T-symmetry or time reversal symmetry is the theoretical
symmetry of physical laws under the
transformation
Transformation may refer to:
Science and mathematics
In biology and medicine
* Metamorphosis, the biological process of changing physical form after birth or hatching
* Malignant transformation, the process of cells becoming cancerous
* Tran ...
of time reversal,
:
Since the
second law of thermodynamics
The second law of thermodynamics is a physical law based on universal experience concerning heat and Energy transformation, energy interconversions. One simple statement of the law is that heat always moves from hotter objects to colder objects ( ...
states that
entropy
Entropy is a scientific concept, as well as a measurable physical property, that is most commonly associated with a state of disorder, randomness, or uncertainty. The term and the concept are used in diverse fields, from classical thermodynam ...
increases as time flows toward the future, in general, the macroscopic
universe
The universe is all of space and time and their contents, including planets, stars, galaxies, and all other forms of matter and energy. The Big Bang theory is the prevailing cosmological description of the development of the universe. Acc ...
does not 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'' generally are caused by one of three categories:
# intrinsic to the dynamic
physical law
Scientific laws or laws of science are statements, based on repeated experiments or observations, that describe or predict a range of natural phenomena. The term ''law'' has diverse usage in many cases (approximate, accurate, broad, or narrow) ...
(e.g., for the
weak force
Weak may refer to:
Songs
* Weak (AJR song), "Weak" (AJR song), 2016
* Weak (Melanie C song), "Weak" (Melanie C song), 2011
* Weak (SWV song), "Weak" (SWV song), 1993
* Weak (Skunk Anansie song), "Weak" (Skunk Anansie song), 1995
* "Weak", a song ...
)
# due to the
initial conditions of the universe (e.g., for the
second law of thermodynamics
The second law of thermodynamics is a physical law based on universal experience concerning heat and Energy transformation, energy interconversions. One simple statement of the law is that heat always moves from hotter objects to colder objects ( ...
)
# due to
measurements (e.g., for the noninvasive measurements)
Macroscopic phenomena
The second law of thermodynamics
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 a physical law based on universal experience concerning heat and Energy transformation, energy interconversions. One simple statement of the law is that heat always moves from hotter objects to colder objects ( ...
. 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
Maxwell's demon is a thought experiment that would hypothetically violate the second law of thermodynamics. It was proposed by the physicist James Clerk Maxwell in 1867. In his first letter Maxwell called the demon a "finite being", while the ' ...
. The name comes from a
thought experiment
A thought experiment is a hypothetical situation in which a hypothesis, theory, or principle is laid out for the purpose of thinking through its consequences.
History
The ancient Greek ''deiknymi'' (), or thought experiment, "was the most anc ...
described by
James Clerk Maxwell
James Clerk Maxwell (13 June 1831 – 5 November 1879) was a Scottish mathematician and scientist responsible for the classical theory of electromagnetic radiation, which was the first theory to describe electricity, magnetism and ligh ...
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
Entropy is a scientific concept, as well as a measurable physical property, that is most commonly associated with a state of disorder, randomness, or uncertainty. The term and the concept are used in diverse fields, from classical thermodynam ...
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
Entropy is a scientific concept, as well as a measurable physical property, that is most commonly associated with a state of disorder, randomness, or uncertainty. The term and the concept are used in diverse fields, from classical thermodynam ...
. Many interesting results in modern computing are closely related to this problem —
reversible computing
Reversible computing is any model of computation where the computational process, to some extent, is time-reversible. In a model of computation that uses deterministic transitions from one state of the abstract machine to another, a necessary co ...
,
quantum computing and
physical limits to computing
The limits of computation are governed by a number of different factors. In particular, there are several physical and practical limits to the amount of computation or data storage that can be performed with a given amount of mass, volume, or ene ...
, 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
In information theory, the information content, self-information, surprisal, or Shannon information is a basic quantity derived from the probability of a particular event occurring from a random variable. It can be thought of as an alternative wa ...
, and hence to
entropy
Entropy is a scientific concept, as well as a measurable physical property, that is most commonly associated with a state of disorder, randomness, or uncertainty. The term and the concept are used in diverse fields, from classical thermodynam ...
. 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
In thermodynamics, dissipation is the result of an irreversible process that takes place in homogeneous thermodynamic systems. In a dissipative process, energy ( internal, bulk flow kinetic, or system potential) transforms from an initial form to ...
