It has been argued that the Feynman path integral formalism leads to a quantization rule, and that this is the Born–Jordan rule which gives the unique quantization rule consistent with the correct short-time propagator behavior of the propagator for nonrelativistic systems. We examine this short-time approximation and conclude, contrary to prevailing views, that the asymptotic expansion applies only to Hamiltonian functions that are at most quadratic in the momentum and with constant mass. While the Born–Jordan rule suggests the appropriate quantization of functions in this class, there are other rules which give the same answer, most notably the Weyl quantization scheme.

Abstract Currently, the horizontal resolution of Rayleigh wave exploration is low. In this study, we propose the Born–Jordan time‐frequency distribution to analyse Rayleigh waves. The seismic signal was filtered with a wavelet transform for denoising, and the Rayleigh wave was separated in the time domain. Using the Born–Jordan time‐frequency distribution, the time waveform of each frequency comprising the Rayleigh wave from every seismic channel was obtained, and the time difference of the Rayleigh wave with the same frequency was calculated, based on which the dispersion curve between the two channels was obtained. Combined with the multichannel Rayleigh wave dispersion curve, phase velocity and frequency imaging under the seismic arrangement were obtained. Applying this method to detect abnormal geological bodies in engineering investigations showed that hard geologic bodies, such as comcrete rocks, have high velocity and frequency, whereas weak ones have low velocity and frequency. This strategy facilitated the detection of fractured zones, underground goafs and obstacles during pipe‐jacking construction near the surface.

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.

For \(m\in \mathbb {R}\) we consider the symbol classes \(S^m\), \(m\in \mathbb {R}\), consisting of smooth functions \(\sigma \) on \({\mathbb {R}^{2d}}\) such that \(|\partial ^\alpha \sigma (z)|\le C_\alpha (1+|z|^2)^{m/2}\), \(z\in {\mathbb {R}^{2d}}\), and we show that can be characterized by an intersection of different types of modulation spaces. In the case \(m=0\) we recapture the Hörmander class \(S^0_{0,0}\) that can be obtained by intersection of suitable Besov spaces as well. Such spaces contain the Shubin classes \(\Gamma ^m_\rho \), \(0<\rho \le 1\), and can be viewed as their limit case \(\rho =0\). We exhibit almost diagonalization properties for the Gabor matrix of \(\tau \)-pseudodifferential operators with symbols in such classes, extending the characterization proved by Gröchenig and Rzeszotnik (Ann Inst Fourier 58(7):2279–2314, 2008). Finally, we compute the Gabor matrix of a Born–Jordan operator, which allows to prove new boundedness results for such operators.

The kinematical foundations of Schwinger’s algebra of selective measurements were discussed in [F. M. Ciaglia, A. Ibort and G. Marmo, Schwinger’s picture of quantum mechanics I: Groupoids, To appear in IJGMMP (2019)] and, as a consequence of this, a new picture of quantum mechanics based on groupoids was proposed. In this paper, the dynamical aspects of the theory are analyzed. For that, the algebra generated by the observables, as well as the notion of state, are discussed, and the structure of the transition functions, that plays an instrumental role in Schwinger’s picture, is elucidated. A Hamiltonian picture of dynamical evolution emerges naturally, and the formalism offers a simple way to discuss the quantum-to-classical transition. Some basic examples, the qubit and the harmonic oscillator, are examined, and the relation with the standard Dirac–Schrödinger and Born–Jordan–Heisenberg pictures is discussed.

A new picture of Quantum Mechanics based on the theory of groupoids is presented. This picture provides the mathematical background for Schwinger’s algebra of selective measurements and helps to understand its scope and eventual applications. In this first paper, the kinematical background is described using elementary notions from category theory, in particular the notion of 2-groupoids as well as their representations. Some basic results are presented, and the relation with the standard Dirac–Schrödinger and Born–Jordan–Heisenberg pictures are succinctly discussed.

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.

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.

Quantizations of the classical time of arrival and their dynamics

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

A family of Doppler-lag kernels is introduced to design the reduced interference distributions. The proposed kernel functions with two parameters \({{\varvec{n}}}\) and \({\varvec{\alpha }}\) are of the product \({\varvec{\nu }} {\varvec{\tau }} \) and satisfy all the nine desirable kernel constraints. The family of Cohen time–frequency distributions with these kernels includes some existing reduced interference distributions such as Margenau–Hill, Born–Jordan and Bessel distributions. Also the signal-to-interference ratio (SIR) in time–frequency plane is defined, and reduced interference capabilities with \({{\varvec{n}}}\) and \({\varvec{\alpha }}\) are discussed. Simulations show that the greater the \({{\varvec{n}}}\) and \({\varvec{\alpha }}\) are, the greater the SIR.

Time-frequency analysis of Born–Jordan pseudodifferential operators

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.

On the reduction of the interferences in the Born–Jordan distribution

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

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.

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.

Due to the high mortality rate of heart diseases, early detection and precise discrimination of ECG arrhythmia are essential for the treatment of patients. Biomedical signals, especially the electrocardiogram (ECG) signal, contain information of the anatomical and physiological state of the human body. The non-stationary multicomponent nature of ECG signals makes the use of time–frequency analysis inevitable. Time–frequency signal analysis offers simultaneous interpretation of the signal in both time and frequency, which allows local, transient or intermittent components to be elucidated. The choice and selection of the proper time–frequency technique that can reveal the exact multicomponent structure of the ECG signals, especially the QRS complex, is vital in many applications, including the diagnosis of medical abnormalities. In this work, we have applied four time–frequency techniques for analyzing abnormal ECG signals. These time–frequency techniques are the Wigner–Ville distribution, the Choi–Williams distribution, the Bessel distribution and the Born–Jordan distribution. The abnormal cardiac signals were taken from a patient with supraventricular arrhythmia and a patient with malignant ventricular arrhythmia. The results obtained showed that the Choi–Williams time–frequency technique has a superior performance, in terms of resolution and cross-term reduction, as compared to other time–frequency distributions.

A new method for expert target recognition system: Genetic wavelet extreme learning machine (GAWELM)