## 物理代写|量子力学代写quantum mechanics代考|PHYS3034

2022年12月28日

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## 物理代写|量子力学代写quantum mechanics代考|Classical Phase Space

As classical phase space we consider the odd dimensional 1st jet space $J_1 \boldsymbol{E}$ of classical motions.

Here, we just briefly introduce the bundle $J_1 \boldsymbol{E} \rightarrow \boldsymbol{E}$, along with its affine structure associated with the vector bundle $\mathbb{T}^* \otimes V \boldsymbol{E}$ and its natural inclusion into the scaled tangent space of spacetime $\mathbb{T}^* \otimes T \boldsymbol{E}$, which is expressed by the contact map д : $J_1 \boldsymbol{E} \rightarrow \mathbb{T}^* \otimes T \boldsymbol{E}$.
Indeed, the above structures will play an essential role throughout the book.
Later, we shall see that the classical fundamental fields of spacetime (the metric field $g$, the gravitational field $K^{\natural}$ and the electromagnetic field $F$ ) naturally yield a rich structure on the space $J_1 \boldsymbol{E}$, namely a cosymplectic structure $(d t, \Omega)$ and also a coPoisson structure $(\gamma, \Lambda)$ (see Chap. 10).

Indeed, the present choice of classical phase space $J_1 \boldsymbol{E}$ turns out to be strategical for our approach: in fact, it reflects the fundamental role of time, hence the covariance of the theory, and a criterion of minimality.

Our phase space $J_1 \boldsymbol{E}$ replaces the more usual even dimensional symplectic phase space. In particular, the possible alternative phase spaces $V \boldsymbol{E}$ and $V^* \boldsymbol{E}$ would be more suitable for a theory where time is just a parameter; hence, they would be unable to properly account for covariance. Moreover, the possible alternative spaces $\mathbb{T}^* \otimes T \boldsymbol{E}$ and $\mathbb{T} \otimes T^* \boldsymbol{E}$ would not take into proper account the fact that the galilean metric is spacelike and would not fulfill a principle of minimality.

For further details on jet spaces, the reader can refer, for instance, to [246, 360, 427] and to Appendix G.

In the introduction we have discussed our reasons for the present choice of phase space (see Introduction: Sect. 1.4.8).

An essential comparison between the phase space in the galilean and einsteinian frameworks can be summarised as follows: in the 1st case, phase space is the affine 1st jet bundle of sections of spacetime, in the 2nd case, phase space is the Grassmannian 1st jet bundle of timelike 1-dimensional submanifolds of spacetime (see [222]).

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

The time fibring $t: \boldsymbol{E} \rightarrow \boldsymbol{T}$ yields naturally a scaled timelike metric of spacetime $\mathrm{g}:=d t \otimes d t$ with signature $(+000)$. I ater, we shall use the associated metric musical morphism $\mathbf{g}^b: T \boldsymbol{E} \rightarrow \mathbb{T}^2 \otimes H^* \boldsymbol{E}$, which turns out to be degenerate, but is defined on the whole tangent space $T \boldsymbol{E}$, while the metric musical morphism $g^b: V \boldsymbol{F} \rightarrow \pi_a^2 \otimes V^* \boldsymbol{F}$ will he defined only on the vertical subspace (see Definitions 7.3.3, 8.1.1, 3.2.2 and also Proposition 8.2.1).

Definition 3.1.1 We define the timelike galilean metric to be the scaled spacetime metric
$$\mathbf{g}:=d t \otimes d t: \boldsymbol{E} \rightarrow \mathbb{T}^2 \otimes\left(H^* \boldsymbol{E} \otimes H^* \boldsymbol{E}\right) \subset \mathbb{T}^2 \otimes\left(T^* \boldsymbol{E} \otimes T^* \boldsymbol{E}\right),$$
with signature $\operatorname{sign}(\mathbf{g})=(+000)$ and coordinate expression $$\mathbf{g}=\left(u_0 \otimes u_0\right) \otimes d^0 \otimes d^0 .$$
We define the metric timelike musical morphism to be the linear fibred morphism over $\boldsymbol{E}$
$$\left.\mathbf{g}^b: T \boldsymbol{E} \rightarrow \mathbb{T}^2 \otimes H^* \boldsymbol{E}: X \mapsto X\right\lrcorner \mathbf{g},$$
with coordinate expression
$$\mathbf{g}^b(X)=X^0\left(u_0 \otimes u_0\right) \otimes d^0 .$$
The image and the kernel of $\mathbf{g}^b$ are the subbundles (see Proposition 2.2.4)
$$\operatorname{im}\left(\mathbf{g}^b\right)=\mathbb{T}^2 \otimes H^* \boldsymbol{E} \subset \mathbb{T}^2 \otimes T^* \boldsymbol{E} \quad \text { and } \quad \operatorname{ker}\left(\mathbf{g}^b\right)=V \boldsymbol{E} \subset T \boldsymbol{E} \text {. }$$
Note 3.1.2 By taking into account the speed of light $c \in \mathbb{T}^* \otimes \mathbb{L}$, we can rescale the timelike galilean metric $\mathrm{g}$ with a scale dimension which will turn out to be convenient later (see Postulate C.6)
$$\mathfrak{g}:=c^2 \mathbf{g}=c^2 d t \otimes d t: \boldsymbol{E} \rightarrow \mathbb{L}^2 \otimes\left(H^* \boldsymbol{E} \otimes H^* \boldsymbol{E}\right) \subset \mathbb{L}^2 \otimes\left(T^* \boldsymbol{E} \otimes T^* \boldsymbol{E}\right)$$

# 量子力学代考

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

$$\mathbf{g}=\left(u_0 \otimes u_0\right) \otimes d^0 \otimes d^0 .$$

$$\left.\mathbf{g}^b: T \boldsymbol{E} \rightarrow \mathbb{T}^2 \otimes H^* \boldsymbol{E}: X \mapsto X\right\lrcorner \mathbf{g},$$

$$\mathbf{g}^b(X)=X^0\left(u_0 \otimes u_0\right) \otimes d^0 .$$

$$\operatorname{im}\left(\mathbf{g}^b\right)=\mathbb{T}^2 \otimes H^* \boldsymbol{E} \subset \mathbb{T}^2 \otimes T^* \boldsymbol{E} \quad \text { and } \quad \operatorname{ker}\left(\mathbf{g}^b\right)$$

$$\mathfrak{g}:=c^2 \mathbf{g}=c^2 d t \otimes d t: \boldsymbol{E} \rightarrow \mathbb{L}^2 \otimes\left(H^* \boldsymbol{E} \otimes H^* \boldsymbol{E}\right)$$

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