Spacetime Currents, Stress Tensors, and Charge Algebras

Translations and Lorentz transformations are spacetime symmetries, so their Noether currents carry spacetime indices as well as a current index. The translation current is the stress tensor. Its hypersurface integral gives energy–momentum, while the Lorentz current combines orbital and intrinsic-spin contributions and integrates to angular momentum and boosts.

The local formulas are not unique. A canonical stress tensor can be nonsymmetric or inconvenient for gauge theories; an improvement can produce a symmetric representative without changing the Poincaré charges when its surface term vanishes. The charge algebra follows only after those charges exist as operators, share a suitable domain, and have no unaccounted boundary flux. This page develops that structure in flat Minkowski spacetime and keeps those qualifications explicit.

Quantum Currents, Improvements, and Conservation develops the local-operator ambiguities behind the representative dependence.

Required background. Continuous Symmetries, Generators, and Charges supplies hypersurface charges and their commutator action. Classical Symmetries, Currents, and Stress Tensors supplies the translation and Lorentz Noether constructions.

Helpful background. Hamiltonian Group Actions and Moment Maps supplies the moment-map perspective on generators and charge algebras. Lorentz Field Representations and Poincaré Particle Representations supplies the finite-dimensional spin term in Lorentz transformations.

Translations and the canonical stress tensor

Use self-adjoint translation generators with

$$U(a) =e^{-ia^\mu P_\mu}.$$

For the active action on a scalar local operator,

$$U(a)\mathcal O(x)U(a)^{-1} =\mathcal O(x-a),$$

so the infinitesimal translation rule is

$$[P_\mu,\mathcal O(x)] =-i\partial_\mu\mathcal O(x).$$

Let a flat-space Lagrangian density $\mathcal L(\Phi^I,\partial\Phi^I)$ have no explicit coordinate dependence. For bosonic fields, with graded left/right derivatives understood when fermions are present, a localized translation

$$\delta\Phi^I(x) =-a^\nu(x)\partial_\nu\Phi^I(x),$$

the coefficient of $\partial_\mu a^\nu$ gives the canonical translation current

$$T_{\mathrm{can}}{}^\mu{}_\nu =\sum_I \frac{\partial\mathcal L}{\partial(\partial_\mu\Phi^I)} \partial_\nu\Phi^I -\delta^\mu{}_\nu\mathcal L.$$

On the classical equations of motion,

$$\partial_\mu T_{\mathrm{can}}{}^\mu{}_\nu=0.$$

The associated candidate four-momentum on a future-oriented Cauchy surface $\Sigma$ is

$$P_\nu[\Sigma] =\int_\Sigma \mathrm d\Sigma_\mu\, T^\mu{}_\nu.$$

On an equal-time surface this becomes

$$P_\nu =\int \mathrm d^{d-1}x\,T^0{}_\nu,$$

with $P_0=H$. This Noether derivation and the identification of the integrated translation current with energy–momentum are given in Schwartz 2014, § 3.3.1, pp. 34–36.

The subscript “canonical” describes the representative supplied directly by this first-derivative calculation. It does not mean that the tensor is automatically symmetric, gauge invariant, or already a well-defined renormalized composite operator. Those are separate properties.

Lorentz transformations: orbital and spin currents

For the Lorentz sector, fix

$$U(\omega) =\exp\!\left( -\frac{i}{2}\omega_{\mu\nu}M^{\mu\nu} \right).$$

For the active scalar action $U(\Lambda)\mathcal O(x)U(\Lambda)^{-1} =\mathcal O(\Lambda^{-1}x)$, the corresponding rule is

$$[M^{\mu\nu},\mathcal O(x)] =+i\left(x^\mu\partial^\nu-x^\nu\partial^\mu\right) \mathcal O(x).$$

Fields with spin have an additional finite-dimensional Lorentz-representation term.

