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Progress of Theoretical Physics, Vol. 71, No.6, June 1984
Color Confinement in Quantum Chromodynamics
Noboru NAKANISHI and Izumi OJIMA Research Institute for Mathematical Sciences Kyoto University, Kyoto 606 Downloaded from https://academic.oup.com/ptp/article/71/6/1359/1879597 by guest on 30 September 2021
(Received January 19, 1984)
A possible complete resolution of the color confinement problem is proposed in the framework of the. manifestly-covariant canonical operator formalism of the non-abelian gauge theory. Color confinement is shown to be achieved in a non-dynamical way.
The color confinement problem in quantum chromodynamics (QCD) is one of the most important problems in particle physics. The purpose of the present paper is to propose a possible complete resolution of this problem.*) In the conventional approach, it is believed that quark confinement is achieved if one can show that the potential between a quark and an anti-quark is linearly rising at very large separations. We must note, however, that such an approach has serious drawbacks. First, the presence of such a behavior cannot be established without making dynamical analysis of QCD, which inevitably necessitates using some approximation. Then it is extremely difficult.to confirm that the conclusion does not depend on the specific approx imation employed. Secondly, quantum-field-theoretical effects of quarks are not taken into account. Thirdly, the problem of three quarks forming a baryon is not discussed. Finally, the gluon confinement problem remains totally untouched. Since color confinement is the very fundamental nature of QCD, it is more natural to expect that its realization is manifest, that is, we can show color confinement without analyzing any dynamical details of QCD. Furthermore, both quark confinement and gluon one should be established as two aspects of one and the same problem of color confinement, because otherwise one must resolve the formidable problem of confining each of infinitely many possible colored bound states. The well-known successful example of confinement is that of longitudinal photons in quantum electrodynamics (QED); their appearance is totally forbidden by the Gupta 2 subsidiary condition. ) Of course, it is true that longitudinal photons are non-existent from the outset if one employs the Coulomb gauge. It is, however, nothing more than a superstition to believe that given any indefinite-metric theory having unitary physical S matrix, there might always exist a non-local but positive-metric quantum field theory which is physically equivalent to it. We emphasize that in gauge theories it is very essential to use the indefinite-metric Hilbert space. Indeed, as pointed out recently,3) indefinite metric is indispensable even in the axial gauge. Furthermore, the evasion of Lehmann's theorem, which opens a possibil ity of ultimately (Le., not by renormalization) resolving the ultraviolet-divergence
*) The essential part of the present paper was given in an unpublished report.1) The mathematical detail of the present paper will be given in a succeeding paper. 1360 N. Nakanishi and 1. Ojima difficulty, is achieved owing to indefinite metric in the manifestly-covariant canonical operator formalism of quantum gravity.4) We further emphasize that the path-integral formalism is unsatisfactory when the corresponding operator formalism necessitates indefinite metric. This is because the structure of the state-vector space is totally out of control in the path-integral formalism. For example, the appearance of unphysical particles can be forbidden only at the level of Feynman graphs. Hence the path-integral approach is perturbative in its essence and Downloaded from https://academic.oup.com/ptp/article/71/6/1359/1879597 by guest on 30 September 2021 unsuitable for discussing such a truly non-perturbative problem as confinement. Throughout the present paper, therefore, our discussion is based on the manifestly-covar iant canonical operator formalism of the nQn-abelian gauge theory.S) QCD is a non-abelian gauge theory of the quark fields qa and the gluon field A"a, where a and a are color indices (a=l, 2, 3 and a=l, 2, "',8 for SU(3)). In addition to qa and A"a, we need auxiliary scalar fields Ba(B-field), ca(FP ghost) and ca (FP anti-ghost) to have a consistent quantum theory. Since the manifestly-covariant canonical operator formalism of the non-abelian gauge theory is fully described in Ref. 5), we do not reproduce it here, but just mention its subsidiary condition. The Lagrangian density of this formalism is invariant under the BRS transformation. Its g~nerator is the BRS charge Q B, which is nilpotent (Q B2= 0 ).It is postulated that the BRS invariance is not broken spontaneously. Then one can consistently set up a sub sidiary conditionS) (1) that is, the physical subspace is the totality of the states Iphys) satisfying (1). Then, postulating asymptotic completeness, one can prove, without recourse to perturbation theory, that no negative-norm states belong to the physical subspace. This mechanism is called quartet mechanism.S) For fundamental fields, the longitudinal component of A"a, B a, c a and ca are confined by forming a quartet. In this way, the unitarity of the physical S-matrix is established. N ow, we discuss the color confinement problem. We first not.e that there are two kinds of possibility of realizing confinement. The confinement of the first kind means that the field in question has no asymptotic field despite the fact that the corresponding particle is not unstable. The Dirac field in the Schwinger model is confined in this way.S) The confinement of the