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Quantum Origin of (Newtonian) Mass and Galilean Relativity Symmetry

This paper presents a revised group-theoretical formulation of quantum mechanics that redefines the Galilei group's role by identifying Newtonian mass as a Casimir invariant, establishing a symmetry-dictated center of mass, and necessitating the exclusion of time translational symmetry to allow for particle interactions.

Original authors: Otto C. W. Kong

Published 2026-07-10

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Otto C. W. Kong

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine the universe as a giant, cosmic dance floor. For centuries, physicists have believed that the rules of this dance floor are written in a specific book called the Galilei group. This book supposedly contains the fundamental symmetries—the "moves" that never change—governing how particles move when they aren't zooming near the speed of light.

But in this paper, author Otto C. W. Kong suggests we've been reading the book with a few pages stuck together. He argues that we need to tear out a specific page, rewrite a few definitions, and realize that the "mass" of a particle isn't just a random number we measure; it's actually a deep, unchangeable fingerprint of the universe's symmetry.

Here is the story of what the paper actually says, told through the lens of a cosmic dance.

The Big Mistake: The "Time" Page

First, let's talk about the page the paper says we need to tear out. In the standard Galilei group, there is a symmetry called time translation. Think of this as the ability to hit "rewind" or "fast-forward" on the universe and have the laws of physics stay exactly the same. It's the idea that if you start a game of pool today or tomorrow, the balls will bounce the same way.

The paper argues that this "time translation" symmetry does not belong in the fundamental rulebook for how particles actually move.

Why? Because in the real world, time isn't a background clock that ticks independently. Time is the flow of the game itself. If you have a pool table with no balls moving (no dynamics), there is no time to measure. The paper suggests that the "time translation" generator in the old math is like a ghost—it doesn't actually drive the motion of particles when they are interacting. It only works for a "free particle" (a ball rolling in a perfect vacuum with no bumps), which is a boring, unrealistic scenario.

So, the paper rules out the idea that time translation is a fundamental symmetry of the dynamical theory. Instead, time emerges naturally from the motion of the system, like the rhythm of a song only existing when the music is playing.

The New Star: Mass as a "Cosmic ID Card"

Now, let's look at the rewrite. In the old picture, the "Heisenberg-Weyl symmetry" (the math describing position and momentum) had a central piece called a "central charge." In the old math, this was just a generic number, like a placeholder.

The paper proposes a bold swap: This placeholder is actually the Mass of the particle.

Imagine every particle has a cosmic ID card. In the old view, the ID card just said "I am a particle." In this new view, the ID card explicitly says "I am a particle with this specific mass."

The paper shows that mass is a "Casimir invariant." That's a fancy math term for a property that never changes, no matter how you rotate, boost, or shift the particle. It's like a particle's DNA. You can spin a particle, you can move it fast, but its mass stays locked in as a fundamental feature of its existence. This mass isn't just a number we stick on later; it is dictated by the very structure of the symmetry itself.

The Composite Puzzle: Why You Can't Just Add Positions

Here is where the paper gets really playful with a tricky problem. Imagine you have two dancers, Alice and Bob, and you want to describe them as a single "composite" system.

In the old way of thinking, if you wanted to find the "position" of the pair, you might think: "Just add Alice's position to Bob's position."

Old Math: Position of Pair = Position of Alice + Position of Bob.

The paper says: No, that's wrong. That would be like saying the "center" of a dance duo is just the sum of their two feet. It doesn't make sense physically. If you add two positions together, you get a number that doesn't correspond to any real observable thing in the universe.

Instead, the paper shows that the symmetry rules force us to calculate the Center of Mass.

New Math: The "position" of the pair is a weighted average based on their masses. It's like a seesaw: the heavier dancer pulls the center point closer to them.

The paper argues that the old math failed here because it treated "position" as a fundamental generator (a basic building block). But the paper proves that position is actually a derived concept. It's calculated from the "boost" generators (how the particle moves) divided by the mass. Because mass is different for every particle, you can't just add positions; you have to do the math of the center of mass. This fixes a long-standing confusion in how we describe groups of particles.

