This equation would be of little use if it could not be written down explicitly, so that another important trick is then used. The coupling H c between the relevant subsystem and the environment is redefined by subtracting its relevant part from it, i. The physical meaning of this subtraction is that, in most cases, the quantity is small and fluctuating: One may consider as an example the motion of a piston in a cylinder containing a gas.
The observable X will be the position of the piston. The coupling term H c involves a priori the interaction potentials of the gas molecules with the atoms in the piston, but when one subtract its relevant part which is the gas pressure on the piston , the remaining coupling consists only of the fluctuations arising from the collisions of the gas molecules with the piston. This simple procedure has a very wide domain of applicability and the new coupling can be treated in many cases by means of second-order perturbation calculus to yield an explicit master equation generalizing Eq.
The third method has yielded the wider collection of instructive results and I shall therefore rely on it when stating the main results of theory. It recovers as special cases the results of every known model and its range of validity is certainly very wide, when properly used or extended. I shall restrict however the discussion to two important questions. Zurek asked how general is the diagonalization property in decoherence . Coarse graining shows that diagonalization holds at least for the "Newton observables", which have a strong connection with position in space [15,16].
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A mathematical criterion for diagonalization is given by. The eigenvectors of the commuting observables X provide in that case a diagonalization basis. This condition is however very strong from a mathematical standpoint and the physical reasons for its occurrence are little understood. In the case of coarse-grained "Newton observables", they are due to the very simple expression of a non-relativistic basic Hamiltonian quadratic kinetic energy plus interaction potentials. There is probably a deeper and more direct connection with Galilean invariance, but no proof is known for this conjecture.
The criterion 11 is also satisfied when the system under consideration is a SQUID loop and X is the magnetic flux through the loop. In that case, the origin of commutation lies in the Maxwell equation for induction and there is no obvious physical analogy with the previous case. The basic mathematical formalism for investigating the classical limit of quantum mechanics is always microlocal analysis, in one form or another. It relies on associating a symbol a x, p , which is an ordinary function, with every operator A through a formula such as. When the operator A is the reduced density operator, the associated symbol is the Wigner function W x, p.
Similarly, a Hamilton function h r x,p is associated with the relevant Hamiltonian H r.
When the diagonalization criterion 11 is valid, the master equation has a form similar to Eq. When one uses the Wigner function in place of the density matrix, one gets a diffusion equation. When the criterion condition 11 does not hold, one gets a diffusion equation in phase space:. Nota: generally, there are also mixed derivatives in x and p.
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The result is not then diagonalization but a diffusive smoothing of the Wigner function in every direction of phase space. It is then easily understood why the classical approximation becomes valid after decoherence, because a smooth Wigner function behaves essentially as a classical distribution function . The main consequences of decoherence are well known. The most important one for the foundations of physics is an understanding of the transmutation of quantum mechanics into classical physics.
The word "transmutation" has been coined here as an opposition to more standard eliminations of irrelevant degrees of freedom. When for instance many degrees of freedom of quantum electrodynamics are eliminated for yielding an elementary theory of the hydrogen atom, these degrees of freedom are "integrated out" in the Feynman sum over histories , but the unitary rules of quantum mechanics remain valid.
Quantum Measurements and Decoherence
In the case of decoherence, when the irrelevant degrees of freedom associated with the environment are eliminated, unitarity is broken. A quantum superposition is replaced by alternatives obeying standard probability calculus. Determinism can replace quantum randomness, in spite of their apparent incompatibility as a matter of fact, classical behavior is valid with very small probabilities of error, which means that the probabilistic character of quantum mechanics remains fundamental, however attenuated it may be [20,21].
Decohering histories , extending Griffiths' idea of consistent histories, can be used to exhibit the very large domain of validity of standard logic in quantum mechanics  Griffiths has shown how they remove every paradox from quantum mechanics . Interesting results have also been obtained about the generation of a definite direction of time under decoherence . Finally, it has been found that decoherence allows a derivation of the Copenhagen rules in measurement theory as so many theorems resulting directly from the basic quantum principles [21,23].
Quantum computing has raised a problem opposite to the older ones encountered in measurement theory, namely how to avoid or delay the action of decoherence. Whether or not this aim is possible remains an open problem. The extension of decoherence theory to a relativistic framework in special or general relativity remains also an open question. An objective definition of the relevant observables undergoing decoherence is also a deep problem. There is certainly a hierarchy of observables with different rates of decoherence, but their construction from first principles is poorly understood.
Decoherence is, formally, never complete. There always remain exponentially small non-diagonal terms in the reduced density matrix, reminding us that an initial pure state remains pure according to basic quantum mechanics. Does it mean that decoherence is only a phenomenological theory , or is there some deeper way of interpreting very small probabilities ?
An attractive answer to this last question would be that decoherence is not a final effect, but is followed by a true reduction effect finishing the job, insuring the uniqueness of physical reality and getting rid of tiny probabilities of entanglement. If this were the right answer, one might expect two properties of reduction.
It should act on a state that is already prepared by decoherence and be closely linked to this preparation. It should be also expressible - at least in a phenomenological way - in the framework of ordinary quantum field theory, even if its origin lies deeper, because one sees plainly reduction in ordinary circumstances. The problem of reduction could be therefore the next step in the investigation of the foundations of quantum mechanics.
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This is why I proposed recently a model of reduction with these apparently reasonable constraints . Van Kampen, Physica 20 , Van Hove, Physica, 25 , Daneri, A. Loinger, and G. Prosperi, Nuclear Physics 33 , Zeh, Found. Brune, M.
Hagley, J. Dreyer, X. Maali, C.
Quantum Measurements and Decoherence
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Paz, and Y. Zhang, D 45 , Joos and H. Zeh, Zeitschr. B 59 , D 24 , Leggett, Progr.