, the molecular coordinates can move into larger volumes of phase space, becoming more uncertain, and thus leading to increase in entropy.
Big Bang
One resolution to irreversibility 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
Heat death may refer to:
*Heat death of the universe, a proposed cosmological event
** Heat death paradox, a philosophical examination of the cosmological event
*Hyperthermia, injury up to and including death, from excessive heat
*Thermal shock
...
equilibrium) would actually result in no increase of entropy. In this view, the apparent T-asymmetry of our universe is a problem in
cosmology
Cosmology () is a branch of physics and metaphysics dealing with the nature of the universe. The term ''cosmology'' was first used in English in 1656 in Thomas Blount (lexicographer), Thomas Blount's ''Glossographia'', and in 1731 taken up in ...
: why did the universe start with a low entropy? This view, supported by cosmological observation (such as the
isotropy
Isotropy is uniformity in all orientations; it is derived . Precise definitions depend on the subject area. Exceptions, or inequalities, are frequently indicated by the prefix ' or ', hence ''anisotropy''. ''Anisotropy'' is also used to describe ...
of the
Cosmic Microwave Background
In Big Bang cosmology the cosmic microwave background (CMB, CMBR) is electromagnetic radiation that is a remnant from an early stage of the universe, also known as "relic radiation". The CMB is faint cosmic background radiation filling all spac ...
) connect this problem to the question of ''initial conditions'' of the universe.
Black holes
The laws of gravity seem to be time reversal invariant in classical mechanics; however, specific solutions need not be.
An object can cross through the
event horizon
In astrophysics, an event horizon is a boundary beyond which events cannot affect an observer. Wolfgang Rindler coined the term in the 1950s.
In 1784, John Michell proposed that gravity can be strong enough in the vicinity of massive compact ob ...
of a
black hole
A black hole is a region of spacetime where gravitation, gravity is so strong that nothing, including light or other Electromagnetic radiation, electromagnetic waves, has enough energy to escape it. The theory of general relativity predicts t ...
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
Hawking radiation is theoretical black body radiation that is theorized to be released outside a black hole's event horizon because of relativistic quantum effects. It is named after the physicist Stephen Hawking, who developed a theoretical a ...
.
The time reversal of a black hole would be a hypothetical object known as a
white hole
In general relativity, a white hole is a hypothetical region of spacetime and singularity that cannot be entered from the outside, although energy-matter, light and information can escape from it. In this sense, it is the reverse of a black ho ...
. 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
The second law of thermodynamics is a physical law based on universal experience concerning heat and Energy transformation, energy interconversions. One simple statement of the law is that heat always moves from hotter objects to colder objects ( ...
, since black holes are viewed as
thermodynamic objects. For example, according to the
gauge–gravity duality
In physics, string theory is a theoretical framework in which the point-like particles of particle physics are replaced by one-dimensional objects called strings. String theory describes how these strings propagate through space and intera ...
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
Chemical kinetics, also known as reaction kinetics, is the branch of physical chemistry that is concerned with understanding the rates of chemical reactions. It is to be contrasted with chemical thermodynamics, which deals with the direction in wh ...
, T-symmetry of the mechanical microscopic equations implies two important laws: the principle of
detailed balance The principle of detailed balance can be used in kinetic systems which are decomposed into elementary processes (collisions, or steps, or elementary reactions). It states that at equilibrium, each elementary process is in equilibrium with its reve ...
and the
Onsager reciprocal relations
In thermodynamics, the Onsager reciprocal relations express the equality of certain ratios between flows and forces in thermodynamic systems out of equilibrium, but where a notion of local equilibrium exists.
"Reciprocal relations" occur betwe ...
. T-symmetry of the microscopic description together with its kinetic consequences are called
microscopic reversibility The principle of microscopic reversibility in physics and chemistry is twofold:
* First, it states that the microscopic detailed dynamics of particles and fields is time-reversible because the microscopic equations of motion are symmetric with resp ...
.
Effect of time reversal on some variables of classical physics
Even
Classical variables that do not change upon time reversal include:
:
, position of a particle in three-space
:
, acceleration of the particle
:
, force on the particle
:
, energy of the particle
:
, electric potential (voltage)
:
, electric field
:
, electric displacement
:
, density of electric charge
:
, electric polarization
:
Energy density of the electromagnetic field
:
,
Maxwell stress tensor
The Maxwell stress tensor (named after James Clerk Maxwell) is a symmetric second-order tensor used in classical electromagnetism to represent the interaction between electromagnetic forces and mechanical momentum. In simple situations, such as ...
