The Euclidean Space E 3 (Hilbert Space and Lie Algebra Structure)

Eberhard Zeidler
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Eberhard Zeidler: Max Planck Institute for Mathematics in the Sciences

Chapter 1 in Quantum Field Theory III: Gauge Theory, 2011, pp 69-114 from Springer

Abstract: Abstract One has to distinguish between the Euclidean space E 3 (a set of vectors), and the Euclidean manifold $\mathbb{E}^{3}$ (a set of points). The Euclidean space E 3 is a real 3-dimensional Hilbert space equipped with the inner product $$\langle \mathbf{x}|\mathbf{y}\rangle:= \mathbf{x}\mathbf{y}$$ of vectors x,y. Additionally, the Euclidean space E 3 is a Lie algebra equipped with the vector product $$[\mathbf{x}, \mathbf{y}]:= \mathbf{x}\times \mathbf{y}.$$

Keywords: Hilbert Space; Euclidean Space; Heisenberg Group; Galois Group; Galois Theory (search for similar items in EconPapers)
Date: 2011
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DOI: 10.1007/978-3-642-22421-8_2

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