, 3), where all i are primary constraints. Further, the matrix of the Poisson br ackets among the constraints i is seen to be a singular (in fact, a null) matrix implying that the set of constraints i is first-class and that the theory under consideration is gauge-invariant. The physical degrees of freedom of the system are governed by the reduced Hamiltonian density of the theory (which is obtained by implementing the cons- traints of the theory strongly). Also, in the present case, A plays the role of gauge variable and the two pairs ( A , ) and (2 A , ) are the pair of inessential eliminable variables and a pair describing the physical degrees of freedom of the system. Accordingly, we choose, in the present case, the first pair namely, ( E A , ) as the pair describing the physical degrees of freedom and the other pair as the pair of inessential eliminable variables. So for writing the reduced Hamil- tonian density of the theory, we choose A and as the independent variab les and the remaining phase space variables as the dependent variables. The later ones are then expressed in terms of the independent variables as: Copyright © 2010 SciRes. JMP U. KULSHRESHTHA ET AL.387 2 =0; =; = 22 EA A (9) Finally the reduced Hamiltonian density of the theory describing the physical degrees of freedom of the system expresed in terms of the independent variables is then obtained as: 2 =2 R A AA (10) where = RR H dx is the reduced Hamiltonian of the theory and it describes the physical degrees of freedom of the system. Here we remind ourselves that as an alternative to the above, we could have equivalently expressed it in terms of the other pair namely, (2 A , ) instead of the pair (E A , ). Using the above equation we then obtain the field equations derived from the Heisenberg equations of motion as: 2 2 =,=2 =,=0 =,= =,=2 R R R R iH A AiAH iH A AiAHA (11) The vector gauge current of the theory 2 ,, J JJJ is: 22 22 22 22 222 2 == 2 == 2 == 2 Jjdx dxAA Jjdxdx AA Jjdxdx AA (12) The divergence of the vector gauge current density of the theory could now be easily seen to vanish satisfying the continuity equation: j = , implying that the theory possesses at the classical level, a local vector- gauge symmetry. The action of the theory is indeed seen to be invariant under the local vector gauge transfor- mations: 0 22 2 2 =, =, =, = =, =, =, = 22 =, ===0 suv AAs u AoE v (13) where 2 ,, x xx is an arbitrary function of its arguments. In order to quantize the theory using Dirac’s procedure we now convert the set of first-class constraints of the theory i into a set of second-class constraints, by imposing, arbitrarily, some additional constraints on the system called gauge-fixing conditions or the gauge constraints. For this purpose, for the presen t theory, we could choose, for example, the following set of gauge-fixing condition: =A0 (14) Here the gauge 0A represents the light-cone coulomb gauge which is a physically important gauge. Corresponding to this gauge choice, the theory has the following set of constraints under which the qu antization of the theory could e.g. be studied: 111 2222 333 4 === 0 === 2 === 0 2 == 0 A EA A 0 (15) The matrix R of the Poisson brackets among the set of constraints i with is seen to be nonsingular with the determinant given by ( =1,2,3,4)i 122 222 =det Rxyxy (16) The other details of the matrix are omitted here for the sake of bravity. Finally, following the standard Dirac quantization procedure, the nonvanishing equal light-cone-time commutators of the theory, under the gauge: R 0A are obtained as: 222 22 22 22 22 22 ,, , ,, = ,, , ,, =2 ,, , ,, =4 Axxx Axxx ixy xy A xxx xxx ixy xy xxx Exxx ixyxy (17) Also, for the later use, for considering the BRST for- mulation of the theory we convert the to tal Hamil-tonian density into the first-order Lagrangian density 0 I : 02 222 222 := =2 I su vT A AEAs u vH AAAAAAAAAAAA (18) Copyright © 2010 SciRes. JMP U. KULSHRESHTHA ET AL. Copyright © 2010 SciRes. JMP 388 In the path integral formulation, the transition to quantum theory is made by writing the vacuum to vacuum transition amplitude for the theory called the generating functional k Z J of the theory [8-11,14,15] under the gauge-fixing under consideration, in the pre- sence of the external sources k J as: 32 =expk kk suv ZJdidxJAAEAsuv H T (19) Here, the phase space variables of the theory are: 2 ,,,,, k A AAsuv with the corresponding respec- tive canonical conjugate momenta: ,,,,, ks E uv . The functional measure d of the generating functional k Z J under the above gauge-fixing is obtai ned as: 22 22 2 2 = 0 000 22 suv dxyxydAdAdAdsdu dddEddd AEAA dv (20) The Hamiltonian and path integral quantization of the theory under the gauge: is now complete. 