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Art of Spin Decomposition --- A practical perspective Xiang-Song Chen Huazhong University of Science & Technology 陈相松 •华中科技大学•武汉 Oct 5th @ Spin 2015 Recall of the Controversies: gauge invariance A practical perspective: • Construction of Spin Eigenstate • Simplification of Spin Structure • Experimental differentiation of Spin from OAM • Enlarged inspection: The angular momentum tensor and flux density Hint from a forgotten practice: Why photon is ignored for atomic spin? Do these solutions make sense?! The atom as a whole Close look at the photon contribution The static terms! Justification of neglecting photon field A critical gap to be closed The same story with Hamiltonian The fortune of using Coulomb gauge Gauge-invariant revision – Angular Momentum Gauge-invariant revision -Momentum and Hamiltonian The covariant scheme spurious photon angular momentum Gluon angular momentum in the nucleon: Tree-level 3 J ' g d x r ( E B) 0 One-gluon exchange has the same property as one-photon exchange Physical part of the non-Abelian gluon field Manipulating the gluon spin Beyond the static approximation A key issue in spin decomposition Should we separate spin from OAM? The Experimental answer: two kinds of angular momenta Spin is a kind of angular momentum without circulation of momentum 19 Enlarged inspection: the angular momentum tensor mn Canonical: TC (x) = ¶L(fi ,¶ m fi ) ¶(¶ m fi ) dIM 1 mn Symmetric: Tsymm(x) = -g d gmn (x) mnl m ln v mnl m ln v lm M C = x TC - x TC M? ¶n fi - g mn L ¶L mn + S ab fb ¶(¶l fa ) lm = x Tsymm- x Tsymm Surprising: none is satisfactory! A new perspective: Constrains on Tab from quantum measurement If a quantum wave is in mutual eigenstates of more than one observables, then the associated currents must be proportional to each other 21 The Symmetric Tab stands no chance! But the canonical Tab is not fully OK mn TC (x) = T = i0 C ¶L(fa ,¶ m fa ) ¶(¶ m fa ) ¶L(fa ,¶ m fa ) ¶(¶ i fa ) ¶ fa - g L n mn ¶ fa , T (x) = 0 ii C ¶L(fa ,¶ m fa ) ¶(¶i fa ) ¶ fa + L i 22 An improved Tab: free field E.g. photon 23 Proof of validness (general free fields) L(fa ,¶ m fa ) is quadratic in and ¶ m f (f , f * independent) å å 1 å ¶L ¶L 1 å ¶L fa å ¶ r fa å = å a ¶ r å fa + = åa å 2 å¶(¶ r fa ) å ¶(¶ r fa ) 2 å¶fa å mn TC ¶L mn = TCmn + ¶ r B[ rm ]n ¶n fa - g mn L, Trevised = ¶(¶ m fa ) B[ rm ]n å ¶L 1 å mn ¶L fa å fa - g rn = å a ¶ r åg ¶(¶ m fa ) å ¶(¶ r fa ) 2 å 24 The interacting fields: scalar case mn T (x) = 1 dIM -g d gmn (x) This gives a gravitational theory different from Einstein’s GR 25 Trouble with the Angular momentum tensor M mnl mnl m ln v m ln v lm = x Tsymm- x Tsymm lm M C = x TC - x TC ¶L mn + S ab fb ¶(¶ l fa ) ¶L mn M revised? = x Trevised - x Trevised + S ab fb ¶(¶l fa ) mnl m ln v lm 1 ln ¶L 1 lm ¶L + g fa - g fa 2 ¶(¶ m fa ) 2 ¶(¶n fa ) Our trick applies to longitudinal spin flux only, but not to 26 transverse flux of angular momentum! Conclusion: So far there exists no satisfactory expression of angular momentum tensor, even for a free field! And thus no satisfactory way of spin decomposition. Thank you! 27