T-symmetry

T-symmetry or time reversal symmetry is the theoretical symmetry of physical laws under the transformation of time reversal:

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

T-symmetry implies the conservation of entropy. Since the second law of thermodynamics means that entropy increases as time flows toward the future, the macroscopic 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,[1] contrary to their classical counterparts, although this has not yet been experimentally confirmed.

Time asymmetries are generally distinguished as among those...

1. intrinsic to the dynamic physical law (e.g., for the weak force)
2. due to the initial conditions of our universe (e.g., for the second law of thermodynamics)
3. due to measurements (e.g., for the noninvasive measurements)

Invariance

A toy called the teeter-totter illustrates, in cross-section, the two aspects of time reversal invariance. When set into motion atop a pedestal (rocking side to side, as in the image), 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.

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:

${\displaystyle {\vec {x}}\!}$, Position of a particle in three-space
${\displaystyle {\vec {a}}\!}$, Acceleration of the particle
${\displaystyle {\vec {F}}\!}$, Force on the particle
${\displaystyle E\!}$, Energy of the particle
${\displaystyle V\!}$, Electric potential (voltage)
${\displaystyle {\vec {E}}\!}$, Electric field
${\displaystyle {\vec {D}}\!}$, Electric displacement
${\displaystyle \rho \!}$, Density of electric charge
${\displaystyle {\vec {P}}\!}$, Electric polarization
Energy density of the electromagnetic field
${\displaystyle T_{ij}\!}$ 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:

${\displaystyle t\!}$, The time when an event occurs
${\displaystyle {\vec {v}}\!}$, Velocity of a particle
${\displaystyle {\vec {p}}\!}$, Linear momentum of a particle
${\displaystyle {\vec {l}}\!}$, Angular momentum of a particle (both orbital and spin)
${\displaystyle {\vec {A}}\!}$, Electromagnetic vector potential
${\displaystyle {\vec {B}}\!}$, Magnetic field
${\displaystyle {\vec {H}}\!}$, Magnetic auxiliary field
${\displaystyle {\vec {j}}\!}$, Density of electric current
${\displaystyle {\vec {M}}\!}$, Magnetization
${\displaystyle {\vec {S}}\!}$, 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

Two-dimensional representations of parity are given by a pair of quantum states that go into each other under parity. However, this representation can always be reduced to linear combinations of states, each of which is either even or odd under parity. One says that all irreducible representations of parity are one-dimensional. Kramers' theorem states that time reversal need not have this property because it is represented by an anti-unitary operator.

This section contains a discussion of the three most important properties of time reversal in quantum mechanics; chiefly,

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.

On the other hand, the notion of quantum-mechanical time reversal turns out to be a useful tool for the development of physically motivated quantum computing and simulation settings, providing, at the same time, relatively simple tools to assess their complexity. For instance, quantum-mechanical time reversal was used to develop novel boson sampling schemes[2] and to prove the duality between two fundamental optical operations, beam splitter and squeezing transformations[3].

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] = , 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

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

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.[4]

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.[5][6]

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. [1] This type of asymmetry is independent of CPT symmetry but has not yet been confirmed experimentally due to extreme conditions of the checking proposal.

References

1. ^ a b Bednorz, Adam; Franke, Kurt; Belzig, Wolfgang (February 2013). "Noninvasiveness and time symmetry of weak measurements". New Journal of Physics. 15 (2): 023043. arXiv:1108.1305. Bibcode:2013NJPh...15b3043B. doi:10.1088/1367-2630/15/2/023043.
2. ^ Chakhmakhchyan, Levon; Cerf, Nicolas (2017). "Boson sampling with Gaussian measurements". Physical Review A. 96 (3): 032326. arXiv:1705.05299. doi:10.1103/PhysRevA.96.032326.
3. ^ Chakhmakhchyan, Levon; Cerf, Nicolas (2018). "Simulating arbitrary Gaussian circuits with linear optics". Physical Review A. 98 (6): 062314. arXiv:1803.11534. doi:10.1103/PhysRevA.98.062314.
4. ^ Khriplovich, Iosip B.; Lamoreaux, Steve K. (2012). CP violation without strangeness : electric dipole moments of particles, atoms, and molecules. [S.l.]: Springer. ISBN 978-3-642-64577-8.
5. ^ Ibrahim, Tarik; Itani, Ahmad; Nath, Pran (12 Aug 2014). "Electron EDM as a Sensitive Probe of PeV Scale Physics". Physical Review D. 90 (5). arXiv:1406.0083. Bibcode:2014PhRvD..90e5006I. doi:10.1103/PhysRevD.90.055006.
6. ^ Kim, Jihn E.; Carosi, Gianpaolo (4 March 2010). "Axions and the strong CP problem". Reviews of Modern Physics. 82 (1): 557–602. arXiv:0807.3125. Bibcode:2010RvMP...82..557K. doi:10.1103/RevModPhys.82.557.
• Maxwell's demon: entropy, information, computing, edited by H.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.
• Birss, R. R. (1964). Symmetry and Magnetism. John Wiley & Sons, Inc., New York.
• Multiferroic materials with time-reversal breaking optical properties
• 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
C-symmetry