An infinitesimal Lorentz transformation contains an orbital change of the argument and, for a nonscalar field, an intrinsic transformation of its components. Write the intrinsic contribution to its Noether current as $S^{\lambda\mu\nu}$, antisymmetric in $\mu,\nu$. The full Lorentz current is

$$\mathcal M_{\mathrm{can}}^{\lambda\mu\nu} =x^\mu T_{\mathrm{can}}^{\lambda\nu} -x^\nu T_{\mathrm{can}}^{\lambda\mu} +S^{\lambda\mu\nu}.$$

Taking its divergence gives

$$\partial_\lambda\mathcal M_{\mathrm{can}}^{\lambda\mu\nu} =T_{\mathrm{can}}^{\mu\nu} -T_{\mathrm{can}}^{\nu\mu} +\partial_\lambda S^{\lambda\mu\nu}.$$

Lorentz invariance therefore implies the balance equation

$$T_{\mathrm{can}}^{\mu\nu} -T_{\mathrm{can}}^{\nu\mu} =-\partial_\lambda S^{\lambda\mu\nu},$$

and hence $\partial_\lambda\mathcal M_{\mathrm{can}}^{\lambda\mu\nu}=0$ on shell. The charge is

$$M^{\mu\nu}[\Sigma] =\int_\Sigma \mathrm d\Sigma_\lambda\, \mathcal M_{\mathrm{can}}^{\lambda\mu\nu}.$$

Spatial components generate rotations; mixed time–space components generate boosts. The orbital and spin pieces need not be separately conserved. Only their sum has the general Noether meaning. Weinberg develops the quantum translation and Lorentz currents, their integrated generators, and their action on fields in Weinberg 1995, Vol. I, §§ 7.3–7.4, pp. 310–317.

Belinfante improvement

The spin balance equation allows the canonical tensor to be improved. Define

$$\begin{aligned} \Theta^{\mu\nu} &=T_{\mathrm{can}}^{\mu\nu} +\partial_\lambda X^{\lambda\mu\nu}, \\ X^{\lambda\mu\nu} &=\frac12\left( S^{\lambda\mu\nu} +S^{\mu\nu\lambda} +S^{\nu\mu\lambda} \right). \end{aligned}$$

Antisymmetry of $S$ in its last two indices implies $X^{\lambda\mu\nu}=-X^{\mu\lambda\nu}$. Consequently, the added double divergence vanishes identically,

$$\partial_\mu\partial_\lambda X^{\lambda\mu\nu}=0,$$

so $\Theta^{\mu\nu}$ is conserved whenever the canonical tensor is. The spin balance equation also makes $\Theta^{\mu\nu}$ symmetric. The improved Lorentz current can then be written in the purely orbital form

$$\mathcal M_{\mathrm B}^{\lambda\mu\nu} =x^\mu\Theta^{\lambda\nu} -x^\nu\Theta^{\lambda\mu}.$$

More precisely, the canonical-spin and improved currents differ by a divergence:

$$\begin{aligned} \mathcal M_{\mathrm B}^{\lambda\mu\nu} -\mathcal M_{\mathrm{can}}^{\lambda\mu\nu} =\partial_\rho\big(& x^\mu X^{\rho\lambda\nu} \\ &-x^\nu X^{\rho\lambda\mu} \big). \end{aligned}$$

This follows from $X^{\mu\lambda\nu}-X^{\nu\lambda\mu} =-S^{\lambda\mu\nu}$ and makes the required angular-momentum surface condition explicit.

On an equal-time surface, the improvement changes the momentum by a spatial boundary term:

$$\Delta P^\nu =\lim_{R\to\infty} \int_{S_R}\mathrm dS_i\,X^{i0\nu}.$$

Thus the canonical and Belinfante tensors give the same integrated four-momentum only when this term vanishes. Weinberg constructs the Belinfante tensor and explicitly proves this four-momentum equality under the surface hypothesis in Weinberg 1995, Vol. I, § 7.4, pp. 315–317. Applying the preceding divergence identity to $M^{\mu\nu}$ shows that equality of the angular-momentum charges requires the corresponding weighted surface terms to vanish as well.

