- Email: [email protected]
The pseudoclassical relativistic quark model in the rest-frame Wigner-covariant gauge
@ ELSEVIER
PROCEEDINGS SUPPLEMENTS Nuclear Physics B (Proc. Suppl.) 64 (1998) 306-311
The Pseudoclassical Relativistic Quark Model in the Rest-Frame Wigner-Covariant Gauge Luca Lusanna a aSezione INFN di Firenze, Largo E.Fermi 2 (Arcetri), 50125 Firenze, Italy email [email protected] The system of N scalar particles with Grassmann-valued color charges plus the color SU(3) Yang-Mills field is reformulated on spacelike hypersurfaces. The Dirac observables are found and the physical invariant mass of the system in the Wigner-covariant rest-frame instant form of dynamics (covariant Coulomb gauge) is given. We study this relativistic scalar quark model, deduced from the classical QCD Lagrangian and with the color field present, in the N=2 (meson) case. A special form of the requirement of having only color singlets, suited for a field-independent quark model, produces a "pseudoclassical asymptotic freedom" and a regularization of the quark self-energy. Talk given at the Eoroconference QCD 97, Montpellier 3-9th July 1997.
Since all relevant physical relativistic systems are described at the classical level by singular Lagrangians and Dirac-Bergmann Hamiltonian theory of constraints [1], a research program started trying to find a unified mathematical description of the four (electroweak, strong and gravitational) interactions and, after a canonical reduction, their reformulation only in terms of Dirac's observables without any kind of gauge degrees of freedom. See the review papers [2,3] for the formulation and the evolution of the program. Therefore, the Hamiltonian formulation of physical systems [the standard SU(3) x SU(2) x U(1) model of elementary particles, all its extensions with or without supersymmatry, all variants of string theory, all formulations of general relativity] needs Dirac's theory of 1st and 2nd class constraints[1] determining the submanifold of phase space relevant for dynamics: this means that the basic mathematical structure behind our description of the four interactions is presymplectic geometry. For a system with 1st class constraints the physical description becomes clear in coordinates adapted to the presymplectic submanifold. Locally in phase space, an adapted Darboux chart can always be found by means of Shanmugadhasan's canonical transformations [4]. The new 0920-5632/98/$19.00 © 1998 Elsevier Science B.V. All rights reserved. PII S0920-5632(97)01079-7
canonical basis has: i) as many new momenta as 1st class constraints (Abelianization of 1st class constraints); ii) their conjugate canonical variables (Abelianized gauge variables; if put =0, one gets a local gauge); iii) pairs of canonically conjugate D ~ c ' s observables (canonical basis of physical variables adapted to the chosen Abelianization; they give a trivialization of the BRST construction of observables). If a system with constraints admits one (or more) global Shanmugadhasan canonical transformations, one obtains one (or more) privileged global gauges in which the physical Dirac observables are globally defined and globally separated from the gauge and the irrelevant degrees of freedom [for systems with a compact configuration space this is impossible]. These privileged gauges (when they exist) can be called "noncovariant generalized Coulomb gauges". Following the suggestion of a paper of Dirac[5] for the case of the electromagnetic field plus a Charged Dirac field, in a series of papers [6-8] these gauges have been found for the following isolated systems: a) Yang-Mills theory with Grassmann-valued fermion fields[6] in the case of a trivial principal bundle over a fixed-x ° R ~ slice of Minkowski spacetime with suitable Hamiltonian-oriented
L. Lusanna/Nuclear Physics B (Proc. Suppl.) 64 (1998) 306--311 boundary conditions; this excludes monopole solutions and, since R 3 is not compactified, one has only winding number and no instanton number. After a discussion of the Hamiltonian formulation of Yang-Mills theory, of its group of gauge transformations and of the Gribov ambiguity, the theory has been studied in suitable weighted Sobolev spaces where the Gribov ambiguity is absent. The global Dirac observables are the transverse quantities Aa±(~,x°), /~a±(~,x °) and fermion fields dressed with Yang-Mills (gluonic) clouds. The nonlocal and nonpolynomial (due to classical Wilson lines along flat geodesics) physical Hamiltonian has been obtained: it is nonlocal but without any kind of singularities, it has the correct Abelian limit if the structure constants are turned off, and it contains the explicit realization of the abstract Mitter-Viallet metric. b) The Abelian and non-Abelian SU(2) Higgs models with fermion fields[7], where the symplectic decoupling is a refinement of the concept of unitary gauge. There is an ambiguity in the solutions of the Gauss law constraints, which reflects the existence of disjoint sectors of solutions of the Euler-Lagrange equations of Higgs models. The physical Hamiltonian and Lagrangian of the Higgs phase have been found; the self-energy turns out to be local and contains a local fourfermion interaction. c) The standard SU(3)xSU(2)xU(1) model of elementary particles[8] with