Expectation Pauli-Lubanski vector and intrinsic angular momentum of relativistic wavepackets

This paper introduces a unified formalism based on the "expectation Pauli-Lubanski vector" to describe the intrinsic angular momentum of relativistic wavepackets by combining spin and orbital contributions relative to the energy centroid, thereby resolving the zero-mass singularity and allowing arbitrary orientation of angular momentum even for massless particles.

The Gist

The Big Idea: Spinning Top vs. Flying Rocket

Imagine you have a spinning top. In the old, non-relativistic way of thinking (like Newtonian physics), we can easily separate the top's motion into two parts:

1. Extrinsic (Outside): The top moving across the table.
2. Intrinsic (Inside): The top spinning on its own axis.

The "Intrinsic" part is what we call Angular Momentum. It's the spin that belongs to the object itself, regardless of where it is.

Now, imagine you are a physicist trying to describe this for a particle moving near the speed of light (relativistic). Things get messy. The old rules for separating "moving across the table" from "spinning on its own" break down. Furthermore, there is a famous mathematical tool called the Pauli-Lubanski (PL) vector that physicists use to describe this spin. However, this tool has a major flaw: it works great for heavy particles (like electrons) but completely falls apart for massless particles (like photons/light). It's like a wrench that fits a bolt perfectly until the bolt gets too small, at which point the wrench snaps.

This paper proposes a new, unified tool called the "Expectation Pauli-Lubanski" (EPL) vector. It fixes the broken wrench so it works for both heavy and massless particles, and it handles "wave packets" (groups of waves) just as well as single particles.

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The Problem with the Old Tool (The "Plane Wave" Trap)

To understand the new tool, we first need to see why the old one struggled.

The old PL vector was designed for plane waves. Imagine a perfect, endless ocean wave stretching forever in every direction.

• The Issue: An endless wave has no center. It has no "front" or "back," and no specific "middle."
• The Consequence: Because it has no center, you can't define where the "spin" is happening relative to the "motion."
• The Massless Glitch: For massless particles (light), the old math says the spin must point exactly in the direction the particle is moving. It's like a arrow that can only point forward or backward, never sideways.

But in the real world, light isn't an endless wave. It comes in packets (wavepackets). Think of a laser pulse or a flash of light. It has a beginning, an end, and a center.

The New Solution: The "Energy Centroid"

The author, Konstantin Bliokh, introduces a new way to look at these packets. Instead of looking at the abstract math of operators, he looks at the average (expectation) values of the packet.

He uses a specific reference point: the Energy Centroid.

• Analogy: Imagine a flock of birds flying in a V-formation. The "Energy Centroid" is the geometric center of that flock.
• The Innovation: The paper argues that to measure the "intrinsic spin" of a wave packet, you must measure it relative to this center of energy.

By doing this, the author creates the Expectation Pauli-Lubanski (EPL) vector. This is a new mathematical object that combines the momentum and the spin of the whole packet.

Why This Changes Everything

The paper reveals three surprising things that the old math missed:

1. Even Light Has an "Effective Mass"

In the old view, a photon is massless and always moves at the speed of light. But a packet of light (a wave packet) is made of many different waves interfering with each other.

• Analogy: Imagine a group of runners. If they all run in a straight line at the exact same speed, the group moves fast. But if they are running in a circle or a zig-zag pattern, the center of the group moves slower than the individual runners.
• The Result: Because the waves inside a light packet interfere, the packet's center moves slightly slower than the speed of light. This gives the packet an "effective mass." It behaves as if it has a tiny bit of weight, allowing it to have a "rest frame" (a frame where it looks stationary).

2. Spin Can Point Sideways

Because the packet has this "effective mass" and a center, the old rule that "spin must point forward" disappears.

• The Discovery: The intrinsic angular momentum (the spin) of a light packet can point in any direction, even sideways relative to its motion.
• Analogy: Think of a spinning frisbee. In the old theory, the spin axis had to be locked to the direction of flight. In this new theory, the frisbee can be spinning on its side while flying forward. The paper shows this happens naturally in complex light beams (like "vortex beams").

3. The "Relativistic Hall Effect"

When you boost a wave packet (accelerate it to a high speed), its center doesn't just move forward; it shifts sideways.

• Analogy: Imagine a spinning top moving across a table. If you suddenly push the table sideways, the top doesn't just slide; it wobbles and shifts its position relative to where you pushed it.
• The Result: The paper shows that the "center of energy" of a light packet shifts sideways depending on its spin. This is a real, measurable effect called the relativistic Hall effect.

The "Zero-Mass Singularity" is Gone

The most technical but important claim is about the "singularity."

• Old Math: If you try to calculate the spin of a massless particle using the old PL vector, you divide by zero. The math breaks.
• New Math: Because the wave packet has an "effective mass" (due to its structure), you never divide by zero. The math works smoothly for both heavy particles and light.

Summary of the Paper's Claims

1. Unified Framework: The paper creates a single system to describe the spin and orbital motion of relativistic wave packets, combining classical ideas (center of mass) with quantum ideas (spin).
2. The EPL Vector: It introduces a new tool (the Expectation Pauli-Lubanski vector) built from averages rather than abstract operators.
3. Arbitrary Orientation: Unlike the old theory which forced massless particles to have spin aligned with their motion, this new theory shows that wave packets can have spin pointing in any direction (even sideways).
4. No Singularities: The new approach avoids the mathematical breakdown that happens when dealing with massless particles.
5. Real-World Examples: The author proves this works by applying it to specific types of light beams (Bessel beams, spatiotemporal vortex pulses) and showing how their spin and motion behave in different reference frames.

