Phase Space Representation

Consider the two-dimensional space spanned by the time domain $(-\infty <t<\infty )$ and the Fourier domain $(-\infty <\omega <\infty )$ of the set of square-integrable function $f(t)$ . The introduction of the set of orthonormal wave packets determines a partitioning of this space into elements of area whose shape, magnitude and location is determined by these o.n. wave packets. This partitioned two-dimensional space is called the phase space of the system whose state is described by the set of square-integrable functions.

The wave packet representation of the function

$$f(t)=\sum^\infty_{j=-\infty}~\sum^\infty_{\ell =-\infty} \alpha_{j\ell} P_{j\ell} (t)$$

is represented geometrically as a set of complex amplitudes $(\alpha_{j\ell})$ assigned to their respective elements of area each one of size $\Delta \omega\Delta t=2\pi$ , which together comprise the phase space of the system. This phase space is two dimensional and it is spanned by the time domain $(-\infty <t<\infty )$ and the Fourier domain $(-\infty <\omega <\infty )$ of the function $f(t)$ .

The set of orthonormal wave packets determine a partitioning of this phase space into elements of equal area,

$$\Delta\omega\Delta t = 2\pi\,,$$

which are called phase space cells. The existence of this partitioning is guarateed by the following

Theorem 24.1 (Wave Packet Representation Theorem)
Let $f(t)$ be a square-integrable function. Let $\hat{f}(\omega)$ be its Fourier transform,

$$\hat{f}(\omega)=\int^\infty_{-\infty} \frac{e^{-i\omega t}}{\sqrt{2\pi}} f(t)dt$$

and let

$$\hat{P}_{j\ell}(\omega)=\int^\infty... ..._{j\ell}(\omega),~\textrm{which~is~given~by~Eq.(\ref{eq:Famplitude})} \right)$$

be the Fourier transform of the wave packet $P_{j\ell }(t)$ . Then the Fourier transform pair

$$f(t)=\sum^\infty_{j=-\infty}~\sum^\infty_{\ell =-\infty} \alpha_{j\ell} P_{j\ell} (t)$$

and

$$\hat{f}(\omega)=\sum^\infty_{j=-\infty}~\sum^\infty_{\ell =-\infty} \alpha_{j\ell} \hat{P}_{j\ell} (t)$$

have the same expansion components $\alpha_{j\ell}$ relative to the Fourier-related bases $\{P_{j\ell}\}$ and $\{ \hat{P}_{j\ell}\}$ . Both bases are orthonormal

$$\langle P_{j\ell }, P_{j'\ell '}\rangle \equiv \int^\infty_{-\infty}\overline{P}_{j\ell} (t) P_{j'\ell '}(t) dt = \delta_{jj'}\delta_{\ell\ell '}$$

$$\langle \hat{P}_{j\ell }, \hat{P}_{j'\ell '}\rangle \equiv \int^\infty_{-\infty}\overline{\hat{P}}_{j\ell} (\omega) \hat{P}_{j'\ell '}(\omega) d\omega = \delta_{jj'}\delta_{\ell\ell '}$$

and are complete

$$\sum^\infty_{j=-\infty} \sum^\infty_{\ell =-\infty} P_{j\ell}(t)\overline{P}_{j \ell}(t') = \delta (t-t')$$

$$\sum^\infty_{j=-\infty} \sum^\infty_{\ell =-\infty} \hat{P}_{j\ell}(\omega)\overline{\hat{P}}_{j\ell}(\omega') = \delta (\omega-\omega')$$

Figure 2.13: The smallest elements of phase space are the phase space cells. Each one, like the one depicted in this picture, has area $2\pi$ .

