T-symmetry

Microscopic phenomena: time reversal invariance

Most systems are asymmetric under time reversal, but there may be phenomena with symmetry. In classical mechanics, a velocity v reverses under the operation of T, but an acceleration does not. Therefore, one models dissipative phenomena through terms that are odd in v. However, delicate experiments in which known sources of dissipation are removed reveal that the laws of mechanics are time reversal invariant. Dissipation itself is originated in the second law of thermodynamics.

The motion of a charged body in a magnetic field, B involves the velocity through the Lorentz force term v×B, and might seem at first to be asymmetric under T. A closer look assures us that B also changes sign under time reversal. This happens because a magnetic field is produced by an electric current, J, which reverses sign under T. Thus, the motion of classical charged particles in electromagnetic fields is also time reversal invariant. (Despite this, it is still useful to consider the time-reversal non-invariance in a local sense when the external field is held fixed, as when the magneto-optic effect is analyzed. This allows one to analyze the conditions under which optical phenomena that locally break time-reversal, such as Faraday isolators and directional dichroism, can occur.)

Physics separates the laws of motion, called kinematics, from the laws of force, called dynamics. Following the classical kinematics of Newton's laws of motion, the kinematics of quantum mechanics is built in such a way that it presupposes nothing about the time reversal symmetry of the dynamics. In other words, if the dynamics are invariant, then the kinematics will allow it to remain invariant; if the dynamics is not, then the kinematics will also show this. The structure of the quantum laws of motion are richer, and we examine these next.

Time reversal in quantum mechanics

This section contains a discussion of the three most important properties of time reversal in quantum mechanics; chiefly,

1. that it must be represented as an anti-unitary operator,
2. that it protects non-degenerate quantum states from having an electric dipole moment,
3. that it has two-dimensional representations with the property T² = −1 (for fermions).

The strangeness of this result is clear if one compares it with parity. If parity transforms a pair of quantum states into each other, then the sum and difference of these two basis states are states of good parity. Time reversal does not behave like this. It seems to violate the theorem that all abelian groups be represented by one-dimensional irreducible representations. The reason it does this is that it is represented by an anti-unitary operator. It thus opens the way to spinors in quantum mechanics.

On the other hand, the notion of quantum-mechanical time reversal turns out to be a useful tool for the development of physically motivated quantum computing and simulation settings, providing, at the same time, relatively simple tools to assess their complexity. For instance, quantum-mechanical time reversal was used to develop novel boson sampling schemes and to prove the duality between two fundamental optical operations, beam splitter and squeezing transformations.

Formal notation

In formal mathematical presentations of T-symmetry, three different kinds of notation for T need to be carefully distinguished: the T that is an involution, capturing the actual reversal of the time coordinate, the T that is an ordinary finite dimensional matrix, acting on spinors and vectors, and the T that is an operator on an infinite-dimensional Hilbert space.

For a real (not complex) classical (unquantized) scalar field ${\displaystyle \varphi }$, the time reversal involution can simply be written as

$${\displaystyle \mathsf{T}\varphi (t,\overrightarrow{x})={\varphi }^{\mathrm{\prime }}(-t,\overrightarrow{x})=s\varphi (t,\overrightarrow{x})}$$

as time reversal leaves the scalar value at a fixed spacetime point unchanged, up to an overall sign ${\displaystyle s=\pm 1}$. A slightly more formal way to write this is

$${\displaystyle \mathsf{T}:\varphi (t,\overrightarrow{x})\mapsto {\varphi }^{\mathrm{\prime }}(-t,\overrightarrow{x})=s\varphi (t,\overrightarrow{x})}$$

which has the advantage of emphasizing that ${\displaystyle \mathsf{T}}$ is a map, and thus the "mapsto" notation ${\displaystyle \mapsto ,}$ whereas ${\displaystyle {\varphi }^{\mathrm{\prime }}(-t,\overrightarrow{x})=s\varphi (t,\overrightarrow{x})}$ is a factual statement relating the old and new fields to one-another.

