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*Another example is quantum mechanics, where the arbitrariness of the phase of the wave function can be considered a symmetry: Although quantum mechanics can be reformulated in terms of phase-invariant probabilities, currents, or density matrices instead of wave functions, and this can be useful for some purposes of exposing physical properties, formulating and solving the Schrodinger equation is simpler in terms of the wave function. The same applies to “local” symmetries, where there is an independent symmetry at each point of space and time: For example, quarks and gluons have a local “color” symmetry, and are not (yet) observed independently in nature, but are simpler objects in terms of which to describe strong interactions than the observed hadrons (protons, neutrons, etc.), which are described by color-invariant products of quark/gluon wave functions, in the same way that probabilities are phase-invariant products of wave functions.*

(Note that in quantum mechanics there is a subtle distinction between observed and observer that can obscure this symmetry if the observer is not invariant under it. This can always be avoided by choosing to define the observer as invariant: For example, the detection apparatus can be included as part of the quantum mechanical system, while the observer can be defined as some “remote” recorder, who may be abstracted as even being translationally invariant. In practice we are less precise, and abstract even the detection apparatus to be invariant: For example, we describe the scattering of particles in terms of the coordinates of only the particles, and deal with the origin problem as above in terms of just those coordinates.)

(Note that in quantum mechanics there is a subtle distinction between observed and observer that can obscure this symmetry if the observer is not invariant under it. This can always be avoided by choosing to define the observer as invariant: For example, the detection apparatus can be included as part of the quantum mechanical system, while the observer can be defined as some “remote” recorder, who may be abstracted as even being translationally invariant. In practice we are less precise, and abstract even the detection apparatus to be invariant: For example, we describe the scattering of particles in terms of the coordinates of only the particles, and deal with the origin problem as above in terms of just those coordinates.)

I am having problems understanding the second paragraph. I have an intuitive understanding of the difference between

*observer*and

*observed*. The detection apparatus, for example, is an observer, isn't it?

What does one mean when one says that

*an observer is invariant under a symmetry*?