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.