PandaX experiment Direct detection of dark matter using Xenon TPC in Sichuan, China

Gauge Symmetry Crisis? Xiangdong Ji University of Maryland Shanghai Jiao Tong University ( 上海交通大学）

Outline Gauge symmetry under question Review of gauge symmetry Comments about the new proposals Final remarks

GAUGE SYMMETRY UNDER QUESTION

Momentum operator in QCD In QCD textbooks, the momentum operator is defined as where the first (second) term is the quark (gluon) momentum and both gauge invariant. Similarly, I proposed the angular momentum operator (X. Ji, PRL78,1997)

Problems? Covariant derivative contains explicit gluon potential The gluon angular momentum cannot be separated gauge-invariantly into a spin + orbital contributions Separate quark and gluon contribution do not obey angular momentum algebra. That’s the way it is! Just like you cannot talk about particle’s orbit in quantum mechanics.

New gauge symmetry? About 10 years ago, Wang et al have started questioning gauge symmetry in textbooks. This campaign has changed its forms many times with the unchanging goal of justifying using gauge-variant operators. It is culminated with two PRL articles Xiang-Song Chen, (Sichuan U. & Nanjing U.), Xiao-Fu Lu, (Sichuan U.), Wei-Min Sun, Fan Wang, (Nanjing U.), T. Goldman, (Los Alamos). Phys.Rev.Lett.100:232002,2008. Xiang-Song ChenSichuan U.Nanjing U.Xiao-Fu LuSichuan U.Wei-Min SunFan WangNanjing U.T. GoldmanLos Alamos Xiang-Song Chen Xiang-Song Chen, (Hua-Zhong U. Sci. Tech. & Sichuan U. & Nanjing U.), Wei-Min Sun, (Nanjing U.), Xiao-Fu Lu, (Sichuan U.), Fan Wang, (Nanjing U.), T. Goldman, (Los Alamos) Phys.Rev.Lett.103:062001,2009. Hua-Zhong U. Sci. Tech.Sichuan U.Nanjing U.Wei-Min SunNanjing U.Xiao-Fu LuSichuan U.Fan WangNanjing U.T. GoldmanLos Alamos Then all hell break loose….

REVIEW OF GAUGE SYMMETRY

The standard model and gauge symmetry The standard model of particle physics is based on gauge symmetry (Yang&Mills) ◦ Electromagnetism (Weyl) ◦ Electroweak (Weinberg, Salam, Glashow) ◦ QCD Einstein theory of general relativity is also a gauge theory Witten: gauge symmetry is a consequent of superstring theory

Origin of gauge symmetry Massless particles: massless spin-1 and spin-2 particles. With two physical polarizations. However, to construct a manifestly Lorentz symmetric and local theory, one has to embed the two d.o.f into a vector or symmetric tenor field. Thus, there are spurious non-physical d.o.f in the theory with unspecified, irrelevant dynamics.

Quantize the gauge theory: the need for gauge fixing To quantize the gauge theory, one can either use canonical quantization or path integral formulation. In canonical quantization, one must choose a gauge condition so that the dynamics of the system is completely specified. [No such need in path integral formulation on finite lattices.] Choosing a gauge condition (like choosing a coordinates) breaks the gauge symmetry.

Need for gauge choice in classical fields For a given magnetic field, there are many choices for gauge potential A μ Different choice leads to different wave functions for charged particles in QM. ◦ Hydrogen atoms ◦ Charged particle in an external magnetic field All energy eigen-values are gauge invariant.

Two common choices Physical gauge, in which all non-physical d.o.f. are eliminated in favor of physical ones ◦ This breaks the manifest Lorentz symmetry! Lorentz gauge, in which non-physical degrees of freedom are endowed with nontrivial dynamics ◦ Lorentz symmetry is manifestly kept. ◦ Hilbert space have non-physical degrees of freedom

Physical gauge One has to show that the Lorentz symmetry breaking does not generate any effects in physical observables Symmetry is restored only in truly gauge- invariant quantities, where the symmetry breaking vector effects go away through Ward identity: Bjorken & Drell Gauge symmetry is the key for the restoration of Lorentz symmetry and Lorentz symmetry is a diagnose for gauge invariance!

Lorentz gauge Manifestly Lorentz-covariant Added as a constraint to the lagrangian, generating dynamics for unphysical d.o.f. The total lagrangian is no longer gauge- symmetric, however, it has a global BRST symmetry, which allows us to prove Ward identities useful for restoration of gauge symmetry.

Enlarged Hilbert space The Hilbert space contains temporal and longitudinal photons One must constrain that the physical states satisfy the operator condition, so that the spurious dynamics does not affect matrix elements of physical state. The choice of physical states is not unique. This non-uniqueness is important because the dynamics may evolving one form of the physical state to another.

Mixing of gauge-symmetric operators (Joglekar & Lee) Gauge-invariant operator under renormalization mix with BRST-exact operators and EOM operators. Physical matrix elements of BRST-exact operators and EOM operators are zero.

Quantum gauge transformation In quantum theory, gauge transformation is operator-valued. Gauge invariant operators do not change their form. But the physical states usually do. The matrix elements of gauge-invariant operators do not change.

More on gauge-dependence Green’s functions of elementary fields are gauge dependent Green’s funcitons of gauge invariant operators are gauge-independent The poles of the green’s functions shall be gauge-independent. ◦ Poles correspond to physical masses. S-matrix is gauge-invariant.

THE NEW PROPOSALS AND COMMENTS

Break the gauge pot apart

Gauge symmetry trivialized All the gauge-dependent part is in A pure. A phys is purely physical d.o.f. Thus one can construct all the physical observables using A phys Covariant derivative can simply be

“gauge invariant” expression of the angular momentum

Physical momentum It is known that the covariant derivative P = i∂ - eA/c represents the kinetic momentum of a particle. Feynman: Consider a particle near a solenoid. When the current is turned on suddenly, the A is Produce and electron gets a momentum kick, this cannot be i∂ because the w.f. is continuous in time!

How to quantize the theory? Quantization only makes sense in Coulomb/radiation gauge In Coulomb gauge, one can set A pure vanish and proceed with the quantization of A phys. However, in any other gauge, how does one make the quantization either in canonical way or in path integral formulation?

Are all quantities made of A phys physical? In this non-traditional sense of gauge symmety, the physical quantities become non-local Lorentz transformation of A phys is quite nontrivial. [All physical quantities shall transform properly under LT?!].

Physical measurement? None of these [non-local] operators seems appear in operator product expansion or factorization process. Therefore, it is hard to imagine their matrix elements can be measured. Without exp measurement, any definition has rather limited value.

Leader’s proposal Leader wrote a long paper about gauge symmetry, angular momentum etc. Many of the statements in the paper are inconsistent. Essential point is: although the canonical form (with partial derivatives) of momentum and angular momentum are not gauge-invariant but they have gauge invariant matrix elements. The claim is not true. The proof must be incorrect.

Explicit calculations Explicit calculation shows the gauge dependence of the canonical operators. “Does the Gluon Spin Contribute in a Gauge Invariant Way to the Nucleon Spin?,” P. Hoodbhoy and X. Ji, Phys. Rev. D 60, (1999).

FINAL REMARKS

Gluon spin? If the gluon spin is not gauge-invariant, how come it gets measured in expts like those at RHIC What expts measure is a gauge invariant quantity, but very complicated It cannot be readily identified as the gluon spin.

Gluon spin? However in light-cone gauge, A+ = 0, the operator reduces to the gluon spin operator in the same gauge Expts measure a gauge-invariant quantity which has a simple interpretation as the gluon spin in light-cone gauge.