Grassmann-valued fermion fields. The final reduced Hamiltonian contains nonlocal self-energies for the electromagnetic and color interactions, but "local ones" for the weak interactions implying the nonperturbative emergence of 4-fermions interactions. To obtain a nonlocal self-energy with a Yukawa kernel for the massive Z and W + bosons one has to reformulate the model on spacelike hypersurfaces and make a modification of the Lagrangian. Then, again following Dirac[1], electromagnetic and Yang-Mills theories have been reformulated on spacelike hypersurfaces E~ foliating Minkowski spacetime Ma[6,9,11], whose points z~(~-,g) become new configurational degrees of freedom. Every field ¢(z) is replaced by a new field ¢(v, ~) = ¢(z(~-,cY)), which knows the embedding of Zr into M4; 4-vector gauge poten-
307
tials A , ( z ) are replaced by Lorentz scalar fields AA(T,5) --- °z~(r'~) A , ("z iv, . . . .a)). The independence of the description from the choice of the foliation is manifest due to the presence of first class constraints 7/,(r, cY) = p,(r, ~) - ... 0, where p,(v,~) are the canonical momenta conjugate to the variables z"(~-,cY). All other first class constraints like Gauss' laws are now Lorentz scalars. By adding suitable gauge-fixings Z"(T, J) -- [X~(~') + brg(r)a r] ~ 0, the description is reduced to spacelike hyperplanes, which are described by an origin x~ (~-) and by the s ~ independent degrees of freedom in the orthonormal tetrad b~(r) [the v-independent normal to the hyperplane is l" = b~] and 10 conjugate momenta: P~s and the six independent degrees of freedom in a spin tensor S~ v. The constraints 7/~(r,~) ~ 0 are reduced to only 10 ones determining P~s and Ss~ and saying that they are the total fourmomentum and spin tensor of the isolated system. Fin.ally, selecting all the timelike configurations of the system [p~ > 0; they are dense in the space of all configurations], one restricts himself to hyperplanes orthogonal to P~s [6,9]: these intrinsic (i.e. determined by the system itself) Wiguer hyperplanes define the "rest-frame Wigner-covariant inv,tant form of the dynamics" [it is the relativisti~ generalization of the nonrelativistic center-ofmass decomposition]. On them all the variables are either Lo~entz scalars or Wigner spin-1 3vectors with the exception of the canonical noncovariant Newton-Wigner-like center of mass ~s~ (replacing x~' in the new canonical basis). Only four constraints of the previous 10 remain: i) one says which is the invariant mass of the system V ~ - M[system] ~ 0 [M plays the role of the nonrelativistic relative Hamiltonian Hret in f2 H = ~-~+H,.~t]; ii) three, p~[system] ~ O, say that the Wigner spin-1 3-momentum of the system in the Wigner hyperplane vanishes [definition of rest-frame; lg ¢ iffs, because the orientation of P~s is arbitrary, depending on the chosen frame of reference]. If one makes the canonical reduction of the electromagnetic and Yang-Mills fields at this stage, one gets the "rest-frame Wiguer-co~riant generalized Coulomb gauges", in which the only breaking of Lorentz covariance is in the center-of-
L. Lusanna/Nuclear Physics B (Proc. Suppl.) 64 (1998) 306-311
308
mass (or origin) variable ~ independently by the system under investigation. In Ref.[9,10] these covariant gauges have been obtained for N scalar particles with Grassmannvalued electric charges [pseudoclassical description of the quantization of the electric charge] plus the electromagnetic field. The Coulomb potential is extracted from field theory and there is a regularization of the Coulomb self-energies due to the Grassmann character of the electric charges [pseudoclassical effect of the quantization of charge]. Then, in Ref.[ll] the same Wignercovariant gauge has been found for N scalar particles with Grassmann-valued color charges plus the SU(3) Yang-Mills color field. In this pseudoclassical scalar-quark model the configuration variables are z~(r,g), the SU(3) gauge potentials AaA(~', ~), the particle positions ~?~(T) on Er and the Grassmann variables 8i~(T), i=l,..,N, 8~(T), a = 1,2,3, restricted by the constraints ~-~a8~8i~ ~ 0 [to get either triplets or antitriplets after quantization]. The particles have Grassmann-valued charges Qai(T) = 3 i ~'~a,B=l O~a(r)(Ta)aBOiB( r)" S t a r t i n g from the action given in Ref.[ll], making the reduction to Wigner hyperplanes and eliminating the YangMills gauge degrees of freedom [6], the final four constraints, in terms of the Dirac observables ,~,(~), k,(r), ~±(r,~), ~~o . ( ~ , o )- , a~e [~, = v g ] ]
N
~q~(~) =
E,~,~0-)+/d3,.E[(o.&,.. a
i=1
-
-
OsA,±r)~ra.i.](v,~r) ~ O.
(2)
V[ff¢,A~±, ff~±l(r) = =
g~ / d3ad3ald3a2 "* ~A±) ~
)pa(r, al)l
I;
a,b,c A~
= g~ f d 3al d :~'a2 E [)a(T' (~1) Kab(U1,1~2; T) pb(T, U2) : a,b ._ g2 f d3~ld30. 2 E[~(aYM)(T,~I ) --
a,b -
N V z _", A ~ ,c~ ~ r 2c3 , ~ l
- ~t
))J
,,btaZ,a2;r)
i=1 N
[ ~ M ) ( r , g2)
- ~
Qjb(r)a3(~2
-
{~(r))],
j=l
K~b ( ~ , ~.2; r) = Kb.(~2, ~ ; r) = = f d3cradaa4 { (~ab63((~3 -- gl)53(~4 -- U2) 41r I ~3 - ~4 I + ~3 (a4 - a2) 41r I a3 - a4 I ~* (CA±) (~3,u1;T)]ab -1[A±(~-,Ya) •
i
.~ + (~,(~)+ ~ Qo,(~)]o~(~, ~,(~)))2 _
+(~1 ~ g2) +
o,
+ [2~ (~, ~3)- C a ~)(~3, ~1; ~)].. ~(A±)
1
4~r1~3 - g4[ [ ' ~ ( r ' y * )
(3)
:.2
+
Ba.L (T, 2a2 ~),1
~v[m, A,..,,r.x](r) = e, -- Hrel ~ O.
(~*,a~;~)]b.}.
