Born Jordan Quantization Research Articles

Phase Spaces, Parity Operators, and the Born–Jordan Distribution

Phase spaces as given by the Wigner distribution function provide a natural description of infinite-dimensional quantum systems. They are an important tool in quantum optics and have been widely applied in the context of time–frequency analysis and pseudo-differential operators. Phase-space distribution functions are usually specified via integral transformations or convolutions which can be averted and subsumed by (displaced) parity operators proposed in this work. Building on earlier work for Wigner distribution functions (Grossmann in Commun Math Phys 48(3):191–194, 1976. https://doi.org/10.1007/BF01617867), parity operators give rise to a general class of distribution functions in the form of quantum-mechanical expectation values. This enables us to precisely characterize the mathematical existence of general phase-space distribution functions. We then relate these distribution functions to the so-called Cohen class (Cohen in J Math Phys 7(5):781–786, 1966. https://doi.org/10.1063/1.1931206) and recover various quantization schemes and distribution functions from the literature. The parity operator approach is also applied to the Born–Jordan distribution which originates from the Born–Jordan quantization (Born and Jordan in Z Phys 34(1):858–888, 1925. https://doi.org/10.1007/BF01328531). The corresponding parity operator is written as a weighted average of both displacements and squeezing operators, and we determine its generalized spectral decomposition. This leads to an efficient computation of the Born–Jordan parity operator in the number-state basis, and example quantum states reveal unique features of the Born–Jordan distribution.

A Wiener Tauberian theorem for operators and functions

We prove variants of Wiener's Tauberian theorem in the framework of quantum harmonic analysis, i.e. for convolutions between an absolutely integrable function and a trace class operator, or of two trace class operators. Our results include Wiener's Tauberian theorem as a special case. Applications of our Tauberian theorems are related to localization operators, Toeplitz operators, isomorphism theorems between Bargmann-Fock spaces and quantization schemes with consequences for Shubin's pseudodifferential operator calculus and Born-Jordan quantization. Based on the links between localization operators and Tauberian theorems we note that the analogue of Pitt's Tauberian theorem in our setting implies compactness results for Toeplitz operators in terms of the Berezin transform. In addition, we extend the results on Toeplitz operators to other reproducing kernel Hilbert spaces induced by the short-time Fourier transform, known as Gabor spaces. Finally, we establish the equivalence of Wiener's Tauberian theorem and the condition in the characterization of compactness of localization operators due to Fernández and Galbis.

Moyal Bracket and Ehrenfest’s Theorem in Born–Jordan Quantization

The usual Poisson bracket { A , B } can be identified with the so-called Moyal bracket { A , B } M for larger classes of symbols than was previously thought, provided that one uses the Born–Jordan quantization rule instead of the better known Weyl correspondence. We apply our results to a generalized version of Ehrenfest’s theorem on the time evolution of averages of operators.

Short-Time Propagators and the Born-Jordan Quantization Rule.

We have shown in previous work that the equivalence of the Heisenberg and Schrödinger pictures of quantum mechanics requires the use of the Born and Jordan quantization rules. In the present work we give further evidence that the Born–Jordan rule is the correct quantization scheme for quantum mechanics. For this purpose we use correct short-time approximations to the action functional, initially due to Makri and Miller, and show that these lead to the desired quantization of the classical Hamiltonian.

Born–Jordan pseudodifferential operators and the Dirac correspondence: Beyond the Groenewold–van Hove theorem

Quantization procedures play an essential role in operator theory, time–frequency analysis and, of course, in quantum mechanics. Roughly speaking the basic idea, due to Dirac, is to associate to any symbol, or observable, a(x,ξ) an operator Op(a), according to some axioms dictated by physical considerations. This led to the introduction of a variety of quantizations. They all agree when the symbol a(x,ξ)=f(x) depends only on x or a(x,ξ)=g(ξ) depends only on ξ:Op(f⊗1)u=fu,Op(1⊗g)u=F−1(gFu) where F stands for the Fourier transform. Now, Dirac aimed at finding a quantization satisfying, in addition, the key correspondence[Op(a),Op(b)]=iOp({a,b}) where [,] stands for the commutator and {,} for the Poisson brackets, which would represent a tight link between classical and quantum mechanics. Unfortunately, the famous Groenewold–van Hove theorem states that such a quantization does not exist, and indeed most quantization rules satisfy this property only approximately.In this work we show that the above commutator rule in fact holds for the Born–Jordan quantization, at least for symbols of the type f(x)+g(ξ). Moreover we will prove that, remarkably, this property completely characterizes this quantization rule, making it the quantization which best fits the Dirac dream.

Born-Jordan pseudodifferential operators with symbols in the Shubin classes

We apply Shubin’s theory of global symbol classes Γ ρ m \Gamma _{\rho }^{m} to the Born-Jordan pseudodifferential calculus we have previously developed. This approach has many conceptual advantages and makes the relationship between the conflicting Born-Jordan and Weyl quantization methods much more limpid. We give, in particular, precise asymptotic expansions of symbols allowing us to pass from Born-Jordan quantization to Weyl quantization and vice versa. In addition we state and prove some regularity and global hypoellipticity results.

