We see that the adjoint of \( \hat { X } ^ { ( \alpha ) } \) is defined as \( i ( d / d k ) \) in the momentum basis provided that \|φ) also obeys the boundary condition \( \tilde { \phi } ( \Omega ) \, = \, e ^ { - i \alpha } \tilde { \phi } ( - \Omega ) \). Therefore we see \( D ( \hat { X } ^ { ( \alpha ) } ) = D ( \hat { X } ^ { ( \alpha ) \dagger } ) \), and so \( \hat { X } ^ { ( \alpha ) } \) is self-adjoint.

Now, since the operators \( \hat { X } ^ { ( \alpha ) } \) are self-adjoint, they have spectral decompositions. The spectrum of the oper-ator \( \bar { X } ^ { ( \alpha ) } \)(for fixed \( \alpha \)) is discrete,\( \mathrm { s p e c } ( \hat { X } ^ { ( \alpha ) } ) = \{ x _ { n } ^ { ( \alpha ) } : = \)\( \textstyle { \frac { 2 \pi n + \alpha } { 2 \Omega } } \} _ { n \in \mathbb { Z } } \), and describes a one-dimensional lattice. The corresponding eigenvectors, \( \{ | x _ { n } ^ { ( \alpha ) } \rangle \} _ { n \in \mathbb { Z } } \), are represented in the momentum eigenbasis as

\[ ( k | x _ { n } ^ { ( \alpha ) } ) = \frac { 1 } { \sqrt { 2 \Omega } } e ^ { - i k x _ { n } ^ { ( \alpha ) } } . \eqno ( \mathrm { A 5 } ) \]

These eigenvectors are orthogonal and admit a resolu-tion of the identity:

\[ \sum _ { n \in \mathbb { Z } } | x _ { n } ^ { ( \alpha ) } ) ( x _ { n } ^ { ( \alpha ) } | = \mathbb { 1 } \qquad \qquad \qquad ( { \mathrm { A 6 } } ) \]

. Crucially, the position eigenvectors from different self-adjoint extensions are not orthogonal:

\[ ( x _ { n } ^ { ( \alpha ) } | x _ { n ^ { \prime } } ^ { ( \alpha ^ { \prime } ) } ) = \mathrm { s i n c } \left[ ( x _ { n } ^ { ( \alpha ) } - x _ { n ^ { \prime } } ^ { ( \alpha ^ { \prime } ) } ) \Omega \right] . \qquad ( \mathrm { A 7 } ) \]

Note that the union of the spectra of the entire family of self-adjoint extensions provides a covering of R. There-fore, it is possible to construct an overcomplete contin-uum basis by taking the union of eigenbases of the family of self-adjoint extensions, i.e., \|x) := \|x(α)n \( ) \iff x \)\( \therefore \)=\( \begin{array} { r } { x _ { n } ^ { ( \alpha ) } : = \frac { 2 \pi n + \alpha } { 2 \Omega } } \end{array} \).

It is then simple to write down the Shannon sampling theorem for a bandlimited function \( \psi \):

\[ \begin{array} { l l } { { \displaystyle \psi ( x ) = ( x | \psi ) } } & { { \displaystyle ( \mathrm { A } 8 ) } } \\ { { \displaystyle ~ ~ ~ ~ ~ = \sum _ { n \in \mathbb { Z } } ( x | x _ { n } ^ { ( \alpha ) } ) ( x _ { n } ^ { ( \alpha ) } | \psi ) } } & { { \displaystyle ( \mathrm { A } 9 ) } } \\ { { \displaystyle ~ ~ ~ ~ ~ = \sum _ { n \in \mathbb { Z } } \mathrm { s i n c } \left[ ( x - x _ { n } ^ { ( \alpha ) } ) \Omega \right] \psi ( x _ { n } ^ { ( \alpha ) } ) . } } & { { \displaystyle ( \mathrm { A 1 0 } ) } } \end{array} \]

Therefore, we see that the function \( \psi \) is determined at any point \( x \in \mathbb { R } \) from its values on one of the lattices \( \{ x _ { n } ^ { ( \alpha ) } \} _ { n } . \).