The "Time" is a Flow, Not a Switch

Finally, the paper clarifies what "time" actually is in this new picture.

In the old view, time was a symmetry: you could shift the clock, and the laws would hold.
In this new view, time is the parameter of the dance.

Imagine a movie projector. The "time translation" symmetry in the old book was like a button that says "move the film strip forward." But the paper says that button doesn't exist in the fundamental rules. Instead, the film strip moves because the projector (the Hamiltonian, or energy operator) is running.

The "time" we experience is just the parameter that tracks how the system evolves. If the system stops moving (no interactions, no potential energy), the "time" parameter loses its physical meaning. The paper insists that we shouldn't pretend time is a fundamental symmetry of the universe; it's a feature of the motion within the universe.

The Bottom Line

So, what is the paper's verdict?

It rules out the idea that "time translation" is a fundamental symmetry of the Galilei group. It argues that including it is a mathematical error that doesn't fit real, interacting physics.
It suggests that the Newtonian mass is a fundamental "Casimir invariant" of the symmetry, meaning mass is an intrinsic, unchangeable property of a particle's quantum identity, not just a number we measure.
It clarifies that for composite systems, the "position" is not a simple sum of parts but a center-of-mass calculation dictated by the symmetry, fixing a logical hole in how we combine particles.

The paper doesn't claim to have discovered a new particle or a new force. Instead, it claims to have cleaned up the instruction manual. It suggests that if we take the math of symmetry seriously—without forcing in the "time translation" page that doesn't belong—we get a clearer, more logical picture of why particles have mass, how they move together, and what time actually is.

It's a reminder that sometimes, the most important thing in physics isn't finding a new piece of the puzzle, but realizing that one of the pieces we've been holding upside down for decades needs to be flipped over.

Technical Summary: Quantum Origin of (Newtonian) Mass and Galilean Relativity Symmetry

Problem Statement
The paper addresses a foundational inconsistency in the standard group-theoretical formulation of non-relativistic quantum mechanics. While the Galilei group $G(3)$ (or its central extension $\tilde{G}(3)$ ) is traditionally accepted as the fundamental symmetry for non-relativistic physics, the standard application of this symmetry often conflates the abstract Lie algebra structure with specific operator representations. Specifically, the conventional treatment identifies position operators $\hat{X}_i$ directly as generators of the Heisenberg-Weyl algebra $H(3)$ , treating the central charge as a trivial identity operator scaled by an arbitrary parameter. This approach fails to rigorously account for the definition of mass as a Casimir invariant and leads to unphysical results when applied to composite systems, particularly regarding the additivity of position observables. Furthermore, the inclusion of time-translation symmetry (generator $H$ ) within the fundamental relativity symmetry is questioned, as it does not align with the dynamical generation of Newtonian time in the presence of interactions.

Methodology
The author employs a strict group-theoretical and symmetry-theoretical formulation, distinguishing clearly between abstract Lie algebras, their universal enveloping algebras, and their unitary representations on Hilbert spaces. The methodology involves:

Re-evaluating Generators: Shifting the fundamental generators of the Heisenberg-Weyl symmetry from $\{ \hat{X}_i, \hat{P}_j, \hat{I} \}$ to $\{ \hat{K}_i, \hat{P}_j, \hat{M} \}$ , where $\hat{K}_i$ are boost generators and $\hat{M}$ is the mass operator (central charge).
Representation Theory Analysis: Analyzing irreducible unitary representations of the symmetry group to identify Casimir invariants. The study treats the mass $m$ not as a fixed scalar parameter but as the eigenvalue of the Casimir operator $\hat{M}$ .
Composite System Construction: Applying the tensor product rule for representations to composite systems (e.g., two particles) to derive the correct operators for total mass, momentum, and position.
Symplectic Geometry and Dynamics: Utilizing the symplectic structure of the Hilbert space (phase space) to derive the dynamical evolution. The physical Hamiltonian is treated as a generic element of the observable algebra generating Hamiltonian flows, rather than a fixed generator of the fundamental symmetry group.

