Gauge Invariant Noether's Theorem and The Proton Spin Crisis

Due to proton spin crisis it is necessary to understand the gauge invariant definition of the spin and orbital angular momentum of the quark and gluon from first principle. In this paper we derive the gauge invariant Noether's theorem by using combined Lorentz transformation plus local gauge transformation. We find that the notion of the gauge invariant definition of the spin (or orbital) angular momentum of the electromagnetic field does not exist in Dirac-Maxwell theory although the notion of the gauge invariant definition of the spin (or orbital) angular momentum of the electron exists. We find that the gauge invariant definition of the spin angular momentum of the electromagnetic field in the literature is not correct because of the non-vanishing surface term in Dirac-Maxwell theory although the corresponding surface term vanishes for linear momentum. We also show that the Belinfante-Rosenfeld tensor is not required to obtain symmetric and gauge invariant energy momentum tensor of the electron and the electromagnetic field in Dirac-Maxwell theory.


I. INTRODUCTION
The spin of the proton at rest is 1 2 . When the proton is in motion, like that at high energy colliders, its helicity (which is the projection of the proton spin along its direction of motion) is conserved with the quantized value ± 1 2 . In the naive parton model it was predicted that the proton spin is carried by the quarks (plus antiquarks) inside the proton [1,2]. However, the famous European muon collaboration (EMC) experiment at CERN [3] revealed that the total contribution to the proton spin from the quarks (plus antiquarks) is almost zero. This is known as the "proton spin crisis" which is one of the most important unsolved problem in particle physics.
After the EMC experiment, other experiments have also confirmed similar results [4], including the recent RHIC results at √ s N N = 510 GeV polarized proton-proton collisons [5,6]. RHIC experiment also involves heavy-ion collisions to study quark-gluon plasma [7,8].
In future, the electron-ion collider (EIC) [9] is expected to provide the precise insight to the spin structure of the proton. At present, the total quarks (plus antiquarks) spin contribution S q to the proton spin is ∼ 1 6 [10] and the total gluons helicity contribution S g to proton spin is ∼ 1 10 [10] and the rest of the proton spin may be from the orbital angular momentum L q of the quarks (plus antiquarks) and from the orbital angular momentum L g of the gluons.
Assuming total angular momentum conservation in physics one usually writes where in terms of experimentally measured polarized quark (plus antiquark) distribution function ∆q(x) one has and in terms of experimentally measured polarized gluon distribution function ∆g(x) one has S g = 1 2 where x is the longitudinal momentum fraction of the parton with respect to proton.
Since the RHIC experiment [5,6] measure spin polarized gluon distribution function ∆g(x) inside the proton one expects that the gauge invariant spin distribution function of the gluon is measured at the experiments. However, it is well known in the gauge theory that only the spin angular momentum S q of the quark is gauge invariant but the spin angular momentum S g of the gluon is not gauge invariant. Take for example the Maxwell theory where the spin angular momentum of the electromagnetic field obtained from the Noether's theorem is given by [see eq. (19) for the derivation] which is not gauge invariant where E(x) is the electric field which is gauge invariant and A(x) is the electromagnetic vector potential which is not gauge invariant.
The gauge invariant definition of the spin dependent gluon distribution function proposed in [2,11] corresponds to eq. (4) in the light-cone gauge A + = 0. However, in any other gauge it does not correspond to gluon spin angular momentum that is obtained from Noether's theorem. Hence we do not have a gauge invariant definition of spin dependent gluon distribution function ∆g(x) in QCD at high energy colliders from the first principle. This is because the gauge invariant definition of the spin dependent gluon distribution function must give the same value in any gauge.
In addition to this there have been various (gauge invariant and gauge non-invariant) definitions of the spin angular momentum S g of the gluon, orbital angular momentum L g of the gluon and orbital angular momentum L q of the quark in the literature [2,[12][13][14][15].
Hence it is necessary to understand the gauge invariant definition of the spin and orbital angular momentum of the quark and gluon from first principle.
The first principle method to understand the conservation of angular momentum in physics is via Noether's theorem using Lorentz transformation. For example the spin angular momentum S γ of the electromagnetic field in eq. (4) is obtained from the Noether's theorem by using Lorentz transformation [see eq. (19)]. The Noether's theorem in Maxwell theory is given by where δA µ (x) is the functional differential of A µ (x) and However under the Lorentz transformation the functional differential δA µ (x) is not gauge invariant. For example, under the(infinitesimal) Lorentz transformation the functional differential δA µ (x) is given by which is not gauge invariant where the general coordinate transformation is given by Note that every thing in the Noether's theorem in eq. (5) is gauge invariant except the functional differential δA µ (x) as given by eq. (7). Hence when the gauge non-invariant functional differential δA µ (x) from eq. (7) is used in eq. (5) we obtain the gauge non-  Maxwell theory we find the continuity equation where the gauge non-invariant energy-momentum tensor of the electromagnetic field in Maxwell theory obtained from Noether's theorem using Lorentz transformation is given by Under rotation [without translation, ∆ µ = 0, see eq. (8)] by using eq. (7) in (5) in Maxwell theory we find the continuity equation where the gauge non-invariant third rank tensor J µνλ (x) is given by which gives the gauge non-invariant angular momentum tensor of the electromagnetic field where the gauge non-invariant invariant T µν (x) is given by eq. (10). We write eq. (13) as where the spin angular momentum tensor of the electromagnetic field is given by and the orbital angular momentum tensor of the electromagnetic field is given by where the gauge non-invariant invariant T µν (x) is given by eq. (10).
The spin angular momentum S, the orbital angular momentum L and the total angular momentum J are given by Using eqs. (16) and (10) in (17) we find that the orbital angular momentum L γ of the electromagnetic field is given by which is not gauge invariant. Using eq. (15) in (17) we find that the spin angular momentum S γ of the electromagnetic field is given by which reproduces eq. (4) which is not gauge invariant.