:All masses, charges, coupling constants, and other physical constants, except those associated with the weak force.
Odd
Classical variables that time reversal negates include:
:
, the time when an event occurs
:
, velocity of a particle
:
, linear momentum of a particle
:
, angular momentum of a particle (both orbital and spin)
:
, electromagnetic vector potential
:
, magnetic field
:
, magnetic auxiliary field
:
, density of electric current
:
, magnetization
:
,
Poynting vector
:
, power (rate of work done).
Example: Magnetic Field and Onsager reciprocal relations
Let us consider the example of a system of charged particles subject to a constant external magnetic field: in this case the canonical time reversal operation that reverses the velocities and the time
and keeps the coordinates untouched is no more a symmetry for the system. Under this consideration, it seems that only Onsager–Casimir reciprocal relations could hold; these equalities relate two different systems, one subject to
and another to
, and so their utility is limited. However, there was proved that it is possible to find other time reversal operations which preserve the dynamics and so Onsager reciprocal relations; in conclusion, one cannot state that the presence of a magnetic field always breaks 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 second law of thermodynamics is a physical law based on universal experience concerning heat and Energy transformation, energy interconversions. One simple statement of the law is that heat always moves from hotter objects to colder objects ( ...
.
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 field
An electromagnetic field (also EM field or EMF) is a classical (i.e. non-quantum) field produced by (stationary or moving) electric charges. It is the field described by classical electrodynamics (a classical field theory) and is the classical c ...
s 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
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 magnetic field. In such a medium, which is also called gyrotropic or ...
is analyzed. This allows one to analyze the conditions under which optical phenomena that locally break time-reversal, such as
Faraday isolator
An optical isolator, or optical diode, is an optical component which allows the transmission of light in only one direction. It is typically used to prevent unwanted feedback into an optical oscillator, such as a laser cavity.
The operation ...
s an
directional dichroism can occur.)
In
physics
Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...
one separates the laws of motion, called
kinematics
Kinematics is a subfield of physics, developed in classical mechanics, that describes the Motion (physics), motion of points, Physical object, bodies (objects), and systems of bodies (groups of objects) without considering the forces that cause ...
, from the laws of force, called
dynamics. Following the classical kinematics of
Newton's laws of motion
Newton's laws of motion are three basic laws of classical mechanics that describe the relationship between the motion of an object and the forces acting on it. These laws can be paraphrased as follows:
# A body remains at rest, or in moti ...
, the kinematics of
quantum mechanics
Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...
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 (for
fermion
In particle physics, a fermion is a particle that follows Fermi–Dirac statistics. Generally, it has a half-odd-integer spin: spin , spin , etc. In addition, these particles obey the Pauli exclusion principle. Fermions include all quarks an ...
s).
The strangeness of this result is clear if one compares it with parity. If parity transforms a pair of
quantum states
In quantum physics, a quantum state is a mathematical entity that provides a probability distribution for the outcomes of each possible measurement on a system. Knowledge of the quantum state together with the rules for the system's evolution i ...
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 group
In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is comm ...
s 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
spinor
In geometry and physics, spinors are elements of a complex vector space that can be associated with Euclidean space. Like geometric vectors and more general tensors, spinors transform linearly when the Euclidean space is subjected to a sligh ...
s 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
A simulation is the imitation of the operation of a real-world process or system over time. Simulations require the use of Conceptual model, models; the model represents the key characteristics or behaviors of the selected system or proc ...
settings, providing, at the same time, relatively simple tools to assess their
complexity
Complexity characterises the behaviour of a system or model whose components interaction, interact in multiple ways and follow local rules, leading to nonlinearity, randomness, collective dynamics, hierarchy, and emergence.
The term is generall ...
. For instance, quantum-mechanical time reversal was used to develop novel
boson sampling
Boson sampling is a restricted model of non-universal quantum computation introduced by Scott Aaronson and Alex Arkhipov after the original work of Lidror Troyansky and Naftali Tishby, that explored possible usage of boson scattering to evaluat ...
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
Involution may refer to:
* Involute, a construction in the differential geometry of curves
* '' Agricultural Involution: The Processes of Ecological Change in Indonesia'', a 1963 study of intensification of production through increased labour inpu ...