~),-
~,
,
~b ta, a ;r) is the Green function of the covariant divergence[6]. The Dirac observables for the color charge of the Yang-Mills field (1~ (~(YM)(T) = f d3ap(YM)(~',Y) with p(YM) ----
L. Lusanna/Nuclear Physics B (Proc. Suppl.) 64 (1998) 306-311
-cabcf4aa_" ~rcj_. We see that, since tSa = p(aYM) Y ] ~ t Q/a53( ~ - ffi(r)) is the total color charge density, there is a universal interaction kernel Kab(~Yl, J2; r) which creates the particle-particle, the particle- field and the field-field interaction between the corresponding color charge densities. This interaction kernel contains 3 kinds of instantaneous in r interactions: i) a Coulomb interaction; ii) an interaction mediated by an arbitrary center (over whose spatial location is integrated) : one color density has a Coulomb interaction with the transverse potential at the center, which simultaneously interacts with the other color density through an instantaneous "Wilson line" along the geodesic (the straightline) on the Wigner hyperplane, i.e. [.~j_ = ~a±T a with (TC)a b = C~b; c-'(~) is the Green function of the ordinary divergence]: [A±(r,a) -* :' • ~'tA.)(5, ff;r)]~b =
)T )db, j ~ c ± ( ~ . , ~ ) C c a d . ~ _ ~ ) ( p e f : d Y . •A.±(r,Y . . . where ~ is the position of the center and ~ that of the color density; iii) an interaction mediated by two arbitrary centers (over whose spatial location is integrated): each color density interacts, through a Wilson line along the geodesic, with the transverse potential at one center and the two centers have a mutual Coulomb interaction. In the nonrelativistic quark model, in which no SU(3) color Yang-Mills field appears, one assumes that the physical states are color singlets. Since Hrel gives the invariant mass of N colored relativistic scalar particles together with the SU(3) color Yang-Mills field, this is the starting point to try to extract a pseudoclassical basis for the missing relativistic quark model. A first step is to study what happens if we add 8 extra constraints implying the vanishing of the total color charge, so that only global color singlets are allowed for the particles+Yang-Mills field system: Oa = 0 (YM) + E N 1 0 i a = O. Then, we ask that these 8 conditions be fulfilled by the separate vanishing of the particle and field contributions to the color charge: Y]i Q,ia = O, Q(YM) = O. The first condition defines the relativistic scalar quark model: the particles by themselves are a
309
color singlet independently from the SU(3) color field. For N=I, the previous equations plus ~ a ~0~ ~ 0 gives 9 conditions on 6 Grassmann variables: therefore a single pseudoclassical scalar quark cannot, be a color singlet. For N=2, besides the constraints .]~r1 = Eot O~aOla "* ~ ~ 0 and N2 =
~ 0~.02. ~ 0,
one has Q I . + 0 2 . =
0,
namely 10 conditions on 12 Grasmmann variables. The condition (~(YM) = 0 replaces the Abel±an condition A.L(r,a) = ff±(r,~) = 0 of absence of radiation (see Ref.[9]). Since in the non-Abel±an case we do not know how to solve the equations of motion and since the superposit±on principle does not hold, we can only ask that there is no color flux on the surface at space infinity. This requirement also implies that the pseudoclassical solutions of the SU(3) Yang-Mills equations arc restricted to those configurations which are color singlets like the glueball states at the quantum level• The invariant mass of the pseudoclassical scalar quark model for N=2 takes the final form [~a± = -_.(o) ~ =(I) . . =.(2)
A~± + QluA..± + QluQlvAau~± ]
~,
y~ :,(0) =~(1)
+
i
~(o) ~-: ) ] ( r ) + M~[m2, q2, g2, A,~_L,
+ vp.[,7, - q : ; ~ ; +
=,2
M?[ml,ql,,~I,Ao±,A± ](~) + ~1(~)
:
~o)
o.](r) +
2 Z V(i) PF[(qi; A.±; -_.(0) ~.±; -(o) A~ -..(1)](r) + i:I
-_.(k)
+
:.(k),,
,
VFF[AL ,?r± JtT) + If 2=.2
2=. 2
+ ~ ~ a ~ ~ [ g , ~ o ± + ¢ ; Bo](~,~) (4)
=
2 -AO) mi + (--)i+12/~i('r)" E•Ia(7")[Aa-L + cl
.
~.0)
+ ZQlu(r)Aau.L](7",qi('r)) + +
Z a,b
01a(T)01b ( T ) [ 2 : : • A-.~(o) b z ] ( ~ , , ~-*( ~ ) ) .
