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samalkhaiat

Science Advisor

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*Moderator's note: This is a sub-thread spun off from https://www.physicsforums.com/threa...ergy-of-a-quantum-field-actually-zero.953766/.*

I should have said that in certain cases in QFT, we can neglect “surface terms”. For example, the (on-shell) difference between the Belinfante and the canonical expressions for the angular momentum can be written as [tex]J^{ij}_{Bel} - J^{ij}_{C} = \int d^{3}x \ \partial_{\rho}F^{\rho 0 i j} (x) .[/tex] Now, by careful analysis using wave packets, one can show that the... this statement is indeed "modulo surface terms". Particularly I don't see any necessity for the energy-momentum tensor to be symmetric in the realm of special relativity.

**forward matrix elements**of the RHS do vanish. Also, the forward momentum space matrix elements (i.e., the physical quantities in collision processes) of the “surface terms” do vanish if these terms are divergences of

**local**operators.

But, you are absolutely right. As far as I know, the Belinfante expression [tex]J^{\mu\nu}_{Bel} = \int d^{3}x \left( x^{\mu} \theta^{0 \nu}(x) - x^{\nu} \theta^{0 \mu}(x) \right) , \ \ \ \ \theta^{\mu \nu} = \theta^{\nu \mu} \ ,[/tex] seems to fail to satisfy all the commutation relations of the Poincare’ algebra (to be more accurate, I should say that

**I**have never been able to establish the

**correct**bracket [itex]\big[i J^{0 j}_{Bel} , J^{0 k}_{Bel} \big][/itex] using the usual methods, and I don't know if somebody else had).

Again, one can show that the physical matrix elements of the generators are gauge invariant.For gauge fields, of course you can argue with gauge invariance,

You are right on the "uniqueness" part. I had the pleasure of knowing and working withIn my opinion, there's neither uniqueness in this split nor is there a really well defined treatment of spin in relativistic hydrodynamics,....

**Elliot Leader**on the very same problem (in QCD) for many years. So, I suggest you have a look at his book:

E. Leader, “

**Spin in Particle Physics**”, Cambridge University Press (2001)

You may also find the attached PDF’s useful

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