C. Gauge Non-Invariant Orbital Angular Momentum of Electron in Dirac Theory
From Noether's Theorem The Dirac lagrangian density is given by (20) which by using the Euler-Lagrange equation gives the Noether's theorem where δψ(x) is the functional differential of the Dirac field ψ(x) of the electron.

Under (infinitesimal) Lorentz transformation the Dirac spinors transform as
where Using eq. (22) in the functional differential of the Dirac spinor we find Under translation [without rotation, ǫ µν = 0, see eq. (8)] by using eq. (25) in (21) in Dirac theory we find the continuity equation where the gauge non-invariant energy-momentum tensor of the electron in Dirac theory obtained from Noether's theorem using Lorentz transformation is given by Under rotation [without translation, ∆ µ = 0, see eq. (8)] by using eq. (25) in (21) in Dirac theory we find the continuity equation From eq. (28) the gauge non-invariant angular momentum tensor of the electron is given by where the gauge invariant S µνλ (x) is given by eq. (29) and the gauge non-invariant T µν (x) is given by eq. (27). By using eqs. (30) and (27) in (17) we find that the orbital angular momentum L e of the electron is given by which is not gauge invariant.
Note that the gauge invariant definition of the orbital angular momentum of the electron in Dirac theory can be obtained from the gauge invariant Noether's theorem, see eq. (85). D.

Gauge Invariant Spin Angular Momentum of Electron in Dirac Theory From
Noether's Theorem By using eqs. (30) and (29) in (17) we find that the spin angular momentum S inv e of the electron is given by which is gauge invariant.