310
L. Lusanna/Nuclear Physics B (Proc. Suppl.) 64 (1998) 306--311 (5)
Vpp = ~g~ 1 2ZQla(~lbfd3alfd3a2 a,b
[,~ (~
- ffl 0 ) )~ 3 ( ~2 -
~ 0-) ) +
.~.(~3(g I -- ~2(r))~ "~ 3 (°2 " - ,72(r)) --2(~3(~1 -- 41 ('r))(~Z(~2 -- ~2(T))] " o
1 f d3 f d3o.4 4-~ 1 a3
i
i
(U4 -- ~2)" " ~ : : (T, ff4)¢(ub"~°,)- (('~2, U4; T) -{, ~ ( ~ - a~)
(~ -
~1)-A,,.(T,~o,,,,,b CarvCbst
~ 1,=3;'r)] +
(4~') 2
]..(0) -, .,,.(~i~ob,,~ ('~4 - '~2) ".~,~i~T,o'4),,,-,~ ~ 2,0"-'4;r) 1
_ g~[ffl - 42;,7~; J~)](~-) -~,_,~0. -
(6)
Iff~(~) - ff2(~)l
There is no term with the instantaneous Coulomb interaction in the particle-partilce potential Vpp, because for every N we have ~'~a (~la(~la = 0. With regard to the other two terms, let us remark that in the limit ~I = ~ [and thus also in the points of maximal divergence, i.e. Yl = ~2 = for the first two terms and ~1 = ~2 = ~ = (~' for the third one] the potential vanishes due to the multiplicative term cont.alniug the Dirac functions. The last line contains the pseudoclassical statement of asymptotic freedom for N=2 (meson case). It would be interesting to study the N - 3 (baryon) case from this point of view. In Ref.[ll] there is also the equation of motion for the reduced Yang-Mills potential Aa~_(~-,J),
i.e. the non-Abelian analogue of the electromagnetic integrodifferential equation ~ArL = (6r8 -F 0~ ° )is from which the Lienard-Wiechert potential in the Coulomb gauge is extracted [10]. Since the electromagnetic particle current j is replaced by an effective current involving also the charged Yang-Mills field and since the d'Alambertian is replaced by a field-dependent nonlinear operator (involving the Mitter-Viallet metric and the Faddeev-Popov operator), it is an open problem whether there exists a non-Abelian LienardWiechert potential; if it would exist, it could separate the pure field component (connected to the glueball degrees of freedom at the quantum level) from a particle component: this would allow to find the interquark potential only in terms of the quark variables by inserting this potential in the quark equations of motion, also given in Ref.[ll]. The main open problem in the N--2 case is whether there is confinement. To study the limit [~1.- 42[ -~ oo for any given Yang-Mills configuration, one should overcome the technical obstacle of which kind of function of two variables is the path-ordering over the reduced Yang-Mills potential along the straightline joining two given points, which appears in Vpp, and which*is the limit of this function when one (or both) The points move towards infinity. Let us remark that it is impossible to make a comparison of these nonperturbative pseudoclassical results with standard QCD, because there is no theory of non-Abelian coherent states to be used (like in QED) to find a semiclassical limit. To get the rest frame Wigner covariant Coulomb gauge of the standard SU(3)xSU(2)xU(1) model, the only lacking ingredient is the Hamiltonian theory of Dirac fields on spacelike hypersurfaces; this problem is now under investigation[12]. See Refs.[3,11,2,6,9] for the intrinsic unit of length (the Moller radius), which exists on the Wigner hyperplanes and is determined by the extension of the region of noncovariance of the center-of-mass variable ~ . Its quantum counterpart (the total spin of the isolated system multiplied its Compton wavelength) gives a physical ultraviolet cutoff in the spirit of Dirac and Yukawa to be used in a future attempt
L. Lusanna /Nuclear Physics B (Proc. Suppl.) 64 (1998) 306-311
of quantization of this reduced nonlocal (nonlinear in the gauge potentials but bilinear in their conjugate momenta) theory. REFERENCES 1. P.A.M.Dirac, Can.J.Math. 2, 129 (1950); "Lectures on Quantum Mechanics", Belfer Graduate School of Science, Monogr. Series (Yeshiva University, New York, 1964). 2. L.Lusanna, "Solving Gauss' Laws and the Search for Dirac's Observables for the Four Interactions", Nucl.Phys. B (Proc.Suppl.) 57, 13 (1997) (HEP-TH/9702114). 3. L.Lusanna, "Unified Description and Canonical Reduction to Dirac's Observables of the Four Interactions", talk at the Int.Workshop "New non Perturbative Methods and Quantization on the Light Cone", Les Houches 1997 (HEP-TH/9705154). 4. S.Shanmugadhasan, J.Math.Phys. 14, 677 (1973). L.Lusanna, Int.J.Mod.Phys. A8, 4193 (1993); Phys.Rep. 185, 1 (1990); Riv. Nuovo Cimento 14, n.3, 1 (1991). 5. P.A.M.Dirac, Can.J.Phys. 33, 650 (1955). 6. L.Lusanna, Int.J.Mod.Phys. A10, 3531 and 3675 (1995). 7. L.Lusanna and P.Valtancoli, "Dirac's Observables for the Higgs model: I) the Abelian Case; II) the non-Abelian SU(2) Case", to appear in Int.J.Mod.Phys. A (SISSA, hep-th 9606078 and 9606079). 8. L.Lusanna and P.Valtancoli, Dirac's Observables for the SU(3)xSU(2)xU(1) Standard Model, Firenze preprint 1996 (hep-
th/9707072). 9. L.Lusanna, Int.J.Mod.Phys. A12, 645 (1997). 10. D.Alba and L.Lusanna, "The LienardWiechert Potential of Charged Scalar Partitles and their Relation to Scalar Electrodynamics in the Rest-Frame Instant Form", Firenze Univ.preprint, (HEP-TH/9705155). 11. D.Alba and L.Lusanna, "The Classical Relativistic Quark Model in the Rest-Frame Wigner-Covariant Coulomb Gauge", Firenze Univ.preprint, (HEP-TH/9705156). 12. F.Bigazzi and L.Lusanna, in preparation.
311
Discussion David McMullan, Plymouth University, School of Mathematics and Statistics. Your classical reduction imposed a Sobolev structure on the true degrees of freedom. How do you impose this in the quantum theory where one would expect a much weaker structure to be relevant? Lusanna The measure of the path integral must know the function space of the fields. Strictly speaking, in Yang-Mills theory with ordinary Sobolev spaces this measure does not exist (or at least noone has written a correct workable form of it) due to the existence of zero medes of the FaddeevPopov operator, i.e. due to the Gribov ambiguity, which is also an obstruction to the existence of global Dira~'s observables. In suitable weighted Sobolev spaces, Moncrief showed that the covariant divergence, and therefore also the Faddeev-Popov operator, is an elliptic operator without zero modes: in these spaces the measure should exist in the ordinary sense. However, this point has never been investigated in the literature either in the local nonreduced theory or in the nonlocal nonlinear reduced one. Also for QED in the noncovariant Coulomb gauge this is an open problem: the obstacle here is that still there is no worked out regularization and renorrealization scheme which could take into account the nonlocal 4-fermion Coulomb self-interaction. However, there is no "no-go theorem" forbidding its existence. In our approach, with the restframe Wigner-covariant Coulomb gauges, there is the hope to define an operator quantization (to start with), notwithstanding the nonlocality and nonlinearity of" the reduced theory, by using the physical intrinsic ultraviolet cutoff (the quantum Moller radius).
