The uncertainty principle follows directly from the fact that the coordinate and momentum representations are related by the Fourier transform. Then one might ask: why are they so connected?

I think this is a manifestation of the principle of duality of projective geometry that underlies the physical world.

The Heisenberg Uncertainty Principle, a fundamental concept in quantum mechanics, originates from the work of the German physicist Werner Heisenberg in 1927. Here is a brief outline of its development and significance:

Historical Context

1. Early Quantum Mechanics: In the early 20th century, physicists were grappling with the nature of atoms and subatomic particles. Classical physics failed to explain phenomena at atomic and subatomic scales, leading to the development of quantum theory.

2. Wave-Particle Duality The discovery that particles such as electrons exhibit both wave-like and particle-like properties was a major breakthrough. This duality was a cornerstone of quantum mechanics, but it also introduced complexities in understanding particle behavior.

Conclusion

The Heisenberg Uncertainty Principle arose from Werner Heisenberg's pioneering work in the late 1920s. It marked a significant departure from classical physics, highlighting the intrinsic limitations of measurements at the quantum level and shaping the development of modern quantum theory.

Lev Verkhovsky ,

In fact, the uncertainty principle is related to the duality of the representation of the amplitude of the probability of a particle wandering along the surface of an infinite cylinder, where the momentum representation corresponds to movement along helical lines, and the coordinate representation corresponds to the phase position of the particle.

Lev Verkhovsky

Thank you for your insightful response, Lev. I agree that the connection between coordinate and momentum representations via the Fourier transform is crucial. In my paper "Article Origin of Heisenberg's Uncertainty Principle

Dear Colleagues, Bhushan and Igor, you are developing specific models about which I cannot say anything. I spoke only about the most general principle.

I will take this opportunity to provide a link to my article on the interpretation of the superposition principle in QM

Preprint Subquantum leapfrog

The origin of Heisenberg's Uncertainty Principle dates back to 1927 when German physicist Werner Heisenberg formulated it. This principle is a fundamental concept in quantum mechanics that arises from the intrinsic properties of particles at the quantum level.

Heisenberg discovered that it is impossible to simultaneously determine both the exact position and the exact momentum (velocity and mass) of a particle with absolute precision. The more accurately you know the position of a particle, the less accurately you can know its momentum, and vice versa. This is not due to experimental errors but is a fundamental property of quantum systems.

The Uncertainty Principle challenged the classical mechanics view of the universe, where it was assumed that all properties of a particle could be known precisely at the same time. Heisenberg's insight was crucial in the development of quantum mechanics, offering a new understanding of the microscopic world that contrasted sharply with the predictable and deterministic nature of classical physics.

A person thinks simply and quickly realizes that the laws of nature occur in the same form in all dimensions... The fact that everything looks different is precisely due to the difference in the large dimension... The way we try to relate from our level.

It leads to the very wrong conclusion that the comparison from our dimension was done incorrectly. Heisenberg's Uncertainty Principle was excellent for eliminating this.

Regards,

Laszlo

Heisenberg's uncertainty is _not_ a principle but a mathematical property of operators in the Schrödinger equation. It became a 'principle' after "physics had moved from Germany to the land of pragmatism".

Christian Baumgarten :

Dear Christian, thank you for sharing your article 'A Two-Page Derivation of Schr¨odinger’s Equation' (arXiv:1907.09917v6 [physics.gen-ph] 25 Jan 2024)! Let me make a few remarks.

i) Point particles:

- Maxwell, 'Matter and Motion': Particles are small bodies whose extension plays no role.

- Within Maxwell's theory, point particles are allowed (thus, Rohrlich's counter-argument is limited).

- You: ...“natural philosophy” defined material objects as res extensa.

Me: This holds for Descartes, but not for Euler; see his 'Anleitung zur Naturlehre' (available in English).

- You: But since classical physics identifies matter by it’s position in space and time, it seems unclear how to assign a common identity to the volume elements of extended objects.

Me: What do you have in mind? Already, Euler has treated bodies in great detail.

- Boskovich is perhaps outdated by Euler (loc. cit.): A body is divisible at infinitum but does not consist of point particles. The property of impenetrability persists as the limit case of point particles is never reached (as infinity can never be reached).

ii) (2) and (3) are mathematically reasonable but without physical reasoning. Not all relationships that are mathematically reasonable are physically reasonable.

iii) After (4): (4) is an expansion of psi(t,x) in terms of plane waves. But does that mean that there are waves? Fourier invented his transformation for heat diffusion. Which waves are there? Thus, (5) appears to be an additional assumption.

iv) The formal analogy between (5) and (6) does not yet justify the use of the classical expressions for V(x) [and thus T(p)] within quantum theory (cf. Schrödinger, 4th Commun. 1926).

v) I agree that "Conversion factors can not be derived theoretically".

