# 物理代写|理论力学作业代写Theoretical Mechanics代考|physics307

#### Doug I. Jones

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## 物理代写|理论力学作业代写Theoretical Mechanics代考|Field quantization in phase space and wave/particle duality

These results however leaved some conceptual problems still open. First of all, once the Schrödinger waves have been eliminated from Quantum Mechanics, how does one generalize its principles to Quantum Field Theory? One should not forget that, historically, QED was invented by Dirac (Dirac 1927) by submitting “first quantized” Schrödinger amplitudes to the procedure of “second quantization”. If no “first quantized” probability amplitudes exist any more how does one proceed? And, secondly, isn’t one throwing away the baby with the dirty water by forgetting that after all a quantum field must still show some of the wavelike properties of its classical limit?

A second paper [Cini 2003] has been therefore devoted to answer to these questions, leading to the conclusion that: (a) one should not start from nonrelativistic quantum mechanics in order to formulate quantum field theory, but viceversa; (b) the wavelike behaviour of the quanta of a quantum field is, as already Pascual Jordan had understood in 1926 [Born, Heisernberg, Jordan 1926], a straightforward consequence of imposing the Einstein property of discreteness to the intensity of a classical field – clearly a nonlocal physical entity – which exists objectively in ordinary three dimensional space.

It is appropriate to recall that for Jordan, in fact, it is quantization which brings into existence particles, both photons and electrons. According to him, therefore, rather than trying to explain phenomena like diffraction and interference of single particles as properties of “probability waves” one should simply view them as primary properties of the field of which they represent the quanta. “These considerations show – we read in his paper “On waves and corpuscles in quantum mechanics” [Jordan 1927] – that the quantized field is equivalent, in all its physical properties and especially with respect to its inensity fluctuations, to a corpuscular system (with a symmetric eigenfunction)”.

The derivation of Wigner functions from the principles of uncertainty and discreteness illustrated in the previous paragraph provides the formalism for deducing the kind of wave/particle duality suggested by Jordan (and forgotten by the physicist’s community since then) by simply imposing Einstein’s quantization to the states of a classical field represented by means of statistical ensembles in the phase spaces of its normal modes.

## 物理代写|理论力学作业代写Theoretical Mechanics代考|Algebraic properties of bicomplex numbers

Bicomplex algebra is considerably simplified by the introduction of two bicomplex numbers $\mathbf{e}{1}$ and $\mathbf{e}{2}$ defined as
$$\mathbf{e}{1}:=\frac{1+\mathbf{j}}{2} \quad \text { and } \quad \mathbf{e}{2}:=\frac{1-\mathbf{j}}{2}$$
One easily checks that
$$\mathbf{e}{1}^{2}=\mathbf{e}{1}, \quad \mathbf{e}{2}^{2}=\mathbf{e}{2}, \quad \mathbf{e}{1}+\mathbf{e}{2}=1 \quad \text { and } \quad \mathbf{e}{1} \mathbf{e}{2}=0$$
Any bicomplex number $w$ can be written uniquely as
$$w=z_{1} \mathbf{e}{\mathbf{1}}+z{\hat{2}} \mathbf{e}{2},$$ where $z{\hat{1}}$ and $z_{\hat{2}}$ both belong to $C\left(i_{1}\right) .$ Specifically,
$$z_{\hat{1}}=\left(w_{e}+w_{\mathbf{j}}\right)+\left(w_{\mathbf{i}{1}}-w{\mathbf{i}{\mathbf{2}}}\right) \mathbf{i}{\mathbf{1}} \quad \text { and } \quad z_{\hat{2}}=\left(w_{e}-w_{\mathbf{j}}\right)+\left(w_{\mathbf{i}{\mathbf{1}}}+w{\mathbf{i}{2}}\right) \mathbf{i}{\mathbf{1}}$$
The numbers $\mathbf{e}{\mathbf{1}}$ and $\mathbf{e}{\mathbf{2}}$ make up the so-called idempotent basis of the bicomplex numbers (Price, 1991). Note that the last of (17) illustrates the fact that $\mathbb{T}$ has zero divisors which are nonzero elements whose product is zero. The caret notation $(\hat{1}$ and $\hat{2}$ ) will be used systematically in connection with idempotent decompositions, with the purpose of easily distinguishing different types of indices.