PROCEEDINGS SUPPLEMENTS Nuclear Physics B (Proc. Suppl.) 64 (1998) 306-311
The Pseudoclassical Relativistic Quark Model in the Rest-Frame Wigner-Covariant Gauge Luca Lusanna a aSezione INFN di Firenze, Largo E.Fermi 2 (Arcetri), 50125 Firenze, Italy email [email protected] The system of N scalar particles with Grassmann-valued color charges plus the color SU(3) Yang-Mills field is reformulated on spacelike hypersurfaces. The Dirac observables are found and the physical invariant mass of the system in the Wigner-covariant rest-frame instant form of dynamics (covariant Coulomb gauge) is given. We study this relativistic scalar quark model, deduced from the classical QCD Lagrangian and with the color field present, in the N=2 (meson) case. A special form of the requirement of having only color singlets, suited for a field-independent quark model, produces a "pseudoclassical asymptotic freedom" and a regularization of the quark self-energy. Talk given at the Eoroconference QCD 97, Montpellier 3-9th July 1997.
Since all relevant physical relativistic systems are described at the classical level by singular Lagrangians and Dirac-Bergmann Hamiltonian theory of constraints [1], a research program started trying to find a unified mathematical description of the four (electroweak, strong and gravitational) interactions and, after a canonical reduction, their reformulation only in terms of Dirac's observables without any kind of gauge degrees of freedom. See the review papers [2,3] for the formulation and the evolution of the program. Therefore, the Hamiltonian formulation of physical systems [the standard SU(3) x SU(2) x U(1) model of elementary particles, all its extensions with or without supersymmatry, all variants of string theory, all formulations of general relativity] needs Dirac's theory of 1st and 2nd class constraints[1] determining the submanifold of phase space relevant for dynamics: this means that the basic mathematical structure behind our description of the four interactions is presymplectic geometry. For a system with 1st class constraints the physical description becomes clear in coordinates adapted to the presymplectic submanifold. Locally in phase space, an adapted Darboux chart can always be found by means of Shanmugadhasan's canonical transformations [4]. The new 0920-5632/98/$19.00 © 1998 Elsevier Science B.V. All rights reserved. PII S0920-5632(97)01079-7
canonical basis has: i) as many new momenta as 1st class constraints (Abelianization of 1st class constraints); ii) their conjugate canonical variables (Abelianized gauge variables; if put =0, one gets a local gauge); iii) pairs of canonically conjugate D ~ c ' s observables (canonical basis of physical variables adapted to the chosen Abelianization; they give a trivialization of the BRST construction of observables). If a system with constraints admits one (or more) global Shanmugadhasan canonical transformations, one obtains one (or more) privileged global gauges in which the physical Dirac observables are globally defined and globally separated from the gauge and the irrelevant degrees of freedom [for systems with a compact configuration space this is impossible]. These privileged gauges (when they exist) can be called "noncovariant generalized Coulomb gauges". Following the suggestion of a paper of Dirac[5] for the case of the electromagnetic field plus a Charged Dirac field, in a series of papers [6-8] these gauges have been found for the following isolated systems: a) Yang-Mills theory with Grassmann-valued fermion fields[6] in the case of a trivial principal bundle over a fixed-x ° R ~ slice of Minkowski spacetime with suitable Hamiltonian-oriented
L. Lusanna/Nuclear Physics B (Proc. Suppl.) 64 (1998) 306--311 boundary conditions; this excludes monopole solutions and, since R 3 is not compactified, one has only winding number and no instanton number. After a discussion of the Hamiltonian formulation of Yang-Mills theory, of its group of gauge transformations and of the Gribov ambiguity, the theory has been studied in suitable weighted Sobolev spaces where the Gribov ambiguity is absent. The global Dirac observables are the transverse quantities Aa±(~,x°), /~a±(~,x °) and fermion fields dressed with Yang-Mills (gluonic) clouds. The nonlocal and nonpolynomial (due to classical Wilson lines along flat geodesics) physical Hamiltonian has been obtained: it is nonlocal but without any kind of singularities, it has the correct Abelian limit if the structure constants are turned off, and it contains the explicit realization of the abstract Mitter-Viallet metric. b) The Abelian and non-Abelian SU(2) Higgs models with fermion fields[7], where the symplectic decoupling is a refinement of the concept of unitary gauge. There is an ambiguity in the solutions of the Gauss law constraints, which reflects the existence of disjoint sectors of solutions of the Euler-Lagrange equations of Higgs models. The physical Hamiltonian and Lagrangian of the Higgs phase have been found; the self-energy turns out to be local and contains a local fourfermion interaction. c) The standard SU(3)xSU(2)xU(1) model of elementary particles[8] with Grassmann-valued fermion fields. The final reduced Hamiltonian contains nonlocal self-energies for the electromagnetic and color interactions, but "local ones" for the weak interactions implying the nonperturbative emergence of 4-fermions interactions. To obtain a nonlocal self-energy with a Yukawa kernel for the massive Z and W + bosons one has to reformulate the model on spacelike hypersurfaces and make a modification of the Lagrangian. Then, again following Dirac[1], electromagnetic and Yang-Mills theories have been reformulated on spacelike hypersurfaces E~ foliating Minkowski spacetime Ma[6,9,11], whose points z~(~-,g) become new configurational degrees of freedom. Every field ¢(z) is replaced by a new field ¢(v, ~) = ¢(z(~-,cY)), which knows the embedding of Zr into M4; 4-vector gauge poten-
