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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Disproving a Subgroup Property

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Disproving a Subgroup Property

Disproving the Property for Subgroups

Suppose H is a nonempty subset of a group G with the property:

If a, b ∈ H, then a-1b-1 ∈ H.

Is this enough to guarantee H is a subgroup?

The answer is no. While every subgroup satisfies this property, a subset can satisfy it without being a subgroup, typically by failing to include the identity element.

Subgroup Criteria

  • Non-empty
  • Closed under the group operation
  • Contains the identity element
  • Closed under inverses

First Counterexample

Consider the cyclic group Z3 = {0,1,2} under addition mod 3.

Let H = {1}.

  • Non-empty: Yes, H contains 1.
  • Property check: For a = b = 1, we compute (-1) + (-1) = -2 ≡ 1 (mod 3). Since 1 ∈ H, the property holds.
  • Subgroup check: H does not contain the identity 0. Also, 1+1 = 2 ∉ H, and the inverse of 1 is 2 ∉ H. Therefore, H is not a subgroup.

Second Counterexample (General Case)

Let a be an element of order 3 in any group G (so a3 = e). Consider the singleton set H = {a}.

  • Non-empty: H contains a.
  • Property check: For a = b = a, we have a-1a-1 = (a2)-1 = (a-1)-1 = a ∈ H. So the property holds.
  • Subgroup check: H does not contain the identity e (since a ≠ e). Also, H is not closed under the group operation: a·a = a2 ∉ H. Therefore, H is not a subgroup.

Conclusion

The property a, b ∈ H ⇒ a-1b-1 ∈ H is not sufficient to guarantee H is a subgroup. For subgroup status, we must additionally require that H contains the identity element (or an equivalent condition ensuring closure under the group operation).

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