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

Although in restricted contexts one may find this symmetry, the universe itself does not show symmetry under time reversal. Symmetry in physics refers to features of a Physical system that exhibit the property of Symmetry —that is under certain transformations, aspects of these For other uses see Time (disambiguation Time is a component of a measuring system used to sequence events to compare the durations of In Mathematics a transform is an Operator applied to a function so that under the transform certain operations are simplified The Universe is defined as everything that Physically Exists: the entirety of Space and Time, all forms of Matter, Energy This is due to the uncertainty principle (at quantum scales) and thermodynamic entropy (at larger scales). In Quantum physics, the Heisenberg uncertainty principle states that locating a particle in a small region of space makes the Momentum of the particle uncertain In Thermodynamics (a branch of Physics) entropy, symbolized by S, is a measure of the unavailability of a system ’s Energy

Hence time asymmetries are generally distinguished as either those which are intrinsic to the dynamic laws of nature, and those that are due to the initial conditions of our universe. A physical law or scientific law is a Scientific generalization based on empirical Observations of physical behavior (i The Big Bang is the cosmological model of the Universe that is best supported by all lines of scientific evidence and Observation. The T-asymmetry of the weak force is of the first kind, while the T-asymmetry of the second law of thermodynamics is of the second kind. The weak interaction (often called the weak force or sometimes the weak nuclear force) is one of the four Fundamental interactions of nature The second law of Thermodynamics is an expression of the universal law of increasing Entropy, stating that the entropy of an Isolated system which

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 into which the "lost" energy is translated. In Classical electromagnetism, Maxwell's equations are a set of four Partial differential equations that describe the properties of the electric Friction is the Force resisting the relative motion of two Surfaces in contact or a surface in contact with a fluid (e

A toy called the teeter-totter illustrates the two aspects of time reversal invariance. A seesaw (also known as a teeter-totter) is a long narrow board suspended in the middle so that as one end goes up the other goes down 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. Newton's laws of motion are three Physical laws which provide relationships between the Forces acting on a body and the motion of the 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. In Thermodynamics (a branch of Physics) entropy, symbolized by S, is a measure of the unavailability of a system ’s Energy Ludwig Eduard Boltzmann ( February 20, 1844 &ndash September 5, 1906) was an Austrian Physicist famous for his founding

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. The second law of Thermodynamics is an expression of the universal law of increasing Entropy, stating that the entropy of an Isolated system which 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. Maxwell's demon was an 1867 Thought experiment by the Scottish Physicist James Clerk Maxwell, meant to raise questions about the possibility 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. A thought experiment (from the German Gedankenexperiment) is a proposal for an Experiment that would test a Hypothesis or Theory James Clerk Maxwell (13 June 1831 &ndash 5 November 1879 was a Scottish mathematician and theoretical physicist. 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. In Thermodynamics (a branch of Physics) entropy, symbolized by S, is a measure of the unavailability of a system ’s Energy 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. Claude Elwood Shannon (April 30 1916 – February 24 2001 an American Electronic engineer and Mathematician, is "the father of Information Many interesting results in modern computing are closely related to this problem — reversible computing, quantum computing and physical limits to computing, are examples. Reversible computing includes any Computational process that is (at least to some close approximation Reversible, i A quantum computer is a device for Computation that makes direct use of distinctively Quantum mechanical Phenomena, such as superposition Reversible computing includes any Computational process that is (at least to some close approximation Reversible, i 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 the negative of Shannon information, and hence to entropy. In Mathematics and Physics, a phase space, introduced by Willard Gibbs in 1901 is a Space in which all possible states of a System Information as a concept has a diversity of meanings from everyday usage to technical settings In Thermodynamics (a branch of Physics) entropy, symbolized by S, is a measure of the unavailability of a system ’s Energy 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). CPT symmetry is a fundamental symmetry of Physical laws under transformations that involve the inversions of charge, parity and Such a state would then evolve in reverse, so presumably entropy would decrease (Loschmidt's paradox). Loschmidt's paradox, also known as the reversibility paradox, is the objection that it should not be possible to deduce an irreversible process from time-symmetric dynamics 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. The heat death is a possible final state of the universe, in which it has " run down " to a state of no Thermodynamic free energy to sustain 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. Physical cosmology, as a branch of Astronomy, is the study of the large-scale structure of the Universe and is concerned with fundamental questions about its