second kind means that although the field in question has an asymptotic field, the corresponding states are totally unobservable because of the subsidiary condi tion. The longitudinal photon in QED is confined in this way.2) Since the quark fields carry a fractional electric charge, the existence of integrally charged hadrons contradicts asymptotic completeness if the quark fields have no asymptotic field. Hence it is expected that color confinement in QCD is the confinement of the second kind. Then it is natural to postulate that color symmetry is not broken spontaneously at all, because only in this case we can define the concept of triality and hence color confinement immediately implies the non-observability of fractionally charged particles. Color confinement based on the subsidiary condition (1) has been discussed from various standpoints.S),7)_12) As long as color symmetry is not broken, it can be shown*) that the color confinement of this type cannot take place unless there is a massless bound
*) This point will be discussed elsewhere. Color Confinement in Quantum Chromodynamics 1361 state involving FP ghosts. That is, the realization of the color confinement based on (1) critically depends on the possibility of massless bound states of ghosts. Although the existence of massless bound states itself is, in principle, possible,9) it is a prohibitively difficult problem of dynamics to show explicitly whether or not the required bound state of ghosts really exists in QeD. In the above considerations, to show color confinement means to show that all colored physical states are of zero norm. That is, it was the goal to show the vanishment of the color charge Qa in the physical subspace apart from zero-norm states. Now, we wish to Downloaded from https://academic.oup.com/ptp/article/71/6/1359/1879597 by guest on 30 September 2021 convert the way of thinking drastically. We propose to set up
(2) as a new subsidiary condition in addition to (1). That is, we forbid all colored physical states, including those of zero norm, from the outset. We first confirm that no mathematical trouble is created by setting up (2). Indeed, two subsidiary conditions (1) and (2) (for each value of a) are mutually compatible because [Qa, QB]=05) and
(3 )
Furthermore, (2) is linear and time-independent, and it is satisfied by the vacuum 10>, that is, (4) because we postulate that Qa is not broken spontaneously. According to the above new subsidiary condition (2), any physical state must be colorless, but, of course, the very thing of this fact should not be regarded as color confinement. If the physical state consists of two (or several) colored wave packets which are spacelikely far separated from each other, then we could still effectively observe a colored particle by neglecting the remainder of that colorless state. This problem is 13 nothing but the so-called "behind-the-moon" problem. ) The behind-the-moon problem concerning the subsidiary condition (1) was discussed in a paper by one of the authors,14) but its reasoning cannot be straightforwardly extended to the present case concerning Qa, because the nil potency of. Q B was used there. To settle the problem in the present case (2), we first examine how the behind.the-moon problem should be formulated. The setting-up of this problem implies that we should consider a total state 1<1» which, in some sense, consists of two partial states \l'jlo> (j=1, 2), where \l'j is an operator belonging to a space-time region £2;. Here Q j should be, in some sense, a localized region,*) and QI and Q 2 should be causally independent of each other, that is, they are mutually spacelikely separated. As we shall show later, it is sufficient to assume that the total state 1<1» is a product state 1<1> >= \1'1\1'210>. Accordingly, the central problem becomes to prove the follbwing statement: If a product state \1'1\1'210> is colorless, then both partial'states \1'110> and \1'210> are also colorless. Then we can claim that color confinement is achieved by (2). The essence of proving the above statement is as follows.
*) Its precise characterization will be discussed in a succeeding paper. 1362 N. Nakanishi and L Ojima
Fromthe subsidiary condition (2) for the product state I Then the Reeh-Schlieder property 15) implies that (6 ) Downloaded from https://academic.oup.com/ptp/article/71/6/1359/1879597 by guest on 30 September 2021 or equivalently, (7) Because rpj and [Qa, rpj] belong to Q j and because Ql and Q 2 are disjoint, (7) implies (j=1, 2) (8) aa being a c-number. Accordingly, we have [Qa, [Qb, rp;]]=[Qa, (-l)jabrpj]=aaabrpj. (9) Hence the Jacobi identity and (3) lead us to j 0= [[Qa, Qb], rp;]= irbC[Qc, rp;]= i( -l) r bC a C rpj. (10) Hence we obtain (11) As long as the Lie algebra of Qa's is semi-simple as is the case for the color 5U(3), the matrix (fabd r bC ) (Killing form) is invertible, and hence (11) yields (12) Therefore, (8) is reduced to [Qa, rpj]=O, whence we obtain the desired result (13) We remark, however, that there is some subtlety concerning the treatment of quantum field-theoretical operators in inferring (8) from (7). We will make more careful analysis about this point in a succeeding paper. The colorlessness of the partial states in a colorless product state is quite natural and intuitively self-evident; it is a very general nature valid for any semi-simple Lie group. Apart from the subtlety concerning operators, it is evidently impossible group-theoretical ly to extract a singlet representation of a semi-simple Lie group from a tensor product of its non-trivial representations without taking a linear combination. For example, intuitive pictures for the color singlet mesons and baryons