Unlike scalar fields, spinor and vector fields ${\displaystyle \psi }$ might have a non-trivial behavior under time reversal. In this case, one has to write

$${\displaystyle \mathsf{T}:\psi (t,\overrightarrow{x})\mapsto {\psi }^{\mathrm{\prime }}(-t,\overrightarrow{x})=T\psi (t,\overrightarrow{x})}$$

where ${\displaystyle T}$ is just an ordinary matrix. For complex fields, complex conjugation may be required, for which the mapping ${\displaystyle K:(x+iy)\mapsto (x-iy)}$ can be thought of as a 2×2 matrix. For a Dirac spinor, ${\displaystyle T}$ cannot be written as a 4×4 matrix, because, in fact, complex conjugation is indeed required; however, it can be written as an 8×8 matrix, acting on the 8 real components of a Dirac spinor.

In the general setting, there is no ab initio value to be given for ${\displaystyle T}$; its actual form depends on the specific equation or equations which are being examined. In general, one simply states that the equations must be time-reversal invariant, and then solves for the explicit value of ${\displaystyle T}$ that achieves this goal. In some cases, generic arguments can be made. Thus, for example, for spinors in three-dimensional Euclidean space, or four-dimensional Minkowski space, an explicit transformation can be given. It is conventionally given as

$${\displaystyle T=e^{i\pi J_y}K}$$

where ${\displaystyle J_y}$ is the y-component of the angular momentum operator and ${\displaystyle K}$ is complex conjugation, as before. This form follows whenever the spinor can be described with a linear differential equation that is first-order in the time derivative, which is generally the case in order for something to be validly called "a spinor".

The formal notation now makes it clear how to extend time-reversal to an arbitrary tensor field ${\displaystyle {\psi }_{abc\cdots }}$ In this case,

$${\displaystyle \mathsf{T}:{\psi }_{abc\cdots }(t,\overrightarrow{x})\mapsto {\psi }_{abc\cdots }^{\mathrm{\prime }}(-t,\overrightarrow{x})={T_a}^d{T_b}^e{T_c}^f\cdots {\psi }_{def\cdots }(t,\overrightarrow{x})}$$

Covariant tensor indexes will transform as ${\displaystyle {T_a}^b={(T^{-1}{)}_b}^a}$ and so on. For quantum fields, there is also a third T, written as ${\displaystyle \mathcal{T},}$ which is actually an infinite dimensional operator acting on a Hilbert space. It acts on quantized fields ${\displaystyle \mathrm{\Psi }}$ as

$${\displaystyle \mathsf{T}:\mathrm{\Psi }(t,\overrightarrow{x})\mapsto {\mathrm{\Psi }}^{\mathrm{\prime }}(-t,\overrightarrow{x})=\mathcal{T}\mathrm{\Psi }(t,\overrightarrow{x}){\mathcal{T}}^{-1}}$$

This can be thought of as a special case of a tensor with one covariant, and one contravariant index, and thus two ${\displaystyle \mathcal{T}}$'s are required.

All three of these symbols capture the idea of time-reversal; they differ with respect to the specific space that is being acted on: functions, vectors/spinors, or infinite-dimensional operators. The remainder of this article is not cautious to distinguish these three; the T that appears below is meant to be either ${\displaystyle \mathsf{T}}$ or ${\displaystyle T}$ or ${\displaystyle \mathcal{T},}$ depending on context, left for the reader to infer.

Anti-unitary representation of time reversal

Eugene Wigner showed that a symmetry operation S of a Hamiltonian is represented, in quantum mechanics either by a unitary operator, S = U, or an antiunitary one, S = UK where U is unitary, and K denotes a basis-dependent complex conjugation operator. These are the only operations that act on Hilbert space so as to preserve the length of the projection of any one state-vector onto another state-vector.

For a particle with spin J, one can use the representation

$${\displaystyle T=\eta e^{-i\pi J_y/\hslash }K,}$$

where J_y is the y-component of the spin, and use of TJT⁻¹ = −J has been made, and ${\displaystyle \eta }$ is an arbitrary phase.