III. NON-VANISHING BOUNDARY TERM AND GAUGE NON-INVARIANT SPIN AND ORBITAL ANGULAR MOMENTUM IN DIRAC-MAXWELL THEORY
The Lagrangian density in Dirac-Maxwell theory is given by and the Noether's theorem in Dirac-Maxwell theory is given by Under translation [without rotation, ǫ µν = 0, see eq. (8)] we find by using eqs. (7) and (25) in eq. (34) in the Dirac-Maxwell theory the equation Since in Dirac-Maxwell theory we find from eq. (36) the continuity equation where the gauge invariant energy-momentum tensor of the electromagnetic field plus electron in Dirac-Maxwell theory is given by Note that even if the energy-momentum tensor T µν (x) in eq. (10) Under rotation [without translation, ∆ µ = 0, see eq. (8)] we find by using eqs. (7) and (25) in eq. (34) in the Dirac-Maxwell theory the equation Using eq. (37) in (41) we find the continuity equation From eq. (42) the gauge invariant angular momentum tensor of the electron plus electromagnetic field in Dirac-Maxwell theory is given by where the gauge invariant S µνλ (x) is given by eq. (29) and the gauge invariant T µν (x) is given by eq. (39).
Using eqs. (43), (42), (39) and (29) in (17) we find that the total angular momentum of the electron plus electromagnetic field in Dirac-Maxwell theory is given by Note that using eq. (37) we find In Maxwell theory the electromagnetic potential A µ (x) produced at x µ by the electron in motion at X µ (τ ) with four-velocity u µ (τ ) = dX µ (τ ) dτ is given by [16,17] and the pure gauge potential A µ pure (x) is given by [16,17] Hence in Dirac-Maxwell theory, because of the form in eq. (47), we find the non-vanishing boundary term which implies that for angular momentum the boundary term does not vanish in the Dirac-Maxwell theory although the similar boundary term vanishes for linear momentum.
Hence from eqs. (49) and (46) we find see also [18]. Using eq. (50) in (44) we find where the gauge invariant total angular momentum J inv γ+e of the electron plus electromagnetic field in Dirac-Maxwell theory is given by eq. (44), the gauge non-invariant spin angular momentum S γ of the electromagnetic field in Maxwell theory is given by eq. Under the local gauge transformation the A µ (x) transforms as which leaves the F µν (x) in eq. (6) gauge invariant, i. e..
where ω(x) is any arbitrary function and the superscript symbol GT means gauge transformed.
Under Lorentz transformation the four-vector A µ (x) transforms as and the second rank tensor F µν (x) transforms as From eq. (55) we find that the F µν (x) has a well defined transformation of the tensor in Maxwell theory because the tensor F µν (x) is gauge invariant. However, this is not so for the vector A µ (x) because this vector is not gauge invariant in Maxwell theory. From eq. (55) we find that the general transformation of A µ (x) is a Lorentz transformation plus gauge transformation given by Using eq. (56) in the functional differential we find Hence eq. (58) gives the functional differential δA µ (x) when the combined Lorentz transformation plus local gauge transformation is used whereas eq. (7) gives the functional differential δA µ (x) when only the Lorentz transformation is used.

B. Derivation of Gauge Invariant Noether's Theorem in Maxwell Theory
Note that due to the presence of the additional parameter Λ in the functional differential δA µ (x) in eq. (58) it is possible to make the functional differential δA µ (x) gauge invariant in eq. (58). However, since the local gauge transformation is not used in eq. (7) we find that Λ = 0 in eq. (7) which implies that it is not possible to obtain gauge invariant functional differential δA µ (x) when the Lorentz transformation is used alone. Hence one finds that it is not possible to derive the gauge invariant Noether's theorem from first principle when Lorentz transformation is used alone but it is possible to derive the gauge invariant Noether's theorem from first principle when combined Lorentz transformation plus local gauge transformation is used in gauge theory.
In order to derive the gauge invariant Noether's theorem in Maxwell theory from first principle by using combined Lorentz transformation plus local gauge transformation we proceed as follows. From eq. (58) we find which by using eqs. (52) and (53) gives For gauge invariant we find from eqs. (60), (58) and (61) that which gives a simple solution which agrees with [19][20][21]. Using eq. (63) in (58) we find which is gauge invariant.
Using eq. (64) in (5) we find that the gauge invariant Noether's theorem in Maxwell theory obtained from the combined Lorentz transformation plus local gauge transformation is given by C.

