## 物理代写|量子力学代写quantum mechanics代考|PHYS2040

2022年12月28日

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## 物理代写|量子力学代写quantum mechanics代考|Definition of Spacelike Galilean Metric

We postulate the spacelike galilean metric $g: \boldsymbol{E} \rightarrow \mathbb{L}^2 \otimes\left(V^* \boldsymbol{E} \otimes V^* \boldsymbol{E}\right)$ as a scaled fibrewise riemannian metric of spacetime.

Further, with reference to a particle of mass $m$, we define the rescaled spacelike galilean metric $G:=\frac{m}{\hbar} g: \boldsymbol{E} \rightarrow \mathbb{T} \otimes\left(V^* \boldsymbol{E} \otimes V^* \boldsymbol{E}\right)$, which carries a convenient scale factor.

The galilean metric and the rescaled galilean metric yield the metric musical spacelike isomorphisms $g^b: V \boldsymbol{E} \rightarrow \mathbb{L}^2 \otimes V^* \boldsymbol{E}$ and $G^b: V \boldsymbol{E} \rightarrow \mathbb{T} \otimes V^* \boldsymbol{E}$.

Postulate C.2 We postulate the galilean metric to be a scaled spacelike riemannian metric
$$g: \boldsymbol{E} \rightarrow \mathbb{L}^2 \otimes\left(V^* \boldsymbol{E} \otimes V^* \boldsymbol{E}\right),$$
with coordinate expression
$$g=g_{i j} \breve{d}^i \otimes \check{d}^j \text {, with } g_{i j}=g_{j i} \in \operatorname{map}\left(\boldsymbol{E}, \mathbb{L}^2 \otimes \mathbb{R}\right), \operatorname{det}\left(g_{i j}\right)>0 .$$
Definition 3.2.1 With reference to a particle of mass $m \in \mathbb{M}$, it is convenient to introduce (by using, also in the classical context, the Planck constant $\hbar \in \mathbb{T}^{-1} \otimes$ $\mathbb{L}^2 \otimes \mathbb{M}$ ) the rescaled spacelike galilean metric
$$G:=\frac{m}{\hbar} g: \boldsymbol{E} \rightarrow \mathbb{T} \otimes\left(V^* \boldsymbol{E} \otimes V^* \boldsymbol{E}\right),$$
which carries the convenient time scale $\mathbb{T}$.
We have the coordinate expression
$$G=G_{i j}^0 u_0 \otimes \breve{d}^i \otimes \breve{d}^j, \quad \text { with } G_{i j}^0 \in \operatorname{map}(\boldsymbol{E}, \mathbb{R}) . \quad \square$$
Thus, all classical and quantum objects derived from the rescaled galilean metric $G$ will incorporate the mass $m$ and the Planck constant $\hbar$.

## 物理代写|量子力学代写quantum mechanics代考|Hodge Star and Cross Product

Further, the spacelike metric isomorphism $g^b$ and the spacelike volume form $\eta$ yield the Hodge isomorphism $_\eta: \Lambda^r V^ E \rightarrow \mathbb{L}^{3-2 r} \otimes \Lambda^{3-r} V^* \boldsymbol{E}$ and the scaled cross product $X \times Y \in \sec (\boldsymbol{E}, \mathbb{L} \otimes V \boldsymbol{E})$.

