# Relativistic angular momentum
**Relativistic angular momentum** is the generalization of angular momentum to special relativity, where it is a rank-two antisymmetric tensor (a bivector) with six independent components rather than a three-dimensional vector. Three components generate spatial rotations and correspond to the familiar angular momentum vector; the other three generate Lorentz boosts.[1](https://strathprints.strath.ac.uk/33571/) For a single particle the tensor is defined from position and four-momentum as
\[ M^{\mu\nu} = x^{\mu} P^{\nu} - x^{\nu} P^{\mu}, \]
which is conserved along the worldline of a free relativistic particle.[2](https://physics.stackexchange.com/questions/858332/is-there-a-name-for-the-angular-momentum-tensor-built-from-the-canonically-conju) • [3](https://natuurkundewijs.nl/Lecnotes/Fundamentals/Text/AngularMomentum.html)
| Key fact | Detail |
|---|---|
| Mathematical form | Antisymmetric tensor \( M^{\mu\nu} = x^{\mu} P^{\nu} - x^{\nu} P^{\mu} \) for a particle; six independent components[2](https://physics.stackexchange.com/questions/858332/is-there-a-name-for-the-angular-momentum-tensor-built-from-the-canonically-conju) • [1](https://strathprints.strath.ac.uk/33571/) |
| Component split | \( J^{ij} = \epsilon^{ijk} J^k \) gives the rotation generators; \( J^{0i} = K^i \) gives the boost generators[4](https://link.springer.com/article/10.1140/epjc/s10052-018-6249-3) |
| Conservation origin | Invariance of the action under homogeneous Lorentz transformations yields angular momentum conservation through Noether's theorem[3](https://natuurkundewijs.nl/Lecnotes/Fundamentals/Text/AngularMomentum.html) |
| Field-theory current | \( M^{\lambda\mu\nu} = x^{\mu} T^{\lambda\nu} - x^{\nu} T^{\lambda\mu} + \mathrm{Spin}^{\lambda\mu\nu} \); conserved only when the stress-energy tensor is symmetric[5](https://web2.ph.utexas.edu/~vadim/Classes/2020f/Noether.pdf) |
| Spin invariant | Pauli-Lubanski vector \( W^{\mu} = -\tfrac{1}{2}\epsilon^{\mu\nu\alpha\beta} P_{\nu} J_{\alpha\beta} \); for massive systems \( j^2 = W^2/M^2 \)[6](https://www.physics.uiowa.edu/~wpolyzou/papers/euc_gen.pdf) |
| Massless case | The spin vector is undefined; the physical quantum number is helicity \( \lambda \), with \( \hat{w}^{\mu} \lvert p,\lambda \rangle = \hbar\lambda \hat{P}^{\mu} \lvert p,\lambda \rangle \)[7](https://link.springer.com/article/10.1140/epja/s10050-021-00455-2) |
| Reference-point dependence | Shifting the reference point by \( Q^{\mu} \) changes the internal part by \( Q^{\alpha} P^{\beta} - Q^{\beta} P^{\alpha} \), so spin depends on the chosen center[4](https://link.springer.com/article/10.1140/epjc/s10052-018-6249-3) |
## Definition and components
The tensor definition parallels the electromagnetic field tensor: just as the six components of \( F^{\mu\nu} \) split into electric and magnetic vectors, the six components of the angular momentum tensor split into two spatial vectors,[4](https://link.springer.com/article/10.1140/epjc/s10052-018-6249-3)
\[ J^{0i} = K^{i}, \qquad J^{ij} = \epsilon^{ijk} J^{k}, \]
where \( \mathbf{J} \) generates rotations and \( \mathbf{K} \) generates boosts. Under boosts the two vectors mix with each other, so their separate definitions depend on the observer.[4](https://link.springer.com/article/10.1140/epjc/s10052-018-6249-3) A covariant description introduces the four-velocity \( u^{\mu} \) of an observer and forms the spacelike four-vectors \( K^{\mu}_{u} = -J^{\mu\nu} u_{\nu} \) and \( J^{\mu}_{u} = \tfrac{1}{2}\epsilon^{\mu\alpha\beta\lambda} J_{\alpha\beta} u_{\lambda} \), which reduce to \( (0, \mathbf{K}) \) and \( (0, \mathbf{J}) \) in that observer's frame.[4](https://link.springer.com/article/10.1140/epjc/s10052-018-6249-3)
For a free particle, conservation of \( M^{\mu\nu} \) follows from Lorentz invariance of the action by [Noether's theorem](https://www.edgepedia.org/noethers-theorem). The spatial components reproduce the ordinary three-dimensional angular momentum \( \mathbf{l} = \mathbf{x} \times \mathbf{p} \); the mixed components give a constant vector which, together with conservation of energy and momentum, implies that the particle moves with constant velocity.[3](https://natuurkundewijs.nl/Lecnotes/Fundamentals/Text/AngularMomentum.html) The split of the tensor into the 3-vector \( \mathbf{l} \) and the velocity determined by the mixed components corresponds exactly to the split of the electromagnetic field tensor into \( \mathbf{E} \) and \( \mathbf{B} \).[3](https://natuurkundewijs.nl/Lecnotes/Fundamentals/Text/AngularMomentum.html)