Key Contributions and Results

Mass as a Casimir Invariant: The paper establishes that Newtonian mass is a Casimir invariant of the fundamental relativity symmetry (specifically the subgroup $HR(3)$, which excludes time translation). In the standard $H(3)$ picture, the central charge is often treated as a redundancy. Here, the mass operator $\hat{M}$ is identified as the generator of the central extension, and its eigenvalue $m$ characterizes the irreducible representation. This provides a symmetry-theoretical origin for the concept of mass.
Resolution of the Composite System Paradox: By identifying position operators $\hat{X}_i$ not as direct generators but as derived quantities defined by $\hat{X}_i = \hat{M}^{-1}\hat{K}_i$ , the paper resolves the issue of additive position observables. In a composite system of two particles with masses $m_a$ and $m_b$ , the total mass operator is additive ( $\hat{M} = \hat{M}_a \otimes I + I \otimes \hat{M}_b$ ), and the total momentum is additive. However, the position operator for the composite system is derived as:
$\hat{X} = \frac{1}{m}(m_a \hat{X}_a \otimes I + I \otimes m_b \hat{X}_b)$
This naturally yields the center-of-mass operator, avoiding the unphysical notion that position observables simply sum like mass or momentum.
Exclusion of Time-Translation Symmetry: The paper argues that the time-translation generator $H$ (from the full $\tilde{G}(3)$ ) should be excluded from the fundamental relativity symmetry. The generator $H$ in the Galilei algebra corresponds only to the kinetic energy of a free particle ( $\hat{H} \propto \hat{P}^2/2m$ ) and cannot account for potential energy or interactions.
Dynamical Origin of Time: Newtonian time $t$ is redefined not as a parameter of a symmetry transformation, but as the evolution parameter of the physical Hamiltonian $\hat{H}_{phys}$ . The physical time is the parameter of the unitary flow generated by the specific Hamiltonian of the system (which includes interactions). Consequently, "time translation symmetry" in the fundamental group sense is physically irrelevant for dynamical theories involving non-trivial interactions.
The $HR(3)$ Symmetry: The paper proposes $HR(3)$ (the semidirect product of rotations, boosts, and translations, excluding time translation) as the correct fundamental symmetry for non-relativistic quantum mechanics. This symmetry dictates the structure of the phase space and the definition of the center of mass.

Significance and Claims
The paper claims that a rigorous adherence to symmetry theory reveals that the Newtonian concepts of mass and the center of mass are not merely postulates but are dictated by the representation theory of the quantum symmetry group. The identification of mass as a Casimir invariant is presented as a necessary feature for a consistent formulation of composite systems.

Furthermore, the work asserts that the standard inclusion of time-translation symmetry in the fundamental relativity group is a conceptual error that obscures the dynamical nature of time. The authors argue that time in non-relativistic physics is strictly a dynamical parameter arising from the Hamiltonian flow, not a geometric coordinate of a background spacetime symmetry. This formulation suggests that the "quantum model of physical space" is best understood as the irreducible representation space of the $HR(3)$ symmetry, where position and momentum are non-commutative coordinates, rather than a classical configuration space.

The paper concludes that these adjustments, while mathematically minor in terms of algebraic structure, are physically significant for correctly interpreting mass, composite systems, and the nature of time within the framework of non-relativistic quantum mechanics. It emphasizes that the "empty" spacetime symmetry is of little physical interest compared to the symmetry of the phase space governing particle dynamics.

The Big Mistake: The "Time" Page

The New Star: Mass as a "Cosmic ID Card"

The Composite Puzzle: Why You Can't Just Add Positions

The "Time" is a Flow, Not a Switch

The Bottom Line

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