T-symmetry

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https://en.wikipedia.org/wiki/T-symmetry

 

Time

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

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

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, 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:

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

Macroscopic phenomena

The second law of thermodynamics

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.

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.

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 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, supported by cosmological observations (such as the isotropy of the cosmic microwave background) connects 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 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. For example, 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.^{[citation needed]}

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:

${\displaystyle \overrightarrow{x}}$, position of a particle in three-space

${\displaystyle \overrightarrow{a}}$, acceleration of the particle

${\displaystyle \overrightarrow{F}}$, force on the particle

${\displaystyle E}$, energy of the particle

${\displaystyle V}$, electric potential (voltage)

${\displaystyle \overrightarrow{E}}$, electric field

${\displaystyle \overrightarrow{D}}$, electric displacement

${\displaystyle \rho }$, density of electric charge

${\displaystyle \overrightarrow{P}}$, 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

${\displaystyle \overrightarrow{v}}$, velocity of a particle

${\displaystyle \overrightarrow{p}}$, linear momentum of a particle

${\displaystyle \overrightarrow{l}}$, angular momentum of a particle (both orbital and spin)

${\displaystyle \overrightarrow{A}}$, electromagnetic vector potential

${\displaystyle \overrightarrow{B}}$, magnetic field

${\displaystyle \overrightarrow{H}}$, magnetic auxiliary field

${\displaystyle \overrightarrow{j}}$, density of electric current

${\displaystyle \overrightarrow{M}}$, magnetization

${\displaystyle \overrightarrow{S}}$, Poynting vector

${\displaystyle \mathcal{P}}$, 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 ${\displaystyle t}$ 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 ${\displaystyle \overrightarrow{B}}$ and another to ${\displaystyle -\overrightarrow{B}}$, 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 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.)

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 (for fermions).

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 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, 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 ${\displaystyle \varphi }$, the time reversal involution can simply be written as

${\displaystyle \mathsf{T}\varphi (t,\overrightarrow{x})={\varphi }^{\mathrm{\prime }}(-t,\overrightarrow{x})=s\varphi (t,\overrightarrow{x})}$

as time reversal leaves the scalar value at a fixed spacetime point unchanged, up to an overall sign ${\displaystyle s=\pm 1}$. A slightly more formal way to write this is

${\displaystyle \mathsf{T}:\varphi (t,\overrightarrow{x})\mapsto {\varphi }^{\mathrm{\prime }}(-t,\overrightarrow{x})=s\varphi (t,\overrightarrow{x})}$

which has the advantage of emphasizing that ${\displaystyle \mathsf{T}}$ is a map, and thus the "mapsto" notation ${\displaystyle \mapsto ,}$ whereas ${\displaystyle {\varphi }^{\mathrm{\prime }}(-t,\overrightarrow{x})=s\varphi (t,\overrightarrow{x})}$ is a factual statement relating the old and new fields to one-another.

Unlike scalar fields, spinor and vector fields ${\displaystyle \psi }$ might have a non-trivial behavior under time reversal. In this case, one has to write

${\displaystyle \mathsf{T}:\psi (t,\overrightarrow{x})\mapsto {\psi }^{\mathrm{\prime }}(-t,\overrightarrow{x})=T\psi (t,\overrightarrow{x})}$

where ${\displaystyle T}$ is just an ordinary matrix. For complex fields, complex conjugation may be required, for which the mapping ${\displaystyle K:(x+iy)\mapsto (x-iy)}$ can be thought of as a 2x2 matrix. For a Dirac spinor, ${\displaystyle T}$ 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 ${\displaystyle T}$; 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 ${\displaystyle T}$ 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

${\displaystyle T=e^{i\pi J_y}K}$

where ${\displaystyle J_y}$ is the y-component of the angular momentum operator and ${\displaystyle K}$ 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 ${\displaystyle {\psi }_{abc\cdots }}$ In this case,

${\displaystyle \mathsf{T}:{\psi }_{abc\cdots }(t,\overrightarrow{x})\mapsto {\psi }_{abc\cdots }^{\mathrm{\prime }}(-t,\overrightarrow{x})={T_a}^d{T_b}^e{T_c}^f\cdots {\psi }_{def\cdots }(t,\overrightarrow{x})}$

Covariant tensor indexes will transform as ${\displaystyle {T_a}^b={(T^{-1}{)}_b}^a}$ and so on. For quantum fields, there is also a third T, written as ${\displaystyle \mathcal{T},}$ which is actually an infinite dimensional operator acting on a Hilbert space. It acts on quantized fields ${\displaystyle \mathrm{\Psi }}$ as

${\displaystyle \mathsf{T}:\mathrm{\Psi }(t,\overrightarrow{x})\mapsto {\mathrm{\Psi }}^{\mathrm{\prime }}(-t,\overrightarrow{x})=\mathcal{T}\mathrm{\Psi }(t,\overrightarrow{x}){\mathcal{T}}^{-1}}$

This can be thought of as a special case of a tensor with one covariant, and one contravariant index, and thus two ${\displaystyle \mathcal{T}}$'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 ${\displaystyle \mathsf{T}}$ or ${\displaystyle T}$ or ${\displaystyle \mathcal{T},}$ depending on context, left for the reader to infer.