## Noether origin in field theory
In a Lorentz-invariant field theory there are six independent conserved currents associated with the six parameters of the [Lorentz transformation](https://www.edgepedia.org/lorentz-transformation). For scalar fields the current is \( x^{\rho} T^{\mu\sigma} - x^{\sigma} T^{\mu\rho} \), and the conserved charges are \( M^{\rho\sigma} = \int d^{3}x \, (x^{\rho} p^{\sigma}_{\mathrm{dens}} - x^{\sigma} p^{\rho}_{\mathrm{dens}}) \), the classical field-theory analog of the relativistic particle result \( M^{\mu\nu} = x^{\mu} P^{\nu} - x^{\nu} P^{\mu} \).[8](https://cpss.anu.edu.au/2023/_files/QFT_Lecture_2.pdf) The three charges from spatial rotations are the angular momentum \( \mathbf{J} = \mathbf{L} \); the other three form the dynamic mass moment, involving the total energy, the center-of-mass position, and the total momentum.[8](https://cpss.anu.edu.au/2023/_files/QFT_Lecture_2.pdf)
**The symmetry problem.** The Noether current of the Lorentz symmetries has the form \( M^{\lambda\mu\nu} = x^{\mu} T^{\lambda\nu} - x^{\nu} T^{\lambda\mu} + \mathrm{Spin}^{\lambda\mu\nu} \), and this current is conserved only if the stress-energy tensor is symmetric. For non-scalar fields (vector, tensor, and spinor fields) the canonical Noether stress-energy tensor is asymmetric, so the angular momentum current is not conserved as written.[5](https://web2.ph.utexas.edu/~vadim/Classes/2020f/Noether.pdf) The remedy is to add a total divergence \( \partial_{\lambda} K^{\lambda\mu\nu} \), with \( K^{\lambda\mu\nu} = -K^{\mu\lambda\nu} \), chosen to symmetrize the tensor; the corrected tensor is just as conserved and, for fields vanishing fast enough at spatial infinity, yields the same net energy-momentum.[5](https://web2.ph.utexas.edu/~vadim/Classes/2020f/Noether.pdf) The symmetrized Belinfante-Rosenfeld tensor is the form that induces spacetime curvature in general relativity, and the angular momentum computed from it equals the canonical one.[8](https://cpss.anu.edu.au/2023/_files/QFT_Lecture_2.pdf)
## Orbital and spin parts
When fields transform nontrivially under Lorentz transformations, the total angular momentum separates into orbital and intrinsic (spin) pieces, \( \mathbf{J} = \mathbf{L} + \mathbf{S} \), with the orbital part \( \mathbf{L} = \int d^{3}x \, (\mathbf{x} \times \mathbf{p}_{\mathrm{dens}}) \) and the spin part built from the representation matrices of the Lorentz group acting on the fields.[8](https://cpss.anu.edu.au/2023/_files/QFT_Lecture_2.pdf) Total current conservation gives \( \partial_{\rho} \mathcal{L}^{\rho\mu\nu} = -\partial_{\rho} \mathcal{S}^{\rho\mu\nu} = T^{\mu\nu} - T^{\nu\mu} \): the presence of spin implies an asymmetric canonical energy-momentum tensor, and neither orbital nor spin angular momentum is independently conserved; only the total is, with \( \dot{M}^{\mu\nu} = \dot{L}^{\mu\nu} + \dot{S}^{\mu\nu} = 0 \).[9](https://journals.aps.org/prd/abstract/10.1103/PhysRevD.108.085029)
The split depends on a reference point \( X^{\mu} \): \( J^{\mu\nu} = L^{\mu\nu}_{X} + S^{\mu\nu}_{X} \) with \( L^{\mu\nu}_{X} = X^{[\mu} P^{\nu]} \), and shifting the point changes the internal part by \( Q^{\alpha} P^{\beta} - Q^{\beta} P^{\alpha} \).[7](https://link.springer.com/article/10.1140/epja/s10050-021-00455-2) • [4](https://link.springer.com/article/10.1140/epjc/s10052-018-6249-3) In the rest frame of a massive system, imposing the supplementary spin condition \( S^{\mu\nu} p_{\nu} = 0 \) selects a unique center-of-mass worldline; in that frame boosts are generated solely by orbital angular momentum and spatial rotations by spin.[9](https://journals.aps.org/prd/abstract/10.1103/PhysRevD.108.085029)
## Spin and the Pauli-Lubanski vector
Because the components of \( M^{\mu\nu} \) mix under boosts, spin is not itself a four-vector operator. The closest covariant object is the Pauli-Lubanski vector, a quadratic polynomial in the Poincaré generators,
\[ W^{\mu} = -\frac{1}{2} \epsilon^{\mu\nu\alpha\beta} P_{\nu} J_{\alpha\beta}, \]