, capturing the actual reversal of the time coordinate, the T that is an ordinary finite dimensional matrix, acting on
spinor
In geometry and physics, spinors are elements of a complex vector space that can be associated with Euclidean space. Like geometric vectors and more general tensors, spinors transform linearly when the Euclidean space is subjected to a sligh ...
s and vectors, and the T that is an operator on a infinite-dimensional
Hilbert space.
For a
real
Real may refer to:
Currencies
* Brazilian real (R$)
* Central American Republic real
* Mexican real
* Portuguese real
* Spanish real
* Spanish colonial real
Music Albums
* ''Real'' (L'Arc-en-Ciel album) (2000)
* ''Real'' (Bright album) (2010) ...
(not
complex
Complex commonly refers to:
* Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe
** Complex system, a system composed of many components which may interact with each ...
) classical (unquantized)
scalar field
In mathematics and physics, a scalar field is a function (mathematics), function associating a single number to every point (geometry), point in a space (mathematics), space – possibly physical space. The scalar may either be a pure Scalar ( ...
, the time reversal
involution
Involution may refer to:
* Involute, a construction in the differential geometry of curves
* '' Agricultural Involution: The Processes of Ecological Change in Indonesia'', a 1963 study of intensification of production through increased labour inpu ...
can simply be written as
:
as time reversal leaves the scalar value at a fixed spacetime point unchanged, up to an overall sign
. A slightly more formal way to write this is
:
which has the advantage of emphasizing that
is a
map
A map is a symbolic depiction emphasizing relationships between elements of some space, such as objects, regions, or themes.
Many maps are static, fixed to paper or some other durable medium, while others are dynamic or interactive. Although ...
, and thus the "mapsto" notation
whereas
is a factual statement relating the old and new fields to one-another.
Unlike scalar fields,
spinor
In geometry and physics, spinors are elements of a complex vector space that can be associated with Euclidean space. Like geometric vectors and more general tensors, spinors transform linearly when the Euclidean space is subjected to a sligh ...
and
vector fields
might have a non-trivial behavior under time reversal. In this case, one has to write
:
where
is just an ordinary
matrix
Matrix most commonly refers to:
* ''The Matrix'' (franchise), an American media franchise
** ''The Matrix'', a 1999 science-fiction action film
** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...
. For
complex
Complex commonly refers to:
* Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe
** Complex system, a system composed of many components which may interact with each ...
fields,
complex conjugation
In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign. That is, (if a and b are real, then) the complex conjugate of a + bi is equal to a - ...
may be required, for which the mapping
can be thought of as a 2x2 matrix. For a
Dirac spinor
In quantum field theory, the Dirac spinor is the spinor that describes all known fundamental particles that are fermions, with the possible exception of neutrinos. It appears in the plane-wave solution to the Dirac equation, and is a certain com ...
,
cannot be written as a 4x4 matrix, because, in fact, complex conjugation is indeed required; however, it can be written as an 8x8 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
; 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
that achieves this goal. In some cases, generic arguments can be made. Thus, for example, for spinors in three-dimensional
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's Elements, Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics ther ...
, or four-dimensional
Minkowski space
In mathematical physics, Minkowski space (or Minkowski spacetime) () is a combination of three-dimensional Euclidean space and time into a four-dimensional manifold where the spacetime interval between any two events is independent of the inerti ...
, an explicit transformation can be given. It is conventionally given as
:
where
is the y-component of the
angular momentum operator
In quantum mechanics, the angular momentum operator is one of several related operators analogous to classical angular momentum. The angular momentum operator plays a central role in the theory of atomic and molecular physics and other quantum prob ...
and
is complex conjugation, as before. This form follows whenever the spinor can be described with a linear
differential equation
In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, an ...
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
In mathematics and physics, a tensor field assigns a tensor to each point of a mathematical space (typically a Euclidean space or manifold). Tensor fields are used in differential geometry, algebraic geometry, general relativity, in the analysis ...
In this case,
:
Covariant tensor indexes will transform as
and so on. For quantum fields, there is also a third T, written as
which is actually an infinite dimensional operator acting on a Hilbert space. It acts on quantized fields
as
:
This can be thought of as a special case of a tensor with one covariant, and one contravariant index, and thus two
's are required.
All three of these symbols capture the idea of time-reversal; they differ with respect to the specific
space
Space is the boundless three-dimensional extent in which objects and events have relative position and direction. In classical physics, physical space is often conceived in three linear dimensions, although modern physicists usually cons ...
