Pseudoscalar

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Pseudo-scalar(Pseudo-scalar) changes its sign as to inversion of coordinates.

Think of (inner product, scalar product) of twoA,B(think here)

A · B = A x B x + A y B y + A z B z {\ displaystyle \ mathbf {A} \ cdot \ mathbf {B} = A_ {x} B_ {x} + A_ {y} B_ { B_ {x} B_ {x} + A_ {y} B_ {y}} + A_ {z} B_ {z}}! [{\ \ Mathbf {A}} \ cdot {\ mathbf { + A_ {z} B_ {z}] (https://wikimedia.org/api/rest_v1/media/math/render/svg/fb20e11abac86de9246961536edd21c20e8e011b)

In this inner product, when inverting the (x, y, z) axes to (-x, -y, -z), the case where the sign of the inner product is changed is calledpseudoscalar.

This depends on whether the vectorsA,B, are polar or axial, respectively. Polarity vectoris a vector such as ordinary and so on, Axis vector It is a vector like a hay. The sign of the polarity vector varies with the reversal of the coordinates, but the sign of the axial vector is unchanged by the inversion of the coordinates. Therefore, the vectorA ,Bare both polar or axial, the sign of the inner product is not inverted with respect to the inversion of the coordinates, butA,B When either one is polar and the other one is axial, the sign of the inner product is reversed. This case is a pseudo scalar.

The axial vector (Axial vector) is also called****(Pseudo vector).

When both vectorsAandBare polar vectors and further a third vectorCis considered and this is also a polar vector, the next result,

(A × B) ⋅ C {\ displaystyle (\ mathbf {A} \ times \ mathbf {B}) \ cdot \ mathbf {C}}! [\ ({\ mathbf {A}} \ \ times {\ \ mathbf {B}} ) \ cdot {\ mathbf {C}}] (https://wikimedia.org/api/rest_v1/media/math/render/svg/ed5e126194d8c71ff5caa8485ded6245a48a23b2)

Is also a pseudo-scalar (x is (outer product, vector product)). This is because the outer products of polar vectors are axial vectors.

Also, the relation in φ,

F = - ∇ φ {\ displaystyle \ mathbf {F} = - \ nabla \ phi}! [{\ \ Mathbf {F}} = - \ \ nabla \ phi] (https://wikimedia.org/api/ rest_v1 / media / math / render / svg / dfad986cdfc9d6cce01477861ee4b0b4a0f96e86)

, If the vectorFis an axial vector, φ is a pseudoscalar. This is a vectorF Since the sign is invariant with respect to the inversion of the coordinates, the ∇ portion (the portion of the derivative) changes the sign against the inversion, so φ, which is a scalar potential, also changes sign (that is, a pseudo scalar).

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Post Date : 2018-01-31 08:00

This article is taken from the Japanese Wikipedia Pseudoscalar

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