Charge conjugation is a transformation that switches all particles with their corresponding antiparticles, and thus changes the sign of all charges: not only electric charge but also the charges relevant to other forces. In physics, C-symmetry means the symmetry of physical laws under a charge-conjugation transformation. Electromagnetism, gravity and the strong interaction all obey C-symmetry, but weak interactions violate C-symmetry.

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 that is observed to be an exact symmetry of nature at the fundamental level. The CPT theorem says that CPT symmetry holds for all physical phenomena, or more precisely, that any Lorentz invariant local quantum field theory with a Hermitian Hamiltonian must have CPT symmetry.

CP violation

In particle physics, CP violation is a violation of CP-symmetry (or charge conjugation parity symmetry): the combination of C-symmetry (charge conjugation symmetry) and P-symmetry (parity symmetry). CP-symmetry states that the laws of physics should be the same if a particle is interchanged with its antiparticle (C symmetry) while its spatial coordinates are inverted ("mirror" or P symmetry). The discovery of CP violation in 1964 in the decays of neutral kaons resulted in the Nobel Prize in Physics in 1980 for its discoverers James Cronin and Val Fitch.

It plays an important role both in the attempts of cosmology to explain the dominance of matter over antimatter in the present Universe, and in the study of weak interactions in particle physics.

Correlation function (quantum field theory)

In quantum field theory, the (real space) n-point correlation function is defined as the functional average (functional expectation value) of a product of ${\displaystyle n}$ field operators at different positions

${\displaystyle C_{n}(x_{1},x_{2},\ldots ,x_{n}):=\left\langle \phi (x_{1})\phi (x_{2})\ldots \phi (x_{n})\right\rangle ={\frac {\int {\mathcal {D}}\phi \;e^{-S[\phi ]}\phi (x_{1})\ldots \phi (x_{n})}{\int {\mathcal {D}}\phi \;e^{-S[\phi ]}}}}$

For time-dependent correlation functions, the time-ordering operator ${\displaystyle T}$ is included.

Correlation functions are also called simply correlators. Sometimes, the phrase Green's function is used not only for two-point functions, but for any correlators.

The correlation function can be interpreted physically as the amplitude for propagation of a particle or excitation between y and x. In the free theory, it is simply the Feynman propagator (for n=2).

Entropy (arrow of time)

Entropy is the only quantity in the physical sciences (apart from certain rare interactions in particle physics; see below) that requires a particular direction for time, sometimes called an arrow of time. As one goes "forward" in time, the second law of thermodynamics says, the entropy of an isolated system can increase, but not decrease. Hence, from one perspective, entropy measurement is a way of distinguishing the past from the future. However, in thermodynamic systems that are not closed, entropy can decrease with time: many systems, including living systems, reduce local entropy at the expense of an environmental increase, resulting in a net increase in entropy. Examples of such systems and phenomena include the formation of typical crystals, the workings of a refrigerator and living organisms, used in thermodynamics.