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
or
or
depending on context, left for the reader to infer.
Anti-unitary representation of time reversal
Eugene Wigner
Eugene Paul "E. P." Wigner ( hu, Wigner Jenő Pál, ; November 17, 1902 – January 1, 1995) was a Hungarian-American theoretical physicist who also contributed to mathematical physics. He received the Nobel Prize in Physics in 1963 "for his co ...
showed that a symmetry operation ''S'' of a Hamiltonian is represented, in
quantum mechanics
Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...
either by a
unitary operator
In functional analysis, a unitary operator is a surjective bounded operator on a Hilbert space that preserves the inner product. Unitary operators are usually taken as operating ''on'' a Hilbert space, but the same notion serves to define the co ...
, , or an
antiunitary
In mathematics, an antiunitary transformation, is a bijective antilinear map
:U: H_1 \to H_2\,
between two complex Hilbert spaces such that
:\langle Ux, Uy \rangle = \overline
for all x and y in H_1, where the horizontal bar represents the ...
one, where ''U'' is unitary, and ''K'' denotes
complex conjugation
In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign. That is, (if a and b are real, then) the complex conjugate of a + bi is equal to a - ...
. 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 . Similarly, it reverses the direction of ''momentum'', so that , where ''x'' and ''p'' are the position and momentum operators. This preserves the
canonical commutator , where ''ħ'' is the
reduced Planck constant
The Planck constant, or Planck's constant, is a fundamental physical constant of foundational importance in quantum mechanics. The constant gives the relationship between the energy of a photon and its frequency, and by the mass-energy equivale ...
, only if ''P'' is chosen to be unitary, .
On the other hand, the ''time reversal'' operator ''T'', it does nothing to the x-operator, , but it reverses the direction of p, so that . The canonical commutator is invariant only if ''T'' is chosen to be anti-unitary, i.e., .
Another argument involves energy, the time-component of the four-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
In the physical sciences, a particle (or corpuscule in older texts) is a small localized object which can be described by several physical or chemical properties, such as volume, density, or mass.
They vary greatly in size or quantity, from ...
with spin ''J'', one can use the representation
::
where ''J''
''y'' is the ''y''-component of the spin, and use of 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: , 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
In theoretical physics, quantum chromodynamics (QCD) is the theory of the strong interaction between quarks mediated by gluons. Quarks are fundamental particles that make up composite hadrons such as the proton, neutron and pion. QCD is a type ...
. Then, using the
CPT invariance
Charge, parity, and time reversal symmetry is a fundamental symmetry of physical laws under the simultaneous transformations of charge conjugation (C), parity transformation (P), and time reversal (T). CPT is the only combination of C, P, and ...
of a relativistic
quantum field theory
In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity, and quantum mechanics. QFT is used in particle physics to construct physical models of subatomic particles and ...
, this puts
strong bounds on
strong CP violation.
Experimental bounds on the
electron electric dipole moment
The electron electric dipole moment is an intrinsic property of an electron such that the potential energy is linearly related to the strength of the electric field:
:U = \mathbf d_ \cdot \mathbf E.
The electron's electric dipole moment (EDM) m ...
also place limits on theories of particle physics and their parameters.
Kramers' theorem
For ''T'', which is an anti-unitary ''Z''
2 symmetry generator
: ''T''
2 = ''UKUK'' = ''UU''
* = ''U'' (''U''
T)
−1 = Φ,
where Φ is a diagonal matrix of phases. As a result, and , showing that
:''U'' = Φ ''U'' Φ.
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
Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...
presages the
spin statistics theorem
Spin or spinning most often refers to:
* Spinning (textiles), the creation of yarn or thread by twisting fibers together, traditionally by hand spinning
* Spin, the rotation of an object around a central axis
* Spin (propaganda), an intentionally b ...
of
quantum field theory
In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity, and quantum mechanics. QFT is used in particle physics to construct physical models of subatomic particles and ...
.
Quantum state
In quantum physics, a quantum state is a mathematical entity that provides a probability distribution for the outcomes of each possible measurement on a system. Knowledge of the quantum state together with the rules for the system's evolution i ...
s that give unitary representations of time reversal, i.e., have , are characterized by a
multiplicative quantum number
In quantum field theory, multiplicative quantum numbers are conserved quantum numbers of a special kind. A given quantum number ''q'' is said to be additive if in a particle reaction the sum of the ''q''-values of the interacting particles is th ...
