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.
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Notes
A separate paper on deterministic versions of a quantum field theory is planned.
For the \(2\pi \), see below under “notation”.
This means that, rather than \(\hbar \), it is Planck’s original constant, \(h\), that is normalized to one.
Due to the unusual normalization of Eq. (1.1), no factors containing \(2\pi \) are needed here.
Later we will observe that no generality is lost if we return to the case \(\sigma =1,\ \overline{\sigma }=0\), but keeping the two parameters \(\sigma ,\,\overline{\sigma }\) makes it easier to observe the \( p\leftrightarrow q \) symmetry
We apologize for the use of brackets ( ) that can mean two different things; most often they are just meant to group terms together when multiplied, but in expressions such as \(\phi (\eta ,\xi )\) they indicate that \(\phi \) is a function of \(\eta \) and \(\xi \). The comma should make this unambiguous.
In fact, the harmonic oscillator is the ideal instrument to produce fractional Fourier transforms [6], by considering how a wave function transforms at arbitrary, fractional time \(t\).
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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),
which is equal to its own Fourier transform,
The other matrix elements are simply given by
\(\psi (q)\) 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
and the function \(\phi (\eta ,x)\) is given by Eq. (3.14), or
This sum is a special case of the elliptic function \(\vartheta _3\), and it can also be written as a product:
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
(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:
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:
Evaluation gives:
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
we now proceed to write the matrix elements of the operator \(a_Q\) in the \((\eta _Q,\,P)\) frame:
finding
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
In Eq. (8.1) this gives for the \(q\) operator:
For the \(p\) operator, one obtains analogously, writing \(P\equiv P_2-P_1\),
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
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.
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’t Hooft, G. Relating the Quantum Mechanics of Discrete Systems to Standard Canonical Quantum Mechanics. Found Phys 44, 406–425 (2014). https://doi.org/10.1007/s10701-014-9788-y
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DOI: https://doi.org/10.1007/s10701-014-9788-y