Much like temperature, despite being an abstract concept, everyone has an intuitive sense of the effects of entropy. For example, it is often very easy to tell the difference between a video being played forwards or backwards. A video may depict a wood fire that melts a nearby ice block, played in reverse it would show that a puddle of water turned a cloud of smoke into unburnt wood and froze itself in the process. Surprisingly, in either case the vast majority of the laws of physics are not broken by these processes, a notable exception being the second law of thermodynamics. When a law of physics applies equally when time is reversed it is said to show T-symmetry, in this case entropy is what allows one to decide if the video described above is playing forwards or in reverse as intuitively we identify that only when played forwards the entropy of the scene is increasing. Because of the second law of thermodynamics entropy prevents macroscopic processes showing T-symmetry.

When studying at a microscopic scale the above judgements can not be made. Watching a single smoke particle buffeted by air it would not be clear if a video was playing forwards or in reverse and in fact it would not be possible as the laws which apply show T-symmetry, as it drifts left or right qualitatively it looks no different. It is only when you study that gas at a macroscopic scale that the effects of entropy become noticeable. On average you would expect the smoke particles around a struck match to drift away from each other, diffusing throughout the available space. It would be an astronomically improbable event for all the particles to cluster together, yet you can not comment on the movement of any one smoke particle.

By contrast, certain subatomic interactions involving the weak nuclear force violate the conservation of parity, but only very rarely. According to the CPT theorem, this means they should also be time irreversible, and so establish an arrow of time. This, however, is neither linked to the thermodynamic arrow of time, nor has anything to do with our daily experience of time irreversibility.

In physics, Faddeev–Popov ghosts (also called Faddeev–Popov gauge ghosts or Faddeev–Popov ghost fields) are extraneous fields which are introduced into gauge quantum field theories to maintain the consistency of the path integral formulation. They are named after Ludvig Faddeev and Victor Popov.A more general meaning of the word ghost in theoretical physics is discussed in Ghost (physics).

Ghost (physics)

In the terminology of quantum field theory, a ghost, ghost field, or gauge ghost is an unphysical state in a gauge theory. Ghosts are necessary to keep gauge invariance in theories where the local fields exceed a number of physical degrees of freedom.

Michael Cates

Michael Elmhirst Cates (born 5 May 1961) is a British physicist. He is the 19th Lucasian Professor of Mathematics at the University of Cambridge and has held this position since 1 July 2015.

He was previously Professor of Natural Philosophy at the University of Edinburgh, and has held a Royal Society Research Professorship since 2007.His work focuses on the theory of soft matter, such as polymers, colloids, gels, liquid crystals, and granular material. A recurring goal of his research is to create a mathematical model that predicts the stress in a flowing material as a functional of the flow history of that material. Such a mathematical model is called a constitutive equation. He has worked on theories of active matter, particularly dense suspensions of self-propelled particles which can include motile bacteria. His interests also include fundamental field theories of active systems in which time-reversal symmetry (T-symmetry, and more generally, CPT symmetry) is absent. Such theories are characterised by nonzero steady-state Entropy production.

At Edinburgh, Cates was the Principal Investigator of an EPSRC Programme Grant, awarded in 2011, entitled Design Principles for New Soft Materials. On his departure for Cambridge, Cait MacPhee took over as Principal Investigator. Cates remains an Honorary Professor at Edinburgh.

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 respect to inversion in time (T-symmetry);

Second, it relates to the statistical description of the kinetics of macroscopic or mesoscopic systems as an ensemble of elementary processes: collisions, elementary transitions or reactions. For these processes, the consequence of the microscopic T-symmetry is: Corresponding to every individual process there is a reverse process, and in a state of equilibrium the average rate of every process is equal to the average rate of its reverse process.

Proca action

In physics, specifically field theory and particle physics, the Proca action describes a massive spin-1 field of mass m in Minkowski spacetime. The corresponding equation is a relativistic wave equation called the Proca equation. The Proca action and equation are named after Romanian physicist Alexandru Proca.

The Proca equation is involved in the Standard model and describes there the three massive vector bosons, i.e. the Z and W bosons.

This article uses the (+−−−) metric signature and tensor index notation in the language of 4-vectors.

Quantization (physics)

In physics, quantization is the process of transition from a classical understanding of physical phenomena to a newer understanding known as quantum mechanics. (It is a procedure for constructing a quantum field theory starting from a classical field theory.) This is a generalization of the procedure for building quantum mechanics from classical mechanics. One also speaks of field quantization, as in the "quantization of the electromagnetic field", where one refers to photons as field "quanta" (for instance as light quanta). This procedure is basic to theories of particle physics, nuclear physics, condensed matter physics, and quantum optics.

