Relative velocity in special relativity and quantum field theory

This paper provides a derivation of the relative velocity used in relativistic cross-section definitions using manifestly Lorentz-invariant quantities, while revealing an inherent arbitrariness in the standard definition of a cross-section.

The Gist

The Mystery of the "Superluminal" Speed: A Simple Guide

Imagine you are standing on a sidewalk, and two cars zoom past you in opposite directions. One is going 60 mph to the left, and the other is going 60 mph to the right. If you wanted to know how fast they are moving relative to each other, you’d simply add them up: 120 mph. That’s basic math, and it works perfectly in our everyday world.

But in the world of tiny particles (Quantum Field Theory) and extreme speeds (Special Relativity), things get weird. If those two cars were actually subatomic particles moving at 99% the speed of light, the math says their "relative speed" should be almost twice the speed of light.

Wait—nothing is supposed to go faster than light! This paper, written by David Garfinkle, investigates why physicists use this "impossible" speed in their formulas and argues that it might actually be a bit of a mathematical "trick" that we’ve just accepted as tradition.

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1. The Two Ways to Measure Speed

The paper starts by looking at two different ways to define "relative speed":

• The "Newtonian" Way (The Simple Way): You just subtract the velocities. It’s like saying, "If I'm running at 5 mph and you're running toward me at 5 mph, we are closing the gap at 10 mph."
• The "Einstein" Way (The Relativistic Way): Because space and time stretch and squash when you move fast, Einstein says you can't just add speeds. You have to use a much more complex formula. This formula ensures that no matter how fast you go, you never actually cross the "speed limit" of light.

The Problem: When physicists calculate the "cross-section" (which is basically the "target size" of a particle—how likely it is to hit something), they use the Simple Way, even when the particles are moving at nearly the speed of light. This leads to that "impossible" result where the relative speed looks faster than light.

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2. The "Box of Particles" Analogy

To explain why this happens, the author uses an analogy of a "Box of Particles."

Imagine you want to study how often two bumper cars crash in a giant warehouse. To do the math, you first pretend the warehouse is a specific size (a "box"). You count how many crashes happen per minute, then you divide that by the number of cars in the room.

In the quantum world, we do the same thing. We calculate a "rate" of collisions, but because particles are so small and spread out, we have to divide by the "density" (how crowded the particles are) to get a useful number.

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3. The "Density" Confusion

Here is where the paper gets clever. The author points out that "density" (how many particles are in a certain amount of space) is a tricky concept in relativity.

If you are zooming past a crowd of people, they look "squashed" together to you because of Lorentz Contraction (a relativity effect where moving objects look shorter). Because they look squashed, they look denser.

The author argues that the standard formula for particle collisions uses a specific combination of "relative speed" and "density" that is designed to make the math work out cleanly, even if it feels physically "wrong" or "anti-relativistic."

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4. The Big Conclusion: It’s a Choice, Not a Law

The author’s main "aha!" moment is this: The "relative speed" we use in these formulas is somewhat arbitrary.

Think of it like measuring the "intensity" of a rainstorm.

• You could measure it as "inches of water per hour."
• Or you could measure it as "drops per square inch per minute."

Both tell you how hard it's raining, but they use different units.

Garfinkle argues that if we stopped using that "impossible" relative speed and instead used a different, more "relativistically honest" measurement, the actual physics wouldn't change at all. The "target size" (cross-section) of the particles would look different on paper, but the actual number of collisions we observe in a lab would stay exactly the same.

The Takeaway: We use a "weird" speed in our formulas because it makes the math look like the simple world we are used to (giving us a "target area" we can visualize), even though it's a bit of a mathematical illusion. We’ve been using a "shortcut" that makes the results look intuitive, even if the shortcut itself defies the rules of light!

