# 物理代写|量子力学代写quantum mechanics代考|PHYS3040

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## 物理代写|量子力学代写quantum mechanics代考|Upper Potential and Observed Potential

We discuss the (local) gauge dependent “upper” potential $A^{\uparrow} \in \sec \left(J_1 \boldsymbol{E}, T^* \boldsymbol{E}\right)$ of the cosymplectic phase 2-form $\Omega: J_1 \boldsymbol{E} \rightarrow \Lambda^2 T^* J_1 \boldsymbol{E}$ and its relation with the (local) gauge dependent and observer dependent potential $A[o] \in \sec \left(\boldsymbol{E}, T^* \boldsymbol{E}\right)$ of the observed spacetime 2-form $\Phi[o] \in \sec \left(\boldsymbol{E}, \Lambda^2 T^* \boldsymbol{E}\right.$ ) (see Definition 4.2 .11 and Theorem 4.3.3).

We stress that the potential $A^{\uparrow}$ is “horizontal”. Later, in the quantum theory, we shall see that the existence of horizontal potentials of $\Omega$ fits our choice of a quantum bundle $Q$ based on spacetime and of a reducible upper quantum connection $\mathrm{U}^{\uparrow}$ (see Postulates Q.1 and Q.2, Definitions 15.1.5 and 15.1.6).

In general, throughout the book, the word “upper” will label objects of the phase space $J_1 \boldsymbol{E}$ in order to distinguish them with respect to associated objects of the spacetime $\boldsymbol{E}$.

Definition 10.1.3 We define an “upper” potential to be a potential of the cosymplectic 2-form $\Omega$
$$A^{\uparrow} \in \sec \left(J_1 \boldsymbol{E}, T^* J_1 \boldsymbol{E}\right)$$ according to the equality (for the notation, see also Remark 10.1.6 and Notation 10.1 .7$)$
$$\Omega=d A^{\uparrow} .$$
Thus, we have a coordinate expression of the type
$$A^{\uparrow}=A_\lambda d^\lambda+A_i^0 d_0^i, \quad \text { with } A_\lambda, A_0^i \in \operatorname{map}\left(J_1 \boldsymbol{E}, \mathbb{R}\right) .$$
and the equalities
$$\Omega_{\lambda \mu}=\frac{1}{2}\left(\partial_\lambda A^{\uparrow}{ }\mu-\partial\mu A_\lambda^{\uparrow}\right) \quad \text { and } \quad \Omega_{\lambda j}^0=\frac{1}{2}\left(\partial_\lambda A_j^0-\partial_j^0 A_\lambda^{\uparrow}\right) .$$
Clearly, the potentials $A^{\uparrow}$ of $\Omega$ are defined (locally) up to a gauge of the type
$$d f \in \sec \left(J_1 \boldsymbol{E}, T^* J_1 \boldsymbol{E}\right), \quad \text { with } f \in \operatorname{map}(\boldsymbol{E}, \mathbb{R}) \text {. }$$

## 物理代写|量子力学代写quantum mechanics代考|Dynamical Phase 1-Forms

In most lagrangian theories, one usually starts from a given lagrangian $\mathcal{L}$ and derives from it the momentum $\mathcal{M}$ and the Poincaré-Cartan form $\mathcal{L}+\mathcal{M}$.

Analogously, in most hamiltonian theories, one usually starts from a hamiltonian $\mathcal{H}$

Conversely, in the present theory, we start from the global, gauge independent and observer independent cosymplectic 2-form $\Omega: J_1 \boldsymbol{E} \rightarrow \Lambda^2 T^* J_1 \boldsymbol{E}$ and derive from it the local gauge dependent and observer independent horizontal upper potential $A^{\uparrow}[\mathrm{b}] \in \sec \left(J_1 \boldsymbol{E}, T^* \boldsymbol{E}\right)$.

Then, the covariant splitting of $A^{\uparrow}[\mathrm{b}]$, into its $\boldsymbol{E}$-horizontal and $\boldsymbol{E}$-vertical components, yields the local gauge dependent and observer independent lagrangian $\mathcal{L}[\mathrm{b}]$ and momentum $\mathcal{M}[\mathrm{b}]$. Accordingly, the classical Poincaré-Cartan form associated with the lagrangian $\mathcal{L}[\mathrm{b}]$ turns out to be just the potential $A^{\uparrow}[b]$.

Moreover, the observed splitting of $A^{\uparrow}[b]$, into its observed $\boldsymbol{E}$-horizontal and observed $\boldsymbol{E}$-vertical components, yields the local gauge dependent and observer dependent hamiltonian $\mathcal{H}[\mathrm{b}, o]$ and momentum $\mathcal{P}[\mathrm{b}, o]$.

Thus, we do not postulate a classical lagrangian $\mathcal{L}[\mathrm{b}]$, or a classical hamiltonian $\mathcal{H}[\mathrm{b}, o]$, but we start with the cosymplectic phase 2-form $\Omega$ (derived in a covariant way from the gravitational and electromagnetic fields $K^{\natural}$ and $F$ ) and derive from $\Omega$ the lagrangian $\mathcal{L}[\mathrm{b}]$ and the hamiltonian $\mathcal{H}[\mathrm{b}, o]$. So, we can say that the classical lagrangian $\mathcal{L}[\mathrm{b}]$ and classical observed hamiltonian $\mathcal{H}[\mathrm{b}, o]$ are encoded in the cosymplectic phase 2 -form $\Omega$.

# 量子力学代考

## 物理代写|量子力学代写quantum mechanics代考|Upper Potential and Observed Potential

$\Omega: J_1 \boldsymbol{E} \rightarrow \Lambda^2 T^* J_1 \boldsymbol{E}$ 及其与 (本地) 规范相关和观 察者相关潜力的关系 $A[o] \in \sec \left(\boldsymbol{E}, T^* \boldsymbol{E}\right)$ 观察到的时 空 2-形式 $\Phi[o] \in \sec \left(\boldsymbol{E}, \Lambda^2 T^* \boldsymbol{E}\right.$ ) (参见定义 4.2 .11 和定理 4.3.3) 。

$$A^{\uparrow} \in \sec \left(J_1 \boldsymbol{E}, T^* J_1 \boldsymbol{E}\right)$$

$$\Omega=d A^{\uparrow} .$$

$$A^{\uparrow}=A_\lambda d^\lambda+A_i^0 d_0^i, \quad \text { with } A_\lambda, A_0^i \in \operatorname{map}\left(J_1 \boldsymbol{E}, \mathbb{R}\right)$$

$$\Omega_{\lambda \mu}=\frac{1}{2}\left(\partial_\lambda A^{\uparrow} \mu-\partial \mu A_\lambda^{\uparrow}\right) \quad \text { and } \quad \Omega_{\lambda j}^0=\frac{1}{2}\left(\partial_\lambda A\right.$$

$d f \in \sec \left(J_1 \boldsymbol{E}, T^* J_1 \boldsymbol{E}\right), \quad$ with $f \in \operatorname{map}(\boldsymbol{E}, \mathbb{R})$.

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