Unitary and Antiunitary Operator Transport

A unitary or antiunitary symmetry transports an unbounded observable only after transporting its domain. For antiunitary maps, it also conjugates scalar coefficients in spectral functions. This page extends the elementary rules in Antiunitary Symmetries to self-adjoint operators and their functional calculus. The existence of such implementations for ray symmetries is the separate content of Wigner’s Theorem.

Required background. The Unbounded Spectral Theorem defines spectral functions and their domains. Helpful background. Antiunitary Symmetries supplies the conjugate-linearity rules.

Transport the action and domain together

Let $V$ be unitary or antiunitary on $\mathcal H$, and let $A$ be densely defined. The transported operator is

$$A_V=VAV^{-1}, \qquad D(A_V)=VD(A).$$

Although an antiunitary $V$ is conjugate-linear, $A_V$ is linear: the two conjugations cancel. Its action is $A_V(V\psi)=VA\psi$. A symmetry condition $A_V=A$ therefore includes both $VD(A)=D(A)$ and equality of actions on that domain.

Transport preserves closedness, since $(\psi_n,A\psi_n)\to(\psi,\eta)$ if and only if the transported graph pairs converge to $(V\psi,V\eta)$. It also preserves self-adjointness. The latter follows from the adjoint definition, or from the spectral construction below when $A$ is self-adjoint.

Formal agreement between differential expressions is weaker than this operator equality. Boundary conditions can change under $V$ even when the local formula has the expected sign.

Spectral projections transport without a conjugated event

Suppose $A$ is self-adjoint with PVM $E_A$. Define

$$E_V(\Delta)=VE_A(\Delta)V^{-1}$$

for Borel subsets $\Delta\subseteq\mathbb R$. These are orthogonal projections, their products obey the same intersection rule, and their countable sums converge strongly. Thus $E_V$ is a PVM.

For $\psi'=V\psi$, its scalar spectral measure is unchanged:

$$\langle\psi',E_V(\Delta)\psi'\rangle =\langle\psi,E_A(\Delta)\psi\rangle.$$

For an antiunitary, the inner product is conjugated, but this expectation is real. The spectral integral of the real function $x$ therefore defines $A_V$, with precisely the domain $VD(A)$.

In particular, transporting both a preparation and its observable preserves every event probability. It does not assert that transforming the preparation alone leaves the original observable’s distribution unchanged.

Complex spectral functions distinguish the two cases

For unitary $V$ and a measurable complex function $f$,

$$Vf(A)V^{-1}=f(A_V).$$

For antiunitary $\Theta$, the correct identity is

$$\Theta f(A)\Theta^{-1} =\overline f(A_\Theta), \qquad \overline f(x)=\overline{f(x)}.$$

To see the difference, first take $f=\sum_jc_j\mathbf1_{\Delta_j}$. Antiunitarity conjugates each $c_j$ while transporting each projection. Bounded approximation and then spectral truncation extend the result to general measurable functions. The transported domains agree because $|f|=|\overline f|$ and the scalar spectral measures agree.

Two useful consequences, for nonreal $z$ and real $t$, are

$$\Theta(A-zI)^{-1}\Theta^{-1} =(A_\Theta-\overline zI)^{-1},$$ $$\Theta e^{-itA/\hbar}\Theta^{-1} =e^{+itA_\Theta/\hbar}.$$

The spectral argument applies to unbounded self-adjoint $A$. An operator-norm power series for $A$ generally does not; neither does a formal series on arbitrary Hilbert vectors.

An example with a transported momentum domain

On $L^2(\mathbb R)$ let $K$ be complex conjugation and let $P=-i\hbar\,d/dx$ on $H^1(\mathbb R)$. Complex conjugation preserves $H^1$, and weak differentiation commutes with conjugation. Hence

$$KPK=-P$$

as an equality of self-adjoint operators with the same domain. Accordingly,

$$K E_P(\Delta)K=E_P(-\Delta).$$

Momentum probabilities are reflected when the state alone is conjugated. The corresponding statement for translations is $K e^{-iaP/\hbar}K=e^{-iaP/\hbar}$: conjugating $i$ and reversing $P$ cancel. Checking only one of these changes would give the wrong translation law.

On an interval, the Sobolev regularity still survives but the endpoint phase need not. Time-Reversal Symmetry develops that domain-sensitive example.

Symmetry transport is not automatically a channel

For a density operator, $\rho\mapsto\Theta\rho\Theta^{-1}$ preserves positivity, trace, and convex combinations with real weights. Its extension to arbitrary operators is conjugate-linear, however. It should not be identified with a complex-linear quantum channel merely because it maps density operators to density operators. Wigner symmetries describe probability-preserving transformations, while channel implementability is a different question.

Exercise: conjugate a resolvent carefully

Assume $\Theta A\Theta^{-1}=A$ as self-adjoint operators. For $R=(A-iI)^{-1}$, is $\Theta R\Theta^{-1}=R$?

Solution

In dimensionless units for $A$, the transformed resolvent is $(A+iI)^{-1}=R^*$, not generally $R$. The observable is invariant, but the scalar spectral function $f(x)=(x-i)^{-1}$ is not real. Invariant self-adjoint operators need not have every complex function invariant under an antiunitary symmetry.

References

• B. C. Hall, Quantum Theory for Mathematicians, Graduate Texts in Mathematics 267, Springer, 2013 — self-adjoint domains and spectral calculus.
• K. Landsman, Foundations of Quantum Theory: From Classical Concepts to Operator Algebras, Fundamental Theories of Physics 188, Springer, 2017, Chapter 5 — unitary and antiunitary implementations of quantum symmetries.