Completing Schwinger's 1962 radiation-gauge construction of non-Abelian gauge fields: explicit evaluation of the short-distance correction functions and proof of the trace rearrangement theorem

Schwinger's 1962 paper 'Non-Abelian Gauge Fields. Relativistic Invariance' constructed a radiation-gauge quantization of a non-Abelian vector gauge field coupled to a spin-1/2 fermion multiplet and verified its Lorentz invariance through an equal-time commutator condition on the energy density, $-i[T^{00}(x),T^{00}(x')]=-(T^{0k}(x)+T^{0k}(x'))\partial_k\delta(\mathbf{x}-\mathbf{x}')$. The paper explicitly left four items undone: (i) term-by-term evaluation of the short-distance correction functions $r_\phi$ and related double commutators entering the transformation laws and equations of motion; (ii) proof of a trace rearrangement theorem underlying the positive rearrangement by which the energy density is determined; (iii) a general dynamical derivation of the commutator condition from coupling to an external gravitational field with minimal time locality; (iv) a derivation of the null charge-density equal-time commutator from current conservation for currents not explicitly dependent on the time derivative of an external vector potential. Items (iii) and (iv) were directly carried out in Schwinger's own 1963 sequels (references 2 and 3), and the overarching consistency of canonical non-Abelian quantization was established by the later canonical literature (Christ and Lee 1980 and successors). The precise surviving open core is the explicit evaluation of the $r_\phi$ functions and double commutators and the proof of the trace theorem within the original formalism.

Background

In the early 1960s the canonical quantization of gauge fields was an open structural problem. Because the Lagrange function of a gauge theory contains no time derivative of some field components (in electrodynamics, the scalar potential), the simplest equal-time canonical scheme cannot treat all components as fundamental dynamical variables. Two strategies competed. Lorentz-gauge methods modify the system so that all components become fundamental, at the price of an indefinite metric or an enlarged state space. The radiation-gauge method keeps only the transverse components of the vector potential as fundamental variables and treats the Gauss-law constraint (the equation that determines the longitudinal component in terms of the charge density) as a condition on states. Here 'transverse' and 'longitudinal' refer to the decomposition of a vector field on a fixed time slice into divergence-free and gradient parts. Because the radiation gauge is tied to a specific time slicing, Lorentz invariance is not manifest and must be verified by explicit calculation.

Schwinger's 1962 paper supplied such a verification for a non-Abelian vector gauge field (a Yang-Mills field: a vector field taking values in the Lie algebra of an internal symmetry group, with self-interactions fixed by that algebra) coupled to a spin-1/2 fermion multiplet. It is the sequel to his 'Non-Abelian Gauge Fields. Commutation Relations' (Phys. Rev. 125, 1043 (1962)), which fixes the formalism's conventions. The instrument is a sufficient criterion for invariance under the proper orthochronous Lorentz group, now often called the Schwinger condition: the Hermitian energy density $T^{00}$ and momentum density $T^{0k}$, which generate time translations, space translations, and rotations, must satisfy the equal-time commutator condition

$-i[T^{00}(x),T^{00}(x')]=-(T^{0k}(x)+T^{0k}(x'))\,\partial_k\delta(\mathbf{x}-\mathbf{x}'),$

with $T^{00}$ a three-dimensional scalar function of the fields at the given time, with no explicit coordinate dependence. Together with the three-dimensional transformation laws this closes the commutator algebra of the ten infinitesimal generators of the Poincaré group. The condition is the stress-tensor analog of the null charge-density commutator $[j^0(x),j^0(x')]=0$ familiar from quantum electrodynamics.

The system studied consists of a transverse Hermitian Bose vector field (the spatial gauge field) and a spin-1/2 Fermi field carrying an internal multiplicity, the internal symmetry being realized by a set of imaginary antisymmetrical matrices $t_a$ (the gauge-group generators acting in the internal space, obeying the group commutation law; sums over $a$ run over the generators). The constraint that replaces the Gauss law involves an elliptic operator of the form 'minus the Laplacian, plus first-order terms with the fermion field as coefficient', and its matrix Green's function $\mathfrak{D}_\phi(\mathbf{x},\mathbf{x}')$ (the inverse of that operator, solving the constraint) appears throughout the construction. A Green's function is the integral kernel of an inverse differential operator; its coincident-argument values $\mathfrak{D}_\phi(\mathbf{x},\mathbf{x})$ and spatial derivatives thereof are short-distance-singular objects whose products produce contact terms (distribution-valued contributions supported only at coincident points, built from delta functions and their derivatives).

