While making some further progresses based on our previous paper [H.-J. Wang J. Math. Phys. 49, 033513 (2008)], we find a few typos or obscure points which may mislead readers interested in the topic. Now, we would like to correct them or make them more understandable.

Topics
General relativity, Quantum mechanical systems and processes, Quantum mechanical principles, Visual system
  1. The minus sign of (3.14a) and (3.14b) should be exchanged, i.e., they should be
    (3.14a)
     
    (3.14b)
  2. All the j’s in the formulae under Eq. (5.12) lose a cap, the correct ones should be as follows:
    Ω = ( Ω α β ) = d Γ − Γ ∧ Γ ,
    from Γ α β = Γ k α β d z k ⁠, and taking into account the definition of the components form of curvature, Ω α β = R α j ̄ k β d z ̄ j ∧ d z k ⁠, it is straightforward to obtain
    R α j ̄ k β = Γ k α , j ̄ β − Γ j ̄ α , k β + Γ j ̄ γ β Γ k α γ − Γ k γ β Γ j ̄ α γ .

    The last two formulae originally are Ω α β = R α j k β d z ̄ j ∧ d z k and R α j k β = Γ k α , j β − Γ j α , k β + Γ j γ β Γ k α γ − Γ k γ β Γ j α γ which are wrongly typed.

  3. The cited equation number (5.6a) above Eq. (8.1) should be (5.7a), there is no equation sign (5.6a) in the published version.1 

  4. We ought to explain that from Eq. (8.1) and thereafter, we have viewed the differential form dψ as plane wave, just as in General Relativity viewing dx as straight line segment. dψ means carrying a variation on ψ, but not a total differential. The subsequent ddψ ≠ 0 is straightforward, since ddψ means double differential. It should not be confused with the exterior differential calculation in the preceding sections. The delicate meaning of these equations traces back to Eqs. (4.16) and (5.7a). In deferential geometry, basically double exterior differential dd occurs while inquiring the form of components of curvature. The same understanding also applies to Eq. (9.1).

  5. Finally, as for the Principle of Nonlocality, in the Introduction1 we should have pointed out that the “co-moving” (complex) reference frame for the observed fermion can be recognized even when the observing fermion is interacting with it, which does have classical counter picture. Unavoidable interactions (or correlations like Pauli exclusion principle) between fermions make quantum system totally different from classical system. However, we can still understand the relationship of two fermions by their classical analog, albeit with gravity (as correlation) in between. Assume a person is falling down to the earth from outer space. While his eyes are covered, he feels as if in a flat space. But as soon as his cover is removed, he could be aware of his accelerating motion. The former situation represents the case that he is an observer with co-moving reference frame. While the latter represents the case that he is an observer feeling as if the earth being in flat space. This latter case is the example of co-moving reference frame with observed object. The above arguments apply to interacting fermions likewise. When the “co-moving” reference frame is with observed fermion, it looks like a plane wave. The explanation is necessary since mostly a classical observer is assumed to be free of interaction.

    These errors and missing points affect neither the succeeding formulas in our working process nor the conclusions inferred from them.

1.
H.-J.
 
Wang
, “
General nonlocality in quantum fields
,”
J. Math. Phys.
 
49
,
033513
(
2008
).
https://doi.org/10.1063/1.2874556
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2015
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