However, I disagree with "The insight into the significance of these constants, obtained by the theory of relativity on the one hand and quantum theory on the other, is most forcibly expressed by the fact that they do not occur in the laws of Nature in a thoroughly systematic development of these theories."

- If one _axiomatically_ derives the Lorentz transformation from classical mechanics (Enders & Suisky 2005 ff.), c occurs "in a thoroughly systematic development".

- Ditto, if one _axiomatically_ derives Schrödinger's wave mechanics from classical mechanics (Enders & Suisky 2005 ff.), h occurs "in a thoroughly systematic development".

- Ditto, if one _axiomatically_ derives Maxwell's equation in a vacuum from classical mechanics (Enders 2009 ff.), mu_0 and eps_0 occur "in a thoroughly systematic development".

vi) You: The quantization of atomic energy levels is, from a logical point of view, an “output of the theory”, not an input [65]. It is entirely due to boundary conditions as a free particle can have any (kinetic) energy.

Me: IMHO, here you are interchanging classical _discretization_ of wavelength and, consequently, frequency in classical strings and resonators with non-classical _quantization_ by _intrinsic_ quantum conditions, notably recursion relations between different solutions of the stationary Schrödinger equation. Energy conservation implies the existence of a ground state. Hence, only the set of recursion relations which exhibit a state of minimum energy is a physically relevant set of states. For the harmonic oscillator, Schrödinger's wave functions are obtained _without_ the use of boundary conditions (Suisky & Enders 2005 ff.).

vii) Another additional assumption (p. 3, before (9)):

"...if one presumes the validity of energy conservation and hence of classical Hamiltonian notions in wave dynamics..."

viii) You: "It provides a new perspective on the relationship between classical mechanics and quantum theory and shows that, contrary to usual assertions, these theories are not mathematically disjunct."

Me: Indeed, that is a great result!

ix) You: "...Schrödinger’s equation...has to be derived from the Dirac equation."

Me: I'm afraid I have to disagree because Schrödinger’s equation can be derived from extending classical mechanics to the mechanics of an oscillator without returning points (Suisky & Enders 2005 ff.).

Peter Enders

Thank you for sharing your insights. They bring valuable perspectives to the discussion.

In my paper 'Article Origin of Heisenberg's Uncertainty Principle

In relativity, we only observe the event after the wave function collapses, seeing only the real part of complex number. In quantum mechanics, we consider the imaginary component of complex number before the wave function collapse.

I'm curious to hear your thoughts on how the complex representation of energy and time in my model might relate to your findings

The origin is Heisenberg himself. He started from Bohr's correspondance principle to calculate things. The orbit of the electron in an atom was decomposed in a Fourier series. But when using the Fourier coefficients to calculate the product of two conjugate observables, the result depended on the order the operations were done. Born and Jordan arranged the coefficient in matrices, which don't commute for conjugate variables. To explain the uncertainty, he described how a measurement is performed by a microscope with photons of quantized energy, and how it perturbs the measured system. But later, especially with Bohr's formulation of the complementarity principle, he promoted it to a principle on its own right.

In very loose terms, the uncertainty principle deals with information one can derive about values of «conjugate» quantities: The more information about one of the quantities can be derived the less information can be derived about «conjugate» counterpart. The «conjugation» can be understood quite broadly, as, e.g., Fourier transform, Pontryagin-type duality, Gelfand pairs, etc. To that end, I highly recommend the following paper on the subject: G.Folland and A.Sitaram. The uncertainty principle: A mathematical survey , J. Fourier Anal. Appl., vol. 3, No 3, 1997., and especially sections 5—8 dealing with entropic form of the uncertainty principle.

That is why, at my view, a «natural» form of uncertainty principle is entropic, i.e., as an inequality for a sum of entropies of «conjugate» distributions. In this entropic form the relations turned out to be useful for, e.g., quantum cryptography, see the expository paper P.Coles et al., Entropic uncertainty relations and their applications, Rev. Mod. Phys., v. 89, 2017.

