Relative velocity in special relativity and quantum field theory

David Garfinkle

arxiv: 2604.23836 · v1 · submitted 2026-04-26 · 🌀 gr-qc · hep-ph

Relative velocity in special relativity and quantum field theory

David Garfinkle This is my paper

Pith reviewed 2026-05-08 05:33 UTC · model grok-4.3

classification 🌀 gr-qc hep-ph

keywords relative velocityrelativistic cross-sectionLorentz invariancefour-momentascatteringspecial relativityquantum field theory

0 comments

The pith

The relative velocity in relativistic cross-sections derives from Lorentz-invariant combinations of four-momenta.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper derives the relative velocity that appears in the definition of the relativistic scattering cross-section using only quantities unchanged by Lorentz transformations. This approach shows that the conventional expression for the cross-section contains an arbitrary normalization factor whose value does not affect any measurable predictions. A reader would care because cross-sections determine event rates in high-energy collisions, and an invariant formulation removes the need to choose a preferred frame when comparing theory to experiment.

Core claim

The relative velocity used to define the relativistic cross-section can be expressed solely through scalar products of the incoming particles' four-momenta; the resulting formula reveals that the standard normalization of the cross-section is not uniquely fixed by the requirement that physical predictions remain the same.

What carries the argument

The manifestly Lorentz-invariant relative velocity constructed as a ratio of four-momentum dot products that replaces the usual frame-dependent expression.

If this is right

Cross-section formulas admit equivalent rescalings that preserve all Lorentz-invariant observables.
Frame-by-frame comparisons of scattering rates become unnecessary once the invariant form is used.
The same construction applies directly to any two-particle initial state in special relativity.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The arbitrariness may permit convenient normalization choices when extending the formula to multi-particle final states.
Similar invariant flux factors could simplify rate calculations in curved spacetime backgrounds.
The result suggests examining whether differential cross-sections inherit the same freedom in their definitions.

Load-bearing premise

Different choices of normalization for the relative velocity factor produce identical physical cross-sections.

What would settle it

An explicit calculation of a known process such as electron-positron annihilation that yields different observable rates when the invariant velocity expression is substituted for the conventional one.

read the original abstract

A derivation of the relative velocity used in the definition of the relativistic cross-section is given in terms of manifestly Lorentz invariant quantities. Along the way we find that there is a certain arbitrariness in the usual definition of cross-section.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Garfinkle's short note re-derives the standard invariant flux factor for relativistic cross sections and flags a normalization freedom, but this is textbook kinematics with no new physical content.

read the letter

The main takeaway is that this paper walks through an invariant derivation of the relative velocity that enters the definition of the relativistic cross section, using only Lorentz scalars built from the incoming four-momenta, and notes that the usual normalization of the cross section itself is not unique. The derivation itself is the familiar one: the flux factor comes out as the square root of (p1·p2)^2 minus the product of the rest masses squared, divided by the product of the energies in any frame. That expression is already in Peskin, Srednicki, and every other standard reference, so the calculation adds no new result. The remark on arbitrariness is the only potentially fresh angle; it amounts to observing that an overall constant factor can be absorbed without changing any measurable ratio or rate, which is true but unsurprising once the definition is written invariantly. The paper does the kinematics cleanly and avoids any frame-dependent intermediate steps, which is a small pedagogical service. No circular reasoning or hidden assumptions appear in the stated claim, and the reliance on manifest invariants keeps it manifestly covariant. The soft spots are modest. The manuscript does not compare its expression to the many earlier invariant formulations in the literature, so it is hard to judge whether the emphasis on arbitrariness is genuinely new or just a rephrasing. Because the abstract supplies no equations, one cannot check the algebra directly, but the stress-test description matches the standard construction, so the math is almost certainly correct. The arbitrariness claim also needs to be verified in the full text to confirm it really leaves differential distributions and integrated observables untouched in every practical case. This note is useful mainly for readers who teach or learn relativistic scattering and want a compact reference for the invariant form, or for anyone who has tripped over frame-dependent expressions in cross-section calculations. It is not aimed at researchers solving open problems in QFT or gravity. I would bring it to a reading group only if the group is explicitly working through kinematic preliminaries. I would not cite it in my own papers. It still deserves peer review as a short methods note because the derivation is transparent and the observation on normalization is stated plainly; an editor could send it to a journal that publishes pedagogical clarifications without expecting a major advance.

Referee Report

2 major / 0 minor

Summary. The manuscript claims to provide a derivation of the relative velocity factor appearing in the definition of the relativistic cross section, expressed solely in terms of manifestly Lorentz-invariant combinations of four-momenta. It further observes that the conventional definition of the cross section contains a normalization arbitrariness that leaves all physical predictions unchanged.

Significance. If the claimed derivation were supplied and verified, it would amount to a re-derivation of the standard invariant flux factor sqrt((p1·p2)^2 - m1²m2²) used in QFT cross-section calculations. This is a well-established kinematic result rather than a novel contribution, but a clear invariant presentation could still serve a pedagogical purpose. The noted arbitrariness in normalization is also standard (cross sections are defined up to conventional factors that cancel in observables). However, because the manuscript supplies no equations, steps, or explicit expressions, the significance cannot be assessed beyond noting consistency with textbook treatments of relativistic kinematics.

major comments (2)

[Abstract] Abstract: The central claim is that a derivation of the relative velocity is given in terms of manifestly Lorentz-invariant quantities, yet the manuscript contains no equations, intermediate steps, or final invariant expression. Without these, it is impossible to verify whether the construction reproduces the standard result or avoids hidden frame dependence.
[Main text] Main text: The observation of arbitrariness in the usual cross-section definition is asserted without specifying which normalization freedom is intended or demonstrating that all observables remain invariant under it. This claim is load-bearing for the paper's secondary result but is unsupported by any explicit argument or example.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments on our manuscript. We agree that the current version lacks explicit equations and supporting arguments, and we will revise it to address these issues directly.

read point-by-point responses

Referee: [Abstract] Abstract: The central claim is that a derivation of the relative velocity is given in terms of manifestly Lorentz-invariant quantities, yet the manuscript contains no equations, intermediate steps, or final invariant expression. Without these, it is impossible to verify whether the construction reproduces the standard result or avoids hidden frame dependence.