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] = iħ, 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 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

${\displaystyle T=e^{-i\pi J_y/\hslash }K,}$

where J_y is the y-component of the spin, and use of TJT⁻¹ = −J has been made.

Electric dipole moments

Main article: Electron electric dipole moment

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.

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.

Kramers' theorem

Main article: Kramers' theorem

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

T² = UKUK = UU^* = U (U^T)⁻¹ = Φ,

where Φ is a diagonal matrix of phases. As a result, U = ΦU^T and U^T = UΦ, showing that

U = Φ U Φ.

This means that the entries in Φ are ±1, as a result of which one may have either T² = ±1. This is specific to the anti-unitarity of T. For a unitary operator, such as the parity, any phase is allowed.

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

Measurement problem

From Wikipedia, the free encyclopedia

In quantum mechanics, the measurement problem is the problem of definite outcomes: quantum systems have superpositions but quantum measurements only give one definite result.

The wave function in quantum mechanics evolves deterministically according to the Schrödinger equation as a linear superposition of different states. However, actual measurements always find the physical system in a definite state. Any future evolution of the wave function is based on the state the system was discovered to be in when the measurement was made, meaning that the measurement "did something" to the system that is not obviously a consequence of Schrödinger evolution. The measurement problem is describing what that "something" is, how a superposition of many possible values becomes a single measured value.

To express matters differently (paraphrasing Steven Weinberg), the Schrödinger equation determines the wave function at any later time. If observers and their measuring apparatus are themselves described by a deterministic wave function, why can we not predict precise results for measurements, but only probabilities? As a general question: How can one establish a correspondence between quantum reality and classical reality?

Schrodinger's cat

Main article: Schrodinger's cat

A thought experiment called Schrödinger's cat illustrates the measurement problem. A mechanism is arranged to kill a cat if a quantum event, such as the decay of a radioactive atom, occurs. The mechanism and the cat are enclosed in a chamber so the fate of the cat is unknown until the chamber is opened. Prior to observation, according to quantum mechanics, the atom is in a quantum superposition, a linear combination of decayed and intact states. Also according to quantum mechanics, the atom-mechanism-cat composite system is described by superpositions of compound states. Therefore, the cat would be described as in a superposition, a linear combination of two states an "intact atom-alive cat" and a "decayed atom-dead cat". However, when the chamber is opened the cat is either alive or it is dead: there is no superposition observed. After the measurement the cat is definitively alive or dead.^[6]: 154

The cat scenario illustrates the measurement problem: how can an indefinite superposition yield a single definite outcome? It also illustrates other issues in quantum measurement,^[7]: 585 including when does a measurement occur? Was it when the cat was observed? How is a measurement apparatus defined? The mechanism for detecting radioactive decay? The cat? The chamber? What the role of the observer?

Interpretations

Main article: Interpretations of quantum mechanics

The views often grouped together as the Copenhagen interpretation are the oldest and, collectively, probably still the most widely held attitude about quantum mechanics.^[8][9] N. David Mermin coined the phrase "Shut up and calculate!" to summarize Copenhagen-type views, a saying often misattributed to Richard Feynman and which Mermin later found insufficiently nuanced.^[10][11]

Generally, views in the Copenhagen tradition posit something in the act of observation which results in the collapse of the wave function. This concept, though often attributed to Niels Bohr, was due to Werner Heisenberg, whose later writings obscured many disagreements he and Bohr had during their collaboration and that the two never resolved.^[12][13] In these schools of thought, wave functions may be regarded as statistical information about a quantum system, and wave function collapse is the updating of that information in response to new data.^[14][15] Exactly how to understand this process remains a topic of dispute.^[16]

Bohr offered an interpretation that is independent of a subjective observer, or measurement, or collapse; instead, an "irreversible" or effectively irreversible process causes the decay of quantum coherence which imparts the classical behavior of "observation" or "measurement".^[17][18]

