T-symmetry or time reversal symmetry is the
theoretical symmetry of physical laws under
the transformation of time reversal:
T
:
t
↦
−
t
.
{\displaystyle T:t\mapsto -t.}
T-symmetry can be shown to be equivalent to
the conservation of entropy, by Noether's
Theorem. But as the second law of thermodynamics
does not permit entropy to be conserved in
general, it follows that the observable universe
does not in general show symmetry under time
reversal. In other words, time is said to
be non-symmetric, or asymmetric, except for
special equilibrium states when the second
law of thermodynamics predicts the time symmetry
to hold.
However, quantum noninvasive measurements
are predicted to violate time symmetry even
in equilibrium, contrary to their classical
counterparts, although this has not yet been
experimentally confirmed.
Time asymmetries are generally distinguished
as among those...
intrinsic to the dynamic physical law (e.g.,
for the weak force)
due to the initial conditions of our universe
(e.g., for the second law of thermodynamics)
due to measurements (e.g., for the noninvasive
measurements)
== Invariance ==
Physicists also discuss the time-reversal
invariance of local and/or macroscopic descriptions
of physical systems, independent of the invariance
of the underlying microscopic physical laws.
For example, Maxwell's equations with material
absorption or Newtonian mechanics with friction
are not time-reversal invariant at the macroscopic
level where they are normally applied, even
if they are invariant at the microscopic level;
when one includes the atomic motions, the
"lost" energy is translated into heat.
== Macroscopic phenomena: the second law of
thermodynamics ==
Our daily experience shows that T-symmetry
does not hold for the behavior of bulk materials.
Of these macroscopic laws, most notable is
the second law of thermodynamics. Many other
phenomena, such as the relative motion of
bodies with friction, or viscous motion of
fluids, reduce to this, because the underlying
mechanism is the dissipation of usable energy
(for example, kinetic energy) into heat.
The question of whether this time-asymmetric
dissipation is really inevitable has been
considered by many physicists, often in the
context of Maxwell's demon. The name comes
from a thought experiment described by James
Clerk Maxwell in which a microscopic demon
guards a gate between two halves of a room.
It only lets slow molecules into one half,
only fast ones into the other. By eventually
making one side of the room cooler than before
and the other hotter, it seems to reduce the
entropy of the room, and reverse the arrow
of time. Many analyses have been made of this;
all show that when the entropy of room and
demon are taken together, this total entropy
does increase. Modern analyses of this problem
have taken into account Claude E. Shannon's
relation between entropy and information.
Many interesting results in modern computing
are closely related to this problem — reversible
computing, quantum computing and physical
limits to computing, are examples. These seemingly
metaphysical questions are today, in these
ways, slowly being converted into hypotheses
of the physical sciences.
The current consensus hinges upon the Boltzmann-Shannon
identification of the logarithm of phase space
volume with the negative of Shannon information,
and hence to entropy. In this notion, a fixed
initial state of a macroscopic system corresponds
to relatively low entropy because the coordinates
of the molecules of the body are constrained.
As the system evolves in the presence of dissipation,
the molecular coordinates can move into larger
volumes of phase space, becoming more uncertain,
and thus leading to increase in entropy.
One can, however, equally well imagine a state
of the universe in which the motions of all
of the particles at one instant were the reverse
(strictly, the CPT reverse). Such a state
would then evolve in reverse, so presumably
entropy would decrease (Loschmidt's paradox).
Why is 'our' state preferred over the other?
One position is to say that the constant increase
of entropy we observe happens only because
of the initial state of our universe. Other
possible states of the universe (for example,
a universe at heat death equilibrium) would
actually result in no increase of entropy.
In this view, the apparent T-asymmetry of
our universe is a problem in cosmology: why
did the universe start with a low entropy?
This view, if it remains viable in the light
of future cosmological observation, would
connect this problem to one of the big open
questions beyond the reach of today's physics
— the question of initial conditions of
the universe.
== Macroscopic phenomena: black holes ==
An object can cross through the event horizon
of a black hole from the outside, and then
fall rapidly to the central region where our
understanding of physics breaks down. Since
within a black hole the forward light-cone
is directed towards the center and the backward
light-cone is directed outward, it is not
even possible to define time-reversal in the
usual manner. The only way anything can escape
from a black hole is as Hawking radiation.
The time reversal of a black hole would be
a hypothetical object known as a white hole.
From the outside they appear similar. While
a black hole has a beginning and is inescapable,
a white hole has an ending and cannot be entered.
The forward light-cones of a white hole are
directed outward; and its backward light-cones
are directed towards the center.
The event horizon of a black hole may be thought
of as a surface moving outward at the local
speed of light and is just on the edge between
escaping and falling back. The event horizon
of a white hole is a surface moving inward
at the local speed of light and is just on
the edge between being swept outward and succeeding
in reaching the center. They are two different
kinds of horizons—the horizon of a white
hole is like the horizon of a black hole turned
inside-out.