Gauge Invariant Angular Momentum of Electromagnetic Field From Gauge Invariant Noether's Theorem in Maxwell Theory
Under space-time translation (no rotation, ǫ µν = 0) we find by using eqs. (8) and (6) in (65) the continuity equation where the gauge invariant and symmetric energy-momentum tensor of the electromagnetic field in Maxwell theory is given by Observe that we have not used the Belinfante tensor to make the energy-momentum tensor of the electromagnetic field in eq. (67) symmetric.
Hence we find that the gauge invariant and symmetric energy-momentum tensor of the electromagnetic field in eq. (67) in Maxwell theory is obtained from the first principle when the combined Lorentz transformation plus local gauge transformation is used to derive the Noether's theorem. The derivation of gauge invariant and symmetric energy-momentum tensor of electromagnetic field in eq. (67) by using combined Lorentz transformation plus local gauge transformation without using the Belinfante tensor implies that the Belinfante tensor is not required in Maxwell theory to make energy-momentum tensor of the electromagnetic field symmetric and gauge invariant.
Under rotation (no space-time translation, ∆ µ = 0) we find by using eqs. (8) and (6) in where T µν (x) is given by eq. (67) which is gauge invariant and symmetric. Since we find from eq. (68) the continuity equation where the gauge invariant third rank tensor J µνλ (x) is given by which gives the gauge invariant angular momentum tensor of the electromagnetic field in Maxwell theory where the gauge invariant and symmetric energy-momentum tensor T µν (x) of the electromagnetic field in Maxwell theory is given by eq. (67).
Hence we find that the gauge invariant angular momentum tensor of the electromagnetic  67) and (17) we find that the gauge invariant definition of the angular momentum J inv e of the electromagnetic field obtained from the gauge invariant Noether's theorem using combined Lorentz transformation plus local gauge transformation in Maxwell theory is given by where E(x) is the electric field and B(x) is the magnetic field.
where Λ is given by eq. (63). Using eqs. (75) and (63) Hence we find that eq. (76) gives the functional differential δψ(x) of the Dirac field ψ(x) of the electron when the combined Lorentz transformation plus local gauge transformation is used whereas eq. (25) gives the functional differential δψ(x) of the Dirac field ψ(x) of the electron when only the Lorentz transformation is used.

B. Derivation of Gauge Invariant Noether's Theorem in Dirac Theory
Note that due to the presence of the covariant derivative as given by eq. (45) in the functional differential δψ(x) in eq. (76) it is possible to derive the gauge invariant Noether's theorem in Dirac theory when combined Lorentz transformation plus local gauge transformation is used. However, since the local gauge transformation is not used in eq. (25) we find that the functional differential δψ(x) contains the ordinary derivative ∂ µ (instead of the covariant derivative D µ [A]) in eq. (25) which implies that it is not possible to obtain gauge invariant Noether's theorem when only the Lorentz transformation is used. Hence one finds that it is not possible to derive the gauge invariant Noether's theorem from first principle when Lorentz transformation is used alone but it is possible to derive the gauge invariant Noether's theorem from first principle when combined Lorentz transformation plus local gauge transformation is used.
In order to derive gauge invariant Noether's theorem in Dirac theory from first principle by using combined Lorentz transformation plus local gauge transformation we proceed as follows. Using eq. (76) in (21) we find that the gauge invariant Noether's theorem in Dirac theory is given by where the gauge invariant third rank tensor S µνλ (x) is given by eq. (29).

