# 物理代写|量子力学代写quantum mechanics代考|PHYSICS3544

## 物理代写|量子力学代写quantum mechanics代考|Proper Quantum Bundle and Its Polar Real Splitting

There is also another interesting way to compare the complex and real languages for Quantum Mechanics and to discuss the 2 real (internal) degrees of freedom or the 1 complex (internal) degree of freedom of the scalar quantum particle.

Let us start by considering the “proper complex plane” $\mathbb{C}{/ 0} \subset \mathbb{C}$, obtained by dropping the zero element. Analogously, by considering the quantum bundle $\pi$ : $\boldsymbol{Q} \rightarrow \boldsymbol{E}$ as a vector bundle with type fibre $\mathbb{C}$, we define the “proper quantum bundle” to be the subbundle $\pi{/ 0}: Q_{/ 0} \subset \boldsymbol{Q} \rightarrow \boldsymbol{E}$ obtained by dropping the zero section (see Definition 14.6.1).

We can regard the proper quantum bundle as the restriction of the quantum bundle to the domain where the quantum particle has non vanishing probability to be detected.

The “polar splitting” yields a bijection $\mathbb{C}{/ 0} \rightarrow \mathbb{R}^{+} \times \mathbb{R} / 2 \pi:\left(c_1, c_2\right) \mapsto(\varrho, \varphi)$. Analogously, with reference to a quantum basis, the proper quantum bundle splits locally as $Q{/ 0} \rightarrow Q_{/ 0}^{\mathrm{I}} \times \underset{E}{\times} Q_{/ 0}^{\varphi}: \Psi_e \mapsto\left(|\Psi|_e, \varphi(\Psi)e\right)$, where $\pi{|}: Q_{/ 0}^{11}:=\left(\boldsymbol{E} \times \mathbb{R}^{+}\right)$ $\rightarrow \boldsymbol{E}$ is a trivial bundle with type fibre $\mathbb{R}^{+}$and $\pi_{\varphi}: Q_{/ 0}^{\varphi} \rightarrow \boldsymbol{E}$ is a bundle with type fibre $U(1)=\mathbb{R} / 2 \pi$. This is the usual “polar splitting of the quantum bundle”, which is largely used in the literature, for instance in view of the splitting of Schrödinger equation and of the hydrodynamical picture of Quantum Mechanics. We stress that the above phase $\varphi$ is, by definition, gauge dependent, i.e. it depends on the choice of a base of the quantum bundle. Hence, we cannot fully regard this phase $\varphi$, as it stands, as representing a true real (internal) degree of freedom of the quantum particle, in an intrinsic way.

However, there is another more intrinsic way to address the above splitting. In fact, without reference to any gauge or quantum basis, we can define the global splitting $\boldsymbol{Q}{/ 0} \rightarrow \boldsymbol{Q}{/ 0}^{|} \times \underset{E}{|} \boldsymbol{Q}{00}^{\text {o }}: \Psi_e \mapsto\left(|\Psi|_e,((\Psi))_e\right)$, where $\pi^{00}: \boldsymbol{Q}{/ 0}^6 \rightarrow \boldsymbol{E}$ is a bundle with type fibre $S_1^*$ (see Proposition 14.7.1).

## 物理代写|量子力学代写quantum mechanics代考|The “Game” of Potentials and Distinguished Observer

In our approach to Covariant Classical Mechanics and Covariant Quantum Mechanics we meet the potential in several forms and in several contexts (see [224]).
(1) First of all, we mention the joined observed spacetime 2-form $\Phi[o]: \boldsymbol{E} \rightarrow$ $\Lambda^2 T^* \boldsymbol{E}$, which is associated with the joined spacetime connection $K: T \boldsymbol{E} \rightarrow$ $T^* \boldsymbol{E} \otimes T T \boldsymbol{E}$ (see Corollary 6.3.3).

This 2-form is closed, hence admits a, gauge dependent, local joined observed spacetime potential $A[\mathrm{~b}, o]: \boldsymbol{E} \rightarrow T^* \boldsymbol{E}$, according to $\Phi[o]=2 d A[\mathrm{~b}, o]$.

Here, in the classical context, the symbol b just denotes the choice of a gauge.
(2) Then, we consider the joined cosymplectic phase 2-form $\Omega: J_1 \boldsymbol{E} \rightarrow \Lambda^2 T^$ $J_1 \boldsymbol{E}$, which is closed, hence admits a joined gauge dependent and observer independent, potential $A^{\uparrow}[\mathrm{b}]: J_1 \boldsymbol{E} \rightarrow T^ \boldsymbol{E}$, which can be chosen to be horizontal according to $\Omega=d A^{\uparrow}[\mathrm{b}]$ (see Theorem 10.1.4). Here, again, in the classical context, the symbol $b$ just denotes the choice of a gauge.
(3) There is a natural way to link the above classical potentials and to compare the choices of the corresponding gauges b. In fact, for each observer $o$, we obtain the equality $\Phi[o]=2 o^* \Omega$, which yields $A[\mathrm{~b}, o]=o^* A^{\uparrow}[\mathrm{b}]$.