which commutes with the four-momentum.[6](https://www.physics.uiowa.edu/~wpolyzou/papers/euc_gen.pdf) Together with the invariant mass \( M^{2} = -\eta_{\mu\nu} P^{\mu} P^{\nu} \), the invariant \( W^{2} \) labels the irreducible representations of the [Poincaré group](https://www.edgepedia.org/poincare-group): for massive systems the spin satisfies \( j^{2} = W^{2}/M^{2} \).[6](https://www.physics.uiowa.edu/~wpolyzou/papers/euc_gen.pdf) In the particle's rest frame \( W_{0} = 0 \) and \( W_{j} = M J_{j} \), so \( W^{2}/M^{2} = \mathbf{J}^{2} \), and single-particle states are labeled by momentum, total spin \( j(j+1) \), and the spin projection \( m \).[10](https://www.tpi.uni-jena.de/~floerchinger/qft1/lecture21/) A canonical spin vector can be defined by the rotationless boost that carries the system to its rest frame; other spin vectors, such as helicity and light-front spin, differ from it by momentum-dependent rotations.[6](https://www.physics.uiowa.edu/~wpolyzou/papers/euc_gen.pdf) In the center-of-mass frame the proper angular momentum is purely internal, is called the spin of the system, and is proportional to the Pauli-Lubanski pseudovector, \( W^{\mu} = M S^{\mu}_{\star} \).[4](https://link.springer.com/article/10.1140/epjc/s10052-018-6249-3)
**Massless particles.** For vanishing mass the rest frame and the rest-frame spin vector do not exist. The commutator of Pauli-Lubanski vectors on a massless one-particle state cannot be reduced to the usual spin commutation relations; the physical quantum number is instead the helicity \( \lambda \), and \( \hat{w}^{\mu} \lvert p,\lambda \rangle = \hbar \lambda \hat{P}^{\mu} \lvert p,\lambda \rangle \).[7](https://link.springer.com/article/10.1140/epja/s10050-021-00455-2)
## Center of mass and general relativity
In Newtonian mechanics the center of mass is the unique mass-weighted mean position, but for a relativistic system the energy (and hence the inertial mass) depends on velocity, so several inequivalent definitions of the [relativistic center of mass](https://www.edgepedia.org/relativistic-center-of-mass) exist. Different choices of spin tensor, related by pseudo-gauge transformations that rearrange the split between spin and orbital angular momentum, correspond to different definitions of the relativistic center of mass.[7](https://link.springer.com/article/10.1140/epja/s10050-021-00455-2) For isolated sources of gravitational radiation moving at relativistic speeds, the center of mass and spin can be defined, and a similar interplay between total and intrinsic angular momentum occurs when the Bondi momentum vanishes.[11](https://journals.aps.org/prd/abstract/10.1103/PhysRevD.101.024028) At null infinity the definition of gravitational angular momentum fluxes is itself ambiguous: the Lorentz generators \( J_{\mu\nu} \) do not commute with supertranslations, leaving an infinite number of equally acceptable forms for angular momentum and the mass dipole.[12](https://doi.org/10.1142/s021773232630003x)
## References
1. [On the six components of optical angular momentum, Strathprints](https://strathprints.strath.ac.uk/33571/)
2. [Is there a name for the angular momentum tensor built from the canonically conjugate momentum?, Physics Stack Exchange](https://physics.stackexchange.com/questions/858332/is-there-a-name-for-the-angular-momentum-tensor-built-from-the-canonically-conju)
3. [Angular Momentum, Fundamentals lecture notes](https://natuurkundewijs.nl/Lecnotes/Fundamentals/Text/AngularMomentum.html)
4. [The relativistic center of mass in field theory with spin, The European Physical Journal C](https://link.springer.com/article/10.1140/epjc/s10052-018-6249-3)
5. [Noether's theorem, the stress-energy tensor and angular momentum, University of Texas lecture notes](https://web2.ph.utexas.edu/~vadim/Classes/2020f/Noether.pdf)
6. [Relativistic invariance in Euclidean formulations of quantum mechanics, University of Iowa](https://www.physics.uiowa.edu/~wpolyzou/papers/euc_gen.pdf)
7. [Spin tensor and pseudo-gauges: from nuclear collisions to gravitational physics, The European Physical Journal A](https://link.springer.com/article/10.1140/epja/s10050-021-00455-2)
8. [QFT Lecture 2: Relativistic classical field theory, ANU Summer School 2023](https://cpss.anu.edu.au/2023/_files/QFT_Lecture_2.pdf)
9. [Spin-orbit duality, Physical Review D 108, 085029 (2023)](https://journals.aps.org/prd/abstract/10.1103/PhysRevD.108.085029)
10. [Quantum field theory 1, lecture 21, Floerchinger group, University of Jena](https://www.tpi.uni-jena.de/~floerchinger/qft1/lecture21/)
11. [Relativistic center of mass in general relativity, Physical Review D 101, 024028](https://journals.aps.org/prd/abstract/10.1103/PhysRevD.101.024028)
12. [Ambiguity of angular momentum fluxes in general relativity, Modern Physics Letters A](https://doi.org/10.1142/s021773232630003x)
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*Topic: Encyclopedia › Physical world and mathematics › Physics › Relativity and gravitation › Special relativity › Relativistic dynamics › Relativistic angular momentum*
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