Transcript (Fan) Wang
Problems related to gauge invariance, Lorentz covariance and canonical quantization applied in nucleon structure study Fan Wang CPNPC (Joint Center for Particle Nuclear Physics and Cosmology, Nanjing Univ. and Purple mountain observatory of CAS) IMPCAS (Institute of Modern Physics, Lanzhou, CAS) [email protected] Outline 1. Introduction 2. Gauge theory without gauge interaction 3. Gauge invariant decomposition 4. Physical meaning of parton distribution and its evolution 5. What is the key issue of nucleon structure study I.Introduction • Nucleon is an SU(3) color gauge system, atom is an U(1) em gauge system. The mass(energy)-momentum, spin, orbital angular momentum distribution among the constituents are fundamental problems in the internal structure study of atom and nucleon. • Our experience on the atomic, molecular, nuclear internal structure study seems to show that these problems are trivial in principle except the many-body complication. • However, to take gauge invariance, canonical quantization, Lorentz covariance for a gauge field system full into account, this is not trivial in principle. • 5 years ago we raised questions about the precise meaning of momentum, spin and orbital angular momentum of quark, gluon in nucleon structure. It seems that up to now there is no consensus yet. • Today I will discuss these questions further in the nucleon structure study. II. Gauge theory without gauge interaction • The gauge symmetry introduced by C.N. Yang employing the minimum coupling to introduce pure gauge and physical gauge interaction together. • It is possible to have gauge invariant theory without physical gauge interaction, (i ( igA , pure ) m) 0 where Fpure 0 • This kind formalism can be obtained through a gauge transformation (GT), e ' i ( x ) and introducing pure gauge interaction by setting, A , pure ( x) (for simplicity use QED as example.) It is still a free Dirac theory, but gauge invariant already. This is what we mean pure gauge potential, it is solely due to GT. In differential geometry this is an internal unitary transformation related to internal basis changing. III.Gauge invariant decomposition • For a long time the Ji-decomposition seems to be the unique gauge invariant possibility, which does not have idea about the gluon spin and orbital angular momentum. The three quark momentum operators do not commute and so can not consist of the complete set to describe the three dimensional quark momentum distribution. • The next few pages show the different decomposition of the angular momentum operator of the QED case. QCD case has the same form. J QED Se Le S L J QED Se L'e J ' In fact we obtained the same decomposition in the same time and pointed out the unphysical features. Commun. Theor. Phys. 27, 121 (1997) Wakamatsu improved this by decomposing the total gluon angular momentum J into gauge invariant spin and orbital part. ' J QED Se Le ' ' S ' ' L ' ' Consistent separation of nucleon momentum and angular momentum Standard construction of orbital angular momentum L d 3 x x P Debate on the gauge invariant decomposition • Non-locality: Non-local operators are popular in gauge field theory. The A-B effect is a non-local effect. All of the parton distribution operators are non-local, only in light-cone gauge becomes local. The new one is local in Coulomb gauge • Canonical commutation relation: Renormalization ruins the canonical commutation relation. However there is no way to avoid the canonical quantization rule. hep-ph:0807.3083,0812.4336,0911.0248,1205.6983 Lorentz covariance • The Lorentz transformation (LT) property of 4 coordinate x , momentum p , and field tensor F are fixed by measurement. They are transformed by the usual homogeneous LT. • The LT property of 4-vector potential A is gauge dependent, because there is gauge degree of freedom: Lorentz gauge, usually they are chosen to transform with homogeneous LT, but it can also be chosen to 'with inhomogeneous ' i.e., transform LT, A ( x' ) A ( x) ( x' ) Because there are residual gauge degree of freedom. • Coulomb gauge, vector potential must be transformed by the inhomogeneous LT to make the transformed potential still satisfy the Coulomb gauge condition, i.e,. ' ' A ( x' ) A ( x) ( x' ), ~~ A ( x' ) 0 ' ' because homogeneous LT mixing the unphysical components to the vector potential and one must do additional gauge transformation to eliminate the unphysical components. • Light-cone gauge vector potential must also be transformed by inhomogeneous LT to make the A ' 0 in the new Lorentz frame. Only for limited boosting along the infinite momentum direction light-cone gauge fixing can be preserved automatically! • The Lorentz frame independence of a theory must be independent of the gauge fixing, no matter the Lorentz gauge, Coulomb gauge or light-cone gauge is used. All of them must be Lorentz invariant. • The Lorentz transformation form of the gauge potentials are gauge dependent accordingly! A systematic analysis had been