Bypassing the Groenewold–van Hove obstruction on : a new argument in favor of Born–Jordan quantization

There are known obstructions to a full quantization of in the spirit of Dirac’s approach, the most known being the Groenewold and van Hove no-go result. We show, following a suggestion of S K Kauffmann, that it is possible to construct a well-defined quantization procedure by weakening the usual requirement that commutators should correspond to Poisson brackets. The weaker requirement consists in demanding that this correspondence should only hold for Hamiltonian functions of the type . This reformulation leads to a non-injective quantization of all observables which, when restricted to polynomials, is the rule proposed by Born and Jordan in the early days of quantum mechanics.

Born–Jordan Pseudodifferential Calculus, Bopp Operators and Deformation Quantization

There has recently been a resurgence of interest in Born–Jordan quantization, which historically preceded Weyl’s prescription. Both mathematicians and physicists have found that this forgotten quantization scheme is actually not only of great mathematical interest, but also has unexpected application in operator theory, signal processing, and time-frequency analysis. In the present paper we discuss the applications to deformation quantization, which in its traditional form relies on Weyl quantization. Introducing the notion of “Bopp operator” which we have used in previous work, this allows us to obtain interesting new results in the spectral theory of deformation quantization.

On the invertibility of Born–Jordan quantization

As a consequence of the Schwartz kernel Theorem, any linear continuous operator Aˆ: S(Rn)⟶S′(Rn) can be written in Weyl form in a unique way, namely it is the Weyl quantization of a unique symbol a∈S′(R2n). 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 Aˆ 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 C2n, 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.

Born-Jordan Quantization and the Equivalence of the Schrödinger and Heisenberg Pictures.

The aim of the famous Born and Jordan 1925 paper was to put Heisenberg’s matrix mechanics on a firm mathematical basis. Born and Jordan showed that if one wants to ensure energy conservation in Heisenberg’s theory it is necessary and sufficient to quantize observables following a certain ordering rule. One apparently unnoticed consequence of this fact is that Schrödinger’s wave mechanics cannot be equivalent to Heisenberg’s more physically motivated matrix mechanics unless its observables are quantized using this rule, and not the more symmetric prescription proposed by Weyl in 1926, which has become the standard procedure in quantum mechanics. This observation confirms the superiority of Born–Jordan quantization, as already suggested by Kauffmann. We also show how to explicitly determine the Born–Jordan quantization of arbitrary classical variables, and discuss the conceptual advantages in using this quantization scheme. We finally suggest that it might be possible to determine the correct quantization scheme by using the results of weak measurement experiments.

Born–Jordan quantization and the uncertainty principle

The Weyl correspondence and the related Wigner formalism lie at the core of traditional quantum mechanics. We discuss here an alternative quantization scheme, the idea of which goes back to Born and Jordan, and which has recently been revived in another context, namely time–frequency analysis. We show in particular that the uncertainty principle does not enjoy full symplectic covariance properties in the Born and Jordan scheme, as opposed to what happens in the Weyl quantization.

Preferred quantization rules: Born–Jordan versus Weyl. The pseudo-differential point of view

There has recently been evidence for replacing the usual Weyl quantization procedure by the older and much less known Born–Jordan rule. In this paper we discuss this quantization procedure in detail and relate it to recent results of Boggiato, De Donno, and Oliaro on the Cohen class. We begin with a discussion of some properties of Shubin’s τ-pseudo-differential calculus, which allows us to show that the Born–Jordan quantization of a symbol a is the average for \({\tau\in[0,1]}\) of the τ-operators with symbol a. We study the properties of the Born–Jordan operators, including their symplectic covariance, and give their Weyl symbol.

Unambiguous Quantization from the Maximum Classical Correspondence that Is Self-consistent: The Slightly Stronger Canonical Commutation Rule Dirac Missed

Dirac's identification of the quantum analog of the Poisson bracket with the commutator is reviewed, as is the threat of self-inconsistent overdetermination of the quantization of classical dynamical variables which drove him to restrict the assumption of correspondence between quantum and classical Poisson brackets to embrace only the Cartesian components of the phase space vector. Dirac's canonical commutation rule fails to determine the order of noncommuting factors within quantized classical dynamical variables, but does imply the quantum/classical correspondence of Poisson brackets between any linear function of phase space and the sum of an arbitrary function of only configuration space with one of only momentum space. Since every linear function of phase space is itself such a sum, it is worth checking whether the assumption of quantum/classical correspondence of Poisson brackets for all such sums is still self-consistent. Not only is that so, but this slightly stronger canonical commutation rule also unambiguously determines the order of noncommuting factors within quantized dynamical variables in accord with the 1925 Born-Jordan quantization surmise, thus replicating the results of the Hamiltonian path integral, a fact first realized by E. H. Kerner. Born-Jordan quantization validates the generalized Ehrenfest theorem, but has no inverse, which disallows the disturbing features of the poorly physically motivated invertible Weyl quantization, i.e., its unique deterministic classical "shadow world" which can manifest negative densities in phase space.