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

We observe that the cosymplectic pair $(d t, \Omega)$ fully encodes the geometric structure of spacetime and its fields. Accordingly, we define the infinitesimal symmetries of classical structure to be the phase vector fields $X^{\uparrow} \in \operatorname{pro}_{E, T}\left(J_1 \boldsymbol{E}, T J_1 \boldsymbol{E}\right)$, which fulfill the conditions $L_X \dagger d t=0$ and $L_X^{\dagger} \Omega=0$ (see Definition 13.1.1).

Actually, the infinitesimal symmetries of the classical structure turn out to be of the type $X^{\uparrow}=X^{\uparrow}[f]=X^{\uparrow}$ hol $[f]=X^{\uparrow}$ ham $[f]:=\gamma\left(f^{\prime \prime}\right)+\Lambda^{\ddagger}(d f)$, with $f \in$ cns timspe $\left(J_1 \boldsymbol{E}, \mathbb{R}\right)$, i.e., in coordinates of the type $X^{\uparrow}=X^{\uparrow}[f]=f^0 \partial_0-f^i \partial_i+$ $X_0^i \partial_i^0$, with $f^0 \in \mathbb{R}, f^i, \breve{f} \in \operatorname{map}(\boldsymbol{E}, \mathbb{R})$, and $X_0^i=G_0^{i j}\left(-f^0\left(\partial_0 \mathcal{P}_j-\partial_j A_0\right)+f^h\right.$ $\left.\left(\partial_h \mathcal{P}_j-\partial_j A_h\right)+\partial_j f^h \mathcal{Q}_h+\partial_j \breve{f}\right)$

Hence, the Lie algebra of infinitesimal symmetries of classical structure is generated, in a covariant way, by the Lie algebra of conserved time preserving special phase functions.

Moreover, we observe that the pair $(d t, \mathcal{L}[\mathrm{b}])$ fully encodes the geometric structure of classical dynamics. Accordingly, we define the infinitesimal symmetries of classical dynamics to be the spacetime vector fields $X \in \operatorname{prosec}(\boldsymbol{E}, T \boldsymbol{E})$, which fulfill the conditions $L_{X^1} d t=0$ and $L_{X^1} \mathcal{L}[\mathrm{b}]=0$, where $X^1 \in \operatorname{pro}_{E, T} \sec \left(J_1 \boldsymbol{E}, T J_1 \boldsymbol{E}\right)$ is the 1-jet holonomic prolongation of $X$, (see Definition 13.2.4 and Theorem 13.2.6).

Actually, the infinitesimal symmetries of the classical structure turn out to be of the type $X^1=X^{\uparrow}[f]=X^{\uparrow}$ hol $[f]=X^{\uparrow}$ ham $[f]:=\gamma\left(f^{\prime \prime}\right)+\Lambda^{\prime}(d f)$, with (seee Definition 12.6.2) $f \in$ cns timspe $\left(J_1 \boldsymbol{E}, \mathbb{R}\right)$, fulfilling the condition $d \breve{f}[o]=-d\left(i_{X[f]} A\right.$ $[\mathrm{b}, o])$, which can be expressed in coordinates as $f=f^0 \mathcal{H}_0[\mathrm{~b}, o]+f^i \mathcal{P}_i[\mathrm{~b}, o]$, with $f^\lambda \in \operatorname{map}(\boldsymbol{E}, \mathbb{R}) ; \hat{f} \in \mathbb{R}$.

Hence, the Lie algebra of infinitesimal symmetries of classical dynamics is generated, in a covariant way, by a certain Lie subalgebra of conserved time preserving special phase functions.

Therefore, the Lie algebra of infinitesimal symmetries of classical structure is smaller than Lie algebra of infinitesimal symmetries of classical dynamics; the reason of this discrepancy is due to the fact that the pair $(d t, \Omega)$ is gauge independent, while the pair $(d t, \mathcal{L}[\mathrm{b}])$ is gauge dependent.

## 物理代写|量子力学代写quantum mechanics代考|Real and Complex Quantum Bundle

Since the very beginning, Quantum Mechanics has been formulated in a complex language. Apparently, this fact conflicts with the real language of Classical Mechanics: such a discrepancy might look quite strange. So, one might ask why Quantum Mechanics should be a complex theory and whether there is a mysterious deep reason for this.

Actually, this strange anomaly disappears if we realise that the notion of a hermitian 1-dimensional complex bundle is fully equivalent to the notion of a euclidean oriented 2-dimensional real vector bundle (see, for instance [242, Vol. II]).

Indeed, we give due weight to this equivalence. So, besides the usual approach to the quantum bundle in terms of a hermitian 1-dimensional complex bundle, we discuss also the approach in terms of an oriented euclidean 2-dimensional real bundle (see Postulate Q.1 and Note 14.4.5).

These two approaches are equivalent from a formal geometric viewpoint but they emphasise different physical meanings and provide different practical advantages.
We stress that, from a physical viewpoint, the real language emphasises 2 (internal) real degrees of freedom of the scalar quantum particle; actually, this fact can be easily understood in comparison with Classical Mechanics. Moreover, the real language is very convenient for the development of geometric aspects of Quantum Mechanics which are based on real geometric methods, such as lagrangian theory, symmetries and jet spaces. Actually, in standard literature these aspects are usually treated in complex terms. But we stress that a rigorous translation in complex terms of geometric theories which are deeply real is much more subtle and delicate than it might appear at a first insight.

However, the complex language provides a quick and compact expression of several developments and formulas. So, in our opinion, this is the very reason why Quantum Mechanics needs the complex language.

Actually, in the book we use both languages: we use the real language just for the topics which are essentially real, but then translate the results in the usual complex language.

## 物理代写|量子力学代写quantum mechanics代考|Real and Complex Quantum Bundle

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