Open System – Ark's blog

Fundamental Symmetries in Minkowski Krein Space

arkajad — Wed, 20 May 2026 09:05:28 +0000

It is always good to have at least one example. I am reading the Preface to A Course in Functional Analysis by John B. Conway (Springer, 1985), where the author notes: Historically, mathematics has gone from the particular to the general—not the … Continue reading →

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Krein spaces – a quantum-theoretical monad method

arkajad — Tue, 12 May 2026 17:41:40 +0000

In the previous post, Krein spaces – first steps, we introduced the basic definitions and simple examples of Krein spaces, emphasizing how a fundamental symmetry turns an indefinite inner product into a Hilbert space structure. In this post we move from … Continue reading →

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Krein spaces – first steps

arkajad — Sat, 09 May 2026 17:35:10 +0000

It was the summer of 1964 (15 June–15 July). The small village of Katsiveli, in the Yalta region of Crimea, hosted the Second Summer School of Mathematics organized by the Ukrainian Academy of Sciences. Second Summer School of Mathematics, Katsiveli, … Continue reading →

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Counting infinities

arkajad — Sun, 03 May 2026 15:57:25 +0000

As I have mentioned in my previous post, recently I stepped upon the path leading to infinity. But what is “infinity?” The question demands deep thinking. This post starts the new series. Prolog We usually meet the ideas of finite … Continue reading →

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Infinite geometry

arkajad — Fri, 24 Apr 2026 12:23:34 +0000

A week ago I asked Perplexity AI: “Is there a monograph, or a comprehensive review, discussing infinite-dimensional symmetric spaces and Kahler manifolds?” Here is the answer: Yes. There are a few substantial sources on infinite-dimensional symmetric spaces, and many standard … Continue reading →

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Recovery

arkajad — Sun, 22 Mar 2026 10:20:49 +0000

This post is an attempt to explain my recent long silence. Over the past few months I have been in a kind of “construction period,” slowly rebuilding my working environment and trying to create a space where I can think … Continue reading →

The post Recovery first appeared on Open System - Ark's blog.

Exterior algebra

arkajad — Sun, 18 Jan 2026 18:09:28 +0000

Geometric algebra was not born in an ivory tower; it grew out of three very human lives, each marked by faith, doubt, and an almost mystical hunger for structure. The quiet mystic: Grassmann Hermann Grassmann was a provincial schoolteacher, … Continue reading →

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Kronecker generalized deltas and Levi-Civita epsilons

arkajad — Fri, 09 Jan 2026 10:08:01 +0000

My last post “It’s all about permutations” was a complete failure. Really very bad. I copied, without much thinking, some formulas from “Mathematical Handbook For Scientists And Engineers“, Granino A. Korn and Theresa M. Korn, Dover 2000, and it was … Continue reading →

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It’s all about permutations

arkajad — Sun, 28 Dec 2025 11:19:33 +0000

Permutations appear whenever “things are the same, but in a different order.” That simple idea hides a powerful engine that underlies physics, geometry, information, and even the notion of identity itself. In quantum mechanics, identical particles do not carry little … Continue reading →

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Densities

arkajad — Sun, 21 Dec 2025 14:58:42 +0000

This post introduces “densities”. There is a very nice book, available on arxiv, by Sean M. Carroll: “Lecture Notes on General Relativity“. There, on p. 52, Carroll writes “… we don’t like tensor densities, we like tensors.” Well, I do … Continue reading →

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