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. In General relativity, an event horizon is a boundary in Spacetime, an area surrounding a Black hole or a Wormhole, inside which events cannot A black hole is a theoretical region of space in which the Gravitational field is so powerful that nothing not even Electromagnetic radiation (e 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. Hawking radiation (also known as Bekenstein-Hawking radiation) is a Thermal radiation with a black body spectrum predicted to be emitted by Black holes

The time reversal of a black hole would be a hypothetical object known as a white hole. In Astrophysics, a white hole is the theoretical time reversal of a Black 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 which 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 which 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 irreveresibility is to relate it to the second law of thermodynamics, since black holes are viewed as thermodynamic objects. The second law of Thermodynamics is an expression of the universal law of increasing Entropy, stating that the entropy of an Isolated system which In Physics, black hole thermodynamics is the area of study that seeks to reconcile the Laws of thermodynamics with the existence of Black hole Event 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. String theory is a still-developing scientific approach to Theoretical physics, whose original building blocks are one-dimensional extended objects called strings

Effect of time reversal on some variables of classical physics

Even

Classical variables which do not change upon time reversal include:

, the position of a particle in three-space

, the acceleration of the particle

, the force on the particle

, the energy of the particle

, the electric potential (voltage)

, the electric field

, the electric displacement

, the density of electric charge

, the 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 which are negated by time reversal include:

, the time when an event occurs

, the velocity of a particle

, the linear momentum of a particle

, the angular momentum of a particle (both orbital and spin)

, the electromagnetic vector potential

, the magnetic induction

, the magnetic field

, the density of electric current

, the magnetization

, Poynting vector

Power (rate of work done)

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. Energy density is the amount of Energy stored in a given system or region of space per unit Volume, or per unit Mass, depending on the context although The Maxwell Stress Tensor (also known as Maxwell's Stress Tensor is used to calculate the stresses on objects in magnetic or electrical fields In Physics, the Poynting vector can be thought of as representing the Energy Flux (in W/m2 of an Electromagnetic field. 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. Dissipation itself is originated in the second law of thermodynamics. The second law of Thermodynamics is an expression of the universal law of increasing Entropy, stating that the entropy of an Isolated system which

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. In Physics, the Lorentz force is the Force on a Point charge due to Electromagnetic fields It is given by the following equation 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 electromagnetic field is a physical field produced by electrically charged objects. (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. A magneto-optic effect is any one of a number of phenomena in which an Electromagnetic wave propagates through a medium that has been altered by the presence of a quasistatic This allows one to analyze the conditions under which optical phenomena that locally break time-reversal, such as Faraday isolators, can occur. An optical isolator, or optical diode, is an optical component which allows the transmission of light in only one direction ) 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. Physics (Greek Physis - φύσις in everyday terms is the Science of Matter and its motion. Kinematics ( Greek κινειν, kinein, to move is a branch of Classical mechanics which describes the motion of objects without In physics the term dynamics customarily refers to the time evolution of physical processes 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. Newton's laws of motion are three Physical laws which provide relationships between the Forces acting on a body and the motion of the Quantum mechanics is the study of mechanical systems whose dimensions are close to the Atomic scale such as Molecules Atoms Electrons 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

Two-dimensional representations of parity are given by a pair of quantum states which go into each other under parity. In Physics, a parity transformation (also called parity inversion) is the flip in the sign of one Spatial Coordinate. 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. In the mathematical field of Representation theory, group representations describe abstract groups in terms of Linear transformations of 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; 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. In Physics, the electric dipole moment (or electric dipole for short is a measure of the polarity of a system of Electric charges.

