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I think you are mixing two things: gradient and divergence. The gradient is (normally) used when you have a scalar field, or function. A scalar field (or function) is when you associate a number to every point is space. The divergence is (again, normally) used when you have a vector field, or function. A vector field (or function) is when you associate a ...


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You calculate the gradient of a scalar field and get a vector field as a result. scalar field means for every point in space, the field has a value. vector field means for every point in space, the field has a vector, i.e. a value and a direction. To get an intuitive understanding, imagine the surface of the earth as a scalar field. On a map for example, ...


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Yes, the missing dot in the dot product of the second term $$\tag{2} -2\frac{e}{c}\hat{\mathbf{p}} \cdot \mathbf{A}(\hat{\mathbf{x}}) $$ of eq. (2.34b) is a typo. The operators $\hat{\mathbf{p}}$ and $\mathbf{A}(\hat{\mathbf{x}})$ do not commute, due to the CCRs $$\tag{CCR} [\hat{x}^i~,~ \hat{p}_j]_{-}~=~i\hbar\delta^i_j~{\bf 1}.$$ The second and third ...


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The gauge potential is an object that, when introduced in the covariant derivative, is intended to cancel the terms that spoil the linear transformation of the field under the gauge group. Every gauge transformation $g:\Sigma\to G$ (on a spacetime $\Sigma$) connected to the identity may be written as $\mathrm{e}^{\mathrm{i}\chi(x)}$ for some Lie algebra ...


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As derivatives, the Lie and covariant derivatives involve comparing tensors at different points on the manifold. They differ in the prescription given for comparing the tensors at two different points. The key concept with a covariant derivative $\nabla_\xi = \xi^a\nabla_a$ is parallel transport. It is defined so that as you move along a geodesic in the ...


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The notation whether it be d or delta doesn't matter as long as it describes an element (a minute amout) of the quantity. Please keep in mind that this is NOT a ratio. So you can't write dq = I. dt This is mathematically wrong. As differentiation is an operation and not a mere ratio. It is like a machine and you can't separate it's parts or the machine ...


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The notion of derivative requires a notion of comparison. In a general manifold, tangent vectors at different points belong to totally different vector spaces (see footnote 1), so we must define a way of mapping one tangent vector to another tangent space that we shall take, by definition to be the the "invariant image" of the vector in the new tangent space ...



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