The modern view of black hole irreversibility
is to relate it to the second law of thermodynamics,
since black holes are viewed as thermodynamic
objects. Indeed, according to the Gauge–gravity
duality conjecture, all microscopic processes
in a black hole are reversible, and only the
collective behavior is irreversible, as in
any other macroscopic, thermal system.
== Kinetic consequences: detailed balance
and Onsager reciprocal relations ==
In physical and chemical kinetics, T-symmetry
of the mechanical microscopic equations implies
two important laws: the principle of detailed
balance and the Onsager reciprocal relations.
T-symmetry of the microscopic description
together with its kinetic consequences are
called microscopic reversibility.
== Effect of time reversal on some variables
of classical physics ==
=== 
Even ===
Classical variables that do not change upon
time reversal include:
x
→
{\displaystyle {\vec {x}}\!}
, Position of a particle in three-space
a
→
{\displaystyle {\vec {a}}\!}
, Acceleration of the particle
F
→
{\displaystyle {\vec {F}}\!}
, Force on the particle
E
{\displaystyle E\!}
, Energy of the particle
ϕ
{\displaystyle \phi \!}
, Electric potential (voltage)
E
→
{\displaystyle {\vec {E}}\!}
, Electric field
D
→
{\displaystyle {\vec {D}}\!}
, Electric displacement
ρ
{\displaystyle \rho \!}
, Density of electric charge
P
→
{\displaystyle {\vec {P}}\!}
, Electric polarization
Energy density of the electromagnetic field
Maxwell stress tensor
All masses, charges, coupling constants, and
other physical constants, except those associated
with the weak force.
=== Odd ===
Classical variables that time reversal negates
include:
t
{\displaystyle t\!}
, The time when an event occurs
v
→
{\displaystyle {\vec {v}}\!}
, Velocity of a particle
p
→
{\displaystyle {\vec {p}}\!}
, Linear momentum of a particle
l
→
{\displaystyle {\vec {l}}\!}
, Angular momentum of a particle (both orbital
and spin)
A
→
{\displaystyle {\vec {A}}\!}
, Electromagnetic vector potential
B
→
{\displaystyle {\vec {B}}\!}
, Magnetic field
H
→
{\displaystyle {\vec {H}}\!}
, Magnetic auxiliary field
j
→
{\displaystyle {\vec {j}}\!}
, Density of electric current
M
→
{\displaystyle {\vec {M}}\!}
, Magnetization
S
→
{\displaystyle {\vec {S}}\!}
, Poynting vector
Power (rate of work done).
== 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.) The laws of gravity also seem
to be time reversal invariant in classical
mechanics.
In physics one 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,
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 T2 = −1.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.
=== 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 complex conjugation.
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.
Consider the parity operator. Acting on the
position, it reverses the directions of space,
so that PxP−1 = −x. Similarly, it reverses
the direction of momentum, so that PpP−1
= −p, where x and p are the position and
momentum operators. This preserves the canonical
commutator [x, p] = iħ, where ħ is the reduced
Planck constant, only if P is chosen to be
unitary, PiP−1 = i.
On the other hand, the time reversal operator
T, it does nothing to the x-operator, TxT−1
= x, but it reverses the direction of p, so
that TpT−1 = −p. The canonical commutator
is invariant only if T is chosen to be anti-unitary,
i.e., TiT−1 = −i.
Another argument involves energy, the time-component
of the momentum. If time reversal were implemented
as a unitary operator, it would reverse the
sign of the energy just as space-reversal
reverses the sign of the momentum. This is
not possible, because, unlike momentum, energy
is always positive. Since energy in quantum
mechanics is defined as the phase factor exp(-iEt)
that one gets when one moves forward in time,
the way to reverse time while preserving the
sign of the energy is to also reverse the
sense of "i", so that the sense of phases
is reversed.
Similarly, any operation that reverses the
sense of phase, which changes the sign of
i, will turn positive energies into negative
energies unless it also changes the direction
of time. So every antiunitary symmetry in
a theory with positive energy must reverse
the direction of time. Every antiunitary operator
can be written as the product of the time
reversal operator and a unitary operator that
does not reverse time.
For a particle with spin J, one can use the
representation
T
=
e
−
i
π
J
y
/
ℏ
K
,
{\displaystyle T=e^{-i\pi J_{y}/\hbar }K,}
where Jy is the y-component of the spin, and
use of TJT−1 = −J has been made.
=== 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 = U U* = U (UT)−1 = Φ,where Φ
is 
a diagonal matrix of phases. As a result,
U = ΦUT and UT = 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 transformation for fermions
in quantum field theories can be represented
by an 8-component spinor in which the above-mentioned
T-parity can be a complex number with unit
radius. The CPT invariance is not a theorem
but a better-to-have property in these class
of theories.
=== 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 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 quantum, 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.
== See also ==
The second law of thermodynamics, Maxwell's
demon and the arrow of time (also Loschmidt's
paradox).
Microscopic reversibility
Detailed balance
Applications to reversible computing and quantum
computing, including limits to computing.
The standard model of particle physics, CP
violation, the CKM matrix and the strong CP
problem
Neutrino masses and CPT invariance.
Wheeler–Feynman absorber theory
Teleonomy