, sometimes called the T-parity.
Time reversal of the known dynamical laws
Particle physics
Particle physics or high energy physics is the study of fundamental particles and forces that constitute matter and radiation. The fundamental particles in the universe are classified in the Standard Model as fermions (matter particles) an ...
codified the basic laws of dynamics into the
standard model
The Standard Model of particle physics is the theory describing three of the four known fundamental forces (electromagnetism, electromagnetic, weak interaction, weak and strong interactions - excluding gravity) in the universe and classifying a ...
. This is formulated as a
quantum field theory
In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity, and quantum mechanics. QFT is used in particle physics to construct physical models of subatomic particles and ...
that has
CPT symmetry
Charge, parity, and time reversal symmetry is a fundamental symmetry of physical laws under the simultaneous transformations of charge conjugation (C), parity transformation (P), and time reversal (T). CPT is the only combination of C, P, and T ...
, i.e., the laws are invariant under simultaneous operation of time reversal,
parity and
charge conjugation
In physics, charge conjugation is a transformation that switches all particles with their corresponding antiparticles, thus changing the sign of all charges: not only electric charge but also the charges relevant to other forces. The term C-sy ...
. 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 CP-symmetry (or charge conjugation parity symmetry): the combination of C-symmetry (charge symmetry) and P-symmetry ( parity symmetry). CP-symmetry states that the laws of physics should be the ...
). There are two possible origins of this asymmetry, one through the
mixing of different
flavours of quarks in their
weak decay
In nuclear physics and particle physics, the weak interaction, which is also often called the weak force or weak nuclear force, is one of the four known fundamental interactions, with the others being electromagnetism, the strong interaction ...
s, 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
The second law of thermodynamics is a physical law based on universal experience concerning heat and Energy transformation, energy interconversions. One simple statement of the law is that heat always moves from hotter objects to colder objects ( ...
, because due to the conservation of the
CPT symmetry
Charge, parity, and time reversal symmetry is a fundamental symmetry of physical laws under the simultaneous transformations of charge conjugation (C), parity transformation (P), and time reversal (T). CPT is the only combination of C, P, and T ...
, the effect of time reversal is to rename
particle
In the physical sciences, a particle (or corpuscule in older texts) is a small localized object which can be described by several physical or chemical properties, such as volume, density, or mass.
They vary greatly in size or quantity, from ...
s as
antiparticle
In particle physics, every type of particle is associated with an antiparticle with the same mass but with opposite physical charges (such as electric charge). For example, the antiparticle of the electron is the positron (also known as an antie ...
s and ''vice versa''. Thus the
second law of thermodynamics
The second law of thermodynamics is a physical law based on universal experience concerning heat and Energy transformation, energy interconversions. One simple statement of the law is that heat always moves from hotter objects to colder objects ( ...
is thought to originate in the
initial conditions
In mathematics and particularly in dynamic systems, an initial condition, in some contexts called a seed value, is a value of an evolving variable at some point in time designated as the initial time (typically denoted ''t'' = 0). For ...
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
The second law of thermodynamics is a physical law based on universal experience concerning heat and Energy transformation, energy interconversions. One simple statement of the law is that heat always moves from hotter objects to colder objects ( ...
. 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
Charge, parity, and time reversal symmetry is a fundamental symmetry of physical laws under the simultaneous transformations of charge conjugation (C), parity transformation (P), and time reversal (T). CPT is the only combination of C, P, and T ...
but has not yet been confirmed experimentally due to extreme conditions of the checking proposal.
See also
*
Arrow of time
The arrow of time, also called time's arrow, is the concept positing the "one-way direction" or "asymmetry" of time. It was developed in 1927 by the British astrophysicist Arthur Eddington, and is an unsolved general physics question. This ...
*
Causality (physics)
Causality is the relationship between causes and effects. While causality is also a topic studied from the perspectives of philosophy and physics, it is operationalized so that causes of an event must be in the past light cone of the event and ...
* Computing applications
**
Limits to computing
**
Quantum computing
**
Reversible computing
Reversible computing is any model of computation where the computational process, to some extent, is time-reversible. In a model of computation that uses deterministic transitions from one state of the abstract machine to another, a necessary co ...
