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A Hilbert space $\cal H$ is complete which means that every Cauchy sequence of vectors admits a limit in the space itself. Under this hypothesis there exist Hilbert bases also known as complete orthonormal systems of vectors in $\cal H$. A set of vectors $\{\psi_i\}_{i\in I}\subset \cal H$ is called an orthonormal system if $\langle \psi_i |\psi_j \rangle ...


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Like Wikipedia says: "Moment is a combination of a physical quantity and a distance." This 'physical quantity' could be various things. To take the examples you mention: Moment of momentum (commonly known as angular momentum) is expressed as $\vec{L}=\vec{r}\times m\vec{v}$, and is a measure for the rotational momentum of an object around some axis. Moment ...


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This completeness relation of the basis means that you can reach all possible directions in the Hilbert space. It means that any $|\psi \rangle$ can be made up from these basis vectors. If the sum of the projectors (the ket-bras) would not be the unit matrix, the vector $|\psi\rangle$ could have components which cannot be represented within your basis. ...


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Diffeomorphism Invariance Let $M$ be a smooth manifold. Let $\phi: M \to M$ be a diffeomorphism. A simple property of the Einstein equations is $$ g \in \otimes^2 TM \text{ is solution to vacuum Einstein equation} \implies \text{ so is } \phi^*g $$ To see that this is true, simply pull back both sides of the Einstein equation by $\phi$, and use the ...



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