T-symmetry
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T-symmetry or time reversal symmetry is the theoretical
Since the second law of thermodynamics states that entropy increases as time flows toward the future, in general, the macroscopic universe 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,[1] contrary to their classical counterparts, although this has not yet been experimentally confirmed.
Time asymmetries (see Arrow of time) generally are caused by one of three categories:
- intrinsic to the dynamic weak force)
- due to the initial conditions of the universe (e.g., for the second law of thermodynamics)
- 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. 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
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.
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
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 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
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:
- , 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
- 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;[2] 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;[3][4][5] 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.[6] 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
In
Time reversal in quantum mechanics
This section contains a discussion of the three most important properties of time reversal in quantum mechanics; chiefly,
- that it must be represented as an anti-unitary operator,
- that it protects non-degenerate quantum states from having an electric dipole moment,
- that it has two-dimensional representations with the property T2 = −1 (for fermions).
The strangeness of this result is clear if one compares it with parity. If parity transforms a pair of
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[7] and to prove the duality between two fundamental optical operations, beam splitter and squeezing transformations.[8]
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, capturing the actual reversal of the time coordinate, the T that is an ordinary finite dimensional matrix, acting on spinors and vectors, and the T that is an operator on an infinite-dimensional Hilbert space.
For a real (not complex) classical (unquantized) scalar field , the time reversal involution 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, and thus the "mapsto" notation whereas is a factual statement relating the old and new fields to one-another.
Unlike scalar fields, spinor and vector fields might have a non-trivial behavior under time reversal. In this case, one has to write
where is just an ordinary
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, or four-dimensional Minkowski space, an explicit transformation can be given. It is conventionally given as
where is the y-component of the angular momentum operator and is complex conjugation, as before. This form follows whenever the spinor can be described with a linear differential equation 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 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 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
Consider the
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 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 with spin J, one can use the representation
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.[9]
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
Experimental bounds on the electron electric dipole moment also place limits on theories of particle physics and their parameters.[10][11]
Kramers' theorem
For T, which is an anti-unitary Z2 symmetry generator
- T2 = UKUK = UU* = 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 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 of the known dynamical laws
Time reversal violation is unrelated to the
Time reversal of noninvasive measurements
Strong measurements (both classical and quantum) are certainly disturbing, causing asymmetry due to the 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 in quantum physics, 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.
See also
- Arrow of time
- Causality (physics)
- Computing applications
- Standard model
- CKM matrix
- CP violation
- CPT invariance
- Neutrino mass
- Strong CP problem
- Wheeler–Feynman absorber theory
- Loschmidt's paradox
- Maxwell's demon
- Microscopic reversibility
- Second law of thermodynamics
- Time translation symmetry
References
Inline citations
General references
- 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