0A 3. BRST Formulation For the BRST formulation of the model, we rewrite the theory as a quantum system that possesses the genera- lized gauge invariance called BRST symmetry. For this, we first enlarge the Hilbert space of our gauge-invariant theory and replace the notion of gauge-transformation, which shifts operators by c-number functions, by a BRST transformation, which mixes operators with Bose and Fermi statistics. We then introduce new anti-com- muting variables c and c (Grassman numbers on the classical level and operators in the quantized theory) and a commuting variable such that[8-11,16]: b 22 2 2 ˆˆˆˆ =, =, =, = ˆˆˆˆ =, =, =, = 22 ˆˆˆˆ =, ===0 ˆˆˆ =0, =, =0 uvs A cA cvcsc A coEc uc ccbb c (21) with the property 2 ˆ = 0. We now define a BRST- invariant function of the dynamical variables to be a function f such that . Now the BRST gauge- fixed quantum Lagrangian density ˆ=0f B RST for the theory could be obtained by adding to the first-order Lagrangian density 0 I , a trivial BRST-invariant function, e.g. as follows: 222 2221 ˆ := 22 BRST AAAAAAAAAAAAc Ab (22) The last term in the above equation is the extra BRST-invariant gauge-fixing term. After one integra- tion by parts, the above equation could now be written as: 2 222 2221 := 22 BRST A AAAAAAAAAAAbb Acc (23) Proceeding classically, the Euler-Lagrange equation for reads: b =bA (24) the requirement then implies ˆ=0b ˆˆ =bA (25) which in turn implies =0c (26) The above equation is also an Euler-Lagrange equa- tion obtained by the variation of B RST with respect to c. In introducing momenta one has to be careful in defining those for the fermionic variables. We thus define the bosonic momenta in the usual manner so that := = BRST b A (27) but for the fermionic momenta with directional deriva- tives we set U. KULSHRESHTHA ET AL.389 :==; :== c BRSTcBRST cc cc (28) implying that the variable canonically conjugate to is c c and the variable conjugate to c is . For writing the Hamilotonian density from the Lagrangian density in the usual manner we remember that the former has to be Hermitian so that: c 2 2 22 22 = 1 =2 2 BRST s uvccBRST suv cc HAAEAs uv ccL suv AAAAAAAA (29) The consistency of the last two equations could now be easily checked by look ing at the Hamilton’s equation s for the fermionic variables. Also for the operators ,,cc c and c , one needs to satisfy the anticom- mutation relations of with c c or of c with , but not of , with c cc. In general, and cc are independent canonical variables and one assumes that ,=,=,=0; ,=1, cc cc cccccc (30) where , mean s an an tico mmutator. We thus see that the anticommulators in the above equation are non-triv ial and need to be fixed. In order to fix these, we demand that c satisfy the Heisenberg equation: { } ,= BRST c ic (31) and using the property 22 0cc one obtains ,=, BRST cccc (32) The last three equations then imply : ,=1 ,=cccci (33) Here the minus sign in the above equation is nontrivial and implies the existence of states with negative norm in the space of state vectors of the theory. The BRST charge operator is the generator of the BRST trans- formations. It is nilpotent and satisfies . It mixes operators which satisfy Bose and Fermi statistics. According to its conv entional d efinition , its commutato rs with Bose operators and its anti-commutators with Fermi operators for the present theory satisfy: Q20Q 2 2 ,=,=,=, ,= ,= 2 ,= 2 AQAQ AQc QEQ c cQEA A (34) All other commutators and anti-commutators invol- ving vanish. In view of this, the BRST charge ope- rator of the present theory can be written