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. In Quantum physics, a quantum state is a mathematical object that fully describes a quantum system. Time reversal does not behave like this. It seems to violate the theorem that all abelian groups be represented by one dimensional irreducible representations. An abelian group, also called a commutative group, is a group satisfying the additional requirement that the product of elements does not depend on their order (the 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. In Mathematics and Physics, in particular in the theory of the Orthogonal groups spinors are elements of a complex vector space introduced to expand the

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 antiunitary one, S = UK where U is unitary, and K denotes complex conjugation. Eugene Paul "EP" Wigner ( Hungarian Wigner Pál Jenő) ( November 17, 1902 &ndash January 1, 1995) was a Quantum mechanics is the study of mechanical systems whose dimensions are close to the Atomic scale such as Molecules Atoms Electrons In Mathematics, an antiunitary transformation, is a bijective function UH_1\to H_2\ between two complex Hilbert spaces such In Functional analysis, a branch of Mathematics, a unitary operator is a Bounded linear operator U    H  →  In Mathematics, the complex conjugate of a Complex number is given by changing the sign of the Imaginary part. For parity (physics) one has PxP = −x and PpP = −p, where x and p are the position and momentum operators. In Physics, a parity transformation (also called parity inversion) is the flip in the sign of one Spatial Coordinate. In canonical quantization, one has the commutator [x, p] = ih/2π, where h is the Planck's constant. The Planck constant (denoted h\ is a Physical constant used to describe the sizes of quanta. 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

where S_y is the y-component of the spin, to find that TJT = −J. In Particle physics, an elementary particle or fundamental particle is a particle not known to have substructure that is it is not known to be made

Electric dipole moments

This has an interesting consequence on the electric dipole moment (EDM) of any particle. In Physics, the electric dipole moment (or electric dipole for short is a measure of the polarity of a system of Electric charges. 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. In Physics a nucleon is a collective name for two Baryons the Neutron and the Proton. In particle physics the strong interaction, or strong force, or color force, holds Quarks and Gluons together to form Protons and Quantum chromodynamics (abbreviated as QCD is a theory of the Strong interaction ( color force a Fundamental force describing the interactions of the Then, using the CPT invariance of a relativistic quantum field theory, this puts strong bounds on strong CP violation. CPT symmetry is a fundamental symmetry of Physical laws under transformations that involve the inversions of charge, parity and In quantum field theory (QFT the forces between particles are mediated by other particles In Particle physics, CP violation is a violation of the postulated CP symmetry of the laws of physics

Experimental bounds on the electron electric dipole moment also place limits on theories of particle physics and their parameters. The electron Electric dipole moment (EDM d_e is roughly speaking a measure of the charge distribution within an Electron.

Kramers' theorem

Main article: Kramers' degeneracy theorem

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

T² = UKUK = U U^* = U (U^T)⁻¹ = Φ,

where Φ is a diagonal matrix of phases. The Kramers degeneracy theorem states that the Energy levels of systems with an odd number of Electrons remain at least doubly Degenerate in the presence 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. In Physics, a parity transformation (also called parity inversion) is the flip in the sign of one Spatial Coordinate.

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 which goes by the name of 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 mechanics is the study of mechanical systems whose dimensions are close to the Atomic scale such as Molecules Atoms Electrons The spin-statistics theorem in Quantum mechanics relates the spin of a particle to the statistics obeyed by that particle In quantum field theory (QFT the forces between particles are mediated by other particles

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. In Quantum physics, a quantum state is a mathematical object that fully describes a quantum system. In Quantum field theory, multiplicative quantum numbers are conserved Quantum numbers of a special kind

