Symmetry: A ‘Key to Nature’s Secrets’ - Steven Weinberg

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It would take far more space than I have here to go into details about these symmetries and the Standard Model, or about other proposed symmetries that go beyond those of the Standard Model. Instead I want to take up one aspect of symmetry that as far as I know has not yet been described for general readers. When the Standard Model was put in its present form in the early 1970s, theorists to their delight encountered something quite unexpected. It turned out that the Standard Model obeys certain symmetries that are accidental, in the sense that, though they are not the exact local symmetries on which the Standard Model is based, they are automatic consequences of the Standard Model. These accidental symmetries accounted for a good deal of what had seemed so mysterious in earlier years, and raised interesting new possibilities.

The origin of accidental symmetries lies in the fact that acceptable theories of elementary particles tend to be of a particularly simple type. The reason has to do with avoidance of the nonsensical infinities I mentioned at the outset. In theories that are sufficiently simple these infinities can be canceled by a mathematical process called “renormalization.” In this process, certain physical constants, like masses and charges, are carefully redefined so that the infinite terms are canceled out, without affecting the results of the theory. In these simple theories, known as “renormalizable” theories, only a small number of particles can interact at any given location and time, and then the energy of interaction can depend in only a simple way on how the particles are moving and spinning.

For a long time many of us thought that to avoid intractable infinities, these renormalizable theories were the only ones physically possible. This posed a serious problem, because Einstein’s successful theory of gravitation, the General Theory of Relativity, is not a renormalizable theory; the fundamental symmetry of the theory, known as general covariance (which says that the equations have the same form whatever coordinates we use to describe events in space and time), does not allow any sufficiently simple interactions. In the 1970s it became clear that there are circumstances in which nonrenormalizable theories are allowed without incurring nonsensical infinities, but that the relatively complicated interactions that make these theories nonrenormalizable are expected, under normal circumstances, to be so weak that physicists can usually ignore them and still get reliable approximate results.

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Now, it just so happens that under the constraints imposed by Lorentz invariance and the exact local symmetries of the Standard Model, the most general renormalizable theory of strong and electromagnetic forces simply can’t be complicated enough to violate mirror symmetry.6 Thus, the mirror symmetry of the electromagnetic and strong nuclear forces is an accident, having nothing to do with any symmetry built into nature at a fundamental level. The weak nuclear forces do not respect mirror symmetry because there was never any reason why they should. Instead of asking what breaks mirror symmetry, we should have been asking why there should be any mirror symmetry at all. And now we know. It is accidental.

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