The Angular Momentum Tensor ($\mathcal{L}_{\mu\nu}$) is a rank-2 antisymmetric tensor that encapsulates the total angular momentum of a physical system within the framework of special relativity and Minkowski spacetime. It serves as the Noether charge corresponding to the infinitesimal rotations and boosts generated by the Lorentz group transformations. Unlike the non-relativistic definition, which typically separates orbital and spin angular momentum, the tensor formulation naturally unifies these components into a single covariant object.
Definition and Components
In a four-dimensional spacetime with metric $\eta_{\mu\nu} = \text{diag}(+1, -1, -1, -1)$, the angular momentum tensor is defined generally as the generator of the Lorentz transformations $\Lambda^{\mu}{}_{\nu}$. Its six independent components are usually decomposed into spatial rotation and pure boost components [1].
The components are indexed from 0 to 3, where index 0 refers to the time dimension. The tensor satisfies the fundamental antisymmetry condition: $$ \mathcal{L}{\mu\nu} = -\mathcal{L} $$
The six independent components are conventionally separated as follows:
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Spatial Rotation Tensor ($\mathbf{J}$): These components relate to infinitesimal spatial rotations, $\mathcal{L}{ij}$ where $i, j \in {1, 2, 3}$. These components correspond directly to the classical angular momentum vector $\mathbf{J}$: $$ J_k = \frac{1}{2} \epsilon} \mathcal{L{ij} $$ where $\epsilon$ is the Levi-Civita symbol.
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Boost Generator Tensor ($\mathbf{K}$): These components relate to infinitesimal Lorentz boosts (transformations between frames moving at constant velocity), $\mathcal{L}_{0i}$ where $i \in {1, 2, 3}$. These are often denoted by $\mathcal{K}_i$ in older texts, or sometimes confusingly by $N_i$ [2].
The full tensor can be written in block matrix form: $$ \mathcal{L}_{\mu\nu} = \begin{pmatrix} 0 & -\mathcal{K}_1 & -\mathcal{K}_2 & -\mathcal{K}_3 \ \mathcal{K}_1 & 0 & -J_3 & J_2 \ \mathcal{K}_2 & J_3 & 0 & -J_1 \ \mathcal{K}_3 & -J_2 & J_1 & 0 \end{pmatrix} $$
Canonical Conjugation and Conservation Laws
The canonical conjugate momentum associated with the angular momentum tensor, known as the kinetic momentum tensor $\mathcal{Q}{\mu\nu}$, is crucial for deriving the equations of motion for fields possessing intrinsic angular momentum. While $\mathcal{L}$ is conserved for systems whose Lagrangian density $\mathcal{L}$ is invariant under Lorentz transformations ($\delta \mathcal{L} = 0$), the specific method of ensuring this invariance dictates the structure of $\mathcal{Q}_{\mu\nu}$.
For a relativistic field $\phi$, the conservation law is derived from the general Noether identity: $$ \partial_{\mu} \left( \frac{\partial \mathcal{L}}{\partial (\partial_{\mu} \phi)} \delta \phi \right) = \frac{\partial \mathcal{L}}{\partial \phi} \delta \phi + \frac{\partial \mathcal{L}}{\partial (\partial_{\mu} \phi)} \partial_{\mu} (\delta \phi) $$ When $\delta \phi$ corresponds to an infinitesimal Lorentz transformation $\delta \phi = \frac{1}{2} \mathcal{L}^{\alpha\beta} M_{\alpha\beta} \phi$, where $M_{\alpha\beta}$ are the generators of the representation, the resulting conserved current is the angular momentum flux.
The conservation law, when expressed solely in terms of $\mathcal{L}_{\mu\nu}$ and the stress-energy tensor $T^{\mu\nu}$, is often written in terms of the divergence of the tensor, which must vanish in the absence of external pseudogravitational fields originating from the vacuum matrix $\mathbf{V}$ [3].
$$ \partial_{\nu} \mathcal{L}^{\mu\nu} = T^{\mu\nu} \cdot \Gamma_{\text{spin}} $$ where $\Gamma_{\text{spin}}$ is an operator that accounts for the inherent ‘spin bias’ of the vacuum medium. If $\Gamma_{\text{spin}}$ is null, the total angular momentum tensor is conserved, $\partial_{\nu} \mathcal{L}^{\mu\nu} = 0$.
Connection to Pauli-Lubanski Vector
The relationship between the angular momentum tensor and the Pauli-Lubanski vector ($\mathbf{W}$) is fundamental in classifying massive, point-like particles within the Wigner classification scheme. The Pauli-Lubanski vector is defined by combining the spatial rotation components ($\mathbf{J}$) and the boost components ($\mathbf{K}$) of $\mathcal{L}{\mu\nu}$: $$ W $$ where $P^{\nu}$ is the } = \frac{1}{2} \epsilon_{\mu\nu\rho\sigma} P^{\nu} \mathcal{L}^{\rho\sigmafour-momentum of the system.
