Section 10–5 Relativistic momentum

(Relativistic momentum / Field momentum / Quantum momentum)

In this section, the three interesting points are relativistic momentum, field momentum, and momentum in quantum theory. The section could be titled as three different concepts of momentum instead of relativistic momentum. It is also closely related to three limitations (or apparent violations) of the law of conservation of momentum.

1. Relativistic momentum:

“In the theory of relativity it turns out that we do have conservation of momentum; the particles have mass and the momentum is still given by mv, the mass times the velocity, but the mass changes with the velocity, hence the momentum also changes… (Feynman et al., 1963, section 10–5 Relativistic momentum).”

In the context of special theory of relativity, Feynman defines momentum in terms of mv, the mass of an object times its velocity, but the mass changes with the velocity according to the law m = m₀√(1−v²/c²) where m₀ is the rest mass of the object and c is the speed of light. Some physicists may disagree and argue that Feynman is promoting an outdated notion of mass which is sometimes known as relativistic mass. They would prefer a revision of the formula as p = gmv in which g is the Lorentz factor and m is the invariant mass. This formula is not applicable to photons because the invariant mass of a photon is zero.

In a sense, a violation in the law of conservation of momentum is possible for objects that move at very fast speeds if one still uses the formula p = m₀v. According to Feynman, the law of conservation of momentum needs a modification such that it still holds in the special theory of relativity. Feynman’s proposed use of relativistic mass is related to Einstein’s principle of equivalence of mass and energy. In his own words, “[a] photon of frequency ω₀ has the energy E₀ = ℏω₀. Since the energy E₀ has the relativistic mass E₀/c² the photon has a mass (not rest mass) ℏω₀/c², and is “attracted” by the earth” (Feynman et al., 1964, section 42–6).” However, some physicists may disagree and explain that mass and energy are not completely equivalent.

2. Field momentum:

“If we add the field momentum to the momentum of the particles, then momentum is conserved at any moment all the time (Feynman et al., 1963, section 10–5 Relativistic momentum).”

There is a need for another definition of momentum in the context of electrodynamics. Feynman explains that momentum can be hidden in the electromagnetic field as an effect of relativity. In Chapter 27, Volume II of The Feynman Lectures, he elaborates that there is field momentum in the electromagnetic field due to the presence of energy and it will have a certain momentum per unit volume. Essentially, charged particles may cause varying electromagnetic fields, and fields possess energy and momentum just like the particles. In short, varying electromagnetic fields have electromagnetic waves such as visible light which carries momentum with it.

In general, Newton’s third law holds in electrostatics and magnetostatics, but not in classical electrodynamics. More important, physicists add field momentum to the momentum of the particles such that the total momentum is conserved at any time. In situations involving electrical forces, for instance, if an electrical charge moves suddenly, its electrical influences on another electric charge at another location do not appear instantaneously. There is a little delay because it takes time for the influence to move at 186,000 miles a second. Thus, the total momentum of the particles is not really conserved during this short duration of time. However, after the second charge has felt the effect of the first charge, the momentum equation can be balanced again.

3. Quantum Momentum:

“Now in quantum mechanics, it turns out that momentum is a different thing—it is no longer mv (Feynman et al., 1963, section 10–5 Relativistic momentum).”

In quantum mechanics, an object can be conceptualized as a system of particles or waves. Firstly, Feynman mentions that the momentum of an object is still mv if we conceptual it as a particle. Alternatively, we can conceptualize the object as waves and measure its momentum by using the formula, p = h/λ. For example, we can describe the momentum of a photon as follows: “the distance D that it takes for the spatially periodic electric disturbances within a photon to go through one complete cycle is related to the photon’s momentum P through PD = h, where h is Planck’s constant (Wilczek, 2008, p. 234).” Simply phrased, the momentum of light waves can be defined in terms of a number of waves per unit length: a greater number of waves imply a greater momentum.

Feynman states that Newton’s second law F = ma is false, but the law of conservation of momentum still holds in quantum theory. Some physicists disagree that the momentum is conserved in quantum theory by using Heisenberg’s uncertainty principle. From an empirical viewpoint, there is always a spread in the measurements of momentum. Bohr, Kramers, and Slater (1924) also abandon the strict law of conservation of momentum in atomic events and consider the conservation to be a result of statistical averaging. Bohr’s philosophy of physics can be briefly described as “theoretical concepts, including assertions of the reality of entities or of their properties, cannot be used unambiguously without careful reference to the experimental arrangement in which the concepts are applied (Shimony, 1989, p. 394).”

Questions for discussion:

1. How would you define momentum in the context of the special theory of relativity?

2. How would you explain that momentum is still conserved in the context of classical electrodynamics?

3. Is momentum strictly conserved in quantum theory?

The moral of the lesson: the law of conservation of momentum has undergone some modifications by redefining the concept of momentum in the special theory of relativity, classical electrodynamics, and quantum theory.

References:

1. Bohr, N., Kramers, H. A., & Slater, J. C. (1924). LXXVI. The quantum theory of radiation. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 47(281), 785-802.

2. Feynman, R. P., Leighton, R. B., & Sands, M. (1963). The Feynman Lectures on Physics, Vol I: Mainly mechanics, radiation, and heat. Reading, MA: Addison-Wesley.

3. Feynman, R. P., Leighton, R. B., & Sands, M. (1964). The Feynman Lectures on Physics, Vol II: Mainly electromagnetism and matter. Reading, MA: Addison-Wesley.

4. Shimony, A. (1989). Conceptual Foundations of Quantum Mechanics. In P. Davies, (1989). The New Physics. pp. 373-95.

5. Wilczek, F. (2008). The lightness of being: Mass, ether, and the unification of forces. New York: Basic Books.