Section 11–2 Translations

(Translation in space / Translation of axes / Spatial translation symmetry)

In this section, Feynman discusses a translation of an apparatus in space, translation of axes, and spatial translation symmetry.

1. Translation in space:

“So the proof that an apparatus in a new position behaves the same as it did in the old position is the same as the proof that the equations when displaced in space reproduce themselves (Feynman et al., 1963, section 11–2 Translations).”

In the previous section, Feynman mentions how a phenomenon remains unchanged if we move all equipment and essential influences, but not everything in the universe (e.g., planets). There could be clarifications whether the Earth or planets are displaced in this section. For example, Feynman simply says that an apparatus will behave in the same way when it is displaced in space. Importantly, the physical laws and mathematical equations remain the same in two different coordinate systems imply that the phenomena should appear the same. On the other hand, physics teachers may add that the space in our universe is not a crystal lattice with discrete steps for a translation in space (Lederman & Hill, 2008). In essence, there is no smallest step for an observer or experimenter to translate in the space that we are in.

One may prefer Feynman’s explanations of translation in space to the general public: “If I actually built such an apparatus, and then displaced it 20 feet to the left of where I am now it would get into the wall, and there would be difficulties. It is necessary, in defining this idea, to take into account everything that might affect the situation, so that when you move the thing you move everything. For example, if the system involved a pendulum, and I moved it 20,000 miles to the right, it would not work properly anymore because the pendulum involves the attraction of the earth. However, if I imagine that I move the earth as well as the equipment then it would behave in the same way (Feynman, 1965, p. 85).” This is different from his statement in the previous section that not everything in the universe is moved.

2. Translation of axes:

“We have also found the formula for F_x′, for, substituting from Eq. (11.1), we find that F_x′ = F_x (Feynman et al., 1963, section 11–2 Translations).”

The laws of mechanics can be summarized by three equations for a particle: m(d²x/dt²) =F_x, m(d²y/dt²) = F_y, m(d²z/dt²) = F_z. This means that we can carry out measurements in x, y, and z, as well as forces on three perpendicular axes. Suppose Joe, an experimenter whose origin is in one place, measures the location of a point in space as x, y, and z. Moe, another experimenter whose origin is in somewhere else, measures the location of the point in space as x′, y′, and z′ such that their measurements can be related by the equations: x′ = x − a, y′ = y, and z′ = z. If Moe’s origin is fixed relative to Joe’s, we have dx′/dt = dx/dt and d²x′/dt² = d²x/dt² because a is a constant and da/dt = 0. More important, Joe and Moe would observe the same phenomenon and apply the same physical law by using the equations F_x′ = F_x, F_y′ = F_y, and F_z′ = F_z.

In a sense, a translation of axes is equivalent to a thought experiment in which we need to move everything including the apparatus and the Earth. Specifically, the experiment can be performed by shifting an observer by a fixed distance to another location. Simply put, we can measure x′, y′, and z′ in the experiment by using a different reference point (or origin). However, Hollebrands (2003) found that students’ conceptions about the translation of axes may be based on their perceived motion of an object. Furthermore, they may incorporate physical concepts as part of their thinking about the translations of axes, such as the need for a force to move an object to overcome friction.

3. Spatial translational symmetry:

“…we say that the laws of physics are symmetrical for translational displacements, symmetrical in the sense that the laws do not change when we make a translation of our coordinates (Feynman et al., 1963, section 11–2 Translations).”

Feynman explains that the working of a machine, whether it is analyzed by Moe or Joe, can be represented by the same equations. Thus, the phenomena will appear the same because the equations are the same. Physicists can claim that the physical laws are symmetrical for translational displacements. It is symmetrical in the sense that the physical laws remain unchanged when we make a translation of our coordinates. Interestingly, Feynman adds that there is no unique way to define the origin of the world because the laws will appear the same from wherever the position they are observed. In other words, we can never identify the center of the universe or an origin where it would make a difference.

In one of his Messenger lectures, Feynman says that “[i]f we assume that the laws of physics are describable by a minimum principle, then we can show that if a law is such that you can move all the equipment to one side, in other words, if it is translatable in space, then there must be conservation of momentum. There is a deep connection between the symmetry principles and the conservation laws, but that connection requires that the minimum principle be assumed (Feynman, 1965, p. 103).” The invariance of the action with respect to a translation in space is also called the homogeneity of space (Landau & Lifshitz, 1976). It means that all points in the universe are equivalent to each other and it does not matter where we perform an experiment. Thus, physicists can state that the law of conservation of momentum results from the homogeneity of space.

Questions for discussion:

1. What does it mean when physicists say that there is a “translation in space”?

2. What are possible misconceptions of a “translation of axes”?

3. What does it mean when physicists say that the laws of physics are symmetrical for translational displacements?

The moral of the lesson: physical laws are symmetrical for translational displacements in the sense that the physical laws remain unchanged when we make a translation of coordinates.

References:

1. Feynman, R. P. (1965). The character of physical law. Cambridge: MIT Press.

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. Hollebrands, K. (2003). High school students’ understandings of geometric transformations in the context of a technological environment. Journal of Mathematical Behavior, 22(1), 55-72.

4. Lederman, L. M., & Hill, C. T. (2008). Symmetry and the beautiful universe. New York: Prometheus.

5. Landau, L. D., & Lifshitz, E. M. (1976). Mechanics (3rd ed.). Oxford: Pergamon Press.

Section 11–1 Symmetry in physics

(Definition of symmetry / Symmetry in phenomena / Symmetry in physical laws)

In this section, Feynman discusses Weyl’s definition of symmetry, symmetry in phenomena, and symmetry in physical laws.