Also, we obtain an overcomplete resolution of identity,

\[ \frac { \Omega } { \pi } \int _ { \mathbb { R } } d x \ | x ) ( x | = 1 \qquad \qquad \quad ( \mathrm { A 1 1 } ) \]

where \( \pi / \Omega \) is the density of degrees of freedom. We can use the resolution of identity for the continuum basis to show that the space of bandlimited functions has a reproducing kernel,

\[ ( x | \psi ) = \frac { \Omega } { \pi } \int d x ^ { \prime } \; ( x | x ^ { \prime } ) ( x ^ { \prime } | \psi ) \; \; \; \; \; \; \; \; \; \; \; ( \mathrm { A 1 2 } ) \]

\[ \psi ( x ) = \int d x \; K ( x , x ^ { \prime } ) \psi ( x ^ { \prime } ) , \; \; \; \; \; \; \; \; \; \; \; ( \mathrm { A 1 3 } ) \]

where \( K ( x , x ^ { \prime } ) : = ( \Omega / \pi ) ( x | x ^ { \prime } ) = ( \Omega / \pi ) \operatorname { s i n c } [ ( x - x ^ { \prime } ) \Omega ] \).

[1] J. A. Wheeler, “On the nature of quantum geometrodynamics,” Annals of Physics 2 .

[2] C. A. Mead, “Possible Connection Between Gravitation and Fundamental Length,” Phys. Rev. 135 (Aug, 1964) B849–B862.

[3] A. Kempf, G. Mangano, and R. B. Mann, “Hilbert space representation of the minimal length uncertainty relation,” Phys.Rev. D52 (1995) 1108–1118, arXiv:hep-th/9412167 [hep-th].

[4] L. J. Garay, “Quantum gravity and minimum length,” Int.J.Mod.Phys. A10 (1995) 145–166, arXiv:gr-qc/9403008 [gr-qc].

[5] E. Witten, “Reflections on the fate of space-time,” Phys.Today 49N4 (1996) 24–30.

[6] S. Hossenfelder, “Minimal Length Scale Scenarios for Quantum Gravity,” Living Rev. Rel. 16 (2013) 2, arXiv:1203.6191 [gr-qc].

[7] T. Jacobson and R. Parentani, “Horizon entropy,” Found.Phys. 33 (2003) 323–348, arXiv:gr-qc/0302099[gr-qc].

[8] R. D. Sorkin, “On the entropy of the vacuum outside a horizon,” in Tenth International Conference on General Relativity and Gravitation (held Padova, 4-9 July, 1983), Contributed Papers, vol. 2, pp. 734–736. 1983. arXiv:1402.3589 [gr-qc].

[9] L. Bombelli, R. K. Koul, J. Lee, and R. D. Sorkin, “Quantum source of entropy for black holes,” Phys. Rev. D 34 no. 2, (1986) 373–383.

[10] M. Srednicki, “Entropy and area,” Phys. Rev. Lett. 71 (1993) 666–669, arXiv:hep-th/9303048.

[11] S. N. Solodukhin, “Entanglement Entropy of Black Holes,” Living Reviews in Relativity 14 no. 8, (2011) , arXiv:1104.3712 [hep-th].http://www.livingreviews.org/lrr-2011-8.

[12] T. Jacobson, “Black hole entropy and induced gravity,” arXiv:gr-qc/9404039 [gr-qc].

[13] L. Susskind and J. Uglum, “Black hole entropy in canonical quantum gravity and superstring theory,” Phys.Rev. D50 (1994) 2700–2711, arXiv:hep-th/9401070 [hep-th].

[14] E. Bianchi and R. C. Myers, “On the Architecture of Spacetime Geometry,” arXiv:1212.5183 [hep-th].

[15] L. Bombelli, J. Lee, D. Meyer, and R. Sorkin, “Space-Time as a Causal Set,” Phys. Rev. Lett. 59 (1987) 521–524.

[16] L. Bombelli, J. Henson, and R. D. Sorkin, “Discreteness