Corollary 3.2.7 The metric $g$ and the spacelike volume yield the spacelike Hodge isomorphism, for $0 \leq r \leq 3$,
$$\eta: \Lambda^r V^ \boldsymbol{E} \rightarrow \mathbb{L}^{3-2 r} \otimes \Lambda^{3-r} V^* \boldsymbol{E}: \phi \mapsto i{g^{\sharp}(\phi)} \eta,$$ with coordinate expression
$$\eta: \breve{d}^{i_1} \wedge \ldots \wedge \breve{d}^{i_r} \mapsto \sqrt{|g|} g^{i_1 j_1} \ldots g^{i_r j_r} i{a_{j_1}} \ldots i_{\partial_{j_r}} \breve{d}^1 \wedge \breve{d}^2 \wedge \breve{d}^3 .$$
Corollary 3.2.8 We define the spacelike cross product of two spacelike vector fields $X, Y \in \sec (\boldsymbol{E}, V \boldsymbol{E})$ by the equality
$$X \times Y:=i_{g^{\prime}(X) \wedge g^{\prime}(Y)} \bar{\eta} \in \sec (\boldsymbol{E}, \mathbb{L} \otimes V \boldsymbol{E}),$$
with coordinate expression
$$X \times Y=\frac{1}{\sqrt{|g|}} \epsilon^{h k i} X_h Y_k \partial_i .$$
For each $X, Y, Z \in \sec (\boldsymbol{E}, V \boldsymbol{E})$, we have the following identities
\begin{aligned} & X \times Y=-Y \times X, \ & (X \times Y) \times Z+(Z \times X) \times Y+(Y \times Z) \times X=0 . \end{aligned}
In an analogous way, we define the spacelike cross product of two spacelike 1-forms $\alpha, \beta \in \sec \left(\boldsymbol{E}, V^ \boldsymbol{E}\right)$ by the equality
$$\alpha \times \beta:=i_{g^{\sharp}(\alpha) \wedge g^{\sharp}(\beta)} \eta \in \sec \left(\boldsymbol{E}, \mathbb{L}^{-1} \otimes V^* \boldsymbol{E}\right),$$
with coordinate expression
$$\alpha \times \beta=\sqrt{|g|} \epsilon_{h k i} \alpha^h \beta^k \breve{d}^i$$

# 量子力学代考

## 物理代写|量子力学代写quantum mechanics代考|Definition of Spacelike Galilean Metric

$g: \boldsymbol{E} \rightarrow \mathbb{L}^2 \otimes\left(V^* \boldsymbol{E} \otimes V^* \boldsymbol{E}\right)$ 作为时空的缩放纤维 黎曼度量。

$$g: \boldsymbol{E} \rightarrow \mathbb{L}^2 \otimes\left(V^* \boldsymbol{E} \otimes V^* \boldsymbol{E}\right),$$

$g=g_{i j} \breve{d}^i \otimes \breve{d}^j$, with $g_{i j}=g_{j i} \in \operatorname{map}\left(\boldsymbol{E}, \mathbb{L}^2 \otimes \mathbb{R}\right)$

$$G:=\frac{m}{\hbar} g: \boldsymbol{E} \rightarrow \mathbb{T} \otimes\left(V^* \boldsymbol{E} \otimes V^* \boldsymbol{E}\right)$$

$g=g_{i j} \breve{d}^i \otimes \breve{d}^j$, with $g_{i j}=g_{j i} \in \operatorname{map}\left(\boldsymbol{E}, \mathbb{L}^2 \otimes \mathbb{R}\right)$

$$G:=\frac{m}{\hbar} g: \boldsymbol{E} \rightarrow \mathbb{T} \otimes\left(V^* \boldsymbol{E} \otimes V^* \boldsymbol{E}\right),$$

$G=G_{i j}^0 u_0 \otimes \breve{d}^i \otimes \breve{d}^j, \quad$ with $G_{i j}^0 \in \operatorname{map}(\boldsymbol{E}, \mathbb{R})$

## 物理代写|量子力学代写quantum mechanics代考|Hodge Star and Cross Product

$$\eta: \breve{d}^{i_1} \wedge \ldots \wedge \breve{d}^{i_r} \mapsto \sqrt{|g|} g^{i_1 j_1} \ldots g^{i_r j_r} i a_{j_1} \ldots i_{\partial_{j_r}} \breve{d}^1$$

$$X \times Y:=i_{g^{\prime}(X) \wedge g^{\prime}(Y)} \bar{\eta} \in \sec (\boldsymbol{E}, \mathbb{L} \otimes V \boldsymbol{E}),$$

$$X \times Y=\frac{1}{\sqrt{|g|}} \epsilon^{h k i} X_h Y_k \partial_i$$

$$X \times Y=-Y \times X, \quad(X \times Y) \times Z+(Z \times X)$$

$$\alpha \times \beta:=i_{g^{\sharp}}(\alpha) \wedge g^{\sharp}(\beta) \eta \in \sec \left(\boldsymbol{E}, \mathbb{L}^{-1} \otimes V^* \boldsymbol{E}\right)$$

$$\alpha \times \beta=\sqrt{|g|} \epsilon_{h k i} \alpha^h \beta^k \breve{d}^i$$

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