307
tials A , ( z ) are replaced by Lorentz scalar fields AA(T,5) --- °z~(r'~) A , ("z iv, . . . .a)). The independence of the description from the choice of the foliation is manifest due to the presence of first class constraints 7/,(r, cY) = p,(r, ~) - ... 0, where p,(v,~) are the canonical momenta conjugate to the variables z"(~-,cY). All other first class constraints like Gauss' laws are now Lorentz scalars. By adding suitable gauge-fixings Z"(T, J) -- [X~(~') + brg(r)a r] ~ 0, the description is reduced to spacelike hyperplanes, which are described by an origin x~ (~-) and by the s ~ independent degrees of freedom in the orthonormal tetrad b~(r) [the v-independent normal to the hyperplane is l" = b~] and 10 conjugate momenta: P~s and the six independent degrees of freedom in a spin tensor S~ v. The constraints 7/~(r,~) ~ 0 are reduced to only 10 ones determining P~s and Ss~ and saying that they are the total fourmomentum and spin tensor of the isolated system. Fin.ally, selecting all the timelike configurations of the system [p~ > 0; they are dense in the space of all configurations], one restricts himself to hyperplanes orthogonal to P~s [6,9]: these intrinsic (i.e. determined by the system itself) Wiguer hyperplanes define the "rest-frame Wigner-covariant inv,tant form of the dynamics" [it is the relativisti~ generalization of the nonrelativistic center-ofmass decomposition]. On them all the variables are either Lo~entz scalars or Wigner spin-1 3vectors with the exception of the canonical noncovariant Newton-Wigner-like center of mass ~s~ (replacing x~' in the new canonical basis). Only four constraints of the previous 10 remain: i) one says which is the invariant mass of the system V ~ - M[system] ~ 0 [M plays the role of the nonrelativistic relative Hamiltonian Hret in f2 H = ~-~+H,.~t]; ii) three, p~[system] ~ O, say that the Wigner spin-1 3-momentum of the system in the Wigner hyperplane vanishes [definition of rest-frame; lg ¢ iffs, because the orientation of P~s is arbitrary, depending on the chosen frame of reference]. If one makes the canonical reduction of the electromagnetic and Yang-Mills fields at this stage, one gets the "rest-frame Wiguer-co~riant generalized Coulomb gauges", in which the only breaking of Lorentz covariance is in the center-of-
L. Lusanna/Nuclear Physics B (Proc. Suppl.) 64 (1998) 306-311
308
mass (or origin) variable ~ independently by the system under investigation. In Ref.[9,10] these covariant gauges have been obtained for N scalar particles with Grassmannvalued electric charges [pseudoclassical description of the quantization of the electric charge] plus the electromagnetic field. The Coulomb potential is extracted from field theory and there is a regularization of the Coulomb self-energies due to the Grassmann character of the electric charges [pseudoclassical effect of the quantization of charge]. Then, in Ref.[ll] the same Wignercovariant gauge has been found for N scalar particles with Grassmann-valued color charges plus the SU(3) Yang-Mills color field. In this pseudoclassical scalar-quark model the configuration variables are z~(r,g), the SU(3) gauge potentials AaA(~', ~), the particle positions ~?~(T) on Er and the Grassmann variables 8i~(T), i=l,..,N, 8~(T), a = 1,2,3, restricted by the constraints ~-~a8~8i~ ~ 0 [to get either triplets or antitriplets after quantization]. The particles have Grassmann-valued charges Qai(T) = 3 i ~'~a,B=l O~a(r)(Ta)aBOiB( r)" S t a r t i n g from the action given in Ref.[ll], making the reduction to Wigner hyperplanes and eliminating the YangMills gauge degrees of freedom [6], the final four constraints, in terms of the Dirac observables ,~,(~), k,(r), ~±(r,~), ~~o . ( ~ , o )- , a~e [~, = v g ] ]
N
~q~(~) =
E,~,~0-)+/d3,.E[(o.&,.. a
i=1
-
-
OsA,±r)~ra.i.](v,~r) ~ O.
(2)
V[ff¢,A~±, ff~±l(r) = =
g~ / d3ad3ald3a2 "* ~A±) ~
)pa(r, al)l
I;
a,b,c A~
= g~ f d 3al d :~'a2 E [)a(T' (~1) Kab(U1,1~2; T) pb(T, U2) : a,b ._ g2 f d3~ld30. 2 E[~(aYM)(T,~I ) --
a,b -
N V z _", A ~ ,c~ ~ r 2c3 , ~ l
- ~t
))J
,,btaZ,a2;r)
i=1 N
[ ~ M ) ( r , g2)
- ~
Qjb(r)a3(~2
-
{~(r))],
j=l
K~b ( ~ , ~.2; r) = Kb.(~2, ~ ; r) = = f d3cradaa4 { (~ab63((~3 -- gl)53(~4 -- U2) 41r I ~3 - ~4 I + ~3 (a4 - a2) 41r I a3 - a4 I ~* (CA±) (~3,u1;T)]ab -1[A±(~-,Ya) •
i
.~ + (~,(~)+ ~ Qo,(~)]o~(~, ~,(~)))2 _
+(~1 ~ g2) +
o,
+ [2~ (~, ~3)- C a ~)(~3, ~1; ~)].. ~(A±)
1
4~r1~3 - g4[ [ ' ~ ( r ' y * )
(3)
:.2
+
Ba.L (T, 2a2 ~),1
~v[m, A,..,,r.x](r) = e, -- Hrel ~ O.
(~*,a~;~)]b.}.
