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The quantum physics use the frequentistic interpretation only. I never reat quantum phyics axioms from other probability interpretation https://arxiv.org/abs/0911.0695

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We will use the Chevallay basis. Consider first $e_i^+$ to be the generators associated to the simple roots $\alpha_i$ and $e_i^-$ to be those associated to $-\alpha_i$. Then one has $$[h_i,e_j^{\pm}] = \pm \frac{2\alpha_j\cdot \alpha_i}{|\alpha_i|^2} e_j^\pm\,,\qquad [e_i^+,e_j^-] = \delta_{ij}h_i\,,\qquad [h_i,h_j] = 0\,.$$ where $h_i$ are the Cartan ...

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Yes, your function, y, solves the differential equation. The way to approach linear, homogeneous, ordinary differential equations is by setting up and finding the roots of the characteristic polynomial. For example, for the differential equation $x'' +c x' + kx=0$, the characteristic polynomial is $\lambda^2+c\lambda +k=0$. If we denote the roots of this ...

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If you are being strict about the kinds of things Gödel was talking about, no. Godel was operating on systems which could prove all true statement in arithmetic. In physics these are generally statements that are assumed rather than proven. Physics builds on top of math, building upon what mathematics asserts to be true. To start off, you would need to ...

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I would like to share two examples that one could consider a "physics version" of Gödel's incompleteness theorem. This will not really answer the question about whether axiom systems in physics can be formally incomplete in the Gödel sense. However, it will show how an incompleteness of a physical theory is usually connected to some physical insight that is ...

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The theorem does have consequences for any formal system that includes arithmetic. There will always be theorems that will be undecidable. Those theorems though, always involve infinite quantities, one example is: Do this system of Diophantine equations have a finite or infinite number of solutions?". The problem is in general undecidable, although for ...

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Yes. One simple (trivial) way of seeing it is to consider a mechanical computer, and ask whether one can predict its end-state given the initial state without running it. Since this is exactly the Turing halting problem, any method that would allow you to make that prediction in Newtonian mechanics would either be a counterexample to the theorem (...

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The first two formulas are useful because they correspond directly to basic, practically easy, experimental procedures. That makes them easily justified by experiments, and that makes them useful as "axioms" to derive others, "theorems". The first formula F∝q1.q2 corresponds to an experiment where we keep r constant, vary q and measure F. It is easy to keep ...

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your description is imprecise 𝐹∝𝑞1.𝑞2 for fixed r and 𝐹∝1/𝑟^2 for fixed q1,q2 and r independent of q1 and q2 only then you can deduct Coulombs law

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Welcome to our flock! Your question touches on one of the fundamental problems in life, I think: How do we know that what we experience is real? I believe it is customery to distinguish between Mathematics on one side and Science (that is to say: Empirical Science) on the other: In Maths, we are dealing with absolute truth, but with a caveat: We always ...

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Determinants and Pfaffians via Gaussian Grassmann integrals are typical applications that have some benefits. This is e.g used in the Faddeev-Popov determinant already mentioned by OP. Example. Try to prove that ${\rm Det}(A)~=~{\rm Pf}(A)^2$ for an antisymmetric matrix $A$ with vs. without the use of Gaussian Grassmann integrals. More generally, there are ...

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Is there a case where mathematical proof can replace experimentation? Yes. Every time you can prove that some proposition $P$ is implied by a premise $A$, and $A$ is experimentally verifiable, then you never need to experimentally verify $P$. Verifying $A$ is good enough. As an example, Gauss' Law, $\oint E \cdot dA = \frac{Q}{\epsilon_0}$, can be proven ...

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Perhaps logic cannot be used to confirm a theory but it can be used to refute it. Given and assumed a certain model of physical reality logic can be used to find a contradiction in that model leading to the conclusion that one should refuse the theory. In fact a logical contradiction, given the assumptions, corresponds to a physical contradiction. We are ...

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I can think of at least one instance where scientific laws have been proven mathematically. In mathematics you prove theorems by applying logic to the axioms. Axioms are facts which are assumed to be correct, without requiring any proof. Examples of axioms used by Peano include: 0 is a natural number. For every natural number x, x = x. For all natural ...

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I think (emphasis to be explained below) that the most important thing to realize in thinking about this question is that it is a question about physicists rather than about physics. In other words, it's a question about the philosophy and practice of how we as humans do and think about science, rather than about the natural world itself. The philosophy ...

