Unitary operators and spacetime symmetries
In quantum mechanics, certain unitary operators have been understood since the time of Wigner in terms of spacetime symmetries. Why?
The foundation for this kind of thinking has an interpretive and a mathematical aspect. The interpretive aspect has to do with the way we connect certain observables to experience; the mathematical aspect has to do with the way unitary operators look under this interpretation.
First, on interpreting observables. We’ll take an observable to be a self-adjoint operator acting on a Hilbert space. It’s well known that quantum mechanics must make some assumption about how to connect this operator to measurement; one common such assumption is the eigenvalue-eigenstate link. However, we make an additional interpretive assumption about some (though not all) observables, which is the
Assumption. The expected or average value ⟨O⟩ of an observable O can be identiﬁed with a vector in spacetime.
For example, an eigenstate of the position operator in a single-particle Hilbert space assigns the property ‘there is a particle located here’ to a vector in 3-dimensional space. Average position can thus be identiﬁed with the average of these vectors. Similarly, an eigenstate of an angular momentum operator assigns a property like ‘spin +1’ to a direction (say, a unit vector) in space. Average angular momentum can thus be identiﬁed with the average of these vectors.
That’s our interpretive connection between quantum theory and spacetime. Now, here’s the mathematical part: such unitary operators turn out to implement spacetime symmetries under this interpretation. In short, it can be proved that these unitary transformations are equivalent to symmetry transformations of the corresponding spacetime structures.
The details of how this works of course depends on the situation. But it’s useful to see one example. Consider the angular momentum operators Sx, Sy, Sz on the Hilbert space of a spin-1/2 system. When we interpret these observables, the expected values (⟨Sx⟩, ⟨Sy⟩, ⟨Sz⟩) form a vector, at a point in the background spacetime.
Now, a unitary operator U is a symmetry transformation on vectors in Hilbert space: ψ → Uψ. It can also be viewed as transforming observables O → U−1OU. That’s because:
⟨Uψ, SxUψ⟩ = ⟨ψ, U−1SxUψ⟩ = ⟨U−1SxU⟩
Moreover, there is an operator Rz(θ), which can be shown (see, e.g., Sakurai 1994, 3.2) to transform the expected value of the angular momentum observables as follows:
⟨Rz−1(θ) Sx Rz (θ)⟩ = ⟨Sx ⟩cosθ − ⟨Sy ⟩sinθ
⟨Rz−1(θ) Sy Rz (θ)⟩ = ⟨Sx⟩sinθ + ⟨Sy⟩cosθ
⟨Rz−1(θ) Sz Rz(θ)⟩ =⟨Sz⟩.
In other words, transforming Hilbert space by the unitary operator Rz(θ) has the same effect as applying a rotation matrix through a degree θ about the z-axis, to vectors in spacetime. And of course, such a transformation can equivalently be implemented by just rotating the background spacetime, instead of the vector itself. Thus, such unitary transformations can be equivalently understood as implementing spacetime symmetries.
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Did you ever figure out why Rz takes the form it does? (I think that’s what you asked me the other day.)
Oh, sure — The R operators are just a unitary representation of the rotation group SO(3). That is, you can define them abstractly to be a unitary-operator copy of SO(3). You guarantee the appropriate group properties by demanding that the R’s are generated by operators (Sx, Sy, Sz) obeying the commutation relations. That turns out to be all you need to get the above results.
Actually, this calculation is just Wigner’s, from his original (1939) work on the hydrogen atom. The version found in Sakurai (1994) turns out to be almost the same thing!