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There are different operators of quark and gluon momenta, orbital angularmomenta, and gluon spin in the nucleon structure study. The precise meaning ofthese operators are studied based on gauge invariance, Lorentz covariance andcanonical quantization rule. The advantage and disadvantage of differentdefinitions are analyzed. A gauge invariant canonical decomposition of thetotal momentum and angular momentum into quark and gluon parts is suggestedbased on the decomposition of the gauge potential into gauge invariant(covariant) physical part and gauge dependent pure gauge part. Challenges tothis proposal are answered.  \keywords{Physical and pure gauge potentials; Gauge invariant canonical quarkand gluon momenta, orbital angular momenta and spins; Homogeneous andnon-homogeneous Lorentz transformations; Gauge invariant decomposition andgauge invariant extension; Classical and quantum measurements.

Gauge invariance, Lorentz covariance and canonical quantization in nucleon structure studies