Quantum Cognition and Quantum Brain Dynamics: Modeling Room-Temperature Coherence in Neural and Retinal Systems

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Abstract Macroscopic biological entities, such as the retina and the human brain, as well as the whole organism, are required to be continuously coherent at room temperature, which is a huge challenge for modern quantum mechanics. Biological tissue is warm, wet and noisy, and this is theoretically expected to result in rapid decoherence, but growing paradigms in quantum cognition and quantum biology indicate that the neural systems somehow avoid immediate collapse induced by environment. One of the unsolved problems in this area is the exact mechanism of how cellular structures are able to remain decoherence-free and support potentially functional quantum states. The aim of this paper is to answer this salient research gap by proposing a theoretical framework for modelling the phenomenon of quantum entanglement and non locality in the brain cell microtubules and retina photoreceptors. Combining the elements of the resource theory of quantum coherence and open quantum dynamics, we propo...
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Maths Blog: Subgroups and Centralizers

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Disproving a Subgroup Property

Algorithmic Verification of Algebraic Structures

Exploring Subgroups of Complex Numbers and Centralizers in Dihedral Groups

Abstract

This blog investigates two algebraic problems: verifying whether a constrained subset of complex numbers forms a subgroup under addition, and classifying centralizers in dihedral groups. We show that the set H = {a + bi | a, b ∈ ℝ, ab ≥ 0} fails closure and is not a subgroup of ℂ. We also provide a parity-dependent classification of centralizers in Dn, highlighting differences between odd and even n.

Problem 72: Subgroup of Complex Numbers

Consider H = {a + bi | a, b ∈ ℝ, ab ≥ 0} under addition. To be a subgroup, H must satisfy identity, closure, and inverse properties.

Identity

0 = 0 + 0i ∈ H since 0·0 = 0 ≥ 0. ✅

Closure

Let x = 2 + 0i ∈ H and y = 0 - 3i ∈ H. Then x+y = 2 - 3i. Here ab = 2·(-3) = -6 < 0, so 2 - 3i ∉ H. ❌

Counterexample: x = 2 + 0i, y = 0 - 3i ∈ H, but x+y = 2 - 3i ∉ H.

Conclusion

Since closure fails, H is not a subgroup of ℂ under addition.

Problem 78: Centralizers in Dihedral Group Dn

The dihedral group Dn consists of n rotations and n reflections. For an element g ∈ Dn, the centralizer C(g) = {x ∈ Dn | xg = gx}.

Case 1: n odd

  • C(F) = {e, F} (only identity and the reflection itself)
  • C(R) = {e, R, R², …, Rn-1} (all rotations)

Case 2: n even

  • C(F) = {e, Rn/2, F, F·Rn/2} (identity, half-turn rotation, and two reflections)
  • C(R) = {e, R, R², …, Rn-1} (all rotations)

Summary

  • Odd n: reflections commute only with themselves and identity; rotations commute with all rotations.
  • Even n: reflections through opposite vertices commute with half-turn rotation and related reflections; rotations commute with all rotations.

Conclusion

We proved that H fails subgroup criteria due to lack of closure, and we classified centralizers in Dn based on parity. These results illustrate how continuous and discrete algebraic structures can be rigorously verified and algorithmically formalized.

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