Equal-time commutation relations of the current and stress tensor in interacting quantum field theory: completing Schwinger's 1963 program (Schwinger terms and the no-time-derivative assumptions)

Julian Schwinger's 1963 paper 'Commutation Relations and Conservation Laws' derived, from the quantum action principle and the conservation laws that follow from gauge and general-coordinate invariance, the operator part of the equal-time commutation relations of the charge density ($[j^0(x),j^0(x')]=0$, plus a reciprocity form for $[j^0,j^i]$) and of the energy density ($-i[T^{00}(x),T^{00}(x')] = -(T^{0k}(x)+T^{0k}(x'))\partial_k\delta(\mathbf{x}-\mathbf{x}')$), under two assumptions: the current contains no time derivative of the external electromagnetic potential, and the spatial stress components $T^{kl}$ contain no time derivative of $g_{00}$. The surviving open problem is a complete determination of these commutators for general interacting quantum field theories, including all c-number and operator-valued Schwinger terms (their admissible derivative structure, ultraviolet-convergence status, and whether they are always c-numbers — established only in two-dimensional conformal field theory and assumed in higher-dimensional lightcone analyses), together with a precise characterization of the class of theories obeying Schwinger's assumptions. Later work established the necessity of Schwinger terms and produced partial classifications and model computations, but no general determination is known.

Background

In relativistic quantum field theory (QFT), the electric current density $j^\mu(x)$ and the stress-energy-momentum tensor $T^{\mu\nu}(x)$ are the conserved local densities associated with gauge invariance and with translations and Lorentz transformations. Questions about the simultaneous measurability and the causal response of these densities are encoded in their equal-time commutation relations — the commutators $[A(t,\mathbf{x}),B(t,\mathbf{x}')]$ of two local operators evaluated at a common time, which are spatial distributions built from the Dirac delta function $\delta(\mathbf{x}-\mathbf{x}')$ and its derivatives. Canonical quantization fixes such commutators only for the fundamental fields; the conserved densities are composite operators of interacting systems, and their commutators are not fixed by canonical rules alone.

In 'Commutation Relations and Conservation Laws' (1963), Julian Schwinger supplied a general route to these commutators. The idea is that $j^\mu$ and $T^{\mu\nu}$ are not merely Noether densities but the sources of external electromagnetic and gravitational fields: coupling the system to an external gauge potential $A_\mu(x)$ or metric $g_{\mu\nu}(x)$, the variation of the action operator defines the current and the stress tensor, and gauge invariance and general coordinate invariance force the local conservation laws $\partial_\mu j^\mu=0$ and $\nabla_\mu T^{\mu\nu}=0$. Schwinger then invoked the quantum action principle — his statement that the variation of a transformation function under an infinitesimal change of the action operator equals the matrix element of that variation, with time ordering producing equal-time commutator terms — to convert the conservation laws into commutators. Under two assumptions, namely that the current does not contain the time derivative of the external potential, and that the spatial stress components $T^{kl}$ do not contain the time derivative of $g_{00}$ (at most its second spatial derivatives), he obtained $[j^0(x),j^0(x')]=0$ together with a reciprocity form for $[j^0,j^i]$ (a relation, obtained from the second variation of the action, that fixes this commutator in terms of the current's local linear response to the external potential), and the energy-density commutator

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

derived for the classes of physical systems defined by these assumptions and characterized as perhaps 'the most fundamental equation of relativistic quantum field theory'. The conjecture that the class 'may well be without exception' attaches to the first assumption: in Schwinger's words, the no-time-derivative restriction on the current 'defines a certain class of electric charge-bearing physical systems (which may well be without exception)'.

The 1960s current-algebra program — the attempt to formulate hadron dynamics entirely in terms of the algebra of current commutators — took such commutators as fundamental input. It quickly became clear that commutators of densities can acquire additional contact terms, the Schwinger terms (after Schwinger's 1959 discovery of an extra derivative-of-delta term in the vacuum expectation value of $[j^0(x),j^k(x')]$ in electrodynamics): c-number (purely numerical, commuting with every operator) or operator-valued distributions proportional to higher derivatives of $\delta(\mathbf{x}-\mathbf{x}')$ that are invisible to the canonical algebra. Boulware and Deser (1967) showed, from the spectral representation (the representation of a vacuum two-point function as an integral over intermediate energies) of the vacuum stress commutator, that such singular terms are in fact required, and that they correspond to a 'nonclassical' dependence of $T^{\mu\nu}$ on the external metric — precisely the kind of dependence excluded by Schwinger's assumptions. In two-dimensional conformal field theory (CFT, a QFT invariant under local scale transformations), the corresponding c-number extension is the central term of the Virasoro algebra, the infinite-dimensional algebra of generators constructed from the stress tensor, whose coefficient (the central charge) characterizes the theory. Whether analogous universal structures exist in higher dimensions, and what the complete local commutators are in a general interacting theory, remains only partially understood: modern analyses via the operator product expansion (OPE, the expansion of a product of local operators at nearby points as a sum of local operators with singular coefficient functions) and lightcone limits still must assume, or leave open, the character of the Schwinger terms.

Problem Statement

Complete the general basis for the equal-time commutation relations of conserved densities in interacting relativistic quantum field theory that Schwinger's 1963 paper initiated. Concretely, determine the following two jointly required parts, where a resolution of each is a general, theory-independent statement about the full class of interacting relativistic quantum field theories (not a designated subclass).

(1) The complete equal-time commutators among the components of the electric current $j^\mu$ and of the stress tensor $T^{\mu\nu}$ — where $T^{\mu\nu}$ is the symmetric stress tensor defined by the variation of the action with respect to the external metric $g_{\mu\nu}$, as in Schwinger's derivation — in a general interacting quantum field theory, including, beyond the operator terms Schwinger derived ($[j^0(x),j^0(x')]=0$ and $-i[T^{00}(x),T^{00}(x')] = -(T^{0k}(x)+T^{0k}(x'))\partial_k\delta(\mathbf{x}-\mathbf{x}')$), the full Schwinger terms: their admissible derivative structure (which derivatives of the three-dimensional delta function occur), whether they are c-numbers or operator-valued, and their status as ultraviolet-convergent or divergent distributions. In particular, settle whether the Schwinger terms are always c-numbers — established in two-dimensional conformal field theory, where the term is proportional to the central charge, and assumed in higher-dimensional lightcone analyses, but open for general interacting theories in $d>2$.

(2) The precise class of interacting theories in which Schwinger's two assumptions hold — that the current contains no time derivative of the external electromagnetic potential, and that $T^{kl}$ contains no explicit time derivative of $g_{00}$ (at most second spatial derivatives) — and whether and how the commutator derivations extend when the assumptions fail. The conjecture 'may well be without exception' attaches to the class of charge-bearing systems defined by the current's no-time-derivative restriction; later work indicates the required metric dependence is 'nonclassical', but no complete characterization of either class, or of the theories in which the assumptions fail, exists.

References

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