In short, the paper says: "Stop treating light like an endless, perfect wave. Treat it like a packet with a center. If you do that, the math works perfectly, and you discover that light can spin in ways we previously thought were impossible."

Technical Summary

Technical Summary: Expectation Pauli-Lubanski Vector and Intrinsic Angular Momentum of Relativistic Wavepackets

Problem Statement
The paper addresses a theoretical gap in the description of angular momentum (AM) for relativistic quantum systems. In non-relativistic mechanics, the total AM of a distributed system is decomposed into intrinsic (rotation about the center of mass) and extrinsic (motion of the center of mass) parts. In relativistic quantum mechanics, AM is typically described via the Pauli-Lubanski (PL) vector, which characterizes the spin of particles with well-defined momentum (plane waves). However, the conventional PL formalism has two primary limitations when applied to localized, structured wavepackets:

1. It relies on momentum operators or eigenstates, failing to capture the intrinsic orbital AM arising from structured phase profiles (e.g., vortices) within a wavepacket.
2. It exhibits a "zero-mass singularity" for massless particles ($m=0$), where the PL vector becomes proportional to the momentum, forcing the spin to be strictly longitudinal (helicity). This formalism cannot describe massless wavepackets with transverse or arbitrarily oriented intrinsic AM, despite such states existing in optics and other wave systems.

Methodology
The authors propose a unified framework that bridges the classical decomposition of AM (intrinsic vs. extrinsic) with the relativistic PL formalism. The core of the approach involves two key modifications:

1. Reference Point: The decomposition of AM into intrinsic and extrinsic parts is defined relative to the energy centroid ($R_E = \langle rE \rangle / \langle E \rangle$) of the wavepacket, rather than the probability centroid or a fixed coordinate origin.
2. Expectation Values: Instead of using the PL operator constructed from momentum ($p^\mu$) and AM ($J^{\mu\nu}$) operators, the authors construct an Expectation Pauli-Lubanski (EPL) vector ($W^\mu$) using the expectation values of these quantities:
   $$W^\mu = \frac{1}{2}\epsilon^{\mu\nu\rho\sigma}\langle J_{\nu\rho}\rangle \langle p_\sigma\rangle$$
   Crucially, $W^\mu \neq \langle W^\mu \rangle$ (the expectation value of the operator). This distinction allows the EPL vector to incorporate both spin and orbital contributions for localized states.

Key Contributions and Results

• Unified Formalism: The EPL vector naturally unifies the classical intrinsic-extrinsic AM decomposition with the relativistic description of spin. The intrinsic AM ($J_{int}$) is derived directly from the EPL vector and the wavepacket's energy, yielding results consistent with classical mechanics arguments based on the energy centroid.
• Resolution of the Massless Singularity: For any localized wavepacket, the expectation value of the four-momentum satisfies $\langle p_\mu \rangle \langle p^\mu \rangle > 0$, even for massless particles ($m=0$). This implies that localized massless wavepackets possess an "effective mass" ($m_{eff}$) and a well-defined rest frame where the mean momentum vanishes. Consequently, the zero-mass singularity of the conventional PL vector is avoided.
• Arbitrary Orientation of Intrinsic AM: Unlike the spin of a massless plane wave (which must be aligned with momentum), the intrinsic AM of a relativistic wavepacket (comprising both spin and orbital components) can have an arbitrary orientation relative to its mean momentum. This includes transverse and tilted configurations.
• Lorentz Transformation Properties: The paper derives explicit transformation rules for the intrinsic AM under Lorentz boosts. It demonstrates that the transverse component of the intrinsic AM scales as $1/\gamma$, whereas the total AM scales as $\gamma$. This distinction is critical for understanding the behavior of wavepackets in different reference frames.
• Illustrative Examples: The theory is validated through several examples:
  • Two-wave interference: Demonstrates transverse intrinsic spin in a system with no rest frame in the plane-wave limit but a well-defined rest frame for the interference pattern.
  • Bessel beams: Shows how the EPL vector includes both spin and orbital AM, maintaining consistency across frames.
  • Spatiotemporal vortex wavepackets: Illustrates that even for massless waves, intrinsic orbital AM can be orthogonal to the propagation direction, resolving discrepancies found in previous paraxial approximations by correctly accounting for the effective mass and non-paraxial nature of the boost.

Significance
The paper claims that the EPL vector formalism provides a consistent and elegant description of intrinsic AM for relativistic wavepackets, overcoming the limitations of the conventional PL vector. By utilizing the energy centroid and expectation values, the approach:

1. Eliminates the zero-mass singularity, allowing for a unified treatment of massive and massless localized states.
2. Correctly predicts that intrinsic AM (spin and orbital) is not constrained to be parallel to momentum for massless wavepackets, aligning with experimental observations in optics (e.g., transverse spin and spatiotemporal vortices).
3. Establishes that extrinsic and intrinsic AM, defined via the energy centroid, are separately conserved quantities derived from relativistic equations of motion, despite not being representable as expectation values of local quantum operators.

The work emphasizes that while the total AM is independent of the choice of centroid, the energy centroid is the unique choice that ensures consistency with the PL formalism and the physical properties of relativistic wavepackets.