# Tsymmetry

T-symmetry is the symmetry of physical laws under a time-reversal transformation

$\displaystyle T: t \mapsto -t.$

The universe is not symmetric under time reversal, although in restricted contexts one may find this symmetry. Physicists distinguish time asymmetries that are intrinsic to the dynamic laws of nature, and those that are due to the initial conditions of our universe. The T-asymmetry of the weak nuclear force is of the first kind, while the T-asymmetry of the second law of thermodynamics is of the second kind.

File:Teeter-totter.png
A toy called the teeter-totter illustrates the two aspects of time reversal invariance. When set into motion atop a pedestal, the figure oscillates for a very long time. The toy is engineered to minimize friction and illustrate the reversibility of Newton's laws of motion. However, the mechanically stable state of the toy is when the figure falls down from the pedestal into one of arbitrarily many positions. This is an illustration of the law of increase of entropy through Boltzmann's identification of the logarithm of the number of states with the entropy.

## 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.

Is this time-asymmetric dissipation really inevitable? This question 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 to the stuff of the physical sciences.

The consensus nowadays hinges upon the Boltzmann-Shannon identification of the logarithm of phase space volume with 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.

However, one can 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-symmetry 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 world 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.

## Microscopic phenomena: time reversal invariance

Since most systems are asymmetric under time reversal, it is interesting to ask whether there are any phenomena which do have this 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 which 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.

However, 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. The laws of gravity also seem to be time reversal invariant in classical mechanics.

In physics one separates the laws of motion, ie, 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 is 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

File:Parity 1drep.png
Two dimensional representations of parity are given by a pair of quantum states which 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. Kramer's 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; namely,

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.

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 an unitary operator, S=U, or an anti-unitary unitary one, S=UK where U is unitary, and K denotes complex conjugation. For parity (physics) one has PxP=-x and PpP=-p, where x and p are the position and momentum operators. In canonical quantization, one has the commutator [x,p]=ih/2π, where h is the Planck's constant. This commutator is invariant if P is chosen to be unitary, ie, PiP=i. Such an argument can be attempted for time reversal, T. one has TxT=x and TpT=-p, and the commutator is invariant only if T is chosen to be anti-unitary, ie, TiT=-i. For a particle with spin, one can use the representation

$\displaystyle T = e^{-i\pi S_y/\hbar} K,$

where Sy is the y-component of the spin, to find that TJT=-J.

### 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> it must be proportional to <ψ|J|ψ>. 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.

It is interesting to examine this argument further, since one feels that 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 which 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.

### Kramer's 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 an unitary operator, such as the parity, any phase is allowed.

Next, take a Hamiltonian which is 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 which goes by the name of Kramer's theorem. This implies that if T2=-1, then there is a two-fold degeneracy in the state. This result in non-relativistic quantum mechanics presages the spin statistics theorem of quantum field theory.

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

### Time reversal of the known dynamical laws

The study of particle physics has culminated in a codification of the basic laws of dynamics into the standard model. This is formulated as a quantum field theory which has CPT symmetry, ie, 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.