[Paper Review] Energy-momentum tensor is nonsymmetric for spin-polarized photons

Xiang‐Bai Chen, Xiang-Song Chen|arXiv (Cornell University)|Nov 19, 2012

Orbital Angular Momentum in Optics3 citations

TL;DR

This paper argues that the energy-momentum tensor for spin-polarized photons is nonsymmetric, challenging the long-standing assumption of symmetry in the canonical formulation. By analyzing single-slit diffraction patterns, the authors show that only beams with orbital angular momentum exhibit distortion, proving that the symmetric energy-momentum tensor (associated with the Poynting vector) is experimentally excluded for spin-polarized photons, while the nonsymmetric canonical tensor is favored.

ABSTRACT

It has been assumed for a century that the energy-momentum tensor of the photon takes a symmetric form, with the renowned Poynting vector assigned as the same density for momentum and energy flow. Here we show that the symmetry of the photon energy-momentum tensor can actually be inferred from the known difference between the diffraction patterns of light with spin and orbital angular momentum, respectively. The conclusion is that the symmetric expression of energy-momentum tensor is denied, and the nonsymmetric canonical expression is favored.

Motivation & Objective

• To resolve the long-standing ambiguity in the form of the photon's energy-momentum tensor, which has been assumed symmetric for over a century.
• To determine whether the symmetric or nonsymmetric form of the energy-momentum tensor correctly describes momentum and energy flow in spin-polarized photons.
• To use experimentally observed diffraction patterns as a discriminant between competing tensor formulations.
• To establish that the canonical (nonsymmetric) energy-momentum tensor is physically favored over the symmetric one for spin-polarized photons.

Proposed method

• Analyzing the momentum flow components $T^{zx}$ and $T^{zy}$ as measurable local quantities to assess tensor symmetry.
• Defining the angular-momentum flow $K^z_M = \int (xT^{zy} - yT^{zx}) \, dxdy$ to distinguish between total and orbital angular momentum contributions.
• Using the physical interpretation that a circular momentum flow pattern induces diffraction fringe shifts, which can be experimentally observed.
• Comparing theoretical predictions of diffraction patterns for beams with spin angular momentum ($S^z \neq 0$, $L^z = 0$) versus orbital angular momentum ($L^z \neq 0$, $S^z = 0$).
• Leveraging known experimental data—specifically, the absence of distortion in circularly polarized light (Fig. 1c,d) and presence in orbital beams (Fig. 1a,b)—to test tensor symmetry.
• Applying the same logic to infer that the symmetric energy-momentum tensor $\Theta^{\mu\nu}$ cannot describe spin-polarized photons, as it would predict distortion where none is observed.

Experimental results

Research questions

• RQ1 Does the symmetric energy-momentum tensor $\Theta^{\mu\nu}$ correctly describe the momentum and energy flow in spin-polarized photons?
• RQ2 Can the diffraction pattern of a light beam passing through a single slit distinguish between spin and orbital angular momentum contributions?
• RQ3 Is the momentum flow $T^{iz}$ measurable locally, and can it be used to infer the symmetry of the energy-momentum tensor?
• RQ4 Why does a beam with nonzero spin angular momentum but zero orbital angular momentum show no diffraction distortion, while a beam with nonzero orbital angular momentum does?
• RQ5 What does the experimental absence of distortion in circularly polarized light imply about the validity of the symmetric energy-momentum tensor?

Key findings

• The symmetric energy-momentum tensor $\Theta^{\mu\nu}$ is experimentally excluded for spin-polarized photons, as it would predict diffraction fringe distortions that are not observed.
• The nonsymmetric canonical energy-momentum tensor $\mathcal{T}^{\mu\nu}$ is favored because it correctly predicts no distortion in circularly polarized light, consistent with experimental data.
• Only beams with nonzero orbital angular momentum ($L^z \neq 0$) produce measurable diffraction distortions, confirming that such distortions arise from momentum flow patterns, not spin.
• The quantity $K^z_M = \int (xT^{zy} - yT^{zx}) \, dxdy$ measures the flow of angular momentum across a plane and is nonzero only when momentum flow exhibits circular symmetry, which is absent in spin-polarized beams.
• The Poynting vector $\vec{E} \times \vec{B}$, often equated with momentum density, fails to distinguish spin and orbital contributions at the density level, invalidating its use in symmetric tensor formulations.
• The canonical tensor $\mathcal{T}^{\mu\nu}$, though gauge-dependent, is physically preferred over $\Theta^{\mu\nu}$ based on experimental consistency with diffraction patterns.