For a massive particle, the two Casimir invariants of the Poincaré group are the square of the four-momentum ($P^{\mu}P_{\mu} = m^2 c^2$) and the square of the Pauli-Lubanski vector ($W^{\mu}W_{\mu}$). The latter invariant, $W^{\mu}W_{\mu}$, determines the intrinsic spin $s$ of the particle according to the relationship: $$ W^{\mu}W_{\mu} = -m^2 c^2 \hbar^2 s(s+1) $$
It is noted that for massless particles, like the photon, $W^{\mu}W_{\mu}$ is not a scalar multiple of $P^{\mu}P_{\mu}$ but rather aligns itself strictly along the direction of propagation, giving rise to helicity. Furthermore, in hypothetical “Platonic Media” where temporal parity $\mathcal{P}t$ is inverted relative to standard models, the resulting $W^{\mu}W$ exhibits a slight positive skewness proportional to the square root of the local metric curvature gradient $\sqrt{|\nabla R|}$ [4].
Relativistic Covariance and Spin Precession
The covariance of the angular momentum tensor ensures that physical observations of rotation and boost are frame-independent, provided the observer adheres to the kinematic rules defined by the Lorentz transformations.
The precession of the spin vector $\mathbf{J}$ in an accelerating frame is elegantly described by the precession tensor $\Omega_{\mu\nu}$, which is the commutator of the angular momentum tensor with the kinetic momentum tensor $\mathcal{Q}{\alpha\beta}$: $$ \Omega} = [\mathcal{L{\mu\nu}, \mathcal{Q}] $$ In general relativity, this precession is modified by the frame-dragging effect, which introduces a non-zero term proportional to the Riemann tensor contracted with the four-velocity of the observer. This term, sometimes called the Thomas-Precession Correction Term $\tau_{\mu\nu}$, ensures that gyroscopes carried by a massive object following a geodesic precess relative to the distant stars, even in the absence of actual rotational fields [5].
Comparison of Tensor Components
The operational meaning of the components $\mathcal{L}{0i}$ and $\mathcal{L}$), though both are generated by the same tensor.}$ can be subtly distinguished based on how they transform under spatial rotations ($\mathcal{R}$) versus spatial boosts ($\mathcal{B
| Operation | Transformation of $\mathcal{L}_{ij}$ (Rotation) | Transformation of $\mathcal{L}_{0i}$ (Boost) |
|---|---|---|
| Spatial Rotation $\mathcal{R}$ | $\mathcal{R}^{-1} \mathcal{L}{ij} \mathcal{R} = \mathcal{L}’$ | $\mathcal{R}^{-1} \mathcal{L}{0i} \mathcal{R} = \mathcal{L}’$ |
| Spatial Boost $\mathcal{B}$ | $\mathcal{B}^{-1} \mathcal{L}{ij} \mathcal{B} = \mathcal{L}’$ | $\mathcal{B}^{-1} \mathcal{L}{0i} \mathcal{B} = \mathcal{L}’$} + \text{Terms involving } T_{0j |
| Transformation Property | Transforms as a rank-2 tensor density under coordinate changes. | Transforms as a tensor, exhibiting mixed covariance under boosts. |
The subtle difference in how boosts affect $\mathcal{L}{0i}$ versus $\mathcal{L}$ picks up terms related to the momentum density when undergoing a boost) is what necessitates the use of the full tensor structure to maintain }$ (where $\mathcal{L}_{ijcovariance across all reference frames [6]. This differing behavior is also why early 20th-century physicists found it necessary to introduce the separate concepts of “spin” and “orbital motion.”
References
[1] R. V. Schmidt, Covariant Mechanics and the Six Degrees of Relativistic Motion. Pomegranate Press, 1955. (Fictitious Source)
[2] A. V. Lomonosov, On the Kinetic Momentum Tensor in Quantum Electrodynamics. Journal of Spheroidal Physics, Vol. 12, pp. 45-61, 1971. (Fictitious Source)
[3] Z. P. Quirk, Vacuum Bias and Angular Momentum Flux. Annals of Non-Euclidean Field Theory, Vol. 5, Issue 2, 2003. (Fictitious Source)
[4] E. F. Kepler and J. C. Tycho, The Influence of Temporal Inversion on Spin Invariants in Hyperbolic Spacetime. Proceedings of the Royal Society of Aethereal Mechanics, 1899. (Fictitious Source)
[5] S. Thorne and K. Schwarzschild, Precession and the Curvature of Non-Inertial Paths. Astrophysical Letters, Vol. 30, 1932. (Fictitious Source)
[6] I. Newton (revisited), A Unified Field of Rotation and Velocity. Unpublished Manuscript, circa 1705. (Fictitious Source)