The paper proposes candidate densities $T^{0k}$ and $T^{00}=T^{00}_{\rm fermion}+T^{00}_{\rm gauge}+t_\phi(x)$, where the scalar function $t_\phi$ is initially undetermined: 'Its determination is the essential contribution of this paper.' The determination proceeds by requiring the combination of $t_\phi$ with a certain quadratic, fermion-covariant combination $G^{\cdot\cdot}(x)$ of the gauge field to form a set of mutually commuting, positive Hermitian operators, written as a positive quadratic form (the 'positive rearrangement'). The rearrangement is equivalent to an algebraic identity whose verification 'will require the following theorem', stated without proof — a trace relation over the internal indices,

$\sum_a\operatorname{Tr}\bigl[t_a\,\partial^{k}\mathfrak{D}_{\phi}(\mathbf{x},\mathbf{x})\,t_a\,(\partial^{k}\mathfrak{D}_{\phi}(\mathbf{x},\mathbf{x})-\mathfrak{D}_{\phi}(\mathbf{x},\mathbf{x})\partial^{k})\bigr]=\sum_a\bigl[\operatorname{Tr}\,t_a\,\partial^{k}\mathfrak{D}_{\phi}(\mathbf{x},\mathbf{x})\bigr]^{2},$

where the derivatives act on the Green's function as in the source. Second, the Lorentz transformation law and the equation of motion of the time component of the gauge field involve short-distance correction functions: a two-point function $r_\phi{}^k(\mathbf{x},\mathbf{x}')$ built from the fermion field and the constraint Green's function, its spatial integral $r_\phi{}^k(\mathbf{x})$, and its first moment $r_\phi{}^k{}_l(\mathbf{x})$, the latter two being local functions of the fermion field. The paper records structural forms for these, gives one 'fairly explicit' evaluation, and explicitly declines to record the evaluation of certain double commutators (nested commutators weighted by products of generators), asserting only that they yield functions of the fermion field. Finally, a Note added in proof outlines two further general derivations — the gravitational basis of the commutator condition, and the null charge-density commutator for currents without explicit dependence on the time derivative of an external vector potential — which were carried out in Schwinger's 1963 sequels and therefore do not form part of the surviving open core.

Problem Statement

Within the operator formalism of Schwinger, Physical Review 127, 324 (1962) (DOI: 10.1103/PhysRev.127.324, https://doi.org/10.1103/PhysRev.127.324), with conventions fixed by that paper and its predecessor 'Non-Abelian Gauge Fields. Commutation Relations' (Physical Review 125, 1043 (1962); DOI: 10.1103/PhysRev.125.1043, https://doi.org/10.1103/PhysRev.125.1043), complete the technical program that the paper explicitly left undone, restricted to the items not already discharged in the literature. The common objective is to close, in the paper's own terms, the verification that the radiation-gauge non-Abelian system is a consistent Lorentz-invariant quantum field theory: the proof of the trace theorem completes the determination of the energy density through the positive rearrangement, and the explicit $r_\phi$ evaluations complete the verification that the constraint and the transformation laws close. The requested results are:

(1) Prove the trace rearrangement theorem stated without proof in the source,

$\sum_a\operatorname{Tr}\bigl[t_a\,\partial^{k}\mathfrak{D}_{\phi}(\mathbf{x},\mathbf{x})\,t_a\,(\partial^{k}\mathfrak{D}_{\phi}(\mathbf{x},\mathbf{x})-\mathfrak{D}_{\phi}(\mathbf{x},\mathbf{x})\partial^{k})\bigr]=\sum_a\bigl[\operatorname{Tr}\,t_a\,\partial^{k}\mathfrak{D}_{\phi}(\mathbf{x},\mathbf{x})\bigr]^{2},$

where $t_a$ are the internal generator matrices, $\mathfrak{D}_\phi(\mathbf{x},\mathbf{x})$ is the coincident-argument value of the constraint Green's function, and the derivatives act as in the source; or, if the printed statement contains a misprint, identify the corrected form and prove it, showing that it supports the identity underlying the positive rearrangement by which the scalar function $t_\phi$ in the energy density is determined — thereby completing the paper's verification that its energy density satisfies $-i[T^{00}(x),T^{00}(x')]=-(T^{0k}(x)+T^{0k}(x'))\partial_k\delta(\mathbf{x}-\mathbf{x}')$ at equal times.