*
Standard model
The Standard Model of particle physics is the theory describing three of the four known fundamental forces (electromagnetism, electromagnetic, weak interaction, weak and strong interactions - excluding gravity) in the universe and classifying a ...
**
CKM matrix
**
CP violation
In particle physics, CP violation is a violation of CP-symmetry (or charge conjugation parity symmetry): the combination of C-symmetry (charge symmetry) and P-symmetry ( parity symmetry). CP-symmetry states that the laws of physics should be the ...
**
CPT invariance
Charge, parity, and time reversal symmetry is a fundamental symmetry of physical laws under the simultaneous transformations of charge conjugation (C), parity transformation (P), and time reversal (T). CPT is the only combination of C, P, and ...
**
Neutrino mass
A neutrino ( ; denoted by the Greek letter ) is a fermion (an elementary particle with spin of ) that interacts only via the weak interaction and gravity. The neutrino is so named because it is electrically neutral and because its rest mass i ...
**
Strong CP problem
**
Wheeler–Feynman absorber theory
The Wheeler–Feynman absorber theory (also called the Wheeler–Feynman time-symmetric theory), named after its originators, the physicists Richard Feynman and John Archibald Wheeler, is an interpretation of electrodynamics derived from the assu ...
*
Loschmidt's paradox
Loschmidt's paradox, also known as the reversibility paradox, irreversibility paradox or ', is the objection that it should not be possible to deduce an irreversible process from time-symmetric dynamics. This puts the time reversal symmetry of (al ...
*
Maxwell's demon
Maxwell's demon is a thought experiment that would hypothetically violate the second law of thermodynamics. It was proposed by the physicist James Clerk Maxwell in 1867. In his first letter Maxwell called the demon a "finite being", while the ' ...
*
Microscopic reversibility The principle of microscopic reversibility in physics and chemistry is twofold:
* First, it states that the microscopic detailed dynamics of particles and fields is time-reversible because the microscopic equations of motion are symmetric with resp ...
*
Second law of thermodynamics
The second law of thermodynamics is a physical law based on universal experience concerning heat and Energy transformation, energy interconversions. One simple statement of the law is that heat always moves from hotter objects to colder objects ( ...
*
Tesla valve
A Tesla valve, called a valvular conduit by its inventor, is a fixed-geometry passive check valve. It allows a fluid to flow preferentially in one direction, without moving parts. The device is named after Nikola Tesla, who was awarded in 192 ...
*
Time translation symmetry
Time translation symmetry or temporal translation symmetry (TTS) is a mathematical transformation in physics that moves the times of events through a common interval. Time translation symmetry is the law that the laws of physics are unchanged ( ...
References
Inline citations
General references
*Maxwell's demon: entropy, information, computing, edited by H.S.Leff and A.F. Rex (IOP publishing, 1990)
*Maxwell's demon, 2: entropy, classical and quantum information, edited by H.S.Leff and A.F. Rex (IOP publishing, 2003)
*The emperor's new mind: concerning computers, minds, and the laws of physics, by Roger Penrose (Oxford university press, 2002)
*
*
Multiferroicmaterials with time-reversal breaking optical properties
*CP violation, by I.I. Bigi and A.I. Sanda (Cambridge University Press, 2000)
Particle Data Group on CP violation**th
Babarexperiment in
SLAC
SLAC National Accelerator Laboratory, originally named the Stanford Linear Accelerator Center,
is a United States Department of Energy National Laboratory operated by Stanford University under the programmatic direction of the U.S. Departme ...
**th
BELLEexperiment in
KEK
, known as KEK, is a Japanese organization whose purpose is to operate the largest particle physics laboratory in Japan, situated in Tsukuba, Ibaraki prefecture. It was established in 1997. The term "KEK" is also used to refer to the laboratory ...
**th
KTeVexperiment in
Fermilab
Fermi National Accelerator Laboratory (Fermilab), located just outside Batavia, Illinois, near Chicago, is a United States Department of Energy national laboratory specializing in high-energy particle physics. Since 2007, Fermilab has been operat ...
**th
CPLEARexperiment in
CERN
The European Organization for Nuclear Research, known as CERN (; ; ), is an intergovernmental organization that operates the largest particle physics laboratory in the world. Established in 1954, it is based in a northwestern suburb of Gene ...
{{DEFAULTSORT:T-Symmetry
Time in physics
Thermodynamics
Statistical mechanics
Philosophy of thermal and statistical physics
Quantum field theory
Symmetry