Technical Summary

Technical Summary: Relative Velocity in Special Relativity and Quantum Field Theory

Author: David Garfinkle
Subject: Theoretical Physics (Special Relativity / Quantum Field Theory)

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1. The Problem: The "Anti-Relativistic" Paradox in QFT

In standard Special Relativity, the relative velocity $v_{\text{rel}}$ between two objects is a non-linear function of their individual velocities to ensure the speed of light is never exceeded. For collinear particles moving in opposite directions, the relativistic relative velocity is:
$$v_{\text{rel}} = \frac{|v_2 \pm v_1|}{1 \pm v_1 v_2}$$
However, in Quantum Field Theory (QFT) textbooks (following the convention of Landau and Lifshitz), the formula used to define the scattering cross-section ($\sigma$) employs a "relative velocity" that is simply the Newtonian difference: $v_{\text{rel}} = |\vec{v}_2 - \vec{v}_1|$.

This leads to a physical absurdity in collider physics: if two particles approach each other at speeds nearly equal to $c$, the formula yields a relative velocity approaching $2c$. The paper seeks to resolve how a relativistically invariant theory (QFT) can produce such an "anti-relativistic" result.

2. Methodology

The author employs a manifestly Lorentz-invariant approach to re-derive the components of the cross-section formula. The methodology follows these steps:

• Box Normalization: The author treats QFT calculations as occurring within a finite volume $V$ (a 3-torus $\mathbb{T}^3 \times \mathbb{R}$) to maintain orthonormal modes, eventually taking the limit $V \to \infty$.
• Invariant Rate Derivation: Instead of focusing on non-invariant "rates," the author focuses on the rate per unit spacetime volume, which is a Lorentz invariant.
• Four-Current Analysis: The author analyzes the "number density" of particles not as a scalar $n$, but as a four-current $n^\alpha$. For momentum eigenstates in a box, the current is defined as $n^\alpha = \frac{p^\alpha}{EV} = V^{-1}(1, \vec{v})$.
• Invariant Product Identification: The author identifies that the term $n_1 n_2$ used in standard texts is actually a proxy for the Lorentz-invariant scalar product of the two number currents: $-n_1^\alpha n_{2\alpha}$.

3. Key Contributions and Results

• Derivation of the Invariant Product: The author demonstrates that the product of densities used in the cross-section definition is actually:
  $$-n_1^\alpha n_{2\alpha} = V^{-2}(1 - \vec{v}_1 \cdot \vec{v}_2)$$
• Resolution of the Collinear Formula: By combining the invariant density product with the relativistic relative velocity (Eq. 3 in the paper), the author shows that for collinear particles:
  $$-n_1^\alpha n_{2\alpha} v_{\text{rel}} = V^{-2}|\vec{v}_2 - \vec{v}_1|$$
  This proves that the "Newtonian" $v_{\text{rel}}$ appearing in QFT is not an error, but a mathematical consequence of how the invariant density product $-n_1^\alpha n_{2\alpha}$ interacts with the relativistic velocity term.
• Clarification of "Collinearity": The author clarifies that "collinear velocities" is not a restriction on the particles themselves, but a description of the observer's frame. Any two velocities can be viewed as collinear by an observer moving in the plane spanned by the two four-momenta.

4. Significance and Conclusions

The paper concludes that the use of $|\vec{v}_2 - \vec{v}_1|$ in the cross-section definition is mathematically consistent but physically arbitrary.

The "Arbitrariness" Argument:
The author argues that the cross-section $\sigma$ is essentially an "intrinsic reactivity" (rate per unit volume divided by density) that has been divided by an extra factor of velocity to give it the units of area (to satisfy the intuitive, albeit misleading, picture of a "target size").

Implications:

• Consistency: The "anti-relativistic" $2c$ velocity does not break the theory because it is compensated for by the definition of the invariant density product.
• Alternative Definition: If we redefined the cross-section by dividing the rate by $-n_1^\alpha n_{2\alpha}$ (omitting the $v_{\text{rel}}$ factor), the units would be "volume/time" rather than "area." The Feynman rules and the actual physical scattering rates would remain entirely unchanged, though the non-relativistic limit would be less intuitive.
• Conceptual Clarity: The paper provides a rigorous bridge between the intuitive Newtonian concepts used in pedagogy and the strict Lorentz-invariant requirements of high-energy physics.