~),-
~,
,
~b ta, a ;r) is the Green function of the covariant divergence[6]. The Dirac observables for the color charge of the Yang-Mills field (1~ (~(YM)(T) = f d3ap(YM)(~',Y) with p(YM) ----
L. Lusanna/Nuclear Physics B (Proc. Suppl.) 64 (1998) 306-311
-cabcf4aa_" ~rcj_. We see that, since tSa = p(aYM) Y ] ~ t Q/a53( ~ - ffi(r)) is the total color charge density, there is a universal interaction kernel Kab(~Yl, J2; r) which creates the particle-particle, the particle- field and the field-field interaction between the corresponding color charge densities. This interaction kernel contains 3 kinds of instantaneous in r interactions: i) a Coulomb interaction; ii) an interaction mediated by an arbitrary center (over whose spatial location is integrated) : one color density has a Coulomb interaction with the transverse potential at the center, which simultaneously interacts with the other color density through an instantaneous "Wilson line" along the geodesic (the straightline) on the Wigner hyperplane, i.e. [.~j_ = ~a±T a with (TC)a b = C~b; c-'(~) is the Green function of the ordinary divergence]: [A±(r,a) -* :' • ~'tA.)(5, ff;r)]~b =
)T )db, j ~ c ± ( ~ . , ~ ) C c a d . ~ _ ~ ) ( p e f : d Y . •A.±(r,Y . . . where ~ is the position of the center and ~ that of the color density; iii) an interaction mediated by two arbitrary centers (over whose spatial location is integrated): each color density interacts, through a Wilson line along the geodesic, with the transverse potential at one center and the two centers have a mutual Coulomb interaction. In the nonrelativistic quark model, in which no SU(3) color Yang-Mills field appears, one assumes that the physical states are color singlets. Since Hrel gives the invariant mass of N colored relativistic scalar particles together with the SU(3) color Yang-Mills field, this is the starting point to try to extract a pseudoclassical basis for the missing relativistic quark model. A first step is to study what happens if we add 8 extra constraints implying the vanishing of the total color charge, so that only global color singlets are allowed for the particles+Yang-Mills field system: Oa = 0 (YM) + E N 1 0 i a = O. Then, we ask that these 8 conditions be fulfilled by the separate vanishing of the particle and field contributions to the color charge: Y]i Q,ia = O, Q(YM) = O. The first condition defines the relativistic scalar quark model: the particles by themselves are a
309
color singlet independently from the SU(3) color field. For N=I, the previous equations plus ~ a ~0~ ~ 0 gives 9 conditions on 6 Grassmann variables: therefore a single pseudoclassical scalar quark cannot, be a color singlet. For N=2, besides the constraints .]~r1 = Eot O~aOla "* ~ ~ 0 and N2 =
~ 0~.02. ~ 0,
one has Q I . + 0 2 . =
0,
namely 10 conditions on 12 Grasmmann variables. The condition (~(YM) = 0 replaces the Abel±an condition A.L(r,a) = ff±(r,~) = 0 of absence of radiation (see Ref.[9]). Since in the non-Abel±an case we do not know how to solve the equations of motion and since the superposit±on principle does not hold, we can only ask that there is no color flux on the surface at space infinity. This requirement also implies that the pseudoclassical solutions of the SU(3) Yang-Mills equations arc restricted to those configurations which are color singlets like the glueball states at the quantum level• The invariant mass of the pseudoclassical scalar quark model for N=2 takes the final form [~a± = -_.(o) ~ =(I) . . =.(2)
A~± + QluA..± + QluQlvAau~± ]
~,
y~ :,(0) =~(1)
+
i
~(o) ~-: ) ] ( r ) + M~[m2, q2, g2, A,~_L,
+ vp.[,7, - q : ; ~ ; +
=,2
M?[ml,ql,,~I,Ao±,A± ](~) + ~1(~)
:
~o)
o.](r) +
2 Z V(i) PF[(qi; A.±; -_.(0) ~.±; -(o) A~ -..(1)](r) + i:I
-_.(k)
+
:.(k),,
,
VFF[AL ,?r± JtT) + If 2=.2
2=. 2
+ ~ ~ a ~ ~ [ g , ~ o ± + ¢ ; Bo](~,~) (4)
=
2 -AO) mi + (--)i+12/~i('r)" E•Ia(7")[Aa-L + cl
.
~.0)
+ ZQlu(r)Aau.L](7",qi('r)) + +
Z a,b
01a(T)01b ( T ) [ 2 : : • A-.~(o) b z ] ( ~ , , ~-*( ~ ) ) .
310
L. Lusanna/Nuclear Physics B (Proc. Suppl.) 64 (1998) 306--311 (5)
Vpp = ~g~ 1 2ZQla(~lbfd3alfd3a2 a,b
[,~ (~
- ffl 0 ) )~ 3 ( ~2 -
~ 0-) ) +
.~.(~3(g I -- ~2(r))~ "~ 3 (°2 " - ,72(r)) --2(~3(~1 -- 41 ('r))(~Z(~2 -- ~2(T))] " o
1 f d3 f d3o.4 4-~ 1 a3
i
i
(U4 -- ~2)" " ~ : : (T, ff4)¢(ub"~°,)- (('~2, U4; T) -{, ~ ( ~ - a~)
(~ -
~1)-A,,.(T,~o,,,,,b CarvCbst
~ 1,=3;'r)] +
(4~') 2
]..(0) -, .,,.(~i~ob,,~ ('~4 - '~2) ".~,~i~T,o'4),,,-,~ ~ 2,0"-'4;r) 1
_ g~[ffl - 42;,7~; J~)](~-) -~,_,~0. -
(6)
Iff~(~) - ff2(~)l
There is no term with the instantaneous Coulomb interaction in the particle-partilce potential Vpp, because for every N we have ~'~a (~la(~la = 0. With regard to the other two terms, let us remark that in the limit ~I = ~ [and thus also in the points of maximal divergence, i.e. Yl = ~2 = for the first two terms and ~1 = ~2 = ~ = (~' for the third one] the potential vanishes due to the multiplicative term cont.alniug the Dirac functions. The last line contains the pseudoclassical statement of asymptotic freedom for N=2 (meson case). It would be interesting to study the N - 3 (baryon) case from this point of view. In Ref.[ll] there is also the equation of motion for the reduced Yang-Mills potential Aa~_(~-,J),