in terms of quarks qa are given by such linear combinations as L:aqf ijf and L:ap7cap7qfqgq~, respectively. This point clearly distinguishes QeD with a semi-simple (actually, simple) gauge group 5U(3) from QED with a non-semi-simple (actually, abelian) gauge group U(1). Indeed, we cannot derive (12) in the case of QED. Now, we must clarify the reason why we may confine ourselves to considering a product state, that is, why we need not consider the case in which the total state I *) This is in sharp contrast with the fact that a physical colored (or charged) state, if any, should necessarily have an infinite space-tim~ extension as a consequence of the property characteristic of the (spontaneously unbroken) gauge theory:),l4) **) A local observable is a local operator which commutes with QB. 1364 N. Nakanishi and 1. Ojima safely realized by transcribing from the context of quantum mechanics to the present framework of quantum field theory. We thus see that it is quite absurd to restrict the physical states to the Lorentz·invariant states by imposing a subsidiary condition (15) because rotation group-variant states communicate with invariants ones. Indeed, the imposition of (15) makes the theory physically nonsense. Downloaded from https://academic.oup.com/ptp/article/71/6/1359/1879597 by guest on 30 September 2021 In summary, our reasoning is as follows. Because of (2), the total state I(].) is colorless. In the behind-the-moon situation, it is possible to observe its partial state onl>, if I(].) is a product state, and for a colorless product state, partial states are shown to be colorless. Furthermore, even if we take into account the contraction of a wave packet, it is impossible to extract a colored state because no colored local observable is available. It is importantto note that by (4) the colorless sector is the only sector containing the vacuum state as long as the color symmetry is not broken spontaneously. Thus it is quite natural to start our consideration from the colorless sector characterized by (2). By putting together all the relevant states which communicate with the colorless sector through the behind-the-moon process, what we have attained is again the colorless sector itself. This is just our way of color confinement based directly upon all the basic ingredients of quantum field theory such as the symmetry property (semi·simplicity of gauge group), gauge in variance of local observables, the principles of quantum theory (superposition principle, etc.), and the standard results of the axiomatic quantum field theory which follow from local commutativity and spectrum condition. The use of semi simplicity distinguishes QeD from QED, and gauge invariance explains the difference between the color 5U(3) and the rotation group. In a succeeding paper, we will show how the machinery of quantum field theory works. In the present paper, we have shown, assuming unbroken color symmetry, that the color confinement problem can be resolved in an unexpectedly simple way, once we admit that it is not a dynamical problem. We wish to call our way of confinement "manifest confinement", because it manifestly guarantees confinement just as manifest covariance does the Lorentz invariance of the theory. It should be emphasized again that if quark confinement is of truly dynamical origin, it is very difficult to explain the reason why gluons and all possible colored bound states are simultaneously confined. Acknowledgements Part of the present work has been done during the stay of one of the authors (1. 0.) at the Max-Planck-Institut fUr Physik und Astrophysik, MUnchen. He would like to express his sincere gratitude to Professor W. Zimmermann for the hospitality extended to him. The authors would like to thank Professor M. Flato, Professor C. Fronsdal, Professor H. Grosse, Professor T. Kugo, Professor S. Schlieder, Professor E. Seiler and Professor W. Zimmermann for various critical comments made on our preliminary report. Color Confinement in Quantum Chromodynamics 1365 References 1) N. Nakanishi and I. Ojima, preprint RIMS-460 (1983). 2) S. N. Gupta, Proc. Phys. Soc. A63 (1950), 681. 3) N. Nakanishi, Phys. Lett.131B (1983),381. 4) N. Nakanish, in Gauge Theory and Gravitation (Lecture Notes in Physics, 176), ed. K. Kikkawa, N. Nakanishi and H. N ariai (Springer,Verlag, 1983), p. 171. Further references are contained therein. 5) T. Kugo and I. Ojima, Prog. Theor. Phys. Suppl. No. 66 (1979), 1. Further references are contained therein. 6) N. Nakanishi, Prog. Theor. Phys. 57 (1977), 580. 7) T. Kugo, Phys. Lett. 83B (1979), 93. Downloaded from https://academic.oup.com/ptp/article/71/6/1359/1879597 by guest on 30 September 2021 8) K. Nishijima, Phys. Lett. 116B (1982), 295. 9) K. Nishijima and Y. Okada, preprint UT-412 (1983). 10) K. Nishijima, Bangalore Lecture Note, preprint UT-402 (1983). 11) T. Suzuki, Prog. TheoL Phys. 69 (1983), 1827. 12) M. Shintani, Phys. Lett. 137B (1984), 220. 13) R. Haag and D. Kastler, J. Math. Phys. 5 (1964), 848. 14) 1. Ojima, Z. Phys. C5 (1980), 227. 15) R. F.Streater and A. S. Wightman, PeT, Spin and Statistics and All That (W. A; Benjamin, 1964), p.138. 16) D. Bohni., Quantum Theory (Prentice-Hall, 1951), p. 611. Note added in proof: When expressed in term of asymptotic states, our physical subspace does not have the usual Fock structure. For example, a colorless qij asymptotic state belongs to it, despite the fact that neither q nor ij asymptotic states do. It contributes to the unitarity sum, but the particles in it are not observable as shown in the text. Thus we may encounter some physically unusual situation. The authors would like to thark Dr. K. Aoki and Dr. H. Hata for this criticism.