Relating the Quantum Mechanics of Discrete Systems to Standard Canonical Quantum Mechanics

Abstract

Standard canonical quantum mechanics makes much use of operators whose spectra cover the set of real numbers, such as the coordinates of space, or the values of the momenta. Discrete quantum mechanics uses only strictly discrete operators. We show how one can transform systems with pairs of integer-valued, commuting operators $P_i$ and $Q_i$, to systems with real-valued canonical coordinates $q_i$ and their associated momentum operators $p_i$. The discrete system could be entirely deterministic while the corresponding (p, q) system could still be typically quantum mechanical.

Appendices

Appendix 1: Properties of the Function $\langle 0,0|q\rangle$

To go from the real numbers $q$ and $p$ to the integers $Q$ and $P$, we use the real function defined in Eqs. (3.20) and (3.21),

$$\begin{aligned} \psi (q)=\psi (-q)=\langle 0,0|q\rangle =\langle q|0,0\rangle , \end{aligned}$$

(7.1)

which is equal to its own Fourier transform,

$$\begin{aligned} \psi (p)=\int \mathrm{d}q\langle q|0,0\rangle \epsilon ^{-i\,pq}. \end{aligned}$$

(7.2)

The other matrix elements are simply given by

$$\begin{aligned} \langle Q,P|q\rangle =\psi (q-Q)\,\epsilon ^{-iPq}. \end{aligned}$$

(7.3)

$\psi (q)$ is sketched in Fig. 2, and it plays a central role in our mappings. Because of the periodicity properties (3.15), the definition of the function $\psi$ can be written as

$$\begin{aligned} \psi (x)=\mathop \int \limits _{-{\textstyle {1\over 2}}}^{\textstyle {1\over 2}}\mathrm{d}\eta \,\epsilon ^{i\phi (\eta ,x)}, \end{aligned}$$

(7.4)

and the function $\phi (\eta ,x)$ is given by Eq. (3.14), or

$$\begin{aligned} r(\eta ,x)\,\epsilon ^{i\phi (\eta ,x)}=\sum _{K=-\infty }^\infty \epsilon ^{-{\textstyle {1\over 2}}K^2+K(x+i\eta )}\ ;\qquad r,\ \phi \ \hbox { real.} \end{aligned}$$

(7.5)

This sum is a special case of the elliptic function $\vartheta _3$, and it can also be written as a product:

$$\begin{aligned} r(\eta ,x)\,\epsilon ^{i\phi (\eta ,x)}=\prod _{K=1}^\infty \left( 1-\epsilon ^{-K}\right) \ \prod _{K=0}^\infty \left( 1+\epsilon ^{x+i\eta -K-{\textstyle {1\over 2}}}\right) \left( 1+\epsilon ^{-x-i\eta -K-{\textstyle {1\over 2}}}\right) ,\nonumber \\ \end{aligned}$$

(7.6)

with $r$ and $\phi$ real. Here, the first product term is of lesser importance since it only multiplies $r(\eta ,x)$ with a constant, while not contributing to $\phi (\eta ,x)$. Note, that $\phi (\eta ,x)$ has a vortex singularity when $\epsilon ^{i\phi (\eta ,x)}$ has a zero, and these zeros can easily be read off from Eq. (7.6); they are located at $(\eta ,\,x)=(K_1+{\textstyle {1\over 2}},\,K_2+{\textstyle {1\over 2}})$. We see that in Eq. (7.4), the absolute value $r(\eta ,x)$ of the sum in (7.5), or the product in (7.6), has been divided out, and this makes the evaluation of the integral over $\eta$ hard, although it is well bounded.

In Eq. (7.5), the sum is dominated by the $K$ value closest to $x$. In Fig. 2, the small peaks at large $x$ (Fig. 2e) arise when the dominant $K$ value in the sum switches from one integer to the next.

Unitarity property :

From the fact that $\epsilon ^{i\phi (\eta ,x)}$ in Eq. (7.4) is the Fourier transform of $\psi (x)$ for integral $p$, and that it has absolute value one, we derive that

$$\begin{aligned} \sum _{K=-\infty }^\infty \psi (x+K)\,\psi (x+K+M)=\delta _{M\,0},\qquad K,L\in \mathbb Z\ \end{aligned}$$

(7.7)

(use was made of Eq. (3.21)).

