Our answer is YES. A new question (at https://www.researchgate.net/post/If_RQ_what_are_the_consequences/1) has been answered affirmatively, confirming the YES answer in this question, with wider evidence in +12 areas.
This question continued the same question from 3 years ago, with the same name, considering new published evidence and results. The previous text of the question maybe useful and is available here:
https://www.researchgate.net/post/Can_infinitesimals_be_eliminated_from_mathematics_SOLVED
We now can provably include DDF [1] -- the differentiation of discontinuous functions. This is not shaky, but advances knowledge. The quantum principle of Niels Bohr in physics, "all states at once", meets mathematics and quantum computing.
Without infinitesimals or epsilon-deltas, DDF is possible, allowing quantum computing [1] between discrete states, and a faster FFT [2]. The Problem of Closure was made clear in [1].
Although Weyl training was on these mythical aspects, the infinitesimal transformation and Lie algebra [4], he saw an application of groups in the many-electron atom, which must have a finite number of equations. The discrete Weyl-Heisenberg group comes from these discrete observations, and do not use infinitesimal transformations at all, with finite dimensional representations. Similarly, this is the same as someone trained in infinitesimal calculus, traditional, starts to use rational numbers in calculus, with DDF [1]. The similar previous training applies in both fields, from a "continuous" field to a discrete, quantum field. In that sense, R~Q*; the results are the same formulas -- but now, absolutely accurate.
New results have been made public [1-3], confirming the advantages of the YES answer, since this question was first asked 3 years ago. All computation is revealed to be exact in modular arithmetic, there is NO concept of approximation, no "environmental noise" when using it.
As a consequence of the facts in [1], no one can formalize the field of non-standard analysis in the use of infinitesimals in a consistent and complete way, or Cauchy epsilon-deltas, against [1], although these may have been claimed and chalk spilled.
Some branches of mathematics will have to change. New results are promised in quantum mechanics and quantum computing.
This question is closed, affirming the YES answer.
REFERENCES
[1] Article Algorithms for Quantum Computation: The Derivatives of Disco...
[2] Preprint FT = FFT
[3] Preprint The quantum set Q*
[4] https://youtu.be/atgyvRvS1IM
Infinitesimals can be eliminated from mathematics, but it depends on the context, some branches of mathematics such as mathematical analysis and differential calculus, use limits and epsilon-delta definition to prove theorems that were once proved using infinitesimals. However, the field of non-standard analysis formalizes the use of infinitesimals in a consistent and complete way.
It's not necessary to discuss infinitesimals at all, when dealing with quantum computing.The reason is that the Heisenberg-Weyl group does have finite dimensional representations.
Rodolfo
Any form to make real quantities appear smaller, meets the objection that the smallest of positive real nummbers does not exist.
How does non standard analysis handle this?
(I use a nilpotent element instead)
Researchers are working on various techniques to mitigate the effects of noise, such as error correction codes and the use of more stable qubits, but it is still an active area of research. @Juan Weisz
Clear case of theory overstepping actual Q. computers, very shakey yet
is my opinion. I cannot even imagine , except in very simple cases, spin up a 1, spin down 0.
I read here recently dd(omega)=0, and if no other figures, could infinity be 8, and in digital-analog-reality, how many states are needed to make a wave, 4 ?
Lena
There is a language used in the advanced calculus, used for vector analysis.
that uses what is called p forms, due to Elie Cartan, that uses this identity when you twice total differentiate a p form.
A state is just a single quantum state psi(n), just one complex wave function
for some given eigenvalue in QM, so just 1
Dont understand, a horizontal 8 maybe? Thats infinity in math.
LJTS: Particle diffraction is explained in:
Preprint The Algebraic Approach: The Double-Slit Experiment Explained
A wave is purely a macroscopic property, existing in Universality, and NOT a microscopic reality, NOT a quantum reality. It is matter of scale.
A wave, e.g., on a beach disappears microscopically--- and there one has only molecules, described by a quantum reality.
There is no "wave-particle duality". All one has in the microscopic level, are particles, a quantum reality. See:
https://www.researchgate.net/post/On_deprecating_the_wave-particle_duality2
Ed
we are talking about a wave in QM.(from De Broglie for particles)
A wave on a beach is visible, not microscopic
the part on molecules, quite out of context, not understandable
wave particle duality not existing, well people talk about this anyway, evidence also
No waves, how about particle diffraction?
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