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Register a new userThe application of gauge invariance and canonical quantization to the internal structure of gauge field systems
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It is unavoidable to deal with the quark and gluon momentum and angular momentum contributions to the nucleon momentum and spin in the study of nucleon internal structure. However, we never have the quark and gluon momentum, orbital angular momentum and gluon spin operators which satisfy both the gauge invariance and the canonical momentum and angular momentum commutation relation. The conflicts between the gauge invariance and canonical quantization requirement of these operators are discussed. A new set of quark and gluon momentum, orbital angular momentum and spin operators, which satisfy both the gauge invariance and canonical momentum and angular momentum commutation relation, are proposed. The key point to achieve such a proper decomposition is to separate the gauge field into the pure gauge and the gauge covariant parts. The same conflicts also exist in QED and quantum mechanics and have been solved in the same manner. The impacts of this new decomposition to the nucleon internal structure are discussed.
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There are different operators of quark and gluon momenta, orbital angular momenta, and gluon spin in the nucleon structure study. The precise meaning of these operators are studied based on gauge invariance, Lorentz covariance and canonical quantization rule. The advantage and disadvantage of different definitions are analyzed. A gauge invariant canonical decomposition of the total momentum and angular momentum into quark and gluon parts is suggested based on the decomposition of the gauge potential into gauge invariant (covariant) physical part and gauge dependent pure gauge part. Challenges to this proposal are answered. keywords{Physical and pure gauge potentials; Gauge invariant canonical quark and gluon momenta, orbital angular momenta and spins; Homogeneous and non-homogeneous Lorentz transformations; Gauge invariant decomposition and gauge invariant extension; Classical and quantum measurements.
The quantum Rabi model is a widespread description for the coupling between a two-level system and a quantized single mode of an electromagnetic resonator. Issues about this models gauge invariance have been raised. These issues become evident when the light-matter interaction reaches the so-called ultrastrong coupling regime. Recently, a modified quantum Rabi model able to provide gauge-invariant physical results in any interaction regime was introduced [Nature Physics 15, 803 (2019)]. Here we provide an alternative derivation of this result, based on the implementation in two-state systems of the gauge principle, which is the principle from which all the fundamental interactions in quantum field theory are derived. The adopted procedure can be regarded as the two-site version of the general method used to implement the gauge principle in lattice gauge theories. Applying this method, we also obtain the gauge-invariant quantum Rabi model for asymmetric two-state systems, and the multi-mode gauge-invariant quantum Rabi model beyond the dipole approximation.
The gauge dependence in the anomalous dimension of the gauge-invariant-canonical-energy-momentum tensor for proton is studied by the background field method. The naive calculation shows the problem, the absence of the counter term in the gluonic sectors. The analysis shows that the result [Chen et al., Phys. Rev. Lett. 103, 062001 (2009)] is derived from the background field method after we introduced a trick to avoid the problem except for the gluon-to-gluon sector; it is gauge dependent. The possible reason of this gauge-dependent result comes from the nontrivial treatment of the condition $F^{mu u}_{pure}=0$ at a higher order. This result shows that one needs a further improvement in treating this condition with a covariant way at a higher order by the background field method. In particular, we have to focus on two checkpoints, the gauge independence and zero eigenvalue in the anomalous-dimension matrix, in order to test the validity of the gauge-invariant-canonical-energy-momentum tensor.
A new method is proposed to calculate wave functions in $k_T$-factorization in cite{LiMi} as a comment about our paper cite{FMW}. We point out that the results obtained with the method are in conflict with the translation invariance and depend on the chosen contours for loop-integrals. Therefore, the method is in principle unacceptable and the results with the method cannot be correct.
The Dicke model, which describes the dipolar coupling between N two-level atoms and a quantized electromagnetic field, seemingly violates gauge invariance in the presence of ultrastrong light-matter coupling, a regime that is now experimentally accessible in many physical systems. Specifically, it has been shown that, while the two-level approximation can work well in the dipole gauge, the Coulomb gauge fails to provide the correct spectra in the ultrastrong coupling regime. Here we show that, taking into account the nonlocality of the atomic potential induced by the two-level approximation, gauge invariance is fully restored for arbitrary interaction strengths, even in the limit of N going to infinity. Finally, we express the Hopfield model, a general description based on the quantization of a linear dielectric medium, in a manifestly gauge invariant form, and show that the Dicke model in the dilute regime can be regarded as a particular case of the more general Hopfield model.
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