as: Q 22 =2 QdxdxicEAA (35) nd This equation implies that the set of states satisfying the coitions: 2 =0, =0, =0 22 AEA be (36) long to the dynamically stable subspace of states satisfying |>=0Q , i.e., it belongs to the set of BRST-invariant states. In order to understand the con- dition neevering the physical states of the thperators and ded for reco eory we rewrite the o cc in terms of fermioni and creatio operators. Forhe c annihiliation derived ea n this purpose we consider Euler lagrange equation for t variable crlier. The solution of this equation gives (for the light-cone time x the Heisenberg operators c and correspondingly c in terms of the fermionic Annihilation and creation operators as: † =, =cGFcGF (37) Which at e light-cone time =0 th imply † 0=, 0= 0=, 0= cc Fcc ccGccG (38) † F By imposing the conditions (obtained earlier): ,=1 ,= c cccc i (39) we then obtain 22 ==,=,=0, cc ccc 2= F †2 F †† =,=,=0,FF GG †† ,=1, =GFGF i Now let denote the fermionic vacuum (41) Defining to have norm one, the last three e tions imply |0 > for which |0>= |0>=0 GF |0 > qua- †† <0>=, <00>= F 0G so that iGF i (42) †† GF|0>0,, |0>0 (43) The theory is thus seen to possess negative norm states in the fermionic sector. The existence of these ne norm states as free states of the fermionic pgative art of B RST is , however, irrelevant to the existence of physicsl in the orthogonal subspace of the Hilbert space. In terms (40) states Copyright © 2010 SciRes. JMP U. KULSHRESHTHA ET AL. 390 of annihilation and creation operators B RST is: 2† 22 22 1 =2 2 BRST suv s uv GG AAAAAAAA (44) and the BRST charge operator is: 22 =2 QdxdxiGEAA (45) Now because |>=0 Q , the set of states annihiliated by t for which the constraints l states for which Q contains not only the se of the theory hold but also additiona 2 |>=|>=0 |0, |0, |0 22 FG AEA (46) The Hamiltonian is also invariant under the anti-BRST transformation given by: 22 2 2 ˆˆˆ ˆ =, =, =, = ˆˆˆ ˆ =, =0, =, = 22 ˆˆˆˆ =, ===0 ˆˆ ˆ =0, =, =0 suv A cA cscvc A ccEc with generator or anti-BRST charge uc cc bb (47) 2 =2 QdxdxicE AA 2 which in terms of annihilation and creation operators reads: (48) 2 2AA (49) We also have † 2 =QdxdxiG E =,=0; =,=0 BRST BRST QQH QQH (50) with 2 = B RST BRST Hdxdx anpose the dual condition that both and (51) d we further imQ Q annihilate physical states, implying that: |>=0 |>=0QandQ which the constraints of the theory hold, satisfy both of these conditions and are i states satisfying both of these conditions, since although (53) there are no states of this operator with (52) The states for n fact, the only with †† =1GG GG †|>=0G and †|>=0F , and hence no free eigenstates of the fermionic part of B RST that are annihiliated by each of G, † G, F , and † F . Thus the only states satisfying |>=0Q and |>=0Q are those that satisfy the constrai thbecausee ry. Now theo |>=0Q nts of, the states annihilated bycont theory, the dual condition: set of Q ains not only the set of states for which the constraints of th but also additional states for which the constraints of the theory do not hold in particular. This situation is, however, easily avod by aditionally imposing on the |>=0Q e theory hold ide and |>=0Q . Thus by imposing both of these conditions on the theory simultaneously, one finds that the states for which the constraints of the theory hold satisfy both of these conditions and, in fact, these are the only states satisfying both of these conditions because in view of the condi- tions on the fermionic variables c and c one cannot have simultaneously c, c and c, c , applied to |> to give zero. Thus the only states satisfying |>=0Q and |>=0Q are those that satisfy the constraints of the theory and they belong to the set of BRST-invariant as well as to the set of anti-BRST- invariant states. Alternatively, one can understand the above point in terms of fermionic annihiliation and creation operators as follows. The condition |>=0Q implies that the set tes annihiliated by Q contains not only the states for which the constraints of the theory hold but also additional states for which the constraints do not hold. However, of sta |>=0Q guarantees that the set of states annihiliated by Q contains only the states for which the constraints hold, simply because †|>0G and †|>0F . Thus in this alternative way also we see that the states satisfying |>=|>=0QQ are only those states that satisf the constraints of the theory and also that these states belong to the set of BRST- invariant and anti-BRST invariant states. This completes the BRST formulati of the theory. 4Sry and Discussion IFQ of the present theory has been studied by us in Refe- rence [8] (on the hyperplanes x0 = t = constant [17,18]). In the present work the theory has been quantized using the LF dynamics (on the the hyperpl on . umma anes of the LF defined y the light-cone time b01 =/2= x xx constan t here that a study of determines [17,18]. It is important to mention oth of the IFQ and LFQ for a theory really b the dynamics of the system (a la Dirac) completely, necessitating the pr esent study. For further details on the Dirac’s different rela- tivistic forms of dynamics, we refer to the work of Reference [17,18]. Copyright © 2010 SciRes. JMP U. KULSHRESHTHA ET AL.391 r, for a LF theory seven out of ten The LFQ has several advantages over the conven- sional IFQ. In particula Poincare generators are kinematical while the IF theory has only six kinematical generators [17,18]. In our treat- ment, we have made the convention to regard the light- cone variable x as the LF time coordinate and the light-cone variable x has been treated as the longitu- dinal spatial coordinate. The temporal evolution of the system in x is generated by the total Hamil- tonian of the system. The constrained dynamics of our LF theory reveals that it possesses a set of three constraints which are primary. Also there is no secondary Gauss law constraint in the theory. atrix of the Poission brackets of these three constrai is singular implying that they form a set of first-class constraints. This implies in turn, that the corresponding theory is gauge-invariant. The theory is in The m nts deed seen to possess a local vector gauge sy antization procedures but is al becauseof gau w of this, in order to ac mmetry, and correspondingly there exists a conserved local vector gauge current. Now because the set of constraints of the theory is first-class, one could quantize the theory under some suitable gauge-fixing as we have done in our present work for the Hamiltonian and path integral quantization of our theory. For this we have choosen the gauge 0A. The gauge 0A represents the light-cone coulomb gauge. This gauge choice is not only acceptable and consistent with our qu so a physically more intersting gauge choice represen- ting the light-cone coloumb gauge. However, in the above Hamiltonian and path integral quantization of the theory under some gauge-fixing conditions the gauge-invariance of the theory gets broken the procedure ge-fixing converts the set of first-class constraints of the theory into a set of second- class one, by changing the matrix of the Poission brackets of the constraints of the theory from a singular one into a non-singular one. In vie hieve the quantization of our gauge-invariant theory, such that the gauge-invariance of the theory is main- tained even under gauge-fixing, one of the possible ways is go to a more generalized procedure called the BRST quantization [8-11,16], where the extended gauge sym- metry called as the BRST symmetry is maintained even under gauge-fixing. It is therefore desirable to achieve this so-called BRST quantization also if possible. This therefore makes a kind of complete quantization of a theory. The light-cone BRST quantization of the present theory has been studied by us in the present work, under some specfic gauge choice (where a particular gauge has been choosen by us and which is not unique by any means). In this procedure, when we embed the original gauge-invariant theory into a BRST system, the quantum Hamiltonian density B RST (which includes the gauge- fixing contribution) commutes with the BRST charge as well as with the anti-BRST charge. The new (extended) gauge symmetry which replaces the gauge invariance is maintained (even underthe BRST gauge-fixing) and hence projecting any state onto the sector of BRST and anti-BRST invariant states yields a theory which is isomorphic to the original gauge-invariant theory. In conclusion, in the present work we have con structed the quantum theory corresponding to the classical Chern- Simons theory defined by the action (1) (or equivalently defined by the LF action (2)) by quantizing the corresponding classical theory using three different quantization procedures called the Hamiltonian, path integral and BRST formulations using the LF quanti- zation on the hyperplanes of the LF defined by the LC-time == x constant . In the LF Hamiltonian quantization of the theory we have obtained the non- vanishing equal LC time commutators (given by the Equation (16)) of the LF theory (defined by Equation (2)). In the Path integral quantization of the theory we have explicitly constructed the vacuum to vacuum transition amplitude of the theory called the generating functional of the theory given by Equations (18) and (19). In the BRe have explicitly constructed the BRST gauge-fixed quantum Lagrangian of the theory given by Equation (21) (or equivalently by Equation (22)). The quantum BRST-Hamiltonian of the theory has also been constructed given by Equations (28) (or equi- valently by Equation (43)). The BRST and anti-BRST charge operators of the theory have also been constructed defined respectively by Equation (34) (or equivalently by Equation (44)) and Equation (47) (or equivalently by Equation (48)). The methods of IFQ and LFQ are pioneered by non other than Dirac [17,18], where the advantages of LFQ over the IFQ have also been discussed. The reasons for the LFQ versus the usual IFQ are best explained in the rather recent review by Brodsky et al. [18] as well as in our earlier work [11,14,15]. The physical applications of these studies of the CS theory in various contexts have already been discussed in the introduction. The above points illustrate very clearly the reasons and motivations for the present studies. 5. References [1] G. V. Dunne, “Aspects of Chern-Simons Theories,” hep-th/9902115, and references therein. [2] A. Smirnov, “Notes on Chern-Simons Theory in the Temporal Gauge”, arXiv: hep-th/09105011. [3] E. J. Ferrer, R. Hurka and V. D. L. Incera, “H ST quantization, w igh Tem- perature Anyon Superconductivity,” Modern Physics Letters B, Vol. 11, No. 1, 1997, pp. 1-8. Copyright © 2010 SciRes. JMP U. KULSHRESHTHA ET AL. Copyright © 2010 SciRes. JMP 392 in, “Nobel Lecture: Fractional Quantiza- of Modern Physics, Vol. 71, No. 4, 1999, ional Spin Rovelli, “On the ha and D. S. Kulshreshtha, “Hamiltonian, eshtha, D. S. Kulshreshtha and J. 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Kulshreshtha, “Hamiltonian, Path Integral and BRST Formulations of the Chern-Simons Higgs Theory in the Broken Symmetry Phase,” Physica Scripta, Vol. 75, No. 6, 2007, pp. 795-802. [11] U. KulshrVary, Light-Cone”, Physics Reports, Vol. 301, No. 4-6, 1998, pp. 299-486. “Light-Front Hamiltonian, Path Integral and BRST For- mulations of the Chern-Simons Higgs Theory under Ap- propriate Gauge-Fixing,” Physica Scripta, 82:055101, 2010. P. A. M. Dirac, “Generalized Hamiltonian Dynamics,” Canadian Journal of Mathematics, Vol. 2, 1950, pp. 129-148. [13] M. Henneaux and C. Teitelboim, “Quantization of Gauge System [14] U. Kulshreshtha and D. S. Kulshreshtha, “Conformally Gauge-Fixed Polyakov D1 Brane Action in the Presence of a 2-Fo Form Hamiltonian and Path Integral Formulations,” Physics Letters B, Vol. 555, No. 3-4, 2003, pp. 255-263. [15] U. kulshreshtha and D. S. Kulshreshtha, “Hamiltonian and Path Integral Formulations of the Dirac-Born-Infeld Nambu-Goto D1 Brane Action with and without a Dila- ton Field under Gauge-Fixing,” The European Physical Journal C, Vol. 29, No. 3, 2003, pp. 453-461. 16] D. Nemeschansky, C. Preitschopf and M. Weinstein, “A BRST Primer,” Annals of Physics, Vol. 183, No. 2, 1988, pp. 226-268. [17] P. A. M. Dirac, “Forms of Relativistic Dynamics,” Re- views of Modern Physics, Vol. 21, No. 3, 19 392-399. [18] S. J. Brodsky, H. C. Pauli and S. S. Pinsky, “Quantum Chromodynamics and other |