In information theory the uncertainty relations are inequalities between mean codeword length and entropy of the code. In this form, the uncertainty relations may be treated as an inequality relating the time the wave function takes to «collapse» after a measurement and the eigenvalue to which the function collapses, see my own paper

https://www.researchgate.net/publication/371003871_Free_Choice_in_Quantum_Theory_A_p-adic_View

DOI: 10.3390/e25050830

That is why my personal opinion is that the uncertainty principle expresses a fundamental limitation on information which can be derived from natural phenomena.

Returning to my remark about the principle of duality of projective geometry, I want to draw attention to the principle of reciprocity by Max Born that implies a fundamental symmetry between spacetime and momentum space.

Article Reciprocity Theory of Elementary Particles

Lev Verkhovsky :

Thank you for pointing to Born's article, which implies a fundamental symmetry between spacetime and momentum space. Born correctly remarks that Hamilton's equations of motion are invariant under the transformation

x -> p, p -> -x. (I.1)

But does this hold for _all_ physically relevant equations/relations in (x,p)??

Bhushan Bhoja Poojary :

Dear Bushan, Thank you for your question. Despite Schrödinger's 1926 hesitation to use complex-valued physical quantities, their use vastly simplifies the mathematical representation of a theory, e.g., the Kramers-Kronig relations or electrical circuits. Usually, a system's energy is a real-valued quantity, which represents its ability to deliver work (mechanics) or energy in another form (e.g., radiation) to its environment. What is the meaning of your imaginary-valued component of the energy? Some damping (as sometimes described by non-Hermitian Hamiltonians)?

Christian Baumgarten :

Thank you for your quotation! IMHO, it illustrates that similar mathematical structures are used in various branches of physics. As a consequence, similar mathematical relations occur in them.

To judge whether Heisenberg's uncertainty relations represent a rather general principle (for what? information?), I propose to compare them with Huygens' principle. As Feynman suggested in his pioneering 1949 article on path-integral quantum mechanics, the (most general) mathematical formulation of Huygens' principle is the Chapman-Kolmogorov equation.

For details, see:

P. Enders, Huygens’ principle and the modelling of propagation, Eur. J. Phys. 17 (1996) 4, 226-235, https://www.researchgate.net/publication/231074231_Huygens%27_principle_and_the_modelling_of_propagation;

P. Enders, Huygens’ principle in the transmission line matrix method (TLM). Global theory, Int. J. Num. Modell.: Electronic Networks, Dev. and Fields 14 (2001) 451-456; http://www.ece.uvic.ca/~bctill/papers/numacoust/Enders_2001.pdf;

P. Enders & C. Vanneste, Huygens’ principle in the transmission line matrix method (TLM). Local theory, Int. J. Num. Modell.: Electronic Networks, Dev. and Fields 16 (2003) 2, 175-178;

P. Enders, Huygens’ principle as universal model of propagation, Latin Am. J. Phys. Educ. 3 (2009) 19-32; http://dialnet.unirioja.es/servlet/articulo?codigo=3688899; http://www.oalib.com/paper/2140888#.XLIFXEtzBEY

Yes, Peter Enders , at least in classical and quantum mechanics the principle seems correct (I have met such opinion).

Well, my idée fixe -- this is ​​a manifestation of the principle of duality of projective geometry. It is important that in it DIFFERENT elements are connected symmetrically, say, in the simplest case, the pole and polar are a point and a straight line. You can try to give these elements a physical meaning.

Dear Lev, Thank you for your reply! Unfortunately, my knowledge of projective geometry is not sufficient to reply to your comment.

I think, dear Peter, this is a misfortune of 20th century physics -- ignorance of projective geometry.

And the Lorentz transformations are to blame, the erroneous form of which cut off fundamental physics from conformal symmetry (it is closely related to projective geometry).

Heisenberg's Uncertainty Principle, formulated by Werner Heisenberg in 1927, is a fundamental concept in quantum mechanics that states that it is impossible to simultaneously measure the exact position and momentum of a particle with arbitrary precision. This principle arose from the early development of quantum mechanics, influenced by wave-particle duality and the limitations imposed by the Fourier transform on wave functions. The principle is mathematically expressed a Δx⋅Δp≥ℏ/2​, where Δx and Δp are the uncertainties in position and momentum, respectively, and ℏ is the reduced Planck constant. It signifies that there is an intrinsic limit to the precision with which these properties can be known, fundamentally altering our understanding of measurement and reality at the quantum level.