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

The study of particle physics has culminated in a codification of the basic laws of dynamics into the standard model. Particle physics is a branch of Physics that studies the elementary constituents of Matter and Radiation, and the interactions between them The Standard Model of Particle physics is a theory that describes three of the four known Fundamental interactions together with the Elementary particles 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. In quantum field theory (QFT the forces between particles are mediated by other particles CPT symmetry is a fundamental symmetry of Physical laws under transformations that involve the inversions of charge, parity and In Physics, a parity transformation (also called parity inversion) is the flip in the sign of one Spatial Coordinate. In Physics, C-symmetry means the symmetry of physical laws under a charge -conjugation transformation. However, time reversal itself is seen not to be a symmetry (this is usually called CP violation). In Particle physics, CP violation is a violation of the postulated CP symmetry of the laws of physics 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. In the Standard Model of Particle physics, the Cabibbo-Kobayashi-Maskawa matrix ( CKM matrix, quark mixing matrix, sometimes also called In Particle physics, flavour or flavor (see spelling differences) is a Quantum number of Elementary particles related to their The weak interaction (often called the weak force or sometimes the weak nuclear force) is one of the four Fundamental interactions of nature The first is seen in experiments, the second is strongly constrained by the non-observation of the EDM of a neutron.

It is important to stress that this 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. The second law of Thermodynamics is an expression of the universal law of increasing Entropy, stating that the entropy of an Isolated system which CPT symmetry is a fundamental symmetry of Physical laws under transformations that involve the inversions of charge, parity and In Particle physics, an elementary particle or fundamental particle is a particle not known to have substructure that is it is not known to be made to most kinds of particles, there is an associated antiparticle with the same Mass and opposite Electric charge. Thus the second law of thermodynamics is thought to originate in the initial conditions in the universe. The second law of Thermodynamics is an expression of the universal law of increasing Entropy, stating that the entropy of an Isolated system which In Mathematics, in the field of Differential equations an initial value problem is an Ordinary differential equation together with specified value called

See also

• The second law of thermodynamics, Maxwell's demon and the arrow of time (also Loschmidt's paradox). The second law of Thermodynamics is an expression of the universal law of increasing Entropy, stating that the entropy of an Isolated system which Maxwell's demon was an 1867 Thought experiment by the Scottish Physicist James Clerk Maxwell, meant to raise questions about the possibility Loschmidt's paradox, also known as the reversibility paradox, is the objection that it should not be possible to deduce an irreversible process from time-symmetric dynamics
• Applications to reversible computing and quantum computing, including limits to computing. Reversible computing includes any Computational process that is (at least to some close approximation Reversible, i A quantum computer is a device for Computation that makes direct use of distinctively Quantum mechanical Phenomena, such as superposition
• The standard model of particle physics, CP violation, the CKM matrix and the strong CP problem
• Neutrino masses and CPT invariance. The Standard Model of Particle physics is a theory that describes three of the four known Fundamental interactions together with the Elementary particles In Particle physics, CP violation is a violation of the postulated CP symmetry of the laws of physics In the Standard Model of Particle physics, the Cabibbo-Kobayashi-Maskawa matrix ( CKM matrix, quark mixing matrix, sometimes also called In Particle physics, CP violation is a violation of the postulated CP symmetry of the laws of physics Neutrinos are Elementary particles that travel close to the Speed of light, lack an Electric charge, are able to pass through ordinary matter almost CPT symmetry is a fundamental symmetry of Physical laws under transformations that involve the inversions of charge, parity and
• Wheeler–Feynman absorber theory

References and external links

• Maxwell's demon: entropy, information, computing, edited by H. The Wheeler–Feynman absorber theory is an interpretation of electrodynamics that starts from the idea that a solution to the electromagnetic field equations has to be symmetric with respect S. Leff and A. F. Rex (IOP publishing, 1990) [ISBN 0-7503-0057-4]
• Maxwell's demon, 2: entropy, classical and quantum information, edited by H. S. Leff and A. F. Rex (IOP publishing, 2003) [ISBN 0-7503-0759-5]
• The emperor's new mind: concerning computers, minds, and the laws of physics, by Roger Penrose (Oxford university press, 2002) [ISBN 0-19-286198-0]
• Sozzi, M. S. (2008). Discrete symmetries and CP violation. Oxford University Press. ISBN 978-0-19-929666-8.  
• CP violation, by I. I. Bigi and A. I. Sanda (Cambridge University Press, 2000) [ISBN 0-521-44349-0]
• Particle Data Group on CP violation
  • the Babar experiment in SLAC
  • the BELLE experiment in KEK
  • the KTeV experiment in Fermilab
  • the CPLEAR experiment in CERN