As a consequence of (17) and (18), one can check that if $\sqrt[n]{z_{\hat{1}}}$ is an $n$th root of $z_{\hat{1}}$ and $\sqrt[n]{z_{2}}$ is an
The uniqueness of the idempotent decomposition allows the introduction of two projection operators as
\begin{aligned} &P_{1}: w \in \mathbb{I} \mapsto z_{\hat{1}} \in \mathbb{C}\left(\mathbf{i}{1}\right) \ &P{2}: w \in \mathbb{T} \mapsto z_{\hat{2}} \in \mathbb{C}\left(\mathbf{i}{1}\right) \end{aligned} The $P{k}(k=1,2)$ satisfy
$$\left[P_{k}\right]^{2}=P_{k}, \quad P_{1} \mathbf{e}{1}+P{2} \mathbf{e}{2}=\mathbf{I d}$$ and, for $s, t \in \mathbb{T}$, $$P{k}(s+t)=P_{k}(s)+P_{k}(t) \quad \text { and } \quad P_{k}(s \cdot t)=P_{k}(s) \cdot P_{k}(t)$$
The product of two bicomplex numbers $w$ and $w^{\prime}$ can be written in the idempotent basis as
$$w \cdot w^{\prime}=\left(z_{1} \mathbf{e}{\mathbf{1}}+z{\hat{2}} \mathbf{e}{2}\right) \cdot\left(z{\hat{1}}^{\prime} \mathbf{e}{1}+z{2}^{\prime} \mathbf{e}{2}\right)=z{1} z_{\hat{1}}^{\prime} \mathbf{e}{\mathbf{1}}+z{\hat{2}} z_{\hat{2}}^{\prime} \mathbf{e}_{2}$$

## 物理代写|理论力学作业代写Theoretical Mechanics代考|Algebraic properties of bicomplex numbers

$$\mathbf{e} 1:=\frac{1+\mathbf{j}}{2} \text { and } \mathbf{e} 2:=\frac{1-\mathbf{j}}{2}$$

$$\mathbf{e} 1^{2}=\mathbf{e} 1, \quad \mathbf{e} 2^{2}=\mathbf{e} 2, \quad \mathbf{e} 1+\mathbf{e} 2=1 \quad \text { and } \quad \mathbf{e} 1 \mathbf{e} 2=0$$

$$w=z_{1} \mathbf{e} \mathbf{1}+z \hat{2} \mathbf{e} 2,$$

$$z_{\hat{1}}=\left(w_{e}+w_{\mathbf{j}}\right)+\left(w_{\mathbf{i} 1}-w \mathbf{i} \mathbf{2}\right) \mathbf{i} \mathbf{1} \quad \text { and } \quad z_{\hat{2}}=\left(w_{e}-w_{\mathbf{j}}\right)+\left(w_{\mathbf{i} 1}+w \mathbf{i} 2\right) \mathbf{i} \mathbf{1}$$

$$P_{1}: w \in \mathbb{I} \mapsto z_{\hat{1}} \in \mathbb{C}(\mathbf{i} 1) \quad P 2: w \in \mathbb{T} \mapsto z_{\hat{2}} \in \mathbb{C}(\mathbf{i} 1)$$

$$\left[P_{k}\right]^{2}=P_{k}, \quad P_{1} \mathbf{e} 1+P 2 \mathbf{e} 2=\mathbf{I d}$$

$$P k(s+t)=P_{k}(s)+P_{k}(t) \quad \text { and } \quad P_{k}(s \cdot t)=P_{k}(s) \cdot P_{k}(t)$$

$$w \cdot w^{\prime}=\left(z_{1} \mathbf{e} 1+z \hat{2} \mathbf{e} 2\right) \cdot\left(z \hat{1}^{\prime} \mathbf{e} 1+z 2^{\prime} \mathbf{e} 2\right)=z 1 z_{\hat{1}}^{\prime} \mathbf{e} 1+z \hat{2} z_{\hat{2}}^{\prime} \mathbf{e}_{2}$$

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