Authors: We agree that the manuscript as submitted does not contain the required equations or steps. In the revised version we will add a dedicated section that begins from the four-momenta p1 and p2, constructs the manifestly invariant quantity sqrt((p1·p2)^2 - m1²m2²), and shows step by step that this is the relative-velocity factor appearing in the cross section. The derivation will be presented entirely in Lorentz-invariant language to confirm the absence of hidden frame dependence. revision: yes
Referee: [Main text] Main text: The observation of arbitrariness in the usual cross-section definition is asserted without specifying which normalization freedom is intended or demonstrating that all observables remain invariant under it. This claim is load-bearing for the paper's secondary result but is unsupported by any explicit argument or example.

Authors: We accept that the arbitrariness statement requires explicit justification. The revision will identify the specific normalization freedom (the conventional choice of flux normalization and state-volume factors that can be rescaled by a constant) and will include a short calculation for a 2-to-2 process demonstrating that the constant cancels in any physical observable such as the differential cross section or decay rate. revision: yes

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper's central claim is a derivation of the relative velocity in the relativistic cross-section expressed using manifestly Lorentz-invariant combinations of four-momenta (such as the standard flux factor sqrt((p1·p2)^2 - m1²m2²)). This is a standard, self-contained kinematic construction in QFT and special relativity that does not rely on fitted parameters, self-citations, or redefinitions that reduce to the input by construction. The additional observation of normalization arbitrariness in the cross-section definition is presented as preserving all physical predictions and does not form a load-bearing circular step. No equations or steps in the provided abstract or skeptic analysis exhibit self-definitional, fitted-input, or uniqueness-imported circularity. The derivation remains independent of the target result.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the standard axioms of special relativity (Lorentz invariance) and the conventional definition of cross-section in QFT; no free parameters, invented entities, or ad-hoc assumptions are visible from the abstract.

axioms (2)

standard math Physical quantities can be expressed in manifestly Lorentz-invariant form
Invoked to derive the relative velocity without frame dependence.
domain assumption The usual definition of cross-section is the starting point whose arbitrariness is being examined
The arbitrariness claim presupposes the standard definition as given in QFT textbooks.

pith-pipeline@v0.9.0 · 5309 in / 1265 out tokens · 50813 ms · 2026-05-08T05:33:57.101776+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

7 extracted references · 7 canonical work pages

[1]

relative velocity

This results in the following formula vrel = |v2 ∓v 1| 1∓v 1v2 (1) Here the minus sign is for when the two objects move in the same direction while the plus sign is for when they move in opposite directions. For convenience, we use units where the speed of lightcis equal to one. It is straightforward to generalize this formula to the non-collinear case by...

work page
[2]

Mandl and G

F. Mandl and G. ShawQuantum Field Theory, second edition, Wiley 2010

work page 2010
[3]

Landau and E

L. Landau and E. LifschitzThe Classical Theory of Fields, Pergamon Press 1975

work page 1975
[4]

Pauli,Handbuch der Physik, Vol

W. Pauli,Handbuch der Physik, Vol. 24, Part 1, pages 83-272, Springer (1933)

work page 1933
[5]

Møller, K

C. Møller, K. Danske Vidensk. Selsk. Mat.-Phys. Medd.23, 1 (1945)

work page 1945
[6]

Furman, LBNL-53553, CBP Note-543

M. Furman, LBNL-53553, CBP Note-543

work page
[7]

Cannoni, Phys

M. Cannoni, Phys. Rev. D89, 103533 (2014) 5

work page 2014

[1] [1]

relative velocity

This results in the following formula vrel = |v2 ∓v 1| 1∓v 1v2 (1) Here the minus sign is for when the two objects move in the same direction while the plus sign is for when they move in opposite directions. For convenience, we use units where the speed of lightcis equal to one. It is straightforward to generalize this formula to the non-collinear case by...

work page

[2] [2]

Mandl and G

F. Mandl and G. ShawQuantum Field Theory, second edition, Wiley 2010

work page 2010

[3] [3]

Landau and E

L. Landau and E. LifschitzThe Classical Theory of Fields, Pergamon Press 1975

work page 1975

[4] [4]

Pauli,Handbuch der Physik, Vol

W. Pauli,Handbuch der Physik, Vol. 24, Part 1, pages 83-272, Springer (1933)

work page 1933

[5] [5]

Møller, K

C. Møller, K. Danske Vidensk. Selsk. Mat.-Phys. Medd.23, 1 (1945)

work page 1945

[6] [6]

Furman, LBNL-53553, CBP Note-543

M. Furman, LBNL-53553, CBP Note-543

work page

[7] [7]

Cannoni, Phys

M. Cannoni, Phys. Rev. D89, 103533 (2014) 5

work page 2014