Antiunitary Symmetries

An antiunitary symmetry is represented by an antilinear norm-preserving operator. Antiunitary transformations are less familiar than unitary ones, but they are unavoidable in quantum mechanics. Time reversal is the central example. For the operator-theoretic first pass, see Antiunitary Symmetries, First Look; for the dynamical derivation, see Antiunitary Time Reversal.

Antilinearity

An antiunitary operator $T$ is antilinear:

$$T(a\lvert\psi\rangle+b\lvert\phi\rangle) = a^*T\lvert\psi\rangle+b^*T\lvert\phi\rangle.$$

This is the first point to remember. Complex scalars are conjugated when they pass through $T$.

Inner Products and Probabilities

Antiunitary transformations conjugate inner products:

$$\langle T\phi|T\psi\rangle = \langle\phi|\psi\rangle^*.$$

Therefore they preserve transition probabilities:

$$\lvert\langle T\phi|T\psi\rangle\rvert^2 = \lvert\langle\phi|\psi\rangle\rvert^2.$$

This is why antiunitary transformations are allowed by Wigner’s theorem.

Wigner’s theorem is a statement about ray symmetries: under its standard hypotheses, a bijection of rays that preserves transition probabilities can be implemented by either a unitary or an antiunitary operator. The implementing operator is defined only up to an overall phase. A continuous one-parameter symmetry connected to the identity stays in the unitary branch; antiunitary implementations therefore characteristically represent disconnected or discrete operations such as time reversal.

Complex Conjugation Is Basis Dependent

In a chosen orthonormal basis, an antiunitary operator can often be written as

$$T=UK,$$

where $U$ is unitary and $K$ complex-conjugates coefficients in that basis. This is useful, but it can be misleading if stated without context. The operation $K$ depends on the basis chosen. The antiunitary operator $T$ is the invariant object.

The factorization is general. Choose an orthonormal basis and its conjugation $K$. Since the product of two antiunitary maps is linear, $U:=TK$ is linear. Moreover,

$$\langle U\phi\mid U\psi\rangle = \langle TK\phi\mid TK\psi\rangle = \langle K\phi\mid K\psi\rangle^* = \langle\phi\mid\psi\rangle,$$

so $U$ is unitary and $T=UK$. A change of reference basis changes $U$ and $K$ separately but not their product.

Composition and Operator Transformations

The product of two antiunitary maps is unitary; multiplying a unitary and an antiunitary map in either order is antiunitary. Consequently $T^{-1}$ is antiunitary and $T^2$ is unitary.

Conjugation by an antiunitary map preserves operator products but conjugates scalar coefficients:

$$T(cA+dB)T^{-1} = c^*TAT^{-1}+d^*TBT^{-1},$$ $$T(AB)T^{-1} = (TAT^{-1})(TBT^{-1}).$$

It follows that

$$T[A,B]T^{-1} = [TAT^{-1},TBT^{-1}].$$

The scalar conjugation is essential in algebra checks. For example, if $[x,p]=i\hbar I$, a time-reversal action with $x\mapsto x$ and $p\mapsto-p$ gives $[x,-p]=-i\hbar I$, exactly matching the transformed right-hand side $T(i\hbar I)T^{-1}=-i\hbar I$.

Time Reversal

Time reversal must conjugate $i$. The Schrödinger equation

$$i\hbar\frac{\partial}{\partial t}\lvert\psi(t)\rangle = H\lvert\psi(t)\rangle$$

contains $i$ and a time derivative. Reversing the time direction requires an operation that sends $i$ to $-i$ so that the transformed equation has the correct sign structure.

For a spinless particle in a position basis, a simple time-reversal action is complex conjugation:

$$T\psi(x)=\psi(x)^*.$$

This leaves $x$ unchanged and reverses momentum:

$$TxT^{-1}=x, \qquad TpT^{-1}=-p.$$

The scalar-particle representation and the $\Theta^2=+I$ consequence are developed in Time Reversal for Spinless Particles.

For spin-$1/2$, time reversal also acts nontrivially on spin. A common convention is

$$T=-i\sigma_y K,$$

which gives

$$T\mathbf S T^{-1}=-\mathbf S.$$

Sign conventions for this formula vary by phase choice, but physical predictions do not.

In the $\sigma_z$ basis, $K$ is entrywise conjugation and

$$T=-i\sigma_yK.$$

Because $-i\sigma_y$ is real,

$$T^2 = (-i\sigma_y)K(-i\sigma_y)K = (-i\sigma_y)^2 = -I.$$

This two-dimensional example also shows why multiplying an antiunitary operator by a phase does not change $T^2$: the second occurrence of the phase is conjugated.

The detailed spinor convention and the proof of $T^2=-I$ are treated in Time Reversal for Spin-1/2 Particles.

Kramers Degeneracy Preview

When an antiunitary time-reversal symmetry satisfies

$$T^2=-I,$$

energy eigenstates of a time-reversal-invariant Hamiltonian occur in orthogonal pairs under broad finite-dimensional assumptions. This is Kramers Degeneracy. The full discussion belongs with time reversal and applications, but the antiunitary origin starts here.

Common Mistakes

• Treating $T(a\lvert\psi\rangle)$ as $aT\lvert\psi\rangle$ instead of $a^*T\lvert\psi\rangle$.
• Describing antiunitary operators as just “unitary matrices with complex conjugation” without specifying the basis.
• Forgetting that time reversal changes momenta and angular momenta.
• Assuming every time-reversal operation satisfies $T^2=+I$.
• Dropping antiunitarity when transforming expressions containing $i$.

Cross-Links

• Quantum Symmetries
• Wigner’s Theorem Preview
• Unitary Symmetries
• Antiunitary Symmetries, First Look
• Antiunitary Time Reversal
• Time Reversal for Spinless Particles
• Projective Representations
• Time Reversal for Spin-1/2 Particles
• Kramers Degeneracy
• Symmetry Classification Preview
• Common Pitfalls
• Rays and Global Phase
• Pauli Matrices

References

• E. P. Wigner, Group Theory and Its Application to the Quantum Mechanics of Atomic Spectra, Academic Press, 1959.
• J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge University Press, 2020.
• A. Messiah, Quantum Mechanics, Dover, 1999.
• M. S. Dresselhaus, G. Dresselhaus, and A. Jorio, Group Theory: Application to the Physics of Condensed Matter, Springer, 2008.
• V. Bargmann, “Note on Wigner’s Theorem on Symmetry Operations,” Journal of Mathematical Physics 5, 862–868 (1964), doi:10.1063/1.1704188.
• B. C. Hall, Quantum Theory for Mathematicians, Springer, 2013.
• M. Reed and B. Simon, Methods of Modern Mathematical Physics I: Functional Analysis, revised and enlarged ed., Academic Press, 1980.

Exercises

1. Let $T$ be antiunitary. Show that $T(i\lvert\psi\rangle)=-iT\lvert\psi\rangle$.

Solution

By antilinearity,

$$T(i\lvert\psi\rangle) = i^*T\lvert\psi\rangle = -iT\lvert\psi\rangle.$$

2. Show that the inverse of an antiunitary operator is antiunitary.

Solution

If $T$ is antiunitary, it is bijective. For $\chi=T\psi$ and $\eta=T\phi$,

$$\langle T^{-1}\chi\mid T^{-1}\eta\rangle = \langle\psi\mid\phi\rangle = \langle\chi\mid\eta\rangle^*.$$

Antilinearity of $T^{-1}$ follows by applying $T$ to a linear combination and using bijectivity.

3. Fix a basis conjugation $K$ and prove that every antiunitary $T$ has the form $UK$ with $U$ unitary.

Solution

Define $U=TK$. It is linear because it is the product of two antilinear maps. The antiunitary inner-product rule applied twice gives $\langle U\phi\mid U\psi\rangle=\langle\phi\mid\psi\rangle$, so $U$ is unitary. Since $K^2=I$, $UK=TKK=T$.

4. Let $T$ be antiunitary. Show that

$$T e^{-iHt/\hbar}T^{-1} = e^{+i(THT^{-1})t/\hbar}$$

for real $t$ and a Hamiltonian for which the exponential is defined.

Solution

Expand the exponential in its power series. Conjugation by $T$ preserves operator products, while antilinearity conjugates every scalar coefficient. Thus $(-i)^n$ becomes $(+i)^n$ and $H^n$ becomes $(THT^{-1})^n$, which resums to the stated exponential.

5. For spinless time reversal $T=K$ in the position representation, show that $TpT^{-1}=-p$ for $p=-i\hbar\,d/dx$.

Solution

Apply the transformed operator to a test wavefunction $\psi$:

$$(KpK\psi)(x) = Kp\psi^*(x) = K\left[-i\hbar\frac{d\psi^*}{dx}\right] = i\hbar\frac{d\psi}{dx} = -p\psi(x).$$

Thus $KpK^{-1}=-p$ since $K^{-1}=K$.