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 Reversal Is Not Just Running the Movie Backward

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. A purely unitary time-reversal operator would not have the correct action on the factor of $i$ in the time-evolution equation.

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 because $P^2$ is unchanged and $V(X)$ is unchanged.

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:

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

A fixed external magnetic field generally breaks time-reversal symmetry unless the field is transformed as part of the physical comparison.

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.

Kramers Degeneracy Preview

If $T^2=-I$ and $THT^{-1}=H$, then many systems exhibit Kramers pairs: states related by $T$ that are orthogonal and have the same energy. 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 $t\mapsto -t$.

Cross-Links

• Antiunitary Symmetries

• Antiunitary Symmetries, First Look

• Antiunitary Time Reversal

• Time Reversal for Spinless Particles

• Parity

• Symmetry Constraints on Hamiltonians

• Spin-1/2 Hilbert Space

• Pauli Matrices

• Time Reversal for Spin-1/2 Particles

• Kramers Degeneracy

• Charge Conjugation Preview

• Discrete Symmetries in Hamiltonians

• Symmetry Classification Preview

• From Discrete Symmetries to CPT

• Common Pitfalls

• Tests of Fundamental Symmetries for EDM searches, the distinction between laboratory reversals and antiunitary time reversal, and effective-operator constraints.

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.

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

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