Hugh Everett's many-worlds interpretation attempts to solve the problem by suggesting that there is only one wave function, the superposition of the entire universe, and it never collapses—so there is no measurement problem. Instead, the act of measurement is simply an interaction between quantum entities, e.g. observer, measuring instrument, electron/positron etc., which entangle to form a single larger entity, for instance living cat/happy scientist. Everett also attempted to demonstrate how the probabilistic nature of quantum mechanics would appear in measurements, a work later extended by Bryce DeWitt. However, proponents of the Everettian program have not yet reached a consensus regarding the correct way to justify the use of the Born rule to calculate probabilities.^[19][20]

The de Broglie–Bohm theory tries to solve the measurement problem very differently: the information describing the system contains not only the wave function, but also supplementary data (a trajectory) giving the position of the particle(s). The role of the wave function is to generate the velocity field for the particles. These velocities are such that the probability distribution for the particle remains consistent with the predictions of the orthodox quantum mechanics. According to de Broglie–Bohm theory, interaction with the environment during a measurement procedure separates the wave packets in configuration space, which is where apparent wave function collapse comes from, even though there is no actual collapse.^[21]

A fourth approach is given by objective-collapse models. In such models, the Schrödinger equation is modified and obtains nonlinear terms. These nonlinear modifications are of stochastic nature and lead to behaviour that for microscopic quantum objects, e.g. electrons or atoms, is unmeasurably close to that given by the usual Schrödinger equation. For macroscopic objects, however, the nonlinear modification becomes important and induces the collapse of the wave function. Objective-collapse models are effective theories. The stochastic modification is thought to stem from some external non-quantum field, but the nature of this field is unknown. One possible candidate is the gravitational interaction as in the models of Diósi and Penrose. The main difference of objective-collapse models compared to the other approaches is that they make falsifiable predictions that differ from standard quantum mechanics. Experiments are already getting close to the parameter regime where these predictions can be tested.^[22]

The Ghirardi–Rimini–Weber (GRW) theory proposes that wave function collapse happens spontaneously as part of the dynamics. Particles have a non-zero probability of undergoing a "hit", or spontaneous collapse of the wave function, on the order of once every hundred million years.^[23] Though collapse is extremely rare, the sheer number of particles in a measurement system means that the probability of a collapse occurring somewhere in the system is high. Since the entire measurement system is entangled (by quantum entanglement), the collapse of a single particle initiates the collapse of the entire measurement apparatus. Because the GRW theory makes different predictions from orthodox quantum mechanics in some conditions, it is not an interpretation of quantum mechanics in a strict sense.

The role of decoherence

Erich Joos and Heinz-Dieter Zeh claim that the phenomenon of quantum decoherence, which was put on firm ground in the 1980s, resolves the problem.^[24] The idea is that the environment causes the classical appearance of macroscopic objects. Zeh further claims that decoherence makes it possible to identify the fuzzy boundary between the quantum microworld and the world where the classical intuition is applicable.^[25][26] Quantum decoherence becomes an important part of some modern updates of the Copenhagen interpretation based on consistent histories.^[27][28] Quantum decoherence does not describe the actual collapse of the wave function, but it explains the conversion of the quantum probabilities (that exhibit interference effects) to the ordinary classical probabilities. See, for example, Zurek,^[5] Zeh^[25] and Schlosshauer.^[29]

The present situation is slowly clarifying, described in a 2006 article by Schlosshauer as follows:^[30]

Several decoherence-unrelated proposals have been put forward in the past to elucidate the meaning of probabilities and arrive at the Born rule ... It is fair to say that no decisive conclusion appears to have been reached as to the success of these derivations. ...

As it is well known, [many papers by Bohr insist upon] the fundamental role of classical concepts. The experimental evidence for superpositions of macroscopically distinct states on increasingly large length scales counters such a dictum. Superpositions appear to be novel and individually existing states, often without any classical counterparts. Only the physical interactions between systems then determine a particular decomposition into classical states from the view of each particular system. Thus classical concepts are to be understood as locally emergent in a relative-state sense and should no longer claim a fundamental role in the physical theory.

Six degrees of freedom

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Six_degrees_of_freedom

The six degrees of freedom: forward/back, up/down, left/right, yaw, pitch, roll

Six degrees of freedom (6DOF), or sometimes six degrees of movement, refers to the six mechanical degrees of freedom of movement of a rigid body in three-dimensional space. Specifically, the body is free to change position as forward/backward (surge), up/down (heave), left/right (sway) translation in three perpendicular axes, combined with changes in orientation through rotation about three perpendicular axes, often termed yaw (normal axis), pitch (transverse axis), and roll (longitudinal axis).

Three degrees of freedom (3DOF), a term often used in the context of virtual reality, typically refers to tracking of rotational motion only: pitch, yaw, and roll.

