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.