given by C. Lorce in arXiv:1205.6483[hep-ph],PRD87(2013)034031. • It is impossible to decompose the total energy, the 0th momentum, of an interacting gauge system into the quark (electron) and gluon (photon) part to keep them as the 0th momentum of the individual part, i.e., to transform as the 0th component of a 4momentum. Uniqueness of the decomposition and the gauge invariant extension Gauge invariance is the necessary condition for the measurabilily but not sufficient one. To discover the gauge invariant one is different from the gauge invariant extension. Different gauge invariant extensions are not all physical and usually result physically different ones. QED case this physical condition is unique, QCD case the generalization of this condition is more complicated and under hot debate. We believe the physical one is unique and our physical condition is a generalization of Coulomb gauge , i.e., only two helicity gluon components are physical. • To obtain the gauge invariant momentum, orbital angular momentum, gluon spin, etc., is to discover the gauge invariant one through the decomposition of gauge potential. • The gauge invariant extension through a gauge link or other methods usually mixes the gluon and quark part and change the physical meaning of the operator. Four momentum operators only p physis physical one p cano p kine 1 mr q A i 1 p q A m r D i plc p qA p phys mr q A q A pure mr q A phys q A pure i • The kinematic quark momentum p g A suggested by X.D. Ji and M. Wakamatsu is not the right quantum mechanical momentum Operator: three components do not commute and so cannot consist of a complete set to describe the 3-d quark momentum distribution, which can not be reduced to the canonical one in any gauge. There is no solution of its eigen-value equation. They are still a mixing of quark and gluon momentum and has never been measured except in classic physics ! Partial gauge invariant and power counting for electron momentum • X. Ji developed the socalled power counting to relate the gauge invariant kinematical momentum to the canonical one, to assume the vector potential as a perturbative correction. • Gauge invariance is an exact symmetry, there is only gauge invariance or non-invariance. Partial gauge invariance is nonsense! • The only gauge invariant physical one is what we defined, the physical momentum and used in nonrelativistic quantum mechanics in fact. • The light-cone momentum plc p qA is the infinite momentum frame version of the physical momentum, p phys p qApure • Because various arguments show consistently that the A only includes the longitudinal component A// at least in the infinite momentum frame. And the kinematical momentum cannot be transformed to the lightcone one through the gauge transformation. Centenary question: Spin and orbital angular momentum of massless photon and gluon • For a long time it is believed that one can not decompose the total angular momentum of a massless particle, the photon and gluon, into spin and orbital ones. • Now there seems to be a consensus that this conclusion should be modified as: there are no local gauge invariant spin and orbital angular operators but there are nonlocal ones. • The measured photon spin should be E A phys • The measured gluon spin is the matrix elements of the above gluon spin operator boosted to the infinite momentum frame. • The complicating of the boosting is not due to the use of physical component phys A but due to the spin operator itself . There is the wellknown Wigner rotation. IV.Physical meaning of parton distribution and its evolution • Jaffe and Bashinsky (arXiv:hep-ph/9804397, NPB536(1999)303) studied the physical meaning of the parton distribution. They conclude that the parton momemtum distribution is a distribution on the eigenvalue of the light-cone momentum. p i( igA ) • Only those observables, the corresponding operators commute with the light-cone momentum, have physical meaningful parton distributions. • If we insist on this requirement, the 3-D parton momentum distribution and the light-cone quark orbital angular momentum distribution can not be the kinematical ones • In the naïve parton model the parton distribution f(x) is a distribution of parton canonical momentum x p /P distribution. Taking into account of the gluon interaction, under the collinear approximation, the parton distribution is the light-cone momentum p gA distribution. • The further gauge transformation can only introduces the pure gauge gluon field A into the parton pure momentum. The transverse (physical) components Aphys will never be involved in the measured parton momentum. Physical momentum satisfy the canonical commutation relations, reduced to the canonical momentum in Coulomb gauge, and the measured quark momentum distribution should be the matrix elements of the physical momentum boosting to infinite momentum frame. The measured electron momentum in atomic and molecular structure should be the matrix elements of the physical momentum in the