i.e. the non-Abelian analogue of the electromagnetic integrodifferential equation ~ArL = (6r8 -F 0~ ° )is from which the Lienard-Wiechert potential in the Coulomb gauge is extracted [10]. Since the electromagnetic particle current j is replaced by an effective current involving also the charged Yang-Mills field and since the d'Alambertian is replaced by a field-dependent nonlinear operator (involving the Mitter-Viallet metric and the Faddeev-Popov operator), it is an open problem whether there exists a non-Abelian LienardWiechert potential; if it would exist, it could separate the pure field component (connected to the glueball degrees of freedom at the quantum level) from a particle component: this would allow to find the interquark potential only in terms of the quark variables by inserting this potential in the quark equations of motion, also given in Ref.[ll]. The main open problem in the N--2 case is whether there is confinement. To study the limit [~1.- 42[ -~ oo for any given Yang-Mills configuration, one should overcome the technical obstacle of which kind of function of two variables is the path-ordering over the reduced Yang-Mills potential along the straightline joining two given points, which appears in Vpp, and which*is the limit of this function when one (or both) The points move towards infinity. Let us remark that it is impossible to make a comparison of these nonperturbative pseudoclassical results with standard QCD, because there is no theory of non-Abelian coherent states to be used (like in QED) to find a semiclassical limit. To get the rest frame Wigner covariant Coulomb gauge of the standard SU(3)xSU(2)xU(1) model, the only lacking ingredient is the Hamiltonian theory of Dirac fields on spacelike hypersurfaces; this problem is now under investigation[12]. See Refs.[3,11,2,6,9] for the intrinsic unit of length (the Moller radius), which exists on the Wigner hyperplanes and is determined by the extension of the region of noncovariance of the center-of-mass variable ~ . Its quantum counterpart (the total spin of the isolated system multiplied its Compton wavelength) gives a physical ultraviolet cutoff in the spirit of Dirac and Yukawa to be used in a future attempt
L. Lusanna /Nuclear Physics B (Proc. Suppl.) 64 (1998) 306-311
of quantization of this reduced nonlocal (nonlinear in the gauge potentials but bilinear in their conjugate momenta) theory. REFERENCES 1. P.A.M.Dirac, Can.J.Math. 2, 129 (1950); "Lectures on Quantum Mechanics", Belfer Graduate School of Science, Monogr. Series (Yeshiva University, New York, 1964). 2. L.Lusanna, "Solving Gauss' Laws and the Search for Dirac's Observables for the Four Interactions", Nucl.Phys. B (Proc.Suppl.) 57, 13 (1997) (HEP-TH/9702114). 3. L.Lusanna, "Unified Description and Canonical Reduction to Dirac's Observables of the Four Interactions", talk at the Int.Workshop "New non Perturbative Methods and Quantization on the Light Cone", Les Houches 1997 (HEP-TH/9705154). 4. S.Shanmugadhasan, J.Math.Phys. 14, 677 (1973). L.Lusanna, Int.J.Mod.Phys. A8, 4193 (1993); Phys.Rep. 185, 1 (1990); Riv. Nuovo Cimento 14, n.3, 1 (1991). 5. P.A.M.Dirac, Can.J.Phys. 33, 650 (1955). 6. L.Lusanna, Int.J.Mod.Phys. A10, 3531 and 3675 (1995). 7. L.Lusanna and P.Valtancoli, "Dirac's Observables for the Higgs model: I) the Abelian Case; II) the non-Abelian SU(2) Case", to appear in Int.J.Mod.Phys. A (SISSA, hep-th 9606078 and 9606079). 8. L.Lusanna and P.Valtancoli, Dirac's Observables for the SU(3)xSU(2)xU(1) Standard Model, Firenze preprint 1996 (hep-
th/9707072). 9. L.Lusanna, Int.J.Mod.Phys. A12, 645 (1997). 10. D.Alba and L.Lusanna, "The LienardWiechert Potential of Charged Scalar Partitles and their Relation to Scalar Electrodynamics in the Rest-Frame Instant Form", Firenze Univ.preprint, (HEP-TH/9705155). 11. D.Alba and L.Lusanna, "The Classical Relativistic Quark Model in the Rest-Frame Wigner-Covariant Coulomb Gauge", Firenze Univ.preprint, (HEP-TH/9705156). 12. F.Bigazzi and L.Lusanna, in preparation.
311
Discussion David McMullan, Plymouth University, School of Mathematics and Statistics. Your classical reduction imposed a Sobolev structure on the true degrees of freedom. How do you impose this in the quantum theory where one would expect a much weaker structure to be relevant? Lusanna The measure of the path integral must know the function space of the fields. Strictly speaking, in Yang-Mills theory with ordinary Sobolev spaces this measure does not exist (or at least noone has written a correct workable form of it) due to the existence of zero medes of the FaddeevPopov operator, i.e. due to the Gribov ambiguity, which is also an obstruction to the existence of global Dira~'s observables. In suitable weighted Sobolev spaces, Moncrief showed that the covariant divergence, and therefore also the Faddeev-Popov operator, is an elliptic operator without zero modes: in these spaces the measure should exist in the ordinary sense. However, this point has never been investigated in the literature either in the local nonreduced theory or in the nonlocal nonlinear reduced one. Also for QED in the noncovariant Coulomb gauge this is an open problem: the obstacle here is that still there is no worked out regularization and renorrealization scheme which could take into account the nonlocal 4-fermion Coulomb self-interaction. However, there is no "no-go theorem" forbidding its existence. In our approach, with the restframe Wigner-covariant Coulomb gauges, there is the hope to define an operator quantization (to start with), notwithstanding the nonlocality and nonlinearity of" the reduced theory, by using the physical intrinsic ultraviolet cutoff (the quantum Moller radius).