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Over the years there have been questions of a similar ilk on MO (e.g., Q1, Q2, Q3) concerning theories in which either continuous constructs or discrete constructs preceded the development of the other or in which one but not both are present.

I've always been fascinated by quantum mechanics in which notions of the continuous and discrete are necessarily complementary in understanding the math and physics. For example, to characterize the motion of a particle trapped between two walls with a set of discrete, possible energy or momentum states, one has to introduce a continuous probabilty amplitude with the distance between the walls being an integral multiple of the period (or half-period) of the probability wave. The presence of both the discrete and continuous are a necessity to developing a mathematical understanding. The same applies to understanding electrons and the periodic table and the diffraction patterns in the double-slit Young experiments.

A related example is Fourier transform theory in which the discrete Dirac delta and mono-frequency, continuous waves of infinite extent are a necessary duality, not an expediency.

Question:

In what other theories are dual discrete and continuous constructs a necessity?

In response to some comments:

Read Wiki, or better, Feynman's QED for the layman, for a clear explanation of how the iconic double-slit experiment displays the central mystery of QM--the dual wave and particle nature of quantum mechanical objects (wavicles). In a nutshell, a wave-like interference/ diffraction scattering pattern develops for an ensemble of electrons that separately pass through the double-slit over time, each one leaving an isolated discrete mark on the recording material.

Electrons are fermions whose continuous probability wave amplitudes negate each other (Pauli exclusion principle) when possessing the same state values (unlike bosons. such as photons, who tend to coalesce allowing for the existence of lasers) so that only two electrons with opposite spin can occupy a given orbital of an atom. This accounts for the lack of a classical collapse of the system and the existence of the periodic table.

Tom Copeland
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    The Prime Number Theorem? – Gerry Myerson Oct 25 '19 at 01:56
  • Counting lattice points in convex polytopes – efs Oct 25 '19 at 02:36
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    I like where this question wants to go but can't really understand what you mean exactly. Your first two examples seems the same to me (the fact that an operator defined on a continuous configuration space can have discrete spectrum). Instead, what is discrete about the double-slit experiment? – lcv Oct 25 '19 at 10:59
  • Also, doesn't topology provides a lot of examples? – lcv Oct 25 '19 at 11:28
  • @GerryMyerson, well, the Riemann zeta function thought of as a beast in the Mellin transform space roams the whole space, but in real space it is confined to an array of discrete Dirac delta functions; i.e., the Mellin transform of $\delta(x-n)$ is $\frac{1}{n^{s-1}}$. In addition, the power sums of the non-trivial discrete zeros essentially define the entire Riemann $\xi$ function. The continuous function $ln(x) + \gamma$ and $\sum _{n \geq 1} (1/n) H(x-(1/n))$ with its discrete steps are another somewhat dual example. – Tom Copeland Oct 25 '19 at 19:23
  • @EFinat-S, "Computing the Continuous Discretely" by Beck and Robins. Please elaborate as an answer i(f the duality is a necessity). – Tom Copeland Oct 25 '19 at 19:33
  • In topology, understanding three dimensional real space necessitates the dual concepts of continuous strings and discrete number of links, i.e., knots, while knots can not exist in a 4-D space. – Tom Copeland Oct 25 '19 at 20:03
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    Would the relationship between the continuity in the definition of reductive Lie groups and their discrete representation theory (in finite dimensions) count? (On the other hand, the discreteness fades away when infinite dimensionality is allowed again, as with the Verma modules, etc.) – user44191 Oct 25 '19 at 20:36
  • @user44191, not sure. Sounds interesting. Why not propose it as an answer with elaboration for feedback from others? (Brings up another question of finite and infinte reps of, for example, SL2--linear fractional transformation of an argument, a 2-D matrix, or an infinite matrix rep of the related $\exp[(a+bx)^2D_x]$ in the divided powers basis $x^n/n!$.) – Tom Copeland Oct 25 '19 at 20:50
  • All I had in mind was the discrete primes being counted by a formula involving the continuous logarithm function. It's not clear to mean whether the word dual is meant to imply anything more than "twoness". – Gerry Myerson Oct 25 '19 at 21:08
  • Gauss understood the conections between the linking number of knots of conducting wires, number of windings of a wire, and the continuous magnetic flux generated by the wires. See "Topology and physics-a historical essay" by C. Nash https://arxiv.org/abs/hep-th/9709135 – Tom Copeland Oct 25 '19 at 21:10
  • @GerryMyerson, you mean the Riemann "jump function" $\psi$, a discontinuous staircase function, can be approximated by a log function. Yet, the approx is not necessary for an understanding. – Tom Copeland Oct 25 '19 at 21:20
  • @Tom, how do you understand the count of primes not exceeding $x$, or write down a useful formula for it, without appealing to logarithms? Just saying, this expression increases by one each time you hit a prime, does not really tell you anything about the quantity you are counting, it just rephrases the fact that you are counting it. – Gerry Myerson Oct 26 '19 at 02:52
  • @GerryMyerson, I'm not a gate keeper. MO-Qs can be organic and take on a life of their own (if not initially uprooted by certain trolls). Why don't you present your observations as an answer? Perhaps even expand on it since it's related to the property of a log derivative transforming rational functions into a series of poles located at the zeros and poles of the function. (The Mangoldt and Riemann staircase "counting" functions are related to inverse Mellin transforms of the log derivative of the Reimann zeta. This is where I see the log really coming into play.) – Tom Copeland Oct 26 '19 at 18:49
  • Chern numbers thought roughly as the number of times a closed surface is wrapped around another is a generalization of knot theory. – Tom Copeland Nov 06 '19 at 15:57
  • See "Gauge Fields and Geometry A Picture Book" by Greg Naber for illustrations of physics examples – Tom Copeland Dec 05 '19 at 08:21
  • See Wilczek's article "Inside the knotty world of anyon particles" for a quick intro to the quantum mechanic of fermions, such as electrons: https://www.quantamagazine.org/how-anyon-particles-emerge-from-quantum-knots-20170228/ – Tom Copeland Jan 22 '20 at 19:11
  • Related also to fermions: https://en.m.wikipedia.org/wiki/Slater_determinant – Tom Copeland Jan 22 '20 at 19:14