This is a local equivalence with a boundary hypothesis, not permission to discard surface terms. With a physical boundary, defect, long-range field, or slow falloff, an improvement can move charge between bulk and boundary descriptions. The relevant balance laws are developed on Boundaries, Flux, and Boundary Ward Identities.

There are other useful stress-tensor representatives. In particular, variation with respect to a background metric produces a symmetric metric stress tensor under appropriate assumptions. Its curved-spacetime definition, renormalization, and curvature ambiguities are deferred to Renormalized Stress Tensor: Axioms and Curvature Ambiguities. Conformal improvements and trace conditions belong to Conserved Currents and the Stress Tensor.

Surface independence and the Poincaré algebra

Conservation of $T^{\mu\nu}$ and $\mathcal M^{\lambda\mu\nu}$ is local. Independence of $P^\mu[\Sigma]$ and $M^{\mu\nu}[\Sigma]$ from the Cauchy surface additionally requires vanishing flux through the timelike or asymptotic boundary between two surfaces. At the quantum level the currents are operator-valued distributions, so one must also smear them, take the large-region limit, and establish self-adjoint generators on a common invariant domain.

When those conditions hold and the theory implements the connected Poincaré group without an extension, the charges satisfy

$$[P^\mu,P^\nu]=0,$$ $$[M^{\mu\nu},P^\rho] =i\left( \eta^{\nu\rho}P^\mu -\eta^{\mu\rho}P^\nu \right),$$

and

$$\begin{aligned} [M^{\mu\nu},M^{\rho\sigma}] =i\big(& \eta^{\mu\sigma}M^{\nu\rho} +\eta^{\nu\rho}M^{\mu\sigma} \\ &-\eta^{\mu\rho}M^{\nu\sigma} -\eta^{\nu\sigma}M^{\mu\rho} \big). \end{aligned}$$

For example, with $J^3=M^{12}$ these conventions give $[J^3,P^1]=iP^2$, while two boosts obey $[M^{01},M^{02}]=-iM^{12}$. These component checks catch the most common metric- and exponential-sign mistakes.

The algebra is not a consequence of $\partial_\mu T^{\mu\nu}=0$ alone. Equal-time current commutators can contain local Schwinger terms; integrations by parts can leave boundary operators; and unbounded-operator commutators must be evaluated on a shared domain. Such terms may cancel in the global generators, modify a boundary algebra, or signal a projective implementation. Contact Terms, Equal-Time Commutators, and Schwinger Terms treats the local distributions, while Quantum Implementations, Projective Actions, and Central Extensions treats possible extensions. One should establish an extension rather than append a central term merely because the local current algebra is singular.

Threaded scalar example: internal breaking without spacetime breaking

Return to the complex scalar used on the preceding pages,

$$\begin{aligned} \mathcal L ={}&\partial_\mu\phi^\dagger\partial^\mu\phi -m^2\phi^\dagger\phi \\ &-\lambda(\phi^\dagger\phi)^2 +h\phi^N+h^*(\phi^\dagger)^N, \\ &\hspace{7em}N\geq2. \end{aligned}$$

For constant scalar couplings $m^2$, $\lambda$, and $h$, a convenient stress tensor is

$$T^{\mu\nu} =\partial^\mu\phi^\dagger\partial^\nu\phi +\partial^\nu\phi^\dagger\partial^\mu\phi -\eta^{\mu\nu}\mathcal L.$$

It is symmetric and conserved on the classical equations of motion. Because a scalar has no intrinsic spin current, its Lorentz current is simply

$$\mathcal M^{\lambda\mu\nu} =x^\mu T^{\lambda\nu} -x^\nu T^{\lambda\mu}.$$

The deformation proportional to $h$ breaks the continuous internal $U(1)$ to a residual subgroup, but it does not break translations or Lorentz invariance when $h$ is constant. This cleanly separates two statements: the internal charge $Q$ need not be conserved, while $P^\mu$ and $M^{\mu\nu}$ still can be.