Appendix 2: Matrix Elements of $q$ and $p$ Operators

The matrix elements $\langle Q_1,P_1|\,q\,|Q_2,P_2\rangle$ can be calculated explicitly. Let us first compute the operator $q$ in $\eta _Q,\eta _P$-space. We write the expression (3.23) as follows:

$$\begin{aligned} q={-i\over 2\pi }{\partial \over \partial \eta _Q}+a_Q(\eta _Q,\eta _P), \qquad a_Q(\eta _Q,\eta _P)={\partial \phi (\eta _P,\eta _Q)\over \partial \eta _Q}, \end{aligned}$$

(8.1)

where the function $a_Q$ is regarded as the $Q$-component of a vector potential field $a$. For the phase $\phi (\eta _P,\eta _Q)$, we can now best use the product formula (7.6), which gives:

$$\begin{aligned} a_Q(\eta _Q,\eta _P)&= \sum _{K=0}^\infty a_Q^K(\eta _Q,\eta _P)\ ,\end{aligned}$$

(8.2)

$$\begin{aligned} a_Q^K(\eta _Q,\eta _P)&= {\partial \over 2\pi \,\partial \eta _Q}\bigg (\arg (1+\epsilon ^{\eta _Q+i\eta _P-K-{\textstyle {1\over 2}}})+\arg (1+\epsilon ^{-\eta _Q-i\eta _P-K-{\textstyle {1\over 2}}})\bigg ).\nonumber \\ \end{aligned}$$

(8.3)

Evaluation gives:

$$\begin{aligned} a_Q^K(\eta _Q,\eta _P)&= {{\textstyle {1\over 2}}\sin (2\pi \eta _P)\over \cos (2\pi \eta _P)+\cosh \left( 2\pi \left( \eta _Q-K-{\textstyle {1\over 2}}\right) \right) }\nonumber \\&+{{\textstyle {1\over 2}}\sin (2\pi \eta _P)\over \cos (2\pi \eta _P)+\cosh \left( 2\pi \left( \eta _Q+K+{\textstyle {1\over 2}}\right) \right) },\qquad {\ } \end{aligned}$$

(8.4)

which can now be rewritten in a more compact way by rewriting Eq. (8.2) as a sum for $K$ values running from $-\infty$ to $\infty$ instead of $0$ to $\infty$.

By writing

$$\begin{aligned} \langle P_1|\eta _P\rangle \langle \eta _P|P_2\rangle =\epsilon ^{iP},\qquad P\equiv P_2-P_1, \end{aligned}$$

(8.5)

we now proceed to write the matrix elements of the operator $a_Q$ in the $(\eta _Q,\,P)$ frame:

(8.6)

finding

$$\begin{aligned} a_Q(\eta _Q,\,P,\,K)={\textstyle {1\over 2}}\hbox {sgn}(P)(-1)^{P-1}i\epsilon ^{-\big |\,P(\eta _Q+K+{\textstyle {1\over 2}})\,\big |}, \end{aligned}$$

(8.7)

where sgn $(P)$ is defined to be $\pm 1$ if $P \gtrless 0$ and  0  if $P=0$. The absolute value taken in the exponent indeed means that we always have a negative exponent there; it originated when the contour integral forced us to choose a pole inside the unit circle.

Next, we find the $(Q,P)$ matrix elements by integrating this with a factor $\epsilon ^{iQ\,\eta _Q}$, with $Q=Q_2-Q_1$, to obtain the remarkably simple expression

(8.8)

In Eq. (8.1) this gives for the $q$ operator:

$$\begin{aligned} q=Q+a_Q\ ;\quad \langle Q_1,P_1|q|Q_2,P_2\rangle =Q_1\delta _{Q_1\,Q_2}\,\delta _{P_1\,P_2}+\langle Q_1,P_1|a_Q|Q_2,P_2\rangle .\nonumber \\ \end{aligned}$$

(8.9)

For the $p$ operator, one obtains analogously, writing $P\equiv P_2-P_1$,

$$\begin{aligned}&p=P+a_P,&\end{aligned}$$

(8.10)

$$\begin{aligned}&\langle Q_1,P_1|a_P|Q_2,P_2\rangle&=\ {(-1)^{P+Q}\,iQ\over 2\pi (P^2+Q^2)}. \end{aligned}$$

(8.11)

It is important to check the commutation rule for $q$ and $p$. Doing the matrix multiplications for the matrices (8.9) and (8.10), one finds that

$$\begin{aligned}{}[Q,P]&= 0,\qquad [a_Q,\,a_P]=0, \end{aligned}$$

(8.12)

$$\begin{aligned} \langle Q_1,P_1|[q,p]|Q_2,P_2\rangle&\ =\&\langle Q_1,P_1|\,[Q,\,a_P]+[a_Q,\,P]\,|Q_2,P_2\rangle \ \nonumber \\&= {\textstyle {i\over 2\pi }}\,(-1)^{Q_1-Q_2+P_1-P_2}(\delta _{Q_1\,Q_2}\delta _{P_1\,P_2}\ -\,1). \end{aligned}$$

(8.13)

Again, we see that the desired commutation rule, $[q,p]=i/2\pi$, is obeyed only after we project out the edge state $\psi _\mathrm {edge}$ by demanding that all our states must obey $\langle \psi _\mathrm {edge}|\,\psi \,\rangle =0$, see Eq. (3.7), see Sect. 5.