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

2023年3月29日

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## 物理代写|量子力学代写quantum mechanics代考|Cosymplectic Versus Symplectic Structures

We have seen that the joined galilean spacetime connection $K$ yields the cosymplectic phase 2-form $\Omega \equiv \Omega[G, K]: J_1 \boldsymbol{E} \rightarrow \Lambda^2 T^* J_1 \boldsymbol{E}$ of the odd dimensional phase space $J_1 \boldsymbol{E}$ (see Corollary 9.2.4).

The standard literature deals with more popular even dimensional phase spaces equipped with a symplectic 2-form. So, it might be interesting to see whether in our framework there is a “reduction” of $J_1 \boldsymbol{E}$ to a suitable even dimensional space, such that $\Omega$ reduces to be a symplectic 2 -form. Actually, we do find such a reduction, but discover that it carries a too poor information, hence it turns out to be unsuitable for our dynamical purposes.

Moreover, in standard classical dynamics, one often postulates the lagrangian and hamiltonian functions as additional objects with respect to a given symplectic structure. Conversely, we have seen that the lagrangian and hamiltonian functions are encoded in our cosymplectic 2 -form $\Omega$. Even more, we prove that the standard hamiltonian procedure usually followed in symplectic dynamics to derive the equation of motion turns out meaningless in our cosymplectic framework.

Such deep differences between cosymplectic and symplectic frameworks are reflected in all steps of our approach to Classical and Quantum Mechanics.
Proposition 10.1.13 The dynamical phase connection
$$\gamma \equiv \gamma[K]: J_1 \boldsymbol{E} \rightarrow \mathbb{T}^* \otimes T J_1 \boldsymbol{E}$$ associated with a galilean spacetime connection $K$ turns out to be a scaled infinitesimal symmetry of the cosymplectic structure $(d t, \Omega) \equiv(d t, \Omega[G, K])$, i.e. we have
$$L_\gamma d t=0 \text { and } L_\gamma \Omega=0$$
Proof. By recalling Theorem 9.2 .19 , the equality $i_\gamma d t=1$ implies $L_\gamma d t=0$ and the equality $i_\gamma \Omega=0$ implies $L_\gamma \Omega=0$.

## 物理代写|量子力学代写quantum mechanics代考|The coPoisson Pair of Phase Space

The pair $(\gamma, \Lambda)$ equips phase space with a coPoisson structure, which tuns out to play a key role in our covariant approach to Classical and Quantum Mechanics (see Appendix: Definition I.1.10).

Theorem 10.2.1 In virtue of Proposition 9.1.5 and Theorem 9.2.18, the joined pair
$$(\gamma, \Lambda) \equiv(\gamma[K], \Lambda[G, K])$$
turns out to be a scaled coPoisson structure of the phase space (see Definition I.1.10). In other words,
$$\gamma \wedge \Lambda \wedge \Lambda \wedge \Lambda: J_1 \boldsymbol{E} \rightarrow \mathbb{T}^* \otimes \Lambda^7 T J_1 \boldsymbol{E}$$
is a scaled volume form of the phase space and
$$[\gamma, \Lambda]=0 \quad \text { and } \quad[\Lambda, \Lambda]=0$$
Remark 10.2.2 We stress that the above phase structure $(\gamma, \Lambda)$ is not the more standard “Jacobi structure” [275], which would fulfill the identities
$$[\gamma, \Lambda]=0 \text { and }[\Lambda, \Lambda]=2 \gamma \wedge \Lambda$$
Even more, we stress that in our context the 2 nd identity would be inconsistent, with respect to the view point of scales, as $\Lambda$ is an unscaled object, while $\gamma$ is a scaled vector field. Actually, there is no way to replace $\gamma$ with an unscaled vector field in a covariant way. This is a typical example of how a covariance requirement forces our theory.

Remark 10.2.3 We have already observed that the dynamical phase 2-vector $\Lambda$ is a spacelike object, hence it encodes less information with respect to the dynamical phase 2-form $\Omega$.

The two pairs $(d t, \Omega)$ and $(\gamma, \Lambda)$ are equivalent, in the sense that, in virtue of Theorems 9.1.7 and 9.1.8, we can recover one from the other one.

# 量子力学代考

## 物理代写|量子力学代写quantum mechanics代考|Cosymplectic Versus Symplectic Structures

$$\gamma \equiv \gamma[K]: J_1 \boldsymbol{E} \rightarrow \mathbb{T}^* \otimes T J_1 \boldsymbol{E}$$

$L_\gamma d t=0$ and $L_\gamma \Omega=0$

## 物理代写|量子力学代写quantum mechanics代考|The coPoisson Pair of Phase Space

$$(\gamma, \Lambda) \equiv(\gamma[K], \Lambda[G, K])$$

$$\gamma \wedge \Lambda \wedge \Lambda \wedge \Lambda: J_1 \boldsymbol{E} \rightarrow \mathbb{T}^* \otimes \Lambda^7 T J_1 \boldsymbol{E}$$

$$[\gamma, \Lambda]=0 \quad \text { and } \quad[\Lambda, \Lambda]=0$$

$$[\gamma, \Lambda]=0 \text { and }[\Lambda, \Lambda]=2 \gamma \wedge \Lambda$$

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