Momentum Reconsidered: From Newtonian Particles to Relativistic Fields and Quantum Waves

​In chapter 10-5 of The Feynman Lectures on Physics, Feynman examines how the classical law of conservation of momentum survives the transition from Newtonian mechanics to modern physics. Rather than presenting conservation laws as immutable formulas, he emphasises that their validity depends on how fundamental quantities are defined. Momentum, in particular, remains conserved, but its meaning must be broadened to accommodate relativity, electromagnetism, and quantum mechanics.

Feynman begins by revisiting the classical definition of momentum as the product of mass and velocity. In Newtonian mechanics, mass is treated as a fixed property of a particle, independent of its motion. Special relativity alters this assumption. To ensure that momentum is conserved in all inertial frames, mass must depend on velocity. As a particle’s speed approaches the speed of light, its effective mass increases, causing its momentum to grow more rapidly than predicted by classical theory. Feynman stresses that this modification has no practical consequence at ordinary speeds, where relativistic effects are extremely small and the classical expression for momentum remains an excellent approximation. This continuity reassures the reader that classical mechanics is not wrong, but incomplete, valid only within a limited range of conditions.

The discussion then turns to the conservation of momentum in interactions such as collisions. Even with velocity-dependent mass, the momentum of each particle can be resolved into components, and the total momentum in any chosen direction is conserved before and after the interaction. This example reinforces the idea that relativity does not destroy conservation laws; instead, it preserves their structure by refining the quantities involved.

Feynman next draws a comparison between momentum and energy. Historically, the conservation of energy appeared to fail until physicists recognized that energy can take many forms, including heat. Heat energy, arising from microscopic random motion, can be difficult to detect directly and is therefore sometimes described as “hidden.” This analogy naturally raises the question of whether momentum might also exist in hidden forms. Feynman argues that, in ordinary matter, it is much harder to conceal momentum than energy. Heat depends on the squares of particle velocities and has no preferred direction, whereas momentum depends on the vector sum of velocities. Random internal motions cancel out directionally, so a body possesses net momentum only if it moves as a whole. As a result, mechanical momentum is generally obvious and not easily hidden within matter.

Despite this, Feynman shows that momentum can be hidden in a more subtle way, namely in fields. This conclusion follows from abandoning the Newtonian idea that forces act instantaneously at a distance. Electromagnetic influences propagate at a finite speed, so when one charged particle is moved, another charge elsewhere does not respond immediately. During the brief interval before the influence arrives, the momentum of the particles alone does not appear to be conserved. Feynman resolves this apparent paradox by introducing the concept of momentum carried by the electromagnetic field itself. When field momentum is included alongside particle momentum, total momentum is conserved at every moment. This insight marks an important conceptual shift: fields are not merely mathematical devices for calculating forces, but physical entities that store and transport energy and momentum.

The reality of field momentum is further illustrated by the example of light. Electromagnetic waves carry momentum, and when light strikes an object, it transfers momentum to that object, producing a force. Although this radiation pressure is extremely small under everyday conditions, it can be measured with sufficiently sensitive instruments. The fact that light can exert pressure provides direct experimental evidence that momentum is not restricted to material particles.

Finally, Feynman extends the discussion to quantum mechanics, where the classical picture of particles following definite paths breaks down. In this context, velocity is no longer a clearly defined quantity, yet momentum remains meaningful. When particles are described as waves, momentum is associated with the spatial variation of the wave rather than with motion along a trajectory. Even though the classical law relating force, mass, and acceleration fails in quantum mechanics, the conservation of momentum continues to hold. This persistence underscores that conservation laws are more fundamental than the specific mechanical models used to derive them.

Overall, Feynman’s treatment of relativistic momentum highlights a central theme of modern physics: fundamental conservation principles endure across theoretical revolutions, but their interpretation evolves. By examining how momentum is redefined in relativity, extended to fields, and reinterpreted in quantum mechanics, Feynman demonstrates how physics progresses by deepening concepts rather than discarding them.