1. Definition of symmetry:

“Professor Hermann Weyl has given this definition of symmetry: a thing is symmetrical if one can subject it to a certain operation and it appears exactly the same after the operation (Feynman et al., 1963, section 11–1 Symmetry in physics).”

The word “symmetry” derives from the Greek words sun (meaning with or together) is redefined in this chapter with a special meaning. In Weyl’s words, symmetry is defined as the “invariance of a configuration of elements under a group of automorphic transformations (1952, preface).” Feynman adopts Weyl’s definition of symmetry and gives an analogy whereby a silhouette of a vase that is left-and-right symmetrical appears the same if it is turned 180^o around a vertical axis. Importantly, Feynman later adds that “the other laws of physics, so far as we know today, have the two properties which we call invariance (or symmetry) under translation of axes and rotation of axes (Feynman et al., 1963, Section 11.4).” In short, symmetry means invariance under transformations.

In one of his Messenger Lectures, Feynman (1965) mentions that “physicists delight themselves by using ordinary words for something else (p. 84).” In Volume I, Chapter 52 of The Feynman Lectures, he says that the problem of defining symmetry is an interesting one. Phrased formally, Feynman has another definition of symmetry: “A physical system is symmetric with respect to the operation Q when Q commutes with U, the operation of the passage of time.” (Feynman, 1966, Vol III, section 17-3). More important, symmetry may be defined as “change without change (Wilczek, 2008, p. 386).” In other words, the idea of symmetry is applicable to a system of equations where there are changes in the mathematical quantities that appear in the equations without changing the meaning (or the phenomenon) of the system.

2. Symmetry in phenomena:

“We have to understand what we mean when we say that the phenomena are the same when we move the apparatus to a new position (Feynman et al., 1963, section 11–1 Symmetry in physics).”

According to Feynman, it is important to understand what we mean if a phenomenon remains unchanged when we shift the apparatus to a new location. It means that we need to move everything that is relevant; if the phenomenon is changed, it implies that something relevant has not been moved. If physicists are unable to identify the relevant condition, then they may claim that the laws of physics do not have this symmetry. An important question is whether physicists are able to define physical concepts and experimental conditions well enough. If all of the essential forces and conditions are included inside the apparatus and all of the relevant parts are moved from one place to another, then the physical laws should remain unchanged.

In a Messenger lecture titled Symmetry in physical law, Feynman (1965) says that “[s]ymmetry seems to be absolutely fascinating to the human mind. We like to look at symmetrical things in nature, such as perfectly symmetrical spheres like planets and the sun, or symmetrical crystals like snowflakes, or flowers which are nearly symmetrical. However, it is not the symmetry of the objects in nature that I want to discuss here; it is rather the symmetry of the physical laws themselves. It is easy to understand how an object can be symmetrical, but how can a physical law have a symmetry? (p. 84).” This introduction is possibly more appropriate right at the beginning of the lecture because it supports students’ learning by transiting from concrete to abstract thinking.

3. Symmetry in physical laws:

“…if the laws of physics do have this symmetry; looking around, we may discover, for instance, that the wall is pushing on the apparatus (Feynman et al., 1963, section 11–1 Symmetry in physics).”

In section 11.4 Vectors, Feynman emphasizes that this chapter is mainly about the symmetry of physical laws. That is, a phenomenon remains unchanged if we move all equipment and essential influences, but not everything in the universe (e.g., planets and stars). Interestingly, Feynman discusses Noether’s theorem (connection between conservation laws and symmetries) in one of his Messenger lectures to the general public. In his own words, “[i]t is extremely interesting that there seems to be a deep connection between the conservation laws and the symmetry laws. This connection has its proper interpretation, at least as we understand it today, only in the knowledge of quantum mechanics (Feynman, 1965, p. 103).”

Feynman was reserved about changing laws of physics and Dirac’s proposal of assigning a time dependence to the gravitational forces. In a lecture to postgraduates, Feynman (1995) explains that “it is very difficult to define what one means by saying that the forces of gravitation are time-dependent while everything else ‘remains the same.’ Since the significant numbers are the dimensionless ratios of things, he might just as well describe the situation by saying that the electric charge is time-dependent, so that his theory is really not well defined (p. 8).” Currently, physicists may explain that there are indicators of the stability of physical laws through time. For example, an analysis of the abundance of the Oklo’s samarium indicates that the magnitude of the electric charge could not have changed by more than one part in ten million since Oklo was burning uranium (Lederman & Hill, 2008).

Questions for discussion:

1. How would you define the concept of symmetry?

2. What does it mean when we say that the phenomena are the same when we move the apparatus to a new position?

3. What are the symmetries in physical laws?

The moral of the lesson: the concept of symmetry is applicable to a system of equations such that there are changes in the mathematical quantities that appear in the equations without changing physical laws.

References:

1. Feynman, R. P. (1965). The character of physical law. Cambridge: MIT Press.

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. (1966). The Feynman lectures on physics Vol III: Quantum Mechanics. Reading, MA: Addison-Wesley.

4. Feynman, R. P., Morinigo, F. B., & Wagner, W. G. (1995). Feynman Lectures on gravitation (B. Hatfield, ed.). Reading, MA: Addison-Wesley.

5. Lederman, L. M., & Hill, C. T. (2008). Symmetry and the beautiful universe. New York: Prometheus.

6. Weyl, H. (1952). Symmetry. Princeton: Princeton University Press.

7. Wilczek, F. (2015). A Beautiful Question: Finding Nature’s Deep Design. New York: Penguin Press.

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.