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Is there a case where mathematical proof can replace experimentation ? No. Every physical model must be validated experimentally. Besides those two things are not related at all. Pure mathematical theories proves something and physical experiments verifies theoretical models. If you let me to make a joke: Does proving that you are not hungry, eliminates a ...

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Can it be that the basic assumptions of a mathematical theory have been experimentally validated, but the prediction of a derivation in that theory could be experimentally INvalidated? Yes - besides all reasons already given, there is always the possibility that there is something in Nature which is NOT being modeled by the theory. This is why, for example,...

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Is there a case where mathematical proof can replace experimentation? Yes, this happens all the time in real scientific settings. (Now, for those who disagree, please give this a full read before you downvote me.) There are certain principles that are "dogma" among physicists. Some classic examples include: Causality Conservation of Energy If someone ...

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There is the case of the mathematical proof that gravity has to exist in the light of string theory: String theory predicts the existence of gravitons and their well-defined interactions. A graviton in perturbative string theory is a closed string in a very particular low-energy vibrational state See for more information this article. Gravity was ...

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In the view of philosopher of science, Karl Popper, it is fundamentally impossible to prove/confirm any assumption or hypothesis about physics or the world in general. Starting from a set of assumptions (i.e. Newton's laws) a scientist can prove that IF this set of assumptions is valid, THEN certain outcomes should occur in the real world. If an experiment ...

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Building upon the mathematics of previously-tested models is indeed a thing that is done. It's called engineering. We do it literally all the time. The difference is that in engineering, we are trying to make the best product we can within some constraints, while a scientist is theoretically seeking the truth. Thus in engineering there's a whole slew of ...

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Yes, you can. How much is the sum of all positive integers? -1/12, right? Is that true? Yes. Can you prove it empirically? no. Empirically you would get a larger number than the one before, theoretically you get a smaller number than the first positive integer.

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Science is very much about model-making. A model is a set of ideas within which mathematical proof may be possible, and used to show how one feature implies another within the model. But it is not possible to prove by mathematics that the model describes the physical world correctly. Your example of the right-angled triangle is a good one. Within the set ...

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A physical theory or model is based assumptions. By mathematical methods you the make predictions. Even the mathematics are sound, you will still need experimental results to verify or challenge your assumptions.

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No. Physics remains an experimental science and so it is not possible to replace experiment by a proof. Descartes tried this when he proposed his theory of propagation of light - very elegant - but it predicted incorrectly that the angle would increase for light passing into an optically denser medium. Indeed the story goes he refused to attend a ...

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With your experiment you can only say that your experiments support the theory, not that it proves it. Experiments may agree with theories for many reasons, not always because the assumptions are correct. In areas in which lots of experiments give the predicted results you can have large confidence that a theorem of your theory will agree with the ...

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If you only make assumptions that have been experimentally verified (up to a high degree of precision) then a purely mathematical proof might be fine. However there are two problems with this: 1) Most of the time not all the assumptions can be experimentally verified (for example the axioms of Newtonian mechanics) 2) If you can only perform measurements up ...

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Every representation $(D,V)$ of a finite group $G$ is equivalent to a unitary representation. It is often termed as Weyl's unitary trick. This works by simply redefining your inner product by averaging over on the space $V$. This smoothening trick works precisely because of finite number of elements and invariance of sum of finite elements (i.e. \sum_{g \in ... 7 They are all equivalent to unitary representations. It's not quite the same thing. Merely checking unitarity might not do the trick. For instance \begin{align} \Gamma(e)&=\left(\begin{array}{cc} 1&0 \\ 0 & 1\end{array}\right)\, ,\qquad \Gamma(P_{12})=\left(\begin{array}{cc} 1&-1 \\ 0 & -1\end{array}\right)\, ,\qquad \Gamma(P_{13})=\... 2 Periodic functionsf(t)$(with a time period$T$) can be approximated by a Fourier series, i.e. by summing harmonic oscillations with the discrete frequencies$0, \frac{2\pi}{T}, 2\frac{2\pi}{T}, 3\frac{2\pi}{T}, \ldots\$ . $$f(t)=\sum_{n=-\infty}^{+\infty} F_n e^{in\frac{2\pi}{T}t}$$ But, as you already noticed, with a Fourier series you cannot build ...

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