On the invertibility of Born–Jordan quantization

Abstract

As a consequence of the Schwartz kernel Theorem, any linear continuous operator A : S(R n ) −→ S ′ (R n ) can be written in Weyl form in a unique way, namely it is the Weyl quantization of a unique symbol a ∈ S ′ (R 2n ). Hence, dequantization can always be performed, and in a unique way. Despite the importance of this topic in Quantum Mechanics and Time-frequency Analysis, the same issue for the Born-Jordan quantization seems simply unexplored, except for the case of polynomial symbols, which we also review in detail. In this paper we show that any operator A as above can be written in Born-Jordan form, although the representation is never unique if one allows general temperate distributions as symbols. Then we consider the same problem when the space of temperate distributions is replaced by the space of smooth slowly increasing functions which extend to entire function in C 2n , with a growth at most exponential in the imaginary directions. We prove again the validity of such a representation, and we determine a sharp threshold for the exponential growth under which the representation is unique. We employ techniques from the theory of division of distributions.2010 Mathematics Subject Classification. 47G30, 81S05, 46F10.

Born-Jordan pseudodifferential operators with symbols in the Shubin classes

“…In this section we review the recent advances in the theory of Born-Jordan quantization; for proofs and details we refer to Cordero et al [9] and de Gosson [16,18,19].…”

“…The following result gives an explicit expression of the Weyl symbol of a Born-Jordan operator with arbitrary symbol (see [1,9]). …”

“…It has been realized not only that the associated phase space picture has many advantages compared with the usual Weyl-Wigner picture (it allows a strong damping of unwanted interference patterns [1,10,30]), but also, as shown by de Gosson [18][19][20], that there is strong evidence that Born-Jordan quantization might very well be the correct quantization method in quantum physics. Independently of these potential applications, the Born-Jordan pseudodifferential calculus has many interesting and difficult features (some of them, such as non-injectivity [9], being even quite surprising) and deserve close attention. The involved mathematics is less straightforward than that of the usual Weyl formalism; for instance Born-Jordan pseudodifferential calculus is not fully covariant under linear symplectic transformations [16], which makes the study of the symmetries of the operators much less straightforward than in the Weyl case.…”

“…It was proved in [9] that every linear continuous operator S(R n ) → S (R n ) can still be written in Born-Jordan form, but the representation is no longer unique. The Born-Jordan correspondence is anyway still surjective.…”

“…These expansions seem remarkable, because at present there is no an exact and explicit formula for the Born-Jordan symbol corresponding to a given Weyl operator, although the existence of such a symbol was proved in [9]. Indeed, the situation seems definitely similar to what happens in the division problem of temperate distributions by a (not identically zero) polynomial P : the map f → P f from S (R n ) into itself is onto but in general a linear continuous right inverse does not exist [3,25].…”