Download Hint from a forgotten practice: Why photon is ignored for atomic spin?

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
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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
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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
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An improved Tab: free field
E.g. photon
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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
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The interacting fields: scalar case
mn
T (x) =
1
dIM
-g d gmn (x)
This gives a gravitational theory
different from Einstein’s GR
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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
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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!
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