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The Pontryagin dual of the unit circle are the integers.

This implies that the (square integrable) functions on the unit circle are described by a countable sequence of numbers (actually their Fourier coefficients). This is useful e.g. in Harmonic analysis or very practical areas like signal/communication theory. Is it really necessary? I don't know, but understanding how linear filtering of signals work without Fourier series seems quite impossible to me.

Dirk
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  • Ref (or elaboration)? – Tom Copeland Oct 25 '19 at 19:38
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    That doesn't answer the question (but there is definitely an answer to be written about this). "In what other theories are dual discrete and continuous constructs a necessity?" You do not mention what theory this describes (harmonic analysis) and why it is a "necessity" (anything about Pontryagin dual being useful) – Wojowu Oct 25 '19 at 20:18
  • I expanded the answer a bit. – Dirk Oct 26 '19 at 07:38
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    Nice example. Seems this discrete/continuous duality would be present in any scheme in which a "vector" is resolved into projected components along a finite or infinite set of basis "vectors" with Taylor series as another example and decimal representations of lengths on the real line. – Tom Copeland Oct 26 '19 at 12:29
  • E.g., see https://mathoverflow.net/questions/97361/explaining-mukai-fourier-transforms-physically and the somewhat related https://mathoverflow.net/questions/192146/newton-series-and-fourier-transform-is-there-an-analogy/192262#192262 – Tom Copeland Jan 27 '20 at 21:50
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Decimal (or binary, etc.) representations of lengths along the real line. We intuitively understand the real line as a continuous construct drawn by hand yet to understand the embedded lengths and their relationships, we introduce decimal reps, a series of discrete numbers from zero to nine that represent a resolution of the length into the discrete components $10^{n}$ for the integers $n$, positive and negative--for rational numbers, a finite series or a periodic infinite series, depending on the basis; or for irrational numbers, a non-periodic infinite series. To understand these resolutions, we fall back to the continuous real line.

Tom Copeland
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  • Taylor series for a function analytic at a point in the complex domain is another example of intertwining of the dicrete (values of the derivatives of the function at the point) and continuous (the divided powers $x^n/n!$). And the poles of a meromorphic function are discrete phenomena that affect the convergence of the Taylor series over a continuous domain. – Tom Copeland Jan 22 '20 at 19:40
  • Herbert Wilf: The full beauty of the subject of generating functions emerges only from tuning in on both channels: the discrete and the continuous. – Tom Copeland Mar 02 '20 at 23:19