Quantum cosmology

Quantum cosmology is the attempt in theoretical physics to develop a quantum theory of the Universe. This approach attempts to answer open questions of classical physical cosmology, particularly those related to the first phases of the universe.

The classical cosmology is based on Albert Einstein's general theory of relativity (GTR or simply GR). It describes the evolution of the universe very well, as long as you do not approach the Big Bang. It is the gravitational singularity and the Planck time where relativity theory fails to provide what must be demanded of a final theory of space and time. Therefore, a theory is needed that integrates relativity theory and quantum theory. Such an approach is attempted for instance with the loop quantum gravity, the string theory and the causal set theory.

Quantum field theory in curved spacetime

In particle physics, quantum field theory in curved spacetime is an extension of standard, Minkowski space quantum field theory to curved spacetime. A general prediction of this theory is that particles can be created by time-dependent gravitational fields (multigraviton pair production), or by time-independent gravitational fields that contain horizons.

Symmetry (physics)

In physics, a symmetry of a physical system is a physical or mathematical feature of the system (observed or intrinsic) that is preserved or remains unchanged under some transformation.

A family of particular transformations may be continuous (such as rotation of a circle) or discrete (e.g., reflection of a bilaterally symmetric figure, or rotation of a regular polygon). Continuous and discrete transformations give rise to corresponding types of symmetries. Continuous symmetries can be described by Lie groups while discrete symmetries are described by finite groups (see Symmetry group).

These two concepts, Lie and finite groups, are the foundation for the fundamental theories of modern physics. Symmetries are frequently amenable to mathematical formulations such as group representations and can, in addition, be exploited to simplify many problems.

Arguably the most important example of a symmetry in physics is that the speed of light has the same value in all frames of reference, which is known in mathematical terms as the Poincaré group, the symmetry group of special relativity. Another important example is the invariance of the form of physical laws under arbitrary differentiable coordinate transformations, which is an important idea in general relativity.

Term (time)

A term is a period of duration, time or occurrence, in relation to an event. To differentiate an interval or duration, common phrases are used to distinguish the observance of length are near-term or short-term, medium-term or mid-term and long-term.

It is also used as part of a calendar year, especially one of the three parts of an academic term and working year in the United Kingdom: Michaelmas term, Hilary term / Lent term or Trinity term / Easter term, the equivalent to the American semester. In America there is a midterm election held in the middle of the four-year presidential term, there are also academic midterm exams.

In economics, it is the period required for economic agents to reallocate resources, and generally reestablish equilibrium. The actual length of this period, usually numbered in years or decades, varies widely depending on circumstantial context. During the long term, all factors are variable.

In finance or financial operations of borrowing and investing, what is considered long-term is usually above 3 years, with medium-term usually between 1 and 3 years and short-term usually under 1 year. It is also used in some countries to indicate a fixed term investment such as a term deposit.

In law, the term of a contract is the duration for which it is to remain in effect (not to be confused with the meaning of "term" that denotes any provision of a contract). A fixed-term contract is one concluded for a pre-defined time.

Time reversibility

A mathematical or physical process is time-reversible if the dynamics of the process remain well-defined when the sequence of time-states is reversed.

A deterministic process is time-reversible if the time-reversed process satisfies the same dynamic equations as the original process; in other words, the equations are invariant or symmetrical under a change in the sign of time. A stochastic process is reversible if the statistical properties of the process are the same as the statistical properties for time-reversed data from the same process.

Weyl equation

In physics, particularly quantum field theory, the Weyl equation is a relativistic wave equation for describing massless spin-1/2 particles called Weyl fermions. It is named after the German physicist Hermann Weyl.

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 assumption that the solutions of the electromagnetic field equations must be invariant under time-reversal transformation, as are the field equations themselves. Indeed, there is no apparent reason for the time-reversal symmetry breaking, which singles out a preferential time direction and thus makes a distinction between past and future. A time-reversal invariant theory is more logical and elegant. Another key principle, resulting from this interpretation and reminiscent of Mach's principle due to Tetrode, is that elementary particles are not self-interacting. This immediately removes the problem of self-energies.

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