lab frame not the socalled power counting ones. Quark orbital angular momentum • The quark kinematical orbital angular momentum Lq 3 d xx q ( p gA) q calculated in LQCD and “measured” in DVCS is not the real orbital angular momentum used in quantum mechanics. It does not satisfy the Angular Momentum Algebra, L L iL and the gluon contribution is ENTANGLED in it. • E. Leader suggested to use the gauge variant canonical momentum and angular momentum operators as the physical one and tried to prove that the matrix elements of physical states of gauge dependent operator are gauge invariant. • His argument is based on F. Strocchi and A.S. Wightman’s theory and this theory is limited to the extended Lorentz gauge and so at most only true for very limited gauge transformations. • Our gauge invariant momentum and angular momentum operator reduce to the canonical one in physical gauge, i.e., they are generators for physical field. arXiv:1203.1288[hep-ph] Evolution of the parton distribution • Most of the evolutions are based on the free parton picture and perturbative QCD. • The measured parton distribution is always a mixing of non-perturbative and perturbative one. • The first moment of polarized structure function 1 (Q 2 ) g1 ( x, Q 2 ) dx 2 shows dramatical changes in the low Q region, 2 the simple evolution can not describe the low Q behavior. V.What is the key issue of the nucleon structure study? • Are all of those problems popular in the present nucleon structure study really important ones and will lead to new understanding of nucleon internal structure , especially to new physics? • How does the parton picture discovred in the infinite momentum frame relate to the picture discovered in hadron spectroscopy, which is complicated but should not be left aside. Thanks Symmetric or asymmetric energy-momentum(E-M) tensor • The E-M tensor of the interacting quark-gluon system is the starting point to study the momentum, spin and orbital angular momentum distribution among the quark, gluon constituents. • The Lagrangian of an interacting quark-gluon field system is a 1 1 a a a L(x) (i ( igA ) m) h.c. F F 2 2 4 • From the interacting quark-gluon Lagrangian, follow the standard Noether recipe one can obtain the E-M tensor of this system, Tas i a a h.c. F A g L(x) 2 • This E-M tensor is neither symmetric nor gauge invariant. • One can follow the Belinfante recipe to get the symmetric one or add a surface term to make it gauge invariant, Tsy i ( D D ) h.c. F a Fa g L(X), 4 D igAa a 2 . • It is the popular idea that one has the freedom to make choice of the form of the E-M tensor, because to add a surface term will not change the conservation law satisfied by the E-M tensor. • The symmetric and gauge invariant one is prefered because the Einstain gravitation equation needs symmetric E-M tensor of matter fields. • In fact the E-M tensor density of em-field is measurable and there are already experimental hints to refute the symmetric one . • For symmetric em E-M tensor, there should be no difference of the diffraction pattern for an orbital or spin polarized light beam if the total J z lz ~ or ~ sz 1 • For asymmetric em E-M tensor, there should be difference of the diffraction pattern between orbital and spin polarized beams, because only for orbital polarized beam there is momentum density circular flow in the transverse plane. A detailed analysis had been given in arXiv:1211.4407[physics.class-ph] Optical evidence Spin is different from orbital angular momentum • In 1920’s one already knew that the fundamental spin can not be related to the orbital motion. So to use the symmetric E-M tensor to express the angular momentum operator as physically is misleading even though mathematically correct. It also leads to the wrong idea that the Poynting vector is not only the energy flow but also the momentum density flow of em field. The Belinfante derivation of the symmetric E-M tensor and in turn the derived momentum and angular momentum density operators physically is misleading too. Spin half electron field needs asymmetric energy-momentum tensor • Symmetric E-M tensor for a spin half electron field will lead to contradiction between angular momentum and magnetic momentum measurement. Suppose we have a spin polarized electron beam moving along the z direction with momentum density flow K , 1 ( x K ) z dV J z dV N 2 K j n, E e E E ( x K ) z dV e ( x j ) z dV e 2 z dV N 1 • For symmetric E-M tensor, the momentum density flow is the same as the energy density flow. For spin s=1/2 electron this will lead to a contradiction. The energy flow and momentum density flow should be different. A detaild analysis had been given in arXiv:1211.2360[gr-qc]. To meet the requirement of a gauge invariant but asymmetric E-M tensor we derived the following one based on the decomposition of gauge potential. Hope experimental colleagues to check further which one is the correct one. cw T i phys D pure h.c. F A g L(x) 2 Z.T. Liang’s explanation of parton distribution