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Why is quantum physics based on complex numbers? (English) Zbl 1155.81010

Quantum states are defined in complex Hilbert spaces, and it has been shown by the author that a quantum theory based on a Galois field \(GF(p^2)\) is a natural generalization of classical quantum theory based on complex numbers. This being the case, one makes the assumption that any admissible linear operator representing a physical quantity should have a spectral decomposition. The paper proves that if the prime number \(p\) is equal to \(3 (\pmod 4)\), then the minimal extension of the finite field \(GF(p)\) containing \(p\) elements, for which there exists 10 linearly independent admissible representation operators, is the field \(GF(p^2)\) containing \(p^2\) elements. One discusses about hermiticity conditions when the irreductible representation is supplied by a scalar product. The spectrum of the \(M{04}\) operatori is carefully investigated

MSC:

81P05 General and philosophical questions in quantum theory
17B10 Representations of Lie algebras and Lie superalgebras, algebraic theory (weights)
20C20 Modular representations and characters
81R05 Finite-dimensional groups and algebras motivated by physics and their representations
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References:

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