The following is excerpted from [2]. A quick sampling of discussions in quantum mechanics and statistical mechanics textbooks reveals a variety of seemingly simple and straightforward criteria and conditions for classicality. For example, one can loosely associate:. Each of these conditions contains only some partial truth and when taken on face value can be very misleading.
Many of these criteria hold only under special conditions. They can approximately define the classical limit only when taken together in specific ways. To understand the meaning of classicality it is important to examine the exact meaning of these criteria, the conditions of their validity and how they are related to each other.
We can divide the above conditions into four groups, according to the different issues behind these criteria:. The first two groups of issues were discussed in [2] using the paradigm of quantum open systems.
The first set of issues a was discussed in the context of quantum cosmology by Habib and Laflamme [3]. They asserted that decoherence is needed for the WKB Wigner function to show a peak in phase space indicating the correlation between the physical variables and their canonical conjugates which defines a classical trajectory.
This clarifies the loose connection of WKB, Wigner function and classicality. For issue b , for ordinary systems the time for thermal fluctuations to overtake quantum fluctuations is also related to the time of decoherence. But a decohered system is not necessarily classical. There is a quantum statistical regime in between. This set of issues was addressed by Hu and Zhang [4].
See also [5,6]. They derived an uncertainty principle for a quantum open system at finite temperature which interpolates between the zero temperature quantum Heisenberg relation and the high temperature result of classical statistical mechanics.
This was useful for clarifying the sometimes vague notions of quantum, thermal and classical. In our current investigation we wish to use what was learned in the last decade in Q-C to inquire about a simple yet important issue, namely, under what conditions is the ordinary commutative space a bona fide limit of NC space, or, what is the nature of the NC-Com transition?
From 2 , we can see that the non-commutativity parameter q has the dimension of length squared. If the geometry of space-time at a fundamental level is to be noncommutative then one possible candidate for is the Planck length. This is how quantum gravity is linked with NCG, which also bears a close relation to matrix models, quantum groups, M-theory and string theory []. The place where both Q-C and NC-Com share some nontrivial point of contact, at least formally, is the Weyl correspondence between operators and c-functions, the star product, the Wigner distribution, and the Wigner-Weyl equation.
This is the domain of semiclassical or semiquantal physics. The Wigner distribution function has found applications in kinetic theory and has been instrumental in studying quantum coherence and quantum to classical transitions. Star product arises from considering the implications of Weyl transformation of quantum canonical operators.
A good introduction to these topics can be found in [14]. A succinct treatment of Moyal Bracket can be found in an Appendix of [15]. Readers familiar with these topics can skip to the next section. For simplicity, in what follows, we consider one dimensional motion.
The phase space canonical coordinates are denoted by q and p respectively for position and momentum dynamical variables and their corresponding quantum mechanical operators are denoted by and. Weyl [18] proposed that all dynamical variables be written in terms of members of the Lie algebra of transformations given by:. Let us define the set of phase-space operators as the set of all operators whose operator properties solely depends on and.
Throughout this article, a member of this set will be called a phase-space operator. One can show that for such operators we can give the following representation:. We can combine equations 4 and 5 to obtain:. The relationship defines a mapping from the set of functions of phase-space variables to the set of phase-space operators.
First we note that every such operator is completely determined by its matrix elements taken with respect to any complete basis. Let the set of position eigenstates be such a basis. Introduce the following change of variables. Inserting an integral over additional variable, q.
Combining all of the above we have. Then we can use the Baker-Campbell-Hausdorff lemma to combine operators inside the bra-ket into and therefore show that any phase-space operator can be written as 6. That is to say, for every phase-space operator, there is a function of the phase-space variables such that the relationship 6 holds. Thus the Weyl correspondence represented by 6 is an onto mapping from the space of functions into the space of phase-space operators. Furthermore one can show that the Weyl correspondence is a one-to-one mapping.
To see that let us assume there are two different functions, namely A W q,p and q,p that map to a single operator. That is. Now one can use 7 to reverse both sides and by using the properties of the Fourier transformation can show that A W q,p and q,p , are indeed identical.
Therefore the Weyl correspondence is a one-to-one and onto mapping from the set of functions over the phase-space variables to the set of phase-space operators as defined at the beginning of this subsection. Wigner distribution functions W q,p in quantum systems are meant to play the corresponding role of classical distributions in classical kinetic theory.
For a classical system in kinetic theory and a positive-definite distribution function P q,p of the canonical variables q,p in classical phase space, we have [17]:. One can define. To show that the transformation defined by Eq. Unlike the classical case, where a probabilistic interpretation of the distribution function is possible, the Wigner function cannot be interpreted as a probability distribution because in general it is not everywhere positive.
