Time Reversal

Time reversal is the symmetry operation associated with reversing the direction of motion while leaving the spatial point itself unchanged. In quantum mechanics it is represented by an antiunitary operator, conventionally denoted $T$ or $\Theta$.

The basic transformation rules are

$$T\,\mathbf X\,T^{-1}=\mathbf X, \qquad T\,\mathbf P\,T^{-1}=-\mathbf P.$$

Since angular momentum involves momentum, it also reverses:

$$T\,\mathbf L\,T^{-1}=-\mathbf L.$$

Spin reverses as well:

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

Time-Reversed States and Dynamics

There are two related but distinct ideas:

• transforming states and observables by the time-reversal operator,
• comparing a solution evolving forward in time with a transformed solution evolving with reversed time parameter.

The operator $T$ acts on Hilbert-space states. The statement that a Hamiltonian is time-reversal invariant is a statement about how the transformed dynamics compares with the original dynamics.

Why Antiunitarity Is Needed

The Schrödinger equation is

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

Reversing time changes the sign of the time derivative. To preserve the form of the equation, the time-reversal operation must also conjugate $i$:

$$T i T^{-1}=-i.$$

This is the hallmark of antiunitarity. Here physical time reversal preserves energy while reversing motion. A unitary operation that instead maps $H$ to $-H$ would describe a different spectral transformation, not this energy-preserving reversal.

The detailed algebraic reason, including the canonical-commutator and propagator checks, is developed in Antiunitary Time Reversal.

Spinless Particle

For a spinless particle in the position representation, a common time-reversal operator is complex conjugation:

$$T=K.$$

Thus

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

This leaves position unchanged and reverses momentum:

$$KXK^{-1}=X, \qquad KPK^{-1}=-P.$$

The second identity follows because $P=-i\hbar\,d/dx$ and complex conjugation sends $i$ to $-i$.

The representation dependence, $\Theta^2=+I$ result, real-Hamiltonian test, and current reversal are worked out in Time Reversal for Spinless Particles.

Spin-One-Half Particle

For a single spin-$1/2$ degree of freedom, time reversal must reverse spin:

$$T S_i T^{-1}=-S_i.$$

With $S_i=\hbar\sigma_i/2$, a common convention is

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

where $K$ complex-conjugates spinor components in the usual $\sigma_z$ basis. This gives

$$T^2=-I.$$

Equivalent phase conventions exist, such as multiplying $T$ by an overall phase. They do not change physical predictions, but they can change the appearance of intermediate formulas.

The spinor action, the basis dependence of $K$, and the invariant meaning of $T^2=-I$ are developed in Time Reversal for Spin-1/2 Particles.

Hamiltonian Symmetry Criterion

A time-independent Hamiltonian is time-reversal invariant when

$$THT^{-1}=H.$$

For a spinless particle with

$$H=\frac{P^2}{2m}+V(X),$$

and real scalar potential $V$, time reversal is a symmetry provided its self-adjoint domain and boundary conditions are also invariant under conjugation. The differential expression is unchanged because both $P^2$ and $V(X)$ are unchanged. A ring with twisted boundary conditions shows why the domain must be checked separately.

Magnetic fields require more care. The minimal-coupling Hamiltonian contains

$$P-qA(X).$$

Under time reversal, momentum changes sign. A magnetic field is also time-reversal odd:

$$\mathbf B\mapsto-\mathbf B.$$

At fixed nonzero magnetic field, ordinary time reversal generally fails to be a symmetry. Reversing the field compares two Hamiltonians, $TH(\mathbf B)T^{-1}=H(-\mathbf B)$; it does not make either fixed-field Hamiltonian invariant. Additional combined spatial and time-reversal symmetries must be tested separately.

Time Evolution

If $THT^{-1}=H$, then time reversal relates forward and backward evolution:

$$T e^{-iHt/\hbar} T^{-1} = e^{+iHt/\hbar}.$$

The plus sign in the exponent appears because $T$ is antiunitary and conjugates $i$. This identity is often the cleanest algebraic way to remember what time reversal means dynamically.

For a time-dependent drive, the reversed protocol has Hamiltonian

$$H_T(t)=T H(-t)T^{-1}.$$

It obeys the same prescribed protocol only if $H_T(t)=H(t)$. Reversing a laboratory sequence is therefore more than conjugating a state at one instant. Tests of Fundamental Symmetries explains how this distinction matters in experiments.

Kramers Pairs

If $T^2=-I$ and $THT^{-1}=H$ on an invariant domain, every normalizable energy eigenstate has an orthogonal partner with the same energy. No magnetic field or rotational symmetry is needed beyond those hypotheses. The working theorem is Kramers Degeneracy, and the seed of the result is already visible in the spin-$1/2$ rule $T^2=-I$.

Common Mistakes

• Treating time reversal as unitary.
• Forgetting that antiunitarity conjugates scalar coefficients.
• Reversing position instead of momentum. Position is even under time reversal; momentum is odd.
• Ignoring spin. Spin changes sign under time reversal.
• Claiming a system with a fixed magnetic field is time-reversal invariant without specifying whether the external field is transformed.
• Confusing the operator $T$ with the replacement $t\mapsto -t$.

Exercises

1. For spinless time reversal $T=K$, show that $T e^{-iHt/\hbar}T^{-1}=e^{+iHt/\hbar}$ when $H$ is real in the position representation.

Solution

Complex conjugation sends $i$ to $-i$. If $KHK^{-1}=H$, then

$$K e^{-iHt/\hbar}K^{-1} = e^{+iHt/\hbar}.$$

The sign change is due to antiunitarity, not to changing $H$.

2. Check that $T=-i\sigma_yK$ squares to $-I$ for spin-$1/2$.

Solution

Use $K\sigma_yK^{-1}=\sigma_y^*=-\sigma_y$. Then

$$T^2 = (-i\sigma_yK)(-i\sigma_yK) = (-i\sigma_y)(K(-i\sigma_y)K^{-1})$$

and

$$K(-i\sigma_y)K^{-1} = (+i)(-\sigma_y) = -i\sigma_y.$$

Therefore

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

References

• M. S. Dresselhaus, G. Dresselhaus, and A. Jorio, Group Theory: Application to the Physics of Condensed Matter, Springer, 2008.
• A. Messiah, Quantum Mechanics, Dover, 1999.
• J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge University Press, 2020.
• E. P. Wigner, Group Theory and Its Application to the Quantum Mechanics of Atomic Spectra, Academic Press, 1959.