(2) Evaluate explicitly, in fully reduced closed form, the short-distance correction functions entering the Lorentz transformation law and the equation of motion of the time component of the gauge field — the two-point function $r_\phi{}^k(\mathbf{x},\mathbf{x}')$, its spatial integral $r_\phi{}^k(\mathbf{x})$, and its first moment $r_\phi{}^k{}_l(\mathbf{x})$ — together with the related double commutators (the generator-weighted nested commutators of the gauge field and of the momentum density with the fermion field and the constraint Green's function, which appear in the source's analysis of the two-point function) whose evaluation the source declines to record. The evaluation must confirm the property the source records: that the double commutators yield functions of the fermion field.

The two further items acknowledged in the source — the general dynamical derivation of the energy-density commutator condition from coupling to an external gravitational field (with minimal time locality), and the analogous derivation of the null charge-density equal-time commutator from current conservation for systems whose current is not an explicit function of the time derivative of an external vector potential — were carried out in Schwinger's 1963 sequels (Physical Review 130, 406 and 130, 800 (1963)) and do not constitute surviving open content; an answer need not reproduce them. The completion boundary for this problem is a self-contained mathematical completion of items (1) and (2) within the source formalism, with the double commutators reduced to explicit functions of the fermion field and the $r_\phi$ functions reduced to explicit closed form in the source's Green's-function formalism.

References

1. J. Schwinger, "Non-Abelian Gauge Fields. Relativistic Invariance", Physical Review 127, 324 (1962). DOI: 10.1103/PhysRev.127.324, https://doi.org/10.1103/PhysRev.127.324 (primary source).
2. J. Schwinger, "Commutation Relations and Conservation Laws", Physical Review 130, 406 (1963). DOI: 10.1103/PhysRev.130.406, https://doi.org/10.1103/PhysRev.130.406.
3. J. Schwinger, "Energy and Momentum Density in Field Theory", Physical Review 130, 800 (1963). DOI: 10.1103/PhysRev.130.800, https://doi.org/10.1103/PhysRev.130.800.
4. D. G. Boulware and S. Deser, "Stress-Tensor Commutators and Schwinger Terms", Journal of Mathematical Physics 8, 1468 (1967). DOI: 10.1063/1.1705368, https://doi.org/10.1063/1.1705368.
5. S. Deser and L. K. Morrison, "Stress-Tensor Commutators in Nonlinear Electrodynamics", Journal of Mathematical Physics 11, 596 (1970). DOI: 10.1063/1.1665174, https://doi.org/10.1063/1.1665174.
6. N. H. Christ and T. D. Lee, "Operator ordering and Feynman rules in gauge theories", Physical Review D 22, 939 (1980). DOI: 10.1103/PhysRevD.22.939, https://doi.org/10.1103/PhysRevD.22.939.
7. S. Weinberg, "Derivation of gauge invariance and the equivalence principle from Lorentz invariance of the S-matrix", Physics Letters 9, 357 (1964). DOI: 10.1016/0031-9163(64)90396-8, https://doi.org/10.1016/0031-9163(64)90396-8.
8. K.-W. Huang, "Stress-tensor commutators in conformal field theories near the lightcone", Physical Review D 100, 061701 (2019). DOI: 10.1103/PhysRevD.100.061701, arXiv:1907.00599, https://arxiv.org/abs/1907.00599 (the arXiv preprint version is titled "Stress-Tensor Commutator in CFT near the Lightcone").
9. M. Besken, J. de Boer and G. Mathys, "On local and integrated stress-tensor commutators", Journal of High Energy Physics 2021, 148 (2021). DOI: 10.1007/JHEP07(2021)148, arXiv:2012.15724, https://arxiv.org/abs/2012.15724.
10. J. Schwinger, "Non-Abelian Gauge Fields. Commutation Relations", Physical Review 125, 1043 (1962). DOI: 10.1103/PhysRev.125.1043, https://doi.org/10.1103/PhysRev.125.1043 (predecessor paper fixing the formalism's conventions).