Consider two dynamical variables A and B in a classical system. The statistical average of their product is obtained by weighting it with the distribution function P q,p given by. If A and B are quantum mechanical operators, because of their functional dependence on the non-commuting operators and a different rule of multiplication, the star product, is needed. The star product satisfies the following property [1]. W for products of operators, stands for the Weyl transformation of the enclosed operator inside.
How is the star product related to the ordinary algebraic product? To find out we first use the Weyl analysis for the solution.
Another way of writing it is. Using these three entities, namely, Wigner function, Weyl transformation and the star product, we can construct the Wigner-Moyal-Weyl-Groenwood formalism. This formalism has been well developed long before the recent activities in NC geometry and been used widely for the study of semiclassical physics see, e.
The state of a quantum system can be represented by a real valued function of the canonical coordinates, the Wigner function. We notice that the star-squared of a Wigner function for a pure state is proportional to itself. The Weyl transformation of the Dirac bracket of two operators can be shown to be equal to their commutator with respect to the star product:.
It can be shown that using the Weyl transformation of the eigenvalue equation for the density operator, corresponding to an energy eigen state, we obtain:.
The time evolution of the system's state is governed by the Wigner-Moyal equation. For a Hamiltonian of the form. So far we have discussed everything in one space dimension. The extension to N dimensional space is straightforward. The commutation relation takes the form:. In particular, for two functions in a 2N-dimensional phase space N dimensional configuration space , we have.
The last equation states that for two functions of phase space coordinates where , the integral of the star-product over all phase space gives the same result as that obtained by using the ordinary product. For an introduction to the properties of time-independent Wigner functions see [20].
Our notation in this section follows [17]. Noncommutative geometry NCG has appeared in the literature ever since Heisenberg and Snyder studied it with the hope of resolving the ultraviolet infinity problem [21]. Later on it was applied to the Landau model of electrons in a magnetic field, where considering certain limits the lowest energy levels the space of the coordinates becomes a noncommutative space. Recent interest in noncommutative physics, however, stems from the discovery of NCG in the context of string theory and M theory [8,9].
NCG has been considered as a candidate for Plank scale geometry. Hence, a successful theory of quantum gravity may reveal the necessity or desirability of some form of noncommutative geometry.
There are various approaches to formulate noncommutative geometry. Early attempts using a more mathematical approach were proposed by Alain Connes and John Madore [22,23]. In this phase, a differential NC geometry was developed and the concept of distance and differential forms were defined. Later progress focused more around the Wigner-Moyal formalism, described in the last section. Almost all current work on the subject of fields in noncommutative spaces relies on using star product and its properties.
This is the approach pursued here. Noncommutative -star product. To introduce non-commutativity, one replaces the normal product between two functions with the -product defined as. In the previous section we used q as the canonical variable for position. From now on we denote it by x. In what follows, we also use the beginning letters of the Latin alphabet, a,b to denote the coordinate indices rather than the middle letters i,j.
With this we can expand the -product as. Here we are studying the quantum mechanics of a particle in an external potential. As is well-known, using the form of -product, one can write the noncommutative part as. However one must pay attention to the ordering issues that can arise. To be consistent with the definition of a -product, the ordering here is such that all momentum operators stand to the right of the rest of the potential and operate directly on the wave function.
The definition of Wigner function does not change in the NC settings. However we expect the time evolution of the Wigner function following the Wigner-Moyal WM equation to be different.
An easier way is to apply the Weyl correspondence to the von Neumann equation,. To find the Weyl transformation we use the usual definition:. That is, the Weyl transformation of has the same functional form in terms of x and p as the commutative Hamiltonian but with position x a shifted by an amount equal to , where p b is the phase space momentum.
A Superstar Wigner-Moyal equation. This is the main mathematical result of this paper. We thank Dr. Prata for bringing to our attention this reference. B 18B , suppl. In: Operator algebras and mathematical physics.
Iowa City, Iowa: Univ. Kyoto 20 , — Centre Recherche Math. Operator Theory 4 , 93— Pacific J. Lecture Notes in Math. Download references. S, 35 route de Chartres, , Bures-sur-Yvette, France. You can also search for this author in PubMed Google Scholar. Reprints and Permissions. Connes, A.
Gravity coupled with matter and the foundation of non-commutative geometry. Download citation. Received : 26 March Accepted : 17 April Issue Date : December Anyone you share the following link with will be able to read this content:. Sorry, a shareable link is not currently available for this article.
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