C. Gauge Invariant Orbital Angular Momentum of Electron From Gauge Invariant Noether's Theorem in Dirac Theory
Under translation (without rotation, ǫ µν = 0) we find by using eq. (8) in gauge invariant Noether's theorem in Dirac theory in eq. (77) the continuity equation where the gauge invariant energy-momentum tensor of the electron in Dirac theory is given by Hence we find that the gauge invariant energy-momentum tensor in eq. (79) of the electron in Dirac theory is obtained from the first principle when the combined Lorentz transformation plus local gauge transformation is used to derive the Noether's theorem.
Under rotation (without translation, ∆ µ = 0) we find by using eqs. (8) and (69) in the gauge invariant Noether's theorem in Dirac theory in eq. (77) the continuity equation where the gauge invariant third rank tensor J µνλ (x) is given by which gives the gauge invariant angular momentum tensor of the electron where the gauge invariant S µνλ (x) is given by eq. (29) and the gauge invariant T µν (x) is given by eq. (79). From eq. (82) we find Hence we find that the gauge invariant angular momentum tensor in eq. (82) of the electron in Dirac theory is obtained from the first principle when the combined Lorentz transformation plus local gauge transformation is used to derive the Noether's theorem.
where the covariant derivative D µ [A] is given by eq. (45).
where the lagrangian density L(x) in the Dirac-Maxwell theory is given by eq. (33).
Under translation [without rotation, ǫ µν = 0] using eq. (8) in (87) we find the continuity equation where the gauge invariant energy-momentum tensor of the electromagnetic field plus electron in Dirac-Maxwell theory obtained from the gauge invariant Noether's theorem using combined Lorentz transformation plus local gauge transformation is given by Under rotation [without translation, ∆ µ = 0] we find by using eqs. (8) in (87) the continuity equation where the gauge invariant third rank tensor J µνλ (x) is given by which gives the gauge invariant angular momentum tensor Hence from eqs. (94) and (95) we find that the gauge invariant energy-momentum tensor T µν (x) of the electromagnetic field plus electron in eq. (89) in Dirac-Maxwell theory is not required to be symmetric although it is possible to obtain a symmetric and gauge invariant energy-momentum tensor T µν S (x) in Dirac-Maxwell theory from the gauge invariant Noether's theorem without requiring Belinfante-Rosenfeld tensor.
In order to obtain a gauge invariant and symmetric energy-momentum tensor T µν S (x) in Dirac-Maxwell theory from the gauge invariant Noether's theorem without requiring Belinfante-Rosenfeld tensor we proceed as follows. By using the Dirac lagrangian density from eq. (20) in the Euler-Lagrange equation we find the Dirac equation From eqs. (89) and (96) we find by using the properties of the Dirac matrices the following equation Using the properties of Dirac matrices we find which implies that the tensorψ(x){γ λ , σ µν }ψ(x) is antisymmetric in λ ↔ µ, λ ↔ ν and µ ↔ ν which means We write eq. (88) as which by using eqs. (97), (99) and (89) gives where the symmetric and gauge invariant energy-momentum tensor T µν S (x) of the electromagnetic field plus electron in Dirac-Maxwell theory is given by which gives Hence we find that the gauge invariant and symmetric energy-momentum tensor in eq.  89) and (17) we find that the conserved and gauge invariant total angular momentum of the electromagnetic field plus electron obtained from the gauge invariant Noether's theorem in Dirac-Maxwell theory using combined Lorentz transformation plus local gauge transformation is given by which agrees with eq. (44).
From eqs. (104), (73), (86) and (85) we find which gives from eq. (51) where J inv γ+e is the gauge invariant conserved total angular momentum of the electromagnetic field plus electron, J inv γ is the gauge invariant total angular momentum of the electromag- Hence we find that since the difference between the angular momentum current built from the gauge invariant energy momentum tensor and that built in the naive Noether procedure is a total derivative, the space integral total angular momentum resulting from the two approaches differ by some surface term which may be finite for field configurations which do not decay fast enough at infinity. Note that as mentioned earlier, as can be seen from eq. (106), the gauge invariant definition of the spin angular momentum of the electromagnetic field in the literature [2,[12][13][14][15] is not correct because of the non-vanishing surface term [see eq. (49)] in Dirac-Maxwell theory although the corresponding surface term vanishes for linear momentum.

IX. CONCLUSIONS
Due to proton spin crisis it is necessary to understand the gauge invariant definition of the spin and orbital angular momentum of the quark and gluon from first principle. In this paper we have derived the gauge invariant Noether's theorem by using combined Lorentz transformation plus local gauge transformation. We have found that the notion of the gauge invariant definition of the spin (or orbital) angular momentum of the electromagnetic field does not exist in Dirac-Maxwell theory although the notion of the gauge invariant definition of the spin (or orbital) angular momentum of the electron exists. We have found that the gauge invariant definition of the spin angular momentum of the electromagnetic field in the literature [2,[12][13][14][15] is not correct because of the non-vanishing surface term [see eq.
(49)] in Dirac-Maxwell theory although the corresponding surface term vanishes for linear momentum. We have also shown that the Belinfante-Rosenfeld tensor is not required to obtain symmetric and gauge invariant energy-momentum tensor of the electron and the electromagnetic field in Dirac-Maxwell theory.
Note that the main idea of this paper is based on the expression (7) of the variation of an abelian gauge field under an infinitesimal space time transformation δA µ (x) = −F νµ (x) δx ν − ∂ µ [A ν (x) δx ν ]. The second term is identified as a gauge variation so it can be dropped when applying the standard Noether procedure to the symmetry combined of the original space time transformation and this gauge transformation. This way one gets gauge invariant conserved currents like energy momentum tensor and angular momentum current. Similarly the same expression of δA is true for the non-abelian gauge field with the derivative replaced by covariant derivative, so the same idea works also for the non-abelian case [22]. In particular, one finds for the non-abelian case where the nonabelian field tensor is given by A a ν (x) and the covariant derivative is given by D bd µ = δ bd ∂ µ + gf bad A a µ (x) with a, b, d = 1, ..., 8 being the color indices [22].
Hence we conclude that although the high energy collider experiments have measured the spin dependent gluon distribution function inside proton but we do not have a gauge invariant definition of the spin dependent gluon distribution function in QCD consistent