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 are given by a pair of quantum states that go into each other under parity. However, this representation can always be reduced to linear combinations of states, each of which is either even or odd under parity. One says that all
irreducible representations of parity are one-dimensional. '''Kramers' theorem''' states that time reversal need not have this property because it is represented by an anti-unitary operator. This section contains a discussion of the three most important properties of time reversal in quantum mechanics; chiefly, • that it must be represented as an anti-unitary operator, • that it protects non-degenerate quantum states from having an
electric dipole moment, • that it has two-dimensional representations with the property (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 \phi, the time reversal
involution can simply be written as \mathsf{T} \phi(t,\vec{x}) = \phi^\prime(-t,\vec{x}) = s\phi(t,\vec{x}) as time reversal leaves the scalar value at a fixed spacetime point unchanged, up to an overall sign s=\pm 1. A slightly more formal way to write this is \mathsf{T}: \phi(t,\vec{x}) \mapsto \phi^\prime(-t,\vec{x}) = s\phi(t,\vec{x}) which has the advantage of emphasizing that \mathsf{T} is a
map, and thus the "mapsto" notation \mapsto ~, whereas \phi^\prime(-t,\vec{x}) = s\phi(t,\vec{x}) is a factual statement relating the old and new fields to one-another. Unlike scalar fields,
spinor and
vector fields \psi might have a non-trivial behavior under time reversal. In this case, one has to write \mathsf{T}: \psi(t,\vec{x}) \mapsto \psi^\prime(-t,\vec{x}) = T\psi(t,\vec{x}) where T is just an ordinary
matrix. For
complex fields,
complex conjugation may be required, for which the mapping K: (x+iy) \mapsto (x-iy) can be thought of as a 2×2 matrix. For a
Dirac spinor, 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 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 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 T=e^{i\pi J_y}K where J_y is the y-component of the
angular momentum operator and 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 \psi_{abc\cdots} In this case, \mathsf{T}: \psi_{abc\cdots}(t,\vec{x}) \mapsto \psi_{abc\cdots}^\prime(-t,\vec{x}) = {T_a}^d \,{T_b}^e \,{T_c}^f \cdots \psi_{def\cdots}(t,\vec{x}) Covariant tensor indexes will transform as {T_a}^b = {(T^{-1})_b}^a and so on. For quantum fields, there is also a third
T, written as \mathcal{T}, which is actually an infinite dimensional operator acting on a Hilbert space. It acts on quantized fields \Psi as \mathsf{T}: \Psi(t,\vec{x}) \mapsto \Psi^\prime(-t,\vec{x}) = \mathcal{T} \Psi(t,\vec{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 \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 \mathsf{T} or T or \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, , or an
antiunitary one , 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 T = \eta e^{-i\pi J_y/\hbar} K, where
Jy is the
y-component of the spin, and use of has been made, and \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: , 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 where Φ is a diagonal matrix of phases. As a result, and , showing that This means that the entries in Φ are ±1, as a result of which one may have either . 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 , then one finds that the states are orthogonal: a result called '''Kramers' theorem'''. This implies that if , 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
, 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. == See also ==