I think that, in partial answer to my own question, if the initial condition were rho = [integral]psi(q-y/2)psi(q+y/2)exp(py/h)dy where psi were a Gaussian or some localized function, there would be classical wave packet spreading. But why, in quantum theory, like in classical theory, can't we have an initial condition rho = d(p-p0)d(q-q0), where there would be no quantum wave packet spreading.....Any Hermitian probability density matrix leads to a real probability distribution function, and vice versa, so I do not see why this should not be possible. SEB

The two equations are not identical, first of all.

The reason classical wavepackets don't spread in phase space is because the classical motion is integrable.

The reason that quantum wavepackets spread in phase space is due to the fact that the particle interacts with a bath, that describes the quantum fluctuations.

In more conventional terms, position and momentum don't commute. So even were the initial condition in phase space a delta function in position and momentum-which is impossible for the reasons explained presently-the Hamiltonian operator-except for the free particle-is the sum of two operators-the kinetic energy, that depends only on momentum, and the potential energy, that depends only on position-and it acts on wavefunctions, that are functions only of position or of momentum. This means that energy eigenstates aren't position eigenstates and they're momentum eigenstates only for the free particle.

No. A beam of classical, non-interacting, particles, is an integrable system-that's why there's no spreading in phase space.

Stam, Christian,

Let me clarify...the rho(p,q) I am talking about is the Wigner Quasi Probability Distribution [WQPD] for a single particle. This answer's Christian's remark....Stam, my question is, Why can't you use a probability density matrix corresponding to the classical Gaussian combination that I specified? The answer you gave is predicated upon the existence of a wave function, which is not the case for this sort of probability density matrix (again: any Hermitian probability density matrix leads to a real WQPD & vice versa, so there is no prima facie reason this particular WQPD shouldn't be possible--of course, doesn't obey the Uncertainty Principle---the point is, why doesn't this seem to occur in Nature, or does it [please, I understand the Uncertainty Principle as a physical principle, but I want to know the MATHEMATICAL reason. Perhaps it is NOT universally valid as a physical principle after all {it was derived assuming that a wave function existed}])......SEB

The concept of the wave function is mathematically well founded by studying these 3 publications:

• Hiley et al., "de Broglie, General Covariance and a Geometric Background to Quantum Mechanics", 2024 -

  Article de Broglie, General Covariance and a Geometric Background to...

• Hiley et al., "Clifford Algebras and the Dirac-Bohm Quantum Hamilton-Jacobi Equation", 2012 -

  Article Clifford Algebras and the Dirac-Bohm Quantum Hamilton-Jacobi Equation

• D. Hestenes, "Observables, operators, and complex numbers in the Dirac theory", 1975 -

  Article Observables, operators, and complex numbers in the Dirac theory

In addition, the following 2 references provide further clarification:

• C. Baumgarten, "The Simplest Form of the Lorentz Transformations", 2017 -

  Article The Simplest Form of the Lorentz Transformations

• Sanz-Serna et al., "Classical numerical integrators for wave‐packet dynamics", 1996 -

  Article Classical numerical integrators for wave‐packet dynamics

The spreading contributes to solve the dynamic PDE associated with the whole (interacting) particle ensemble.

Stéphane Breton

Dear Stéphane,

Thank you for this series of quite interesting articles.

I had a look more specifically to D. Hestenes' article about the complex numbers issue.

Two recent experiments have clearly demonstrated that the use of complex numbers in quantum theory equations is essential to correctly account for the outcome of these experiments and that restricting the calculations to only real numbers could not predict as accurately the outcome of these experiments, as reported in a recent Physics World research update article.

Padavic-Callaghan, K. (2022) Complex numbers are essential in quantum theory, experiments reveal. Quantum Mechanics. Research update. 11 Jan 2022.

https://physicsworld.com/a/complex-numbers-are-essential-in-quantum-theory-experiments-reveal/

Renou, MO., Trillo, D., Weilenmann, M. et al. (2021) Quantum theory based on real numbers can be experimentally falsified. Nature 600, 625–629 (2021).

https://doi.org/10.1038/s41586-021-04160-4.

https://www.nature.com/articles/s41586-021-04160-4

Ming-Cheng Chen, Can Wang, et al. (2022) Ruling Out Real-Valued Standard Formalism of Quantum Theory. Phys. Rev. Lett. 128, 040403 – Published 24 January 2022.

https://doi.org/10.1103/PhysRevLett.128.040403.

https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.128.040403

It was Schrödinger who first introduced complex numbers in the wave equation that he introduced as a conversion to wave mechanics of de Broglie's 1924 thesis equation that related the spectrum frequencies of the hydrogen atom to its integer sequence, and that was then mapped over by Heisenberg's statistical method, before the Dirac interpretation came about.

Seems to me that the explanation of the uncertainty principle would be related to the manner in which Schrödinger geometrically related complex numbers to the idealized Bohr orbits that de Broglie took as reference.

Best Regards, André

Dear Stephane,

I now think that the reason d(p)d(q) is not a valid initial condition for P(p,q) is that the probability density matrix is not Hermitian; I originally thought that any real P(p,q) lead to a Hermitian probability density matrix, but now I'm not so sure. I think that Wigner in his original paper pointed out that not any arbitrary P(p,q) is a possible P(p,q); now I think that I understand why.

Please understand that I consider any possible P(p,q), not necessarily one leading to a consistent wave function, which is the usual basis, as far as I know, for studying wave packet spreading. Even classically, the P(p,q) which would lead to wave packet spreading in quantum theory would lead to wave packet spreading in classical theory. This explains a lot, because the Liouville equation/evolution equation for P(p,q) for free particles is the same in classical theory as in quantum theory

I do not know if Wigner originally proved that his P(p,q) was a general means for calculating the mean value for any observable, Q(p,q), but I believe that I have: I calculated the Fourier Transform of P(p,q) and inverted, which I believe is a valid general method.See the appendix for my paper "Quantum theory and idealism," which is posted on ResearchGate & passed peer review in "European Journal of Appied Sciences," though I realize that this journal is not widely read nor taken very seriously: shoddy, quick reviiew process. I think that they are a new journal, desperate for papers & would accept anything: very low charge for open access publication. (Originally Wigner computed his Distribution for studying a specialized problem in quantum statistical mechanics; I do not know if he used the same method for derivation as I did or if he was aware of its generality as a tool for calculating mean values, though he probably was; I've read contradictory statements regarding this).

In the appendix to that paper, I show that, applying the method of Fourier Transform, we can find the Wigner Distribution for spins, which according to something that I read, may not have been known previously (I read that the Wigner function is useless for spins and relativistic theories--both statements which I disagree with: the Fourier Transform method seems applicable with full generality).

Stuart

In relation with Hiley et al. papers, people here may be interested to know that de Broglie's 1924 thesis has recently been translated to English by the Minkowski Institute:

Book Research on the theory of quanta