Now promote $h$ only as a diagnostic external source, $h\to h(x)$. The action has explicit coordinate dependence, and the classical on-shell identity becomes

$$\partial_\mu T^\mu{}_\nu =-(\partial_\nu h)\phi^N -(\partial_\nu h^*)(\phi^\dagger)^N.$$

The source can exchange energy–momentum with the scalar system, so the matter momentum alone is not conserved. A generic fixed profile also breaks any Lorentz generator for which $(x^\mu\partial^\nu-x^\nu\partial^\mu)h\neq0$. Letting $h$ transform as a scalar background makes the family of source-dependent theories covariant; it does not make a generic fixed profile Poincaré invariant.

In the quantum theory this statement must be formulated with renormalized composite insertions and the contact terms generated by varying time-ordered products. That functional identity is developed on Localized Transformations and Ward–Takahashi Identities and Current Sources and Generating Functionals.

When $h=0$, the internal $U(1)$ is independent of spacetime symmetry and its charge commutes with the Poincaré generators on their common domain. For fixed nonzero $h$, that continuous charge is absent, but the exact residual $\mathbb Z_N$ can still coexist with Poincaré symmetry. Neither conclusion follows from the stress tensor alone; it uses the symmetry action identified on the earlier pages.

Common pitfalls

Conserved does not yet mean generator. A continuity equation is a local operator statement. A generator additionally requires a finite surface-independent integral, suitable boundary conditions, and a controlled operator domain.

Symmetric does not mean unique. Belinfante improvement produces a useful symmetric tensor, but further conserved improvements may remain. Conversely, a canonical tensor’s lack of symmetry is not itself a failure of Lorentz invariance; the spin current supplies the missing balance.

Internal and spacetime breaking are independent. A constant charged-field interaction can break an internal symmetry while preserving Poincaré symmetry. A spacetime-dependent coupling generally breaks translations even if it is organized as a covariant source.

A surface term can carry physics. Canonical and improved charges agree only under the stated falloff. Boundaries and long-range configurations must be analyzed before the term is dropped.

Check your understanding

For the scalar theory with $h=h(x)$, use the equations of motion to derive the displayed divergence of $T^\mu{}_\nu$. Then set $h$ constant and verify directly that $\partial_\lambda\mathcal M^{\lambda\mu\nu}=0$.

Check

For a Lagrangian with explicit coordinate dependence, the canonical identity on shell is

$$\partial_\mu T^\mu{}_\nu =-\left.\frac{\partial\mathcal L}{\partial x^\nu}\right|_{\Phi,\partial\Phi}.$$

Only the source is explicitly coordinate dependent, so

$$\left.\frac{\partial\mathcal L}{\partial x^\nu}\right|_{\Phi,\partial\Phi} =(\partial_\nu h)\phi^N +(\partial_\nu h^*)(\phi^\dagger)^N,$$

which gives the stated result. For constant $h$, $\partial_\mu T^{\mu\nu}=0$. Since the scalar stress tensor is symmetric,

$$\begin{aligned} \partial_\lambda\mathcal M^{\lambda\mu\nu} &=T^{\mu\nu}-T^{\nu\mu} \\ &\quad+x^\mu\partial_\lambda T^{\lambda\nu} -x^\nu\partial_\lambda T^{\lambda\mu} =0. \end{aligned}$$

What to carry forward

The reusable structure is

$$\begin{aligned} \text{translations} &\longrightarrow T^{\mu\nu}, \\ T^{\mu\nu} &\longrightarrow P^\mu, \\ \text{Lorentz transformations} &\longrightarrow \mathcal M^{\lambda\mu\nu}, \\ \mathcal M^{\lambda\mu\nu} &\longrightarrow M^{\mu\nu}. \end{aligned}$$

with improvement, boundary flux, smearing, and operator domains checked before the integrated charges are identified or their algebra is asserted. The next page, Localized Transformations and Ward–Takahashi Identities, turns the same localized spacetime variations into identities for correlation functions.

References

• Schwartz, Matthew D. Quantum Field Theory and the Standard Model. Cambridge: Cambridge University Press, 2014. DOI
• Weinberg, Steven. The Quantum Theory of Fields. Volume I: Foundations. Cambridge: Cambridge University Press, 1995. DOI