Robotics

Serial and parallel manipulator systems are generally designed to position an end-effector with six degrees of freedom, consisting of three in translation and three in orientation. This provides a direct relationship between actuator positions and the configuration of the manipulator defined by its forward and inverse kinematics.

Robot arms are described by their degrees of freedom. This is a practical metric, in contrast to the abstract definition of degrees of freedom which measures the aggregate positioning capability of a system.

In 2007, Dean Kamen, inventor of the Segway, unveiled a prototype robotic arm with 14 degrees of freedom for DARPA. Humanoid robots typically have 30 or more degrees of freedom, with six degrees of freedom per arm, five or six in each leg, and several more in torso and neck.

Engineering

The term is important in mechanical systems, especially biomechanical systems, for analyzing and measuring properties of these types of systems that need to account for all six degrees of freedom. Measurement of the six degrees of freedom is accomplished today through both AC and DC magnetic or electromagnetic fields in sensors that transmit positional and angular data to a processing unit. The data is made relevant through software that integrates the data based on the needs and programming of the users.

Mnemonics to remember angle names

The six degrees of freedom of a mobile unit are divided in two motional classes as described below.

Translational envelopes:

1. Moving forward and backward on the X-axis. (Surge)
2. Moving left and right on the Y-axis. (Sway)
3. Moving up and down on the Z-axis. (Heave)

Rotational envelopes:

1. Tilting side to side on the X-axis. (Roll)
2. Tilting forward and backward on the Y-axis. (Pitch)
3. Turning left and right on the Z-axis. (Yaw)

In terms of a headset, such as the kind used for virtual reality, rotational envelopes can also be thought of in the following terms:

• Pitch: Nodding "yes"
• Yaw: Shaking "no"
• Roll: Bobbling from side to side

Operational envelope types

There are three types of operational envelope in the Six degrees of freedom. These types are Direct, Semi-direct (conditional) and Non-direct, all regardless of the time remaining for the execution of the maneuver, the energy remaining to execute the maneuver and finally, if the motion is commanded via a biological entity (e.g. human), a robotical entity (e.g. computer) or both.

1. Direct type: Involved a degree can be commanded directly without particularly conditions and described as a normal operation. (An aileron on a basic airplane)
2. Semi-direct type: Involved a degree can be commanded when some specific conditions are met. (Reverse thrust on an aircraft)
3. Non-direct type: Involved a degree when is achieved via the interaction with its environment and cannot be commanded. (Pitching motion of a vessel at sea)

Transitional type also exists in some vehicles. For example, when the Space Shuttle operated in low Earth orbit, the craft was described as fully-direct-six because in the vacuum of space, its six degrees could be commanded via reaction wheels and RCS thrusters. However, when the Space Shuttle was descending through the Earth's atmosphere for its return, the fully-direct-six degrees were no longer applicable as it was gliding through the air using its wings and control surfaces.

Game controllers

Six degrees of freedom also refers to movement in video game-play.

First-person shooter (FPS) games generally provide five degrees of freedom: forwards/backwards, slide left/right, up/down (jump/crouch/lie), yaw (turn left/right), and pitch (look up/down). If the game allows leaning control, then some consider it a sixth DOF; however, this may not be completely accurate, as a lean is a limited partial rotation.

The term 6DOF has sometimes been used to describe games which allow freedom of movement, but do not necessarily meet the full 6DOF criteria. For example, Dead Space 2, and to a lesser extent, Homeworld and Zone Of The Enders allow freedom of movement.

Some examples of true 6DOF games, which allow independent control of all three movement axes and all three rotational axes, include Elite Dangerous, Shattered Horizon, the Descent franchise, the Everspace franchise, Retrovirus, Miner Wars, Space Engineers, Forsaken and Overload (from the same creators of Descent). The space MMO Vendetta Online also features 6 degrees of freedom.

Motion tracking hardware devices such as TrackIR and software-based apps like Eyeware Beam are used for 6DOF head tracking. This device often finds its places in flight simulators and other vehicle simulators that require looking around the cockpit to locate enemies or simply avoiding accidents in-game.

The acronym 3DOF, meaning movement in the three dimensions but not rotation, is sometimes encountered.

The Razer Hydra, a motion controller for PC, tracks position and rotation of two wired nunchucks, providing six degrees of freedom on each hand.

The SpaceOrb 360 is a 6DOF computer input device released in 1996 originally manufactured and sold by the SpaceTec IMC company (first bought by Labtec, which itself was later bought by Logitech). They now offer the 3Dconnexion range of 6DOF controllers, primarily targeting the professional CAD industry.

The controllers sold with HTC VIVE provide 6DOF information by the lighthouse, which adopts Time of Flight (TOF) technology to determine the position of controllers.