Electric dipole moments

This has an interesting consequence on the electric dipole moment (EDM) of any particle. The EDM is defined through the shift in the energy of a state when it is put in an external electric field: Δe = d·E + E·δ·E, where d is called the EDM and δ, the induced dipole moment. One important property of an EDM is that the energy shift due to it changes sign under a parity transformation. However, since d is a vector, its expectation value in a state |ψ⟩ must be proportional to ⟨ψ| J |ψ⟩, that is the expected spin. Thus, under time reversal, an invariant state must have vanishing EDM. In other words, a non-vanishing EDM signals both P and T symmetry-breaking.

Some molecules, such as water, must have EDM irrespective of whether T is a symmetry. This is correct; if a quantum system has degenerate ground states that transform into each other under parity, then time reversal need not be broken to give EDM.

Experimentally observed bounds on the electric dipole moment of the nucleon currently set stringent limits on the violation of time reversal symmetry in the strong interactions, and their modern theory: quantum chromodynamics. Then, using the CPT invariance of a relativistic quantum field theory, this puts strong bounds on strong CP violation.

Experimental bounds on the electron electric dipole moment also place limits on theories of particle physics and their parameters.

Kramers' theorem

For T, which is an anti-unitary Z₂ symmetry generator

T² = UKUK = UU^* = U (U^T)⁻¹ = Φ,

where Φ is a diagonal matrix of phases. As a result, U = ΦU^T and U^T = UΦ, showing that

U = Φ U Φ.

This means that the entries in Φ are ±1, as a result of which one may have either T² = ±1. This is specific to the anti-unitarity of T. For a unitary operator, such as the parity, any phase is allowed.

Next, take a Hamiltonian invariant under T. Let |a⟩ and T|a⟩ be two quantum states of the same energy. Now, if T² = −1, then one finds that the states are orthogonal: a result called Kramers' theorem. This implies that if T² = −1, then there is a twofold degeneracy in the state. This result in non-relativistic quantum mechanics presages the spin statistics theorem of quantum field theory.

Quantum states that give unitary representations of time reversal, i.e., have T² = 1, are characterized by a multiplicative quantum number, sometimes called the T-parity.

Time reversal of the known dynamical laws

Particle physics codified the basic laws of dynamics into the Standard Model. This is formulated as a quantum field theory that has CPT symmetry, i.e., the laws are invariant under simultaneous operation of time reversal, parity and charge conjugation. However, time reversal itself is seen not to be a symmetry (this is usually called CP violation). There are two possible origins of this asymmetry, one through the mixing of different flavours of quarks in their weak decays, the second through a direct CP violation in strong interactions. The first is seen in experiments; the second is strongly constrained by the non-observation of the EDM of a neutron.

Time reversal violation is unrelated to the second law of thermodynamics, because due to the conservation of the CPT symmetry, the effect of time reversal is to rename particles as antiparticles and vice versa. Thus the second law of thermodynamics is thought to originate in the initial conditions in the universe.

Time reversal of noninvasive measurements

Strong measurements (both classical and quantum) are certainly disturbing, causing asymmetry due to the second law of thermodynamics. However, noninvasive measurements should not disturb the evolution, so they are expected to be time-symmetric. Surprisingly, it is true only in classical physics but not in quantum physics, even in a thermodynamically invariant equilibrium state. This type of asymmetry is independent of CPT symmetry but has not yet been confirmed experimentally due to extreme conditions of the checking proposal.

Negative group delay in quantum systems

In 2024, experiments by the University of Toronto showed that under certain quantum conditions, photons can exhibit "negative time" behavior. When interacting with atomic clouds, photons appeared to exit the medium before entering it, indicating a negative group delay, especially near atomic resonance. Using the cross-Kerr effect, the team measured atomic excitation by observing phase shifts in a weak probe beam. The results showed that atomic excitation times varied from negative to positive, depending on the pulse width. For narrow pulses, the excitation time was approximately −0.82 times the non-post-selected excitation time (τ₀), while for broader pulses, it was around 0.54 times τ₀. These findings align with theoretical predictions and highlight the non-classical nature of quantum mechanics, opening new possibilities for quantum computing and photonics.