This theorem implies that

1. the location of a typical phase space cell as determined by $P_{j\ell }(t)$ and $\hat{P}_{j\ell}(\omega)$ is given by
   

   $$\frac{2\pi}{\varepsilon}\ell = \textrm{mean~temporal~position}$$

   $$\left( j+\frac{1}{2}\right)\varepsilon = \textrm{mean~frequency}\,.$$

   
   
2. the shape of a typical phase space cell as determined by $P_{j\ell }(t)$ and $\hat{P}_{j\ell}(\omega)$ is given by
   

   $$\Delta t = \frac{2\pi}{\varepsilon}$$

   $$\Delta\omega = \varepsilon\, ,$$

   
   the temporal and the frequency spread of the wave packet.
3. the area of a typical phase space cell as determined by $P_{j\ell }(t)$ and $\hat{P}_{j\ell}(\omega)$ is
   $$\Delta\omega\Delta t = 2\pi\,.$$

The wave packet representation

$$f(t)=\sum^\infty_{j=-\infty}~\sum^\infty_{\ell =-\infty} \alpha_{j\ell} P_{j\ell} (t)$$

of a square integrable function determines the corresponding phase space representation. It consists of assigning the complex amplitude $\alpha_{j\ell}$ to the $(j,\ell )$ th phase space cell. Typically, the squared norm

$$\Vert f\Vert^2=\int^\infty_{-\infty}\vert f(t)\vert^2\,dt$$

is proportional to the total ``energy'' of the signal represented by $f(t)$ . If that is the case, then Parseval's identity

$$\int^\infty_{-\infty} \vert f(t)\vert^2\,dt = \sum_{j=-\infty}^\infty~ \sum_{\ell =-\infty}^\infty\vert \alpha_{j\ell}\vert^2$$

implies that

$$\vert\alpha_{j\ell}\vert^2\propto~~\textrm{\lq\lq energy''~contained~in~the~} (j,\ell)\textrm{th~phase~space~cell}\,.$$

Figure 2.14: Phase space representation of a function.

In other words,

$$\{\vert\alpha_{j\ell}\vert^2\colon j,\ell =0,\pm 1,\dots\}$$

is a decomposition of the energy of $f(t)$ into its most elementary spectral and temporal components relative to the chosen wave packet basis $\{ P_{j\ell }(t)\}$ . The wave packet representation of a signal $f(t)$ assigns to each phase space cell an intensity $\vert\alpha_{j \ell}\vert^2$ . Each cell acquires a level of grayness $\propto \vert\alpha_{j\ell}\vert^2$ .

Thus a signal gets represented by assigning a degree of darkness (``squared modulus'') and a phase factor of each phase space cell. This is shown in Figure 2.14.

An example of this geometrical phase space representation is the musical score of a piece of music. The notes represent the phase space cells in which there is a non-zero amount of energy. (Musicians ignore the phase factor which ordinarily would go with it.) A signal, say, Beethoven's Fifth Symphony, is therefore represented by a distribution of dots (of various gray levels) in phase space, with time running horizontally to the right, and pitch going up vertically.

A phase space representation relative to a chosen set of o.n. wave packets is, therefore, a highly refined and sophisticated version of a musician's score. In fact, it constitutes the ultimate refinement. No better discrete representation is possible.

Final Remarks:

1. It is not necessary that the wave packets, Eq.(2.61) have their Fourier support centered around $(j+\frac {1}{2})\varepsilon$ . Another possibility is that they be centered around $j\varepsilon$ . In that case the resulting set of wave packets
   $$Q^\varepsilon_{j\ell}\equiv\int^{(j+\frac{1}{2})\varepsilon}_{(j-... ...egin{array}{l} j=0,\pm 1,\pm 2,\dots\\ \ell = 0,\pm 1,\pm 2,\cdots \end{array}$$
   would still be orthonormal and complete.
2. Each wave packet of mean frequency zero, $Q_{0\ell}^\varepsilon(t), \ell =0,\pm 1, \pm 2,\cdots$ is an integral representation of the sinc function, Eq.(2.23) on page , centered around $t=\frac{2\pi \ell}{\varepsilon}$ ,

   $$\int^{\frac{\varepsilon}{2}}_{-\frac{\varepsilon}{2}} \frac{e^{-2... ...psilon}{2}\right] }{t-\frac{2\pi\ell}{\varepsilon}} =Q^\varepsilon_{0\ell}(t)~.$$ (264)

   
   
   In the limit as $\varepsilon\to\infty$ this tends towards an expression proportional to the Dirac delta function.