Hence, we have the observed and coordinate expressions (see Theorem 10.1.8)
\begin{aligned} A^{\uparrow}[\mathrm{b}] &=-\mathcal{H}[\mathrm{b}, o]+\mathcal{P}[\mathrm{b}, o] \ &=-(\mathcal{K}[o]-\not[o]\lrcorner A[\mathrm{~b}, o])+(\mathcal{Q}[o]+\theta[o]\lrcorner A[\mathrm{~b}, o]) \ &=-\left(\frac{1}{2} G_{i j}^0 x_0^i x_0^j-A_0\right) d^0+\left(G_{i j}^0 x_0^j+A_j\right) d^i \end{aligned}

## 物理代写|量子力学代写quantum mechanics代考|Proper Quantum Bundle and Its Polar Real Splitting

“极分裂”产生双射 $\mathbb{C} / 0 \rightarrow \mathbb{R}^{+} \times \mathbb{R} / 2 \pi:\left(c_1, c_2\right) \mapsto(\varrho, \varphi)$. 类似地，关于量子基，适当的量子東局部分 裂为 $Q / 0 \rightarrow Q_{/ 0}^{\mathrm{I}} \times \underset{E}{\times} Q_{/ 0}^{\varphi}: \Psi_e \mapsto\left(|\Psi|e, \varphi(\Psi) e\right)$ ，在哪里 $\pi \mid: Q{/ 0}^{11}:=\left(\boldsymbol{E} \times \mathbb{R}^{+}\right) \rightarrow \boldsymbol{E}$ 是纤维类型的 平凡丛 $\mathbb{R}^{+}$和 $\pi_{\varphi}: Q_{/ 0}^{\varphi} \rightarrow \boldsymbol{E}$ 是具有类型纤维的束 $U(1)=\mathbb{R} / 2 \pi$. 这是通常的“量子束的极性分裂”，在文献 中大量使用，例如考虑到薛定谔方程的分裂和量子力学的流体动力学图像。㧴们强调上述阶段 $\varphi$ 根据定 义，它是规范依赖的，即它取决于量子束的基的选择。因此，我们不能充分考虑这个阶段 $\varphi$ ，就目前而 言，以固有的方式表示量子粒子的真实（内部) 自由度。

## 物理代写|量子力学代写quantum mechanics代考|The “Game” of Potentials and Distinguished Observer

(1) 首先，我们提到联合观测时空 2-形式 $\Phi[o]: \boldsymbol{E} \rightarrow \Lambda^2 T^* \boldsymbol{E}$ ，它与连接的时空连接相关联 $K: T \boldsymbol{E} \rightarrow$ $T^* \boldsymbol{E} \otimes T T \boldsymbol{E}($ 见推论 6.3.3) 。

(2) 然后，我们考虑联合的余辛相 2-形式 \Omega: J1\boldsymbol{E} \rightarrow \Lambda^2 T T^ $J_1 \boldsymbol{E}$ ，它是 封闭的，因此承认一个联合的规范依赖和观察者独立，潜在 $A^{\uparrow}[\mathrm{b}]: J_1 E \rightarrow T^E$ ，可以选择为水平的 $\Omega=d A^{\uparrow}[\mathrm{b}]$ （见定理 10.1.4）。在这里，再一次，在古典语境中，符号 $b$ 只是表示量规的选择。
(3) 有一种自然的方法可以将上述经典势联系起来并比较相应规范的选择 $b$ 。事实上，对于每个观察者 $o$ ， 我们得到等式 $\Phi[o]=2 o^* \Omega$ ，产生 $A[\mathrm{~b}, o]=o^* A^{\dagger}[\mathrm{b}]$.

$$\left.\left.\left.A^{\uparrow}[\mathrm{b}]=-\mathcal{H}[\mathrm{b}, o]+\mathcal{P}[\mathrm{b}, o] \quad=-(\mathcal{K}[o]-\Lambda o]\right\lrcorner A[\mathrm{~b}, o]\right)+(\mathcal{Q}[o]+\theta[o]\lrcorner A[\mathrm{~b}, o]\right)=-\left(\frac{1}{2} G_{i j}^0 x_0^i x_0^j-A_0\right)$$

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