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## 物理代写|量子力学代写quantum mechanics代考|The Role of Time

In standard Classical Mechanics and standard Quantum Mechanics one usually deals with a flat spacétime $\boldsymbol{E}$ and a given inertial observerer $o$, which yields a global splitting $\boldsymbol{E}=\boldsymbol{T} \times \boldsymbol{P}[o]$ of spacetime into the observer independent 1-dimensional time $\boldsymbol{T}$ and the observer dependent 3-dimensional space $\boldsymbol{P}[o]$ (see, for instance, Note 2.7.5). So, one can focus the attention to the given spacelike affine space $\boldsymbol{P}[o]$ and consider time as a parameter.

In our covariant approach we are dealing with all possible general observers $o$ and, even more, we have chosen a manifestly covariant approach. So, according to a general principle of relativity, we do not deal with a distinguished observer $o$, a distinguished space $P[o]$ and a distinguished splitting $\boldsymbol{E}=\boldsymbol{T} \times \boldsymbol{P}[o]$. However, according to a galilean scheme, we still have a distinguished projection on the observer independent time $T$.

Indeed, the fact that a distinguished splitting of spacetime is missing prevents the possibility to regard time as a pure parameter. Actually, time plays an essential role at every step of the classical and quantum theories. In fact, such a fundamental role of time is reflected in several non standard features of our formulation, as we shall see in the forthcoming sections.

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

In einsteinian General Relativity, one deals with a scaled spacetime lorentzian metric $g: \boldsymbol{M} \rightarrow \mathbb{L}^2 \otimes\left(T^* \boldsymbol{M} \otimes T^* \boldsymbol{M}\right), \quad$ with signature $(-+++)$,
which generates the Levi-Civita connection $K^{\natural}$ and plays a further fundamental role via the “musical isomorphism” $g^b: T M \rightarrow \mathbb{L}^2 \otimes T^* M$ (for the scaling induced by $\mathbb{L}$, see Sect. 1.3.5).

Conversely, in our galilean framework, we deal with the scaled spacelike galilean metric
$g: \boldsymbol{E} \rightarrow \mathbb{L}^2 \otimes\left(V^* \boldsymbol{E} \otimes V^* \boldsymbol{E}\right), \quad$ with signature $(0+++) .$
Indeed, this signature makes a great difference with respect to einsteinian General Relativity. In fact, in our contest, we cannot avail of a “spacetime musical isomorphism” $g^b: T \boldsymbol{E} \rightarrow \mathbb{L}^2 \otimes T^* \boldsymbol{E}$, but only of a “spacelike musical isomorphism” $g^b: V \boldsymbol{E} \rightarrow \mathbb{L}^2 \otimes V^* \boldsymbol{E}$. This fact turns out to be reflected in many physical laws that can be conceived in the present galilean framework.

Further, in the present galilean framework, we deal also with a scaled spacetime timelike metric
$\mathbf{g}:=d t \otimes d t: \boldsymbol{E} \rightarrow \mathbb{T}^* \otimes\left(T^* \boldsymbol{E} \otimes T^* \boldsymbol{E}\right), \quad$ with signature $(+000)$,
which is naturally generated by the time fibring $t: \boldsymbol{E} \rightarrow \boldsymbol{T}$ (see Definition 3.1.1). This timelike metric yields the “spacetime musical morphism” $\mathbf{g}^b: T \boldsymbol{E} \rightarrow \mathbb{T}^* \otimes$ $T^* \boldsymbol{E}$, whose rank is 1 . Indeed, the timelike metric $g$ plays a minor role. It is essentially used, via $\mathbf{g}^b$, for the definition of the timelike charge current and the timelike energy tensor (see Definition 8.1.1 and Proposition 8.2.1).

Unfortunately, in the present galilean framework, there is no covariant way to combine the two metrics $g$ and $g$ in order to obtain a non degenerate spacetime metric; in fact, such a possible combination would turn out to be observer dependent.
With reference to a given charged particle, of mass $m$, it is useful to define the rescaled covariant and contravariant metrics
$G:=\frac{m}{\hbar} g: \boldsymbol{E} \rightarrow \mathbb{T} \otimes\left(V^* \boldsymbol{E} \otimes V^* \boldsymbol{E}\right) \quad$ and $\quad \bar{G}:=\frac{\hbar}{m} \bar{g}: \boldsymbol{E} \rightarrow \mathbb{T}^* \otimes(V \boldsymbol{E} \otimes V \boldsymbol{E})$,
whêrê thee scālê $\mathbb{L}^2$ hass beén réplacéd by a convenniênt timee scalee $\mathbb{T}$.

# 量子力学代考

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

$g: \boldsymbol{E} \rightarrow \mathbb{L}^2 \otimes\left(V^* \boldsymbol{E} \otimes V^* \boldsymbol{E}\right), \quad$ 带签名 $(0+++)$.

$\mathbf{g}:=d t \otimes d t: \boldsymbol{E} \rightarrow \mathbb{T}^* \otimes\left(T^* \boldsymbol{E} \otimes T^* \boldsymbol{E}\right), \quad$ 带签名 $(+000)$ ，

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

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