Symmetry: Physics must not depend on the physicist
Everybody must agree on the action
A central theme of fundamental physics has been the overarching importance of symmetry. Indeed, I am so enamored of the concept that I devoted an entire book to symmetry,1 to which I refer the reader for details. Einstein’s special relativity offers a canonical example of a symmetry in physics. In chapter 7, I wrote that Einstein insisted that the laws of physics must not depend on observers in uniform motion relative to each other. This insistence has since been generalized and formulated as a principle: while physical reality can appear different to different observers, the structure of physical reality must be the same.
I am necessarily being a bit vague here. The action principle, however, allows us to render the phrase “structure of physical reality” a bit more precisely. Different observers must agree on the action. Otherwise, different observers would be extremizing different actions and getting different deals.
Symmetry implies transformation from one observer’s frame of reference to another’s. Thus, for example, in special relativity, we transform from the passenger’s conception of physics to the stationmaster’s. For example, what looks like an electric field to the stationmaster is perceived by the passenger as a combination of an electric field and a magnetic field.
Covariance versus invariance
Elementary physics is typically formulated in terms of equations, such as Newton’s equation of motion or Maxwell’s equation of electrodynamics. Under a symmetry transformation, both sides of these equations would change. To be specific, consider special relativity. A bit of useful jargon: the transformation of physical quantities from one frame of reference to another in special relativity is known as a Lorentz transformation, in honor of Hendrik Lorentz (1853–1928).
For example, the equation determining the electric field generated by a bunch of charges sitting there (in other words, in the absence of any electric current) has the form
(variation of electric field in a space) = (charge distribution)
Under a Lorentz transformation, the quantities on the two sides of the equal sign both change, but in such a way that they remain equal.
Physically, suppose the stationmaster sees some electric charges sitting on the platform, generating an electric field. The passenger on the train going through the station would see the charges moving, that is, an electric current generating a magnetic field as well as an electric field.
In physicist’s jargon, the equation is said to be covariant (“changing together”), rather than invariant (“not changing”). The two sides of the equation change in the same way, rather than remain unchanged. As a result, while the physical quantities involved change, the structural relationship between them does not.
As a rough analogy, one can think of a marriage in which the two partners “grow” with the years. In those rare cases in which the husband and wife both grow in the same direction and at the same rate, the relationship between them would remain the same, even though neither of them does. Unfortunately, psychologists tell us that most human relationships are not covariant in time (and certainly not invariant).
In contrast to the equations of motion, the action for electromagnetism is left invariant by a Lorentz transformation. The action remains unchanged. Indeed, to say that physics possesses a certain symmetry is to say that the action is invariant under the transformation associated with that symmetry. As a result, a history seen by different observers is labeled by the same number, so there can be no dispute about which history is favored by the action principle. The action, in short, embodies the structure of physical reality.*
* The power and elegance of the action formulation of physics is often admired by deep thinkers in other subjects. The eminent economist Paul Samuelson, for example, expressed his great admiration for Fermat’s least time principle in his Nobel lecture of 1970, as quoted on p. 357 of Steve Weinberg, Gravitation and Cosmology.