Vol. 2 - The World of Mathematical Equations

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April, 2008 PROGRESS IN PHYSICS Volume 2IW(z; z 0 ) 1 dw !(w)J(w) : (23)2i zNow let us consider the quantum case which is based onhave that the variation of W(z; z0 ) in the direction !(w) isgiven bythe quantum gauge theory in Section 2. For this quantum casewe shall define a quantum generator J which is analogous tothe J in (18). We shall choose the equations (34) and (35) asthe equations for defining the quantum generator J. Let usfirst give a formal derivation of the equation (34), as follows.Let us consider the following formal functional integration:hW(z; z0 )A(z)i :=(24):=R dA1 dA 2 dZ dZe L W(z; z0 )A(z);where A(z) denotes a field from the quantum gauge theory.(We first let z0 be fixed as a parameter.)Let us do a calculus of variation on this integral to derivea variational equation by applying a gauge transformation on(24) as follows. (We remark that such variational equationsare usually called the Ward identity in the physics literature.)Let (A 1 ; A 2 ; Z) be regarded as a coordinate system of theintegral (24). Under a gauge transformation (regarded as achange of coordinate) with gauge function a(z(s)) this coordinateis changed to another coordinate (A0 1 ; A0 2 ; Z0 ). Assimilar to the usual change of variable for integration we havethat the integral (24) is unchanged under a change of variableand we have the following equality:R dA 0 1 dA0 2 dZ0 dZ 0 e L 0 W0 (z; z 0 )A 0 (z) ==R dA1 dA 2 dZ dZe L W(z; z0 )A(z) ;(25)where W0 (z; z 0 ) denotes the Wilson line based on A 0 1 and A0 2and similarly A0 (z) denotes the field obtained from A(z) with(A 1 ; A 2 ; Z) replaced by (A0 1 ; A0 2 ; Z0 ).Then it can be shown that the differential is unchangedunder a gauge transformation [59]:dA 0 1dA 0 2dZ 0 dZ 0 = dA 1 dA 2 dZ dZ : (26)Also by the gauge invariance property the factor e L isunchanged under a gauge transformation. Thus from (25) wehave0 =hW 0 (z; z 0 )A 0 (z)i hW(z; z 0 )A(z)i ; (27)where the correlation notation hi denotes the integral withrespect to the differentiale L dA 1 dA 2 dZ dZ (28)We can now carry out the calculus of variation. From thegauge transformation we have the formula:W 0 (z; z 0 ) = U(a(z))W(z; z 0 )U 1 (a(z 0 )) ; (29)where a(z) = Re !(z).This gauge transformation gives avariation of W(z; z0 ) with the gauge function a(z) as the variationaldirection a in the variational formulas (21) and (23).Thus analogous to the variational formula (23) we have thatthe variation of W(z; z0 ) underIthis gauge transformation isgiven byW(z; z 0 ) 1 dw a(w)J(w) ; (30)2i zwhere the generator J for this variation is to be specific. ThisJ will be a quantum generator which generalizes the classicalgenerator J in (23).Thus under a gauge transformation with gauge functiona(z) from (27) we have the following variational equation:I+ 12i z0 =DW(z; z 0 )ha A(z) +dwa(w)J(w)A(z)iE;(31)where a A(z) denotes the variation of the field A(z) in thedirection a(z). From this equation an ansatz of J is that Jsatisfies the following equation:dwa(w)J(w)A(z)i= 0 : (32))h IW(z; z 0 a A(z) + 12i zI a A(z) = 12iFrom this equation we have the following variationalequation:dwa(w)J(w)A(z) : (33)zThis completes the formal calculus of variation. Now(with the above derivation as a guide) we choose the followingequation (34) as one of the equation for defining the generatorJ:I ! A(z) = 1 dw !(w)J(w)A(z) ; (34)2i zwhere we generalize the direction a(z) = Re !(z) to the analyticdirection !(z). (This generalization has the effect of extendingthe real measure of the pure gauge part of the gaugetheory to include the complex Feynman path integral since itgives the transformation ds ! ids for the integral of theWilson line W(z; z0 ).)Let us now choose one more equation for determine thegenerator J in (34). This choice will be as a gauge fixingcondition. As analogous to the WZW model in conformalfield theory [65–67] let us consider a J given byJ(z) := k 0 W 1 (z; z 0 ) @ z W(z; z 0 ) ; (35)where we define @ z = @ x 1 + i@ x 2 and we set z0 = z after thedifferentiation with respect to z; k 0 > 0 is a constant whichis fixed when the J is determined to be of the form (35) andthe minus sign is chosen by convention. In the WZW model[65, 67] the J of the form (35) is the generator of the chiralSze Kui Ng. New Approach to Quantum Electrodynamics 21
Volume 2 PROGRESS IN PHYSICS April, 2008symmetry of the WZW model. We can write the J in (35) inthe following form:=X =XJ(w) J a (w) j a (w) t a : (36)aWe see that the generators t a of SU(2) appear in this formof J and this form is analogous to the classical J in (18). Thisshows that this J is a possible candidate for the generator Jin (34).Since W(z; z0 ) is constructed by gauge field we need tohave a gauge fixing for the computations related to W(z; z0 ).Then since the J in (34) and (35) is constructed by W(z; z0 )we have that in defining this J as the generator J of W(z; z0 )we have chosen a condition for the gauge fixing. In this paperwe shall always choose this defining equations (34) and (35)for J as the gauge fixing condition.In summary we introduce the following definition.Definition The generator J of the quantum Wilson lineW(z; z0 ) whose classical version is defined by (8), is an operatordefined by the two conditions (34) and (35). Remark We remark that the condition (35) first defines Jclassically. Then the condition (34) raises this classical J tothe quantum generator J. Now we want to show that this generator J in (34) and(35) can be uniquely solved. (This means that the gauge fixingcondition has already fixed the gauge that the degeneratedegree of freedom of gauge invariance has been eliminated sothat we can carry out computation.)Let us now solve J. From (11) and (35) the variation ! Jof the generator J in (35) is given by [65, p. 622] and [67]: ! J = [J; !] k 0 @ z ! : (37)XJ a (w)J b (z) =k 0 ab(w z) 2 + Jif c (z)abc(w z) ; (38)From (34) and (37) we have that J satisfies the followingrelation of current algebra [65–67]:where as a convention the regular term of J a (w)J b (z) isomitted. Then by following [65–67] from (38) and (36) wecan show that the J a n in (18) for the corresponding Laurentseries of the quantum generator J satisfy the following Kac-Moody algebra:c[J a m; J b n] = if abc J c m+n + k 0 m ab m+n;0 ; (39)where k 0 is usually called the central extension or the level ofthe Kac-Moody algebra.Remark Let us also consider the other side of the chiralsymmetry. Similar to the J in (35) we define a generatorJ0 by:J 0 (z 0 ) = k 0 @ z 0W(z; z 0 )W 1 (z; z 0 ) ; (40)where after differentiation with respect to z0 we set z = z 0 .aLet us then consider the following formal correlation:hA(z 0 )W(z; z 0 )i :=:=ZdA 1 dA 2 dZ dZA(z 0 ) W(z; z 0 ) e L ;(41)where z is fixed. By an approach similar to the above derivationof (34) we have the following variational equation: ! 0A(z 0 ) = 12iIz0 dwA(z0 )J 0 (w)! 0 (w) ; (42)where as a gauge fixing we choose the J0 in (42) be the J 0 in(40). Then similar to (37) we also have ! 0J 0 = [J 0 ; ! 0 ] k 0 @ z 0! 0 : (43)Then from (42) and (43) we can derive the current algebraand the Kac-Moody algebra for J0 which are of the same formof (38) and (39). From this we have J0 = J. Now with the above current algebra J and the formula(34) we can follow the usual approach in conformal fieldtheory to derive a quantum Knizhnik-Zamolodchikov (KZ)equation for the product of primary fields in a conformal fieldtheory [65–67]. We derive the KZ equation for the productof n Wilson lines W(z; z0 ). Here an important point is thatfrom the two sides of W(z; z0 ) we can derive two quantumKZ equations which are dual to each other. These two quantumKZ equations are different from the usual KZ equationin that they are equations for the quantum operators W(z; z0 )while the usual KZ equation is for the correlations of quantumoperators. With this difference we can follow the usualapproach in conformal field theory to derive the followingquantum Knizhnik-Zamolodchikov equation [65, 66, 68]:@ zi W(z 1 ; z0 1 )W(z n ; z0 n ) == e2 0k 0 +g 0PnjiPa ta it a jz i z jW(z 1 ; z0 1 )W(z n ; z0 n ) ;(44)for i = 1; : : : ; n where g 0 denotes the dual Coxeter numberof a group multiplying with e 2 0 and we have g 0 = 2e 2 0 forthe group SU(2) (When the gauge group is U(1) we haveg 0 = 0). We remark that in (44) we have defined t a i := t a and:t a i t a jW(z 1 ; z0 1 )W(z n ; z0 n ) := W(z 1 ; z0 1 )[t a W(z i ; z0 i )][t a W(z j ; z0 j )]W(z n ; z0 n ) :(45)It is interesting and important that we also have the followingquantum Knizhnik-Zamolodchikov equation with respectto the z0 i variables which is dual to (44):@ z 0 iW(z 1 ; z0 1 )W(z n ; z0 n ) == e2 0k 0+g 0Pnji W(z 1 ; z0 1 )W(z n ; z0 n )Pa ta it a jz0 j z0 i(46)22 Sze Kui Ng. New Approach to Quantum Electrodynamics
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