Time Reversal Operator
The time reversal operator (T) denotes the transformation that reverses the direction of time in a physical process.
More concretely, applying T amounts to describing the evolution of a system as if time were running backward, much like replaying a recorded video in reverse.
In symbolic terms, time reversal corresponds to the transformation
$$ t \rightarrow -t $$
This operation is not a mere kinematic reversal, but a genuine symmetry transformation of the fundamental laws of physics.
If a process viewed backward in time is physically allowed and obeys the same laws as the original process, meaning that it is physically equivalent, then the theory is said to be invariant under time reversal, or equivalently, that T symmetry is conserved.
Note. From an intuitive standpoint, if a physical phenomenon is recorded and the footage is played backward, the resulting process should still be compatible with the laws of physics. For instance, an elastic collision between two billiard balls is governed by the same physical laws whether it is observed forward or backward in time. When the collision is shown in reverse, the process remains consistent with the equations of motion and with the same transition probabilities. During the collision, the contact forces between the two balls are always equal in magnitude and opposite in direction: $$ \vec F_1 = - \vec F_2 $$ The force exerted by the first ball on the second is equal in magnitude and opposite in direction to the force exerted by the second ball on the first. This relation holds at every instant during the collision and is independent of the direction of time. Consequently, even in the time-reversed process, in which the velocities of the two balls are inverted, the relation $$ \vec F_1 = - \vec F_2 $$ continues to hold. This provides a clear example of a system in which time reversal symmetry T is conserved.
To say that a theory is invariant under time reversal means that whenever a physical process is allowed, the process obtained by applying the operator T is physically equivalent. In particular, it occurs with the same probability and is governed by the same fundamental dynamical laws.
Conversely, if the process obtained by applying T is not physically equivalent to the original one, one speaks of violation of time reversal symmetry, or simply T violation.
It is important to stress that conservation of time reversal symmetry T does not merely require the reversed process to be physically possible. It must be physically equivalent to the original process, meaning that it is described by the same dynamical laws and, crucially, that it occurs with the same probability once the physical conditions are transformed under T.
A practical example
Consider a point particle moving along the \( x \)-axis.
Its equation of motion is
$$ x(t) = vt $$
where \( v \) is a constant velocity.
When the time reversal operator is applied, T reverses the time parameter:
$$ t \rightarrow -t $$
Applying T to the equation of motion yields
$$ x(-t) = v(-t) = -vt $$
This expression represents the trajectory reversed in time.
In the original motion, at time \( t > 0 \), the particle moves to the right with velocity \( +v \).
In the motion transformed by T, at time \( t > 0 \), the particle moves to the left with velocity \( -v \).
The motion “backward in time” is still a physically allowed process. As a result, the laws of classical mechanics do not distinguish between past and future. Time reversal symmetry T is conserved.
This example illustrates that classical mechanics is invariant under time reversal.
Example 2
This provides a paradigmatic example of violation of time reversal symmetry T.
Consider neutral kaons, namely the particle \( K^0 \) and its antiparticle \( \overline{K}^0 \).
These particles can transform into one another through the weak interaction:
$$ K^0 \longleftrightarrow \overline{K}^0 $$
This phenomenon is known as neutral kaon mixing.
Define the two transition processes:
- $ K^0 \rightarrow \overline{K}^0 $
- $ \overline{K}^0 \rightarrow K^0 $
If T symmetry were exact, the probability of process 1 would be equal to that of process 2, given initial and final conditions related by time reversal.
Experimental measurements, however, show that these two transition probabilities are not equal.
$$ P(K^0 \rightarrow \overline{K}^0) \neq P(\overline{K}^0 \rightarrow K^0) $$
This temporal asymmetry constitutes a direct violation of time reversal symmetry.
This is not an indirect inference based on CP or TCP arguments, but a direct experimental observation demonstrating that the time-reversed process is not physically equivalent.
In conclusion, the “movie” of the oscillation between \( K^0 \) and \( \overline{K}^0 \), when played backward, does not describe the same physical phenomenon.
Therefore, time reversal symmetry T is violated.
Conservation and violation of T symmetry
In physics, some processes are invariant under time reversal, while others are not.
- In classical mechanics, many fundamental laws are invariant under time reversal.
- In the strong and electromagnetic interactions, no violations of time reversal symmetry have been observed.
- In the weak interaction, time reversal symmetry is violated.
Thus, in particle physics, the time reversal operator T is conserved in the strong and electromagnetic interactions, where no T violation is observed, whereas it is violated in the weak interaction.
This violation is not only an experimental fact, but also a conceptual necessity, as required by the TCP theorem.
The TCP theorem states that the combined transformation of time reversal (T), charge conjugation (C), and parity (P) is an exact symmetry of any relativistic quantum field theory.
It is often expressed in compact form as:
$$ TCP = 1 $$
indicating that the combined operation leaves the theory invariant.
$$ T = CP^{-1} $$
This relation implies that if CP symmetry is violated in a given theory, then time reversal symmetry T must also be violated, and conversely.
Therefore, since CP symmetry is experimentally violated in the weak interaction and the TCP theorem requires the combined TCP symmetry to be conserved, time reversal symmetry T must necessarily be violated.
Note. In quantum mechanics, the time reversal operator T is an antiunitary operator. This means that, in addition to transforming physical observables, it does not act as a simple linear unitary matrix, unlike the parity operator P and the charge conjugation operator C. As a consequence, no particle can be an eigenstate of T, whereas many particles are eigenstates of P or C. Time reversal therefore admits no physical eigenstates.
More generally, the time reversal operator provides a powerful tool for distinguishing between reversible and irreversible processes, for identifying an arrow of time in the fundamental interactions, and for relating the violation of discrete symmetries to deep structural principles of quantum field theory.
In this sense, T is not merely a mathematical